1 header {* The main lemma about NUM and the Second Fixed Point Theorem *} |
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2 |
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3 theory Theorem imports Consts begin |
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4 |
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5 lemmas [simp] = b3[OF bI] b1 b4 b5 supp_NUM[unfolded NUM_def supp_ltgt] NUM_def lam.fresh[unfolded fresh_def] fresh_def b6 |
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6 lemmas app = Ltgt1_app |
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7 |
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8 lemma NUM: |
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9 shows "NUM \<cdot> \<lbrace>M\<rbrace> \<approx> \<lbrace>\<lbrace>M\<rbrace>\<rbrace>" |
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10 proof (induct M rule: lam.induct) |
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11 case (Var n) |
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12 have "NUM \<cdot> \<lbrace>Var n\<rbrace> = NUM \<cdot> (VAR \<cdot> Var n)" by simp |
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13 also have "... = \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright> \<cdot> (VAR \<cdot> Var n)" by simp |
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14 also have "... \<approx> VAR \<cdot> Var n \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using app . |
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15 also have "... \<approx> \<guillemotleft>[A1,A2,A3]\<guillemotright> \<cdot> Umn 2 2 \<cdot> Var n \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using VAR_app . |
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16 also have "... \<approx> A1 \<cdot> Var n \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using U_app by simp |
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17 also have "... \<approx> F1 \<cdot> Var n" using A_app(1) . |
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18 also have "... \<approx> APP \<cdot> \<lbrace>VAR\<rbrace> \<cdot> (VAR \<cdot> Var n)" using F_app(1) . |
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19 also have "... = \<lbrace>\<lbrace>Var n\<rbrace>\<rbrace>" by simp |
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20 finally show "NUM \<cdot> \<lbrace>Var n\<rbrace> \<approx> \<lbrace>\<lbrace>Var n\<rbrace>\<rbrace>". |
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21 next |
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22 case (App M N) |
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23 assume IH: "NUM \<cdot> \<lbrace>M\<rbrace> \<approx> \<lbrace>\<lbrace>M\<rbrace>\<rbrace>" "NUM \<cdot> \<lbrace>N\<rbrace> \<approx> \<lbrace>\<lbrace>N\<rbrace>\<rbrace>" |
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24 have "NUM \<cdot> \<lbrace>M \<cdot> N\<rbrace> = NUM \<cdot> (APP \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace>)" by simp |
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25 also have "... = \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright> \<cdot> (APP \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace>)" by simp |
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26 also have "... \<approx> APP \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using app . |
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27 also have "... \<approx> \<guillemotleft>[A1,A2,A3]\<guillemotright> \<cdot> Umn 2 1 \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using APP_app . |
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28 also have "... \<approx> A2 \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using U_app by simp |
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29 also have "... \<approx> F2 \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace> \<cdot> NUM" using A_app(2) by simp |
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30 also have "... \<approx> APP \<cdot> (APP \<cdot> \<lbrace>APP\<rbrace> \<cdot> (NUM \<cdot> \<lbrace>M\<rbrace>)) \<cdot> (NUM \<cdot> \<lbrace>N\<rbrace>)" using F_app(2) . |
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31 also have "... \<approx> APP \<cdot> (APP \<cdot> \<lbrace>APP\<rbrace> \<cdot> (\<lbrace>\<lbrace>M\<rbrace>\<rbrace>)) \<cdot> (NUM \<cdot> \<lbrace>N\<rbrace>)" using IH by simp |
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32 also have "... \<approx> \<lbrace>\<lbrace>M \<cdot> N\<rbrace>\<rbrace>" using IH by simp |
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33 finally show "NUM \<cdot> \<lbrace>M \<cdot> N\<rbrace> \<approx> \<lbrace>\<lbrace>M \<cdot> N\<rbrace>\<rbrace>". |
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34 next |
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35 case (Lam x P) |
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36 assume IH: "NUM \<cdot> \<lbrace>P\<rbrace> \<approx> \<lbrace>\<lbrace>P\<rbrace>\<rbrace>" |
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37 have "NUM \<cdot> \<lbrace>\<integral> x. P\<rbrace> = NUM \<cdot> (Abs \<cdot> \<integral> x. \<lbrace>P\<rbrace>)" by simp |
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38 also have "... = \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright> \<cdot> (Abs \<cdot> \<integral> x. \<lbrace>P\<rbrace>)" by simp |
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39 also have "... \<approx> Abs \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using app . |
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40 also have "... \<approx> \<guillemotleft>[A1,A2,A3]\<guillemotright> \<cdot> Umn 2 0 \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using Abs_app . |
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41 also have "... \<approx> A3 \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> \<guillemotleft>[A1,A2,A3]\<guillemotright>" using U_app by simp |
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42 also have "... \<approx> F3 \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright>" using A_app(3) . |
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43 also have "... = F3 \<cdot> (\<integral> x. \<lbrace>P\<rbrace>) \<cdot> NUM" by simp |
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44 also have "... \<approx> APP \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> \<integral> x. (NUM \<cdot> ((\<integral> x. \<lbrace>P\<rbrace>) \<cdot> Var x)))" by (rule F3_app) simp_all |
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45 also have "... \<approx> APP \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> \<integral> x. (NUM \<cdot> \<lbrace>P\<rbrace>))" using beta_app by simp |
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46 also have "... \<approx> APP \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> \<integral> x. \<lbrace>\<lbrace>P\<rbrace>\<rbrace>)" using IH by simp |
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47 also have "... = \<lbrace>\<lbrace>\<integral> x. P\<rbrace>\<rbrace>" by simp |
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48 finally show "NUM \<cdot> \<lbrace>\<integral> x. P\<rbrace> \<approx> \<lbrace>\<lbrace>\<integral> x. P\<rbrace>\<rbrace>" . |
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49 qed |
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50 |
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51 lemmas [simp] = Ap NUM |
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52 lemmas [simp del] = fresh_def NUM_def |
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53 |
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54 theorem SFP: |
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55 fixes F :: lam |
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56 shows "\<exists>X. X \<approx> F \<cdot> \<lbrace>X\<rbrace>" |
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57 proof - |
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58 obtain x :: name where [simp]:"atom x \<sharp> F" using obtain_fresh by blast |
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59 def W \<equiv> "\<integral>x. (F \<cdot> (APP \<cdot> Var x \<cdot> (NUM \<cdot> Var x)))" |
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60 def X \<equiv> "W \<cdot> \<lbrace>W\<rbrace>" |
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61 have a: "X = W \<cdot> \<lbrace>W\<rbrace>" unfolding X_def .. |
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62 also have "... = (\<integral>x. (F \<cdot> (APP \<cdot> Var x \<cdot> (NUM \<cdot> Var x)))) \<cdot> \<lbrace>W\<rbrace>" unfolding W_def .. |
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63 also have "... \<approx> F \<cdot> (APP \<cdot> \<lbrace>W\<rbrace> \<cdot> (NUM \<cdot> \<lbrace>W\<rbrace>))" by simp |
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64 also have "... \<approx> F \<cdot> (APP \<cdot> \<lbrace>W\<rbrace> \<cdot> \<lbrace>\<lbrace>W\<rbrace>\<rbrace>)" by simp |
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65 also have "... \<approx> F \<cdot> \<lbrace>W \<cdot> \<lbrace>W\<rbrace>\<rbrace>" by simp |
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66 also have "... = F \<cdot> \<lbrace>X\<rbrace>" unfolding X_def .. |
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67 finally show ?thesis by blast |
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68 qed |
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69 |
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70 end |
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