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1 (* Title: FOLP/ex/nat.ML |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1992 University of Cambridge |
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5 |
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6 Examples for the manual "Introduction to Isabelle" |
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7 |
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8 Proofs about the natural numbers |
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9 |
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10 To generate similar output to manual, execute these commands: |
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11 Pretty.setmargin 72; print_depth 0; |
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12 *) |
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13 |
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14 open Nat; |
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15 |
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16 goal Nat.thy "?p : ~ (Suc(k) = k)"; |
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17 by (res_inst_tac [("n","k")] induct 1); |
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18 by (rtac notI 1); |
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19 by (etac Suc_neq_0 1); |
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20 by (rtac notI 1); |
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21 by (etac notE 1); |
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22 by (etac Suc_inject 1); |
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23 val Suc_n_not_n = result(); |
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24 |
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25 |
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26 goal Nat.thy "?p : (k+m)+n = k+(m+n)"; |
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27 prths ([induct] RL [topthm()]); (*prints all 14 next states!*) |
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28 by (rtac induct 1); |
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29 back(); |
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30 back(); |
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31 back(); |
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32 back(); |
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33 back(); |
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34 back(); |
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35 |
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36 goalw Nat.thy [add_def] "?p : 0+n = n"; |
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37 by (rtac rec_0 1); |
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38 val add_0 = result(); |
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39 |
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40 goalw Nat.thy [add_def] "?p : Suc(m)+n = Suc(m+n)"; |
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41 by (rtac rec_Suc 1); |
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42 val add_Suc = result(); |
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43 |
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44 (* |
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45 val nat_congs = mk_congs Nat.thy ["Suc", "op +"]; |
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46 prths nat_congs; |
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47 *) |
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48 val prems = goal Nat.thy "p: x=y ==> ?p : Suc(x) = Suc(y)"; |
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49 by (resolve_tac (prems RL [subst]) 1); |
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50 by (rtac refl 1); |
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51 val Suc_cong = result(); |
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52 |
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53 val prems = goal Nat.thy "[| p : a=x; q: b=y |] ==> ?p : a+b=x+y"; |
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54 by (resolve_tac (prems RL [subst]) 1 THEN resolve_tac (prems RL [subst]) 1 THEN |
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55 rtac refl 1); |
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56 val Plus_cong = result(); |
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57 |
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58 val nat_congs = [Suc_cong,Plus_cong]; |
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59 |
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60 |
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61 val add_ss = FOLP_ss addcongs nat_congs |
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62 addrews [add_0, add_Suc]; |
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63 |
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64 goal Nat.thy "?p : (k+m)+n = k+(m+n)"; |
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65 by (res_inst_tac [("n","k")] induct 1); |
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66 by (SIMP_TAC add_ss 1); |
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67 by (ASM_SIMP_TAC add_ss 1); |
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68 val add_assoc = result(); |
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69 |
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70 goal Nat.thy "?p : m+0 = m"; |
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71 by (res_inst_tac [("n","m")] induct 1); |
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72 by (SIMP_TAC add_ss 1); |
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73 by (ASM_SIMP_TAC add_ss 1); |
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74 val add_0_right = result(); |
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75 |
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76 goal Nat.thy "?p : m+Suc(n) = Suc(m+n)"; |
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77 by (res_inst_tac [("n","m")] induct 1); |
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78 by (ALLGOALS (ASM_SIMP_TAC add_ss)); |
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79 val add_Suc_right = result(); |
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80 |
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81 (*mk_typed_congs appears not to work with FOLP's version of subst*) |
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82 |