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1 (* Title: Substitutions/setplus.ML |
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2 Author: Martin Coen, Cambridge University Computer Laboratory |
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3 Copyright 1993 University of Cambridge |
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4 |
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5 For setplus.thy. |
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6 Properties of subsets and empty sets. |
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7 *) |
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8 |
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9 open Setplus; |
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10 |
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11 (*********) |
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12 |
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13 (*** Rules for subsets ***) |
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14 |
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15 goal Set.thy "A <= B = (! t.t:A --> t:B)"; |
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16 by (fast_tac set_cs 1); |
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17 qed "subset_iff"; |
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18 |
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19 goalw Setplus.thy [ssubset_def] "A < B = ((A <= B) & ~(A=B))"; |
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20 by (rtac refl 1); |
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21 qed "ssubset_iff"; |
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22 |
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23 goal Setplus.thy "((A::'a set) <= B) = ((A < B) | (A=B))"; |
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24 by (simp_tac (set_ss addsimps [ssubset_iff]) 1); |
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25 by (fast_tac set_cs 1); |
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26 qed "subseteq_iff_subset_eq"; |
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27 |
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28 (*Rule in Modus Ponens style*) |
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29 goal Setplus.thy "A < B --> c:A --> c:B"; |
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30 by (simp_tac (set_ss addsimps [ssubset_iff]) 1); |
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31 by (fast_tac set_cs 1); |
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32 qed "ssubsetD"; |
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33 |
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34 (*********) |
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35 |
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36 goalw Setplus.thy [empty_def] "~ a : {}"; |
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37 by (fast_tac set_cs 1); |
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38 qed "not_in_empty"; |
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39 |
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40 goalw Setplus.thy [empty_def] "(A = {}) = (ALL a.~ a:A)"; |
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41 by (fast_tac (set_cs addIs [set_ext]) 1); |
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42 qed "empty_iff"; |
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43 |
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44 |
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45 (*********) |
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46 |
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47 goal Set.thy "(~A=B) = ((? x.x:A & ~x:B) | (? x.~x:A & x:B))"; |
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48 by (fast_tac (set_cs addIs [set_ext]) 1); |
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49 qed "not_equal_iff"; |
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50 |
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51 (*********) |
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52 |
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53 val setplus_rews = [ssubset_iff,not_in_empty,empty_iff]; |
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54 |
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55 (*********) |
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56 |
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57 (*Case analysis for rewriting; P also gets rewritten*) |
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58 val [prem1,prem2] = goal HOL.thy "[| P-->Q; ~P-->Q |] ==> Q"; |
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59 by (rtac (excluded_middle RS disjE) 1); |
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60 by (etac (prem2 RS mp) 1); |
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61 by (etac (prem1 RS mp) 1); |
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62 qed "imp_excluded_middle"; |
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63 |
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64 fun imp_excluded_middle_tac s = res_inst_tac [("P",s)] imp_excluded_middle; |