1 (* Title: HOL/ex/Induction_Schema.thy |
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2 Author: Alexander Krauss, TU Muenchen |
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3 *) |
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4 |
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5 section \<open>Examples of automatically derived induction rules\<close> |
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6 |
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7 theory Induction_Schema |
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8 imports Main |
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9 begin |
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10 |
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11 subsection \<open>Some simple induction principles on nat\<close> |
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12 |
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13 lemma nat_standard_induct: (* cf. Nat.thy *) |
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14 "\<lbrakk>P 0; \<And>n. P n \<Longrightarrow> P (Suc n)\<rbrakk> \<Longrightarrow> P x" |
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15 by induction_schema (pat_completeness, lexicographic_order) |
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16 |
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17 lemma nat_induct2: |
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18 "\<lbrakk> P 0; P (Suc 0); \<And>k. P k ==> P (Suc k) ==> P (Suc (Suc k)) \<rbrakk> |
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19 \<Longrightarrow> P n" |
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20 by induction_schema (pat_completeness, lexicographic_order) |
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21 |
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22 lemma minus_one_induct: |
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23 "\<lbrakk>\<And>n::nat. (n \<noteq> 0 \<Longrightarrow> P (n - 1)) \<Longrightarrow> P n\<rbrakk> \<Longrightarrow> P x" |
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24 by induction_schema (pat_completeness, lexicographic_order) |
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25 |
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26 theorem diff_induct: (* cf. Nat.thy *) |
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27 "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==> |
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28 (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n" |
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29 by induction_schema (pat_completeness, lexicographic_order) |
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30 |
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31 lemma list_induct2': (* cf. List.thy *) |
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32 "\<lbrakk> P [] []; |
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33 \<And>x xs. P (x#xs) []; |
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34 \<And>y ys. P [] (y#ys); |
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35 \<And>x xs y ys. P xs ys \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk> |
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36 \<Longrightarrow> P xs ys" |
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37 by induction_schema (pat_completeness, lexicographic_order) |
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38 |
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39 theorem even_odd_induct: |
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40 assumes "R 0" |
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41 assumes "Q 0" |
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42 assumes "\<And>n. Q n \<Longrightarrow> R (Suc n)" |
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43 assumes "\<And>n. R n \<Longrightarrow> Q (Suc n)" |
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44 shows "R n" "Q n" |
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45 using assms |
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46 by induction_schema (pat_completeness+, lexicographic_order) |
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47 |
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48 end |
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