author | paulson |
Fri, 05 Mar 2004 15:19:55 +0100 | |
changeset 14436 | 77017c49c004 |
parent 14430 | 5cb24165a2e1 |
child 14443 | 75910c7557c5 |
permissions | -rw-r--r-- |
14430
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(* Title: Parity.thy |
5cb24165a2e1
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Author: Jeremy Avigad |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
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License: GPL (GNU GENERAL PUBLIC LICENSE) |
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|
5cb24165a2e1
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Last modified 2 March 2004 |
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*) |
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|
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header {* Parity: Even and Odd for ints and nats*} |
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9 |
|
5cb24165a2e1
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theory Parity = Divides + IntDiv + NatSimprocs: |
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|
5cb24165a2e1
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axclass even_odd < type |
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|
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instance int :: even_odd .. |
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instance nat :: even_odd .. |
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|
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consts |
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even :: "'a::even_odd => bool" |
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|
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syntax |
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odd :: "'a::even_odd => bool" |
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|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
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translations |
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"odd x" == "~even x" |
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|
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defs (overloaded) |
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even_def: "even (x::int) == x mod 2 = 0" |
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even_nat_def: "even (x::nat) == even (int x)" |
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|
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|
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subsection {* Casting a nat power to an integer *} |
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32 |
|
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lemma zpow_int: "int (x^y) = (int x)^y" |
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apply (induct_tac y) |
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apply (simp, simp add: zmult_int [THEN sym]) |
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done |
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|
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subsection {* Even and odd are mutually exclusive *} |
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39 |
|
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lemma int_pos_lt_two_imp_zero_or_one: |
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41 |
"0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1" |
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42 |
by auto |
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|
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lemma neq_one_mod_two [simp]: "((x::int) mod 2 ~= 0) = (x mod 2 = 1)" |
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45 |
apply (subgoal_tac "x mod 2 = 0 | x mod 2 = 1", force) |
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parents:
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46 |
apply (rule int_pos_lt_two_imp_zero_or_one, auto) |
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47 |
done |
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48 |
|
5cb24165a2e1
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49 |
subsection {* Behavior under integer arithmetic operations *} |
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50 |
|
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51 |
lemma even_times_anything: "even (x::int) ==> even (x * y)" |
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52 |
by (simp add: even_def zmod_zmult1_eq') |
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53 |
|
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54 |
lemma anything_times_even: "even (y::int) ==> even (x * y)" |
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55 |
by (simp add: even_def zmod_zmult1_eq) |
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56 |
|
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lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)" |
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58 |
by (simp add: even_def zmod_zmult1_eq) |
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|
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lemma even_product: "even((x::int) * y) = (even x | even y)" |
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61 |
apply (auto simp add: even_times_anything anything_times_even) |
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62 |
apply (rule ccontr) |
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63 |
apply (auto simp add: odd_times_odd) |
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64 |
done |
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65 |
|
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66 |
lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)" |
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67 |
by (simp add: even_def zmod_zadd1_eq) |
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68 |
|
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lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)" |
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70 |
by (simp add: even_def zmod_zadd1_eq) |
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71 |
|
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72 |
lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)" |
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73 |
by (simp add: even_def zmod_zadd1_eq) |
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74 |
|
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lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" |
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76 |
by (simp add: even_def zmod_zadd1_eq) |
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77 |
|
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78 |
lemma even_sum: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))" |
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79 |
apply (auto intro: even_plus_even odd_plus_odd) |
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parents:
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80 |
apply (rule ccontr, simp add: even_plus_odd) |
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parents:
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81 |
apply (rule ccontr, simp add: odd_plus_even) |
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82 |
done |
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83 |
|
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84 |
lemma even_neg: "even (-(x::int)) = even x" |
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parents:
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85 |
by (auto simp add: even_def zmod_zminus1_eq_if) |
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86 |
|
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lemma even_difference: |
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88 |
"even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" |
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parents:
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89 |
by (simp only: diff_minus even_sum even_neg) |
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|
90 |
|
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91 |
lemma even_pow_gt_zero [rule_format]: |
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92 |
"even (x::int) ==> 0 < n --> even (x^n)" |
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93 |
apply (induct_tac n) |
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parents:
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94 |
apply (auto simp add: even_product) |
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parents:
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95 |
done |
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96 |
|
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97 |
lemma odd_pow: "odd x ==> odd((x::int)^n)" |
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parents:
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|
98 |
apply (induct_tac n) |
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parents:
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99 |
apply (simp add: even_def) |
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parents:
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100 |
apply (simp add: even_product) |
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parents:
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|
101 |
done |
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parents:
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102 |
|
5cb24165a2e1
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parents:
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103 |
lemma even_power: "even ((x::int)^n) = (even x & 0 < n)" |
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parents:
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|
104 |
apply (auto simp add: even_pow_gt_zero) |
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parents:
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105 |
apply (erule contrapos_pp, erule odd_pow) |
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parents:
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106 |
apply (erule contrapos_pp, simp add: even_def) |
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parents:
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|
107 |
done |
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parents:
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|
108 |
|
5cb24165a2e1
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parents:
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|
109 |
lemma even_zero: "even (0::int)" |
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parents:
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|
110 |
by (simp add: even_def) |
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parents:
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|
111 |
|
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parents:
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112 |
lemma odd_one: "odd (1::int)" |
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parents:
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|
113 |
by (simp add: even_def) |
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parents:
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|
114 |
|
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|
115 |
lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero |
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parents:
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116 |
odd_one even_product even_sum even_neg even_difference even_power |
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|
117 |
|
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parents:
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118 |
|
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parents:
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|
119 |
subsection {* Equivalent definitions *} |
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120 |
|
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parents:
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|
121 |
lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" |
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122 |
by (auto simp add: even_def) |
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parents:
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|
123 |
|
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124 |
lemma two_times_odd_div_two_plus_one: "odd (x::int) ==> |
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|
125 |
2 * (x div 2) + 1 = x" |
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parents:
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|
126 |
apply (insert zmod_zdiv_equality [of x 2, THEN sym]) |
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paulson
parents:
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|
127 |
by (simp add: even_def) |
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parents:
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|
128 |
|
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parents:
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|
129 |
lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" |
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parents:
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130 |
apply auto |
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paulson
parents:
diff
changeset
|
131 |
apply (rule exI) |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
132 |
by (erule two_times_even_div_two [THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
133 |
|
5cb24165a2e1
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paulson
parents:
diff
changeset
|
134 |
lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
135 |
apply auto |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
136 |
apply (rule exI) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
137 |
by (erule two_times_odd_div_two_plus_one [THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
138 |
|
5cb24165a2e1
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paulson
parents:
diff
changeset
|
139 |
|
5cb24165a2e1
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paulson
parents:
diff
changeset
|
140 |
subsection {* even and odd for nats *} |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
141 |
|
5cb24165a2e1
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paulson
parents:
diff
changeset
|
142 |
lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)" |
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paulson
parents:
diff
changeset
|
143 |
by (simp add: even_nat_def) |
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paulson
parents:
diff
changeset
|
144 |
|
5cb24165a2e1
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paulson
parents:
diff
changeset
|
145 |
lemma even_nat_product: "even((x::nat) * y) = (even x | even y)" |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
146 |
by (simp add: even_nat_def zmult_int [THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
147 |
|
5cb24165a2e1
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paulson
parents:
diff
changeset
|
148 |
lemma even_nat_sum: "even ((x::nat) + y) = |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
149 |
((even x & even y) | (odd x & odd y))" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
150 |
by (unfold even_nat_def, simp) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
151 |
|
5cb24165a2e1
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paulson
parents:
diff
changeset
|
152 |
lemma even_nat_difference: |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
153 |
"even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))" |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
154 |
apply (auto simp add: even_nat_def zdiff_int [THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
155 |
apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
156 |
apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
157 |
done |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
158 |
|
14436 | 159 |
lemma even_nat_Suc: "even (Suc x) = odd x" |
14430
5cb24165a2e1
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paulson
parents:
diff
changeset
|
160 |
by (simp add: even_nat_def) |
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paulson
parents:
diff
changeset
|
161 |
|
14436 | 162 |
text{*Compatibility, in case Avigad uses this*} |
163 |
lemmas even_nat_suc = even_nat_Suc |
|
164 |
||
14430
5cb24165a2e1
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paulson
parents:
diff
changeset
|
165 |
lemma even_nat_power: "even ((x::nat)^y) = (even x & 0 < y)" |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
166 |
by (simp add: even_nat_def zpow_int) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
167 |
|
5cb24165a2e1
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paulson
parents:
diff
changeset
|
168 |
lemma even_nat_zero: "even (0::nat)" |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
169 |
by (simp add: even_nat_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
170 |
|
5cb24165a2e1
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paulson
parents:
diff
changeset
|
171 |
lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard] |
14436 | 172 |
even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power |
14430
5cb24165a2e1
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paulson
parents:
diff
changeset
|
173 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
174 |
|
5cb24165a2e1
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paulson
parents:
diff
changeset
|
175 |
subsection {* Equivalent definitions *} |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
176 |
|
5cb24165a2e1
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paulson
parents:
diff
changeset
|
177 |
lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==> |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
178 |
x = 0 | x = Suc 0" |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
179 |
by auto |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
180 |
|
5cb24165a2e1
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paulson
parents:
diff
changeset
|
181 |
lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0" |
5cb24165a2e1
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paulson
parents:
diff
changeset
|
182 |
apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
183 |
apply (drule subst, assumption) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
184 |
apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0") |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
185 |
apply force |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
186 |
apply (subgoal_tac "0 < Suc (Suc 0)") |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
187 |
apply (frule mod_less_divisor [of "Suc (Suc 0)" x]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
188 |
apply (erule nat_lt_two_imp_zero_or_one, auto) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
189 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
190 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
191 |
lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
192 |
apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
193 |
apply (drule subst, assumption) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
194 |
apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0") |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
195 |
apply force |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
196 |
apply (subgoal_tac "0 < Suc (Suc 0)") |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
197 |
apply (frule mod_less_divisor [of "Suc (Suc 0)" x]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
198 |
apply (erule nat_lt_two_imp_zero_or_one, auto) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
199 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
200 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
201 |
lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
202 |
apply (rule iffI) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
203 |
apply (erule even_nat_mod_two_eq_zero) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
204 |
apply (insert odd_nat_mod_two_eq_one [of x], auto) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
205 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
206 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
207 |
lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
208 |
apply (auto simp add: even_nat_equiv_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
209 |
apply (subgoal_tac "x mod (Suc (Suc 0)) < Suc (Suc 0)") |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
210 |
apply (frule nat_lt_two_imp_zero_or_one, auto) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
211 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
212 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
213 |
lemma even_nat_div_two_times_two: "even (x::nat) ==> |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
214 |
Suc (Suc 0) * (x div Suc (Suc 0)) = x" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
215 |
apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
216 |
apply (drule even_nat_mod_two_eq_zero, simp) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
217 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
218 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
219 |
lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==> |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
220 |
Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
221 |
apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
222 |
apply (drule odd_nat_mod_two_eq_one, simp) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
223 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
224 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
225 |
lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
226 |
apply (rule iffI, rule exI) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
227 |
apply (erule even_nat_div_two_times_two [THEN sym], auto) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
228 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
229 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
230 |
lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
231 |
apply (rule iffI, rule exI) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
232 |
apply (erule odd_nat_div_two_times_two_plus_one [THEN sym], auto) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
233 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
234 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
235 |
subsection {* Powers of negative one *} |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
236 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
237 |
lemma neg_one_even_odd_power: |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
238 |
"(even x --> (-1::'a::{number_ring,ringpower})^x = 1) & |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
239 |
(odd x --> (-1::'a)^x = -1)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
240 |
apply (induct_tac x) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
241 |
apply (simp, simp add: power_Suc) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
242 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
243 |
|
14436 | 244 |
lemma neg_one_even_power [simp]: |
245 |
"even x ==> (-1::'a::{number_ring,ringpower})^x = 1" |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
246 |
by (rule neg_one_even_odd_power [THEN conjunct1, THEN mp], assumption) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
247 |
|
14436 | 248 |
lemma neg_one_odd_power [simp]: |
249 |
"odd x ==> (-1::'a::{number_ring,ringpower})^x = -1" |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
250 |
by (rule neg_one_even_odd_power [THEN conjunct2, THEN mp], assumption) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
251 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
252 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
253 |
subsection {* Miscellaneous *} |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
254 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
255 |
lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
256 |
apply (subst zdiv_zadd1_eq) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
257 |
apply (simp add: even_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
258 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
259 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
260 |
lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
261 |
apply (subst zdiv_zadd1_eq) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
262 |
apply (simp add: even_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
263 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
264 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
265 |
lemma div_Suc: "Suc a div c = a div c + Suc 0 div c + |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
266 |
(a mod c + Suc 0 mod c) div c" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
267 |
apply (subgoal_tac "Suc a = a + Suc 0") |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
268 |
apply (erule ssubst) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
269 |
apply (rule div_add1_eq, simp) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
270 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
271 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
272 |
lemma even_nat_plus_one_div_two: "even (x::nat) ==> |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
273 |
(Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
274 |
apply (subst div_Suc) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
275 |
apply (simp add: even_nat_equiv_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
276 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
277 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
278 |
lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
279 |
(Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
280 |
apply (subst div_Suc) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
281 |
apply (simp add: odd_nat_equiv_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
282 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
283 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
284 |
end |