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 |
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7 Goal "?p : ~ (Suc(k) = k)"; |
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8 by (res_inst_tac [("n","k")] induct 1); |
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9 by (rtac notI 1); |
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10 by (etac Suc_neq_0 1); |
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11 by (rtac notI 1); |
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12 by (etac notE 1); |
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13 by (etac Suc_inject 1); |
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14 val Suc_n_not_n = result(); |
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15 |
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16 |
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17 Goal "?p : (k+m)+n = k+(m+n)"; |
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18 prths ([induct] RL [topthm()]); (*prints all 14 next states!*) |
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19 by (rtac induct 1); |
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20 back(); |
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21 back(); |
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22 back(); |
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23 back(); |
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24 back(); |
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25 back(); |
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26 |
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27 Goalw [add_def] "?p : 0+n = n"; |
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28 by (rtac rec_0 1); |
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29 val add_0 = result(); |
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30 |
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31 Goalw [add_def] "?p : Suc(m)+n = Suc(m+n)"; |
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32 by (rtac rec_Suc 1); |
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33 val add_Suc = result(); |
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34 |
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35 (* |
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36 val nat_congs = mk_congs (the_context ()) ["Suc", "op +"]; |
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37 prths nat_congs; |
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38 *) |
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39 val prems = goal (the_context ()) "p: x=y ==> ?p : Suc(x) = Suc(y)"; |
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40 by (resolve_tac (prems RL [subst]) 1); |
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41 by (rtac refl 1); |
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42 val Suc_cong = result(); |
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43 |
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44 val prems = goal (the_context ()) "[| p : a=x; q: b=y |] ==> ?p : a+b=x+y"; |
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45 by (resolve_tac (prems RL [subst]) 1 THEN resolve_tac (prems RL [subst]) 1 THEN |
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46 rtac refl 1); |
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47 val Plus_cong = result(); |
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48 |
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49 val nat_congs = [Suc_cong,Plus_cong]; |
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50 |
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51 |
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52 val add_ss = FOLP_ss addcongs nat_congs |
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53 addrews [add_0, add_Suc]; |
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54 |
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55 Goal "?p : (k+m)+n = k+(m+n)"; |
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56 by (res_inst_tac [("n","k")] induct 1); |
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57 by (SIMP_TAC add_ss 1); |
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58 by (ASM_SIMP_TAC add_ss 1); |
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59 val add_assoc = result(); |
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60 |
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61 Goal "?p : m+0 = m"; |
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62 by (res_inst_tac [("n","m")] induct 1); |
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63 by (SIMP_TAC add_ss 1); |
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64 by (ASM_SIMP_TAC add_ss 1); |
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65 val add_0_right = result(); |
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66 |
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67 Goal "?p : m+Suc(n) = Suc(m+n)"; |
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68 by (res_inst_tac [("n","m")] induct 1); |
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69 by (ALLGOALS (ASM_SIMP_TAC add_ss)); |
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70 val add_Suc_right = result(); |
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71 |
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72 (*mk_typed_congs appears not to work with FOLP's version of subst*) |
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