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1 |
1 |
2 header {* \title{}\subsection{Example: The ordered ring of integers} *} |
2 header {* \title{}\subsection{Example: The ordered ring of integers} *} |
3 |
3 |
4 theory Ring_and_Field_Example = Ring_and_Field: |
4 theory Ring_and_Field_Example = Main: |
5 |
5 |
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6 text{*The Integers Form an Ordered Ring*} |
6 instance int :: ordered_ring |
7 instance int :: ordered_ring |
7 proof |
8 proof |
8 fix i j k :: int |
9 fix i j k :: int |
9 show "(i + j) + k = i + (j + k)" by simp |
10 show "(i + j) + k = i + (j + k)" by simp |
10 show "i + j = j + i" by simp |
11 show "i + j = j + i" by simp |
13 show "i - j = i + (-j)" by simp |
14 show "i - j = i + (-j)" by simp |
14 show "(i * j) * k = i * (j * k)" by simp |
15 show "(i * j) * k = i * (j * k)" by simp |
15 show "i * j = j * i" by simp |
16 show "i * j = j * i" by simp |
16 show "1 * i = i" by simp |
17 show "1 * i = i" by simp |
17 show "(i + j) * k = i * k + j * k" by (simp add: int_distrib) |
18 show "(i + j) * k = i * k + j * k" by (simp add: int_distrib) |
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19 show "0 \<noteq> (1::int)" by simp |
18 show "i \<le> j ==> k + i \<le> k + j" by simp |
20 show "i \<le> j ==> k + i \<le> k + j" by simp |
19 show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: zmult_zless_mono2) |
21 show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: zmult_zless_mono2) |
20 show "\<bar>i\<bar> = (if i < 0 then -i else i)" by (simp only: zabs_def) |
22 show "\<bar>i\<bar> = (if i < 0 then -i else i)" by (simp only: zabs_def) |
21 qed |
23 qed |
22 |
24 |