author | fleuriot |
Thu, 21 Sep 2000 12:17:11 +0200 | |
changeset 10045 | c76b73e16711 |
child 10201 | b52140d1a7bc |
permissions | -rw-r--r-- |
10045
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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(* Title : HyperNat.ML |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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Author : Jacques D. Fleuriot |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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Copyright : 1998 University of Cambridge |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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Description : Explicit construction of hypernaturals using |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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ultrafilters |
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New theories: construction of hypernaturals, nonstandard extensions,
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*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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|
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New theories: construction of hypernaturals, nonstandard extensions,
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(*------------------------------------------------------------------------ |
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New theories: construction of hypernaturals, nonstandard extensions,
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Properties of hypnatrel |
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New theories: construction of hypernaturals, nonstandard extensions,
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------------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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|
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New theories: construction of hypernaturals, nonstandard extensions,
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(** Proving that hyprel is an equivalence relation **) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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(** Natural deduction for hypnatrel - similar to hyprel! **) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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Goalw [hypnatrel_def] |
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New theories: construction of hypernaturals, nonstandard extensions,
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"((X,Y): hypnatrel) = ({n. X n = Y n}: FreeUltrafilterNat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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by (Fast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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18 |
qed "hypnatrel_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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19 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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20 |
Goalw [hypnatrel_def] |
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New theories: construction of hypernaturals, nonstandard extensions,
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"{n. X n = Y n}: FreeUltrafilterNat ==> (X,Y): hypnatrel"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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22 |
by (Fast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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qed "hypnatrelI"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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24 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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25 |
Goalw [hypnatrel_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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"p: hypnatrel --> (EX X Y. \ |
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New theories: construction of hypernaturals, nonstandard extensions,
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\ p = (X,Y) & {n. X n = Y n} : FreeUltrafilterNat)"; |
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New theories: construction of hypernaturals, nonstandard extensions,
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28 |
by (Fast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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29 |
qed "hypnatrelE_lemma"; |
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New theories: construction of hypernaturals, nonstandard extensions,
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30 |
|
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New theories: construction of hypernaturals, nonstandard extensions,
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val [major,minor] = goal thy |
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"[| p: hypnatrel; \ |
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\ !!X Y. [| p = (X,Y); {n. X n = Y n}: FreeUltrafilterNat\ |
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New theories: construction of hypernaturals, nonstandard extensions,
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\ |] ==> Q |] ==> Q"; |
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New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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35 |
by (cut_facts_tac [major RS (hypnatrelE_lemma RS mp)] 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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36 |
by (REPEAT (eresolve_tac [asm_rl,exE,conjE,minor] 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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qed "hypnatrelE"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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38 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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39 |
AddSIs [hypnatrelI]; |
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New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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40 |
AddSEs [hypnatrelE]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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41 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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Goalw [hypnatrel_def] "(x,x): hypnatrel"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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43 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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qed "hypnatrel_refl"; |
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New theories: construction of hypernaturals, nonstandard extensions,
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|
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New theories: construction of hypernaturals, nonstandard extensions,
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Goalw [hypnatrel_def] "(x,y): hypnatrel --> (y,x):hypnatrel"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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47 |
by (auto_tac (claset() addIs [lemma_perm RS subst],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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qed_spec_mp "hypnatrel_sym"; |
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New theories: construction of hypernaturals, nonstandard extensions,
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49 |
|
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New theories: construction of hypernaturals, nonstandard extensions,
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Goalw [hypnatrel_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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"(x,y): hypnatrel --> (y,z):hypnatrel --> (x,z):hypnatrel"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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52 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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53 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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54 |
qed_spec_mp "hypnatrel_trans"; |
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New theories: construction of hypernaturals, nonstandard extensions,
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55 |
|
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New theories: construction of hypernaturals, nonstandard extensions,
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56 |
Goalw [equiv_def, refl_def, sym_def, trans_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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"equiv {x::nat=>nat. True} hypnatrel"; |
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New theories: construction of hypernaturals, nonstandard extensions,
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58 |
by (auto_tac (claset() addSIs [hypnatrel_refl] addSEs |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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59 |
[hypnatrel_sym,hypnatrel_trans] delrules [hypnatrelI,hypnatrelE], |
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New theories: construction of hypernaturals, nonstandard extensions,
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60 |
simpset())); |
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New theories: construction of hypernaturals, nonstandard extensions,
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61 |
qed "equiv_hypnatrel"; |
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New theories: construction of hypernaturals, nonstandard extensions,
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62 |
|
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New theories: construction of hypernaturals, nonstandard extensions,
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val equiv_hypnatrel_iff = |
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New theories: construction of hypernaturals, nonstandard extensions,
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64 |
[TrueI, TrueI] MRS |
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New theories: construction of hypernaturals, nonstandard extensions,
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([CollectI, CollectI] MRS |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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(equiv_hypnatrel RS eq_equiv_class_iff)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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67 |
|
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New theories: construction of hypernaturals, nonstandard extensions,
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68 |
Goalw [hypnat_def,hypnatrel_def,quotient_def] "hypnatrel^^{x}:hypnat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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69 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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70 |
qed "hypnatrel_in_hypnat"; |
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New theories: construction of hypernaturals, nonstandard extensions,
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71 |
|
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New theories: construction of hypernaturals, nonstandard extensions,
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Goal "inj_on Abs_hypnat hypnat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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|
73 |
by (rtac inj_on_inverseI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
diff
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|
74 |
by (etac Abs_hypnat_inverse 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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75 |
qed "inj_on_Abs_hypnat"; |
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New theories: construction of hypernaturals, nonstandard extensions,
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|
76 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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77 |
Addsimps [equiv_hypnatrel_iff,inj_on_Abs_hypnat RS inj_on_iff, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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78 |
hypnatrel_iff, hypnatrel_in_hypnat, Abs_hypnat_inverse]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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79 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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80 |
Addsimps [equiv_hypnatrel RS eq_equiv_class_iff]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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|
81 |
val eq_hypnatrelD = equiv_hypnatrel RSN (2,eq_equiv_class); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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82 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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83 |
Goal "inj(Rep_hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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|
84 |
by (rtac inj_inverseI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
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|
85 |
by (rtac Rep_hypnat_inverse 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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86 |
qed "inj_Rep_hypnat"; |
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New theories: construction of hypernaturals, nonstandard extensions,
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87 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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88 |
Goalw [hypnatrel_def] "x: hypnatrel ^^ {x}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
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|
89 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
90 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
diff
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|
91 |
qed "lemma_hypnatrel_refl"; |
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New theories: construction of hypernaturals, nonstandard extensions,
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|
92 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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|
93 |
Addsimps [lemma_hypnatrel_refl]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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diff
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|
94 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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95 |
Goalw [hypnat_def] "{} ~: hypnat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
96 |
by (auto_tac (claset() addSEs [quotientE],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
diff
changeset
|
97 |
qed "hypnat_empty_not_mem"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
diff
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|
98 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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99 |
Addsimps [hypnat_empty_not_mem]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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100 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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101 |
Goal "Rep_hypnat x ~= {}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
diff
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|
102 |
by (cut_inst_tac [("x","x")] Rep_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
103 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
diff
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|
104 |
qed "Rep_hypnat_nonempty"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
diff
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|
105 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
diff
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|
106 |
Addsimps [Rep_hypnat_nonempty]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
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|
107 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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|
108 |
(*------------------------------------------------------------------------ |
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New theories: construction of hypernaturals, nonstandard extensions,
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109 |
hypnat_of_nat: the injection from nat to hypnat |
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New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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110 |
------------------------------------------------------------------------*) |
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New theories: construction of hypernaturals, nonstandard extensions,
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parents:
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|
111 |
Goal "inj(hypnat_of_nat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
112 |
by (rtac injI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
113 |
by (rewtac hypnat_of_nat_def); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
114 |
by (dtac (inj_on_Abs_hypnat RS inj_onD) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
115 |
by (REPEAT (rtac hypnatrel_in_hypnat 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
116 |
by (dtac eq_equiv_class 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
117 |
by (rtac equiv_hypnatrel 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
118 |
by (Fast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
119 |
by (rtac ccontr 1 THEN rotate_tac 1 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
120 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
121 |
qed "inj_hypnat_of_nat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
122 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
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|
123 |
val [prem] = goal thy |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
124 |
"(!!x. z = Abs_hypnat(hypnatrel^^{x}) ==> P) ==> P"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
125 |
by (res_inst_tac [("x1","z")] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
126 |
(rewrite_rule [hypnat_def] Rep_hypnat RS quotientE) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
127 |
by (dres_inst_tac [("f","Abs_hypnat")] arg_cong 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
128 |
by (res_inst_tac [("x","x")] prem 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
129 |
by (asm_full_simp_tac (simpset() addsimps [Rep_hypnat_inverse]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
130 |
qed "eq_Abs_hypnat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
131 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
132 |
(*----------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
133 |
Addition for hyper naturals: hypnat_add |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
134 |
-----------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
135 |
Goalw [congruent2_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
136 |
"congruent2 hypnatrel (%X Y. hypnatrel^^{%n. X n + Y n})"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
137 |
by (safe_tac (claset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
138 |
by (ALLGOALS(Fuf_tac)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
139 |
qed "hypnat_add_congruent2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
140 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
141 |
Goalw [hypnat_add_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
142 |
"Abs_hypnat(hypnatrel^^{%n. X n}) + Abs_hypnat(hypnatrel^^{%n. Y n}) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
143 |
\ Abs_hypnat(hypnatrel^^{%n. X n + Y n})"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
144 |
by (asm_simp_tac |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
145 |
(simpset() addsimps [[equiv_hypnatrel, hypnat_add_congruent2] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
146 |
MRS UN_equiv_class2]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
147 |
qed "hypnat_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
148 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
149 |
Goal "(z::hypnat) + w = w + z"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
150 |
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
151 |
by (res_inst_tac [("z","w")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
152 |
by (asm_simp_tac (simpset() addsimps (add_ac @ [hypnat_add])) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
153 |
qed "hypnat_add_commute"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
154 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
155 |
Goal "((z1::hypnat) + z2) + z3 = z1 + (z2 + z3)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
156 |
by (res_inst_tac [("z","z1")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
157 |
by (res_inst_tac [("z","z2")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
158 |
by (res_inst_tac [("z","z3")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
159 |
by (asm_simp_tac (simpset() addsimps [hypnat_add,add_assoc]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
160 |
qed "hypnat_add_assoc"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
161 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
162 |
(*For AC rewriting*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
163 |
Goal "(x::hypnat)+(y+z)=y+(x+z)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
164 |
by (rtac (hypnat_add_commute RS trans) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
165 |
by (rtac (hypnat_add_assoc RS trans) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
166 |
by (rtac (hypnat_add_commute RS arg_cong) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
167 |
qed "hypnat_add_left_commute"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
168 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
169 |
(* hypnat addition is an AC operator *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
170 |
val hypnat_add_ac = [hypnat_add_assoc,hypnat_add_commute, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
171 |
hypnat_add_left_commute]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
172 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
173 |
Goalw [hypnat_zero_def] "(0::hypnat) + z = z"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
174 |
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
175 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
176 |
[hypnat_add]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
177 |
qed "hypnat_add_zero_left"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
178 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
179 |
Goal "z + (0::hypnat) = z"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
180 |
by (simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
181 |
[hypnat_add_zero_left,hypnat_add_commute]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
182 |
qed "hypnat_add_zero_right"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
183 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
184 |
Addsimps [hypnat_add_zero_left,hypnat_add_zero_right]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
185 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
186 |
(*----------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
187 |
Subtraction for hyper naturals: hypnat_minus |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
188 |
-----------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
189 |
Goalw [congruent2_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
190 |
"congruent2 hypnatrel (%X Y. hypnatrel^^{%n. X n - Y n})"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
191 |
by (safe_tac (claset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
192 |
by (ALLGOALS(Fuf_tac)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
193 |
qed "hypnat_minus_congruent2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
194 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
195 |
Goalw [hypnat_minus_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
196 |
"Abs_hypnat(hypnatrel^^{%n. X n}) - Abs_hypnat(hypnatrel^^{%n. Y n}) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
197 |
\ Abs_hypnat(hypnatrel^^{%n. X n - Y n})"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
198 |
by (asm_simp_tac |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
199 |
(simpset() addsimps [[equiv_hypnatrel, hypnat_minus_congruent2] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
200 |
MRS UN_equiv_class2]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
201 |
qed "hypnat_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
202 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
203 |
Goalw [hypnat_zero_def] "z - z = (0::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
204 |
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
205 |
by (asm_full_simp_tac (simpset() addsimps [hypnat_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
206 |
qed "hypnat_minus_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
207 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
208 |
Goalw [hypnat_zero_def] "(0::hypnat) - n = 0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
209 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
210 |
by (asm_full_simp_tac (simpset() addsimps [hypnat_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
211 |
qed "hypnat_diff_0_eq_0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
212 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
213 |
Addsimps [hypnat_minus_zero,hypnat_diff_0_eq_0]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
214 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
215 |
Goalw [hypnat_zero_def] "(m+n = (0::hypnat)) = (m=0 & n=0)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
216 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
217 |
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
218 |
by (auto_tac (claset() addIs [FreeUltrafilterNat_subset] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
219 |
addDs [FreeUltrafilterNat_Int], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
220 |
simpset() addsimps [hypnat_add] )); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
221 |
qed "hypnat_add_is_0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
222 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
223 |
AddIffs [hypnat_add_is_0]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
224 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
225 |
Goal "!!i::hypnat. i-j-k = i - (j+k)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
226 |
by (res_inst_tac [("z","i")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
227 |
by (res_inst_tac [("z","j")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
228 |
by (res_inst_tac [("z","k")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
229 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
230 |
[hypnat_minus,hypnat_add,diff_diff_left]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
231 |
qed "hypnat_diff_diff_left"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
232 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
233 |
Goal "!!i::hypnat. i-j-k = i-k-j"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
234 |
by (simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
235 |
[hypnat_diff_diff_left, hypnat_add_commute]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
236 |
qed "hypnat_diff_commute"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
237 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
238 |
Goal "!!n::hypnat. (n+m) - n = m"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
239 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
240 |
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
241 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
242 |
[hypnat_minus,hypnat_add]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
243 |
qed "hypnat_diff_add_inverse"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
244 |
Addsimps [hypnat_diff_add_inverse]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
245 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
246 |
Goal "!!n::hypnat.(m+n) - n = m"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
247 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
248 |
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
249 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
250 |
[hypnat_minus,hypnat_add]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
251 |
qed "hypnat_diff_add_inverse2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
252 |
Addsimps [hypnat_diff_add_inverse2]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
253 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
254 |
Goal "!!k::hypnat. (k+m) - (k+n) = m - n"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
255 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
256 |
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
257 |
by (res_inst_tac [("z","k")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
258 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
259 |
[hypnat_minus,hypnat_add]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
260 |
qed "hypnat_diff_cancel"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
261 |
Addsimps [hypnat_diff_cancel]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
262 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
263 |
Goal "!!m::hypnat. (m+k) - (n+k) = m - n"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
264 |
val hypnat_add_commute_k = read_instantiate [("w","k")] hypnat_add_commute; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
265 |
by (asm_simp_tac (simpset() addsimps ([hypnat_add_commute_k])) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
266 |
qed "hypnat_diff_cancel2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
267 |
Addsimps [hypnat_diff_cancel2]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
268 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
269 |
Goalw [hypnat_zero_def] "!!n::hypnat. n - (n+m) = (0::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
270 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
271 |
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
272 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
273 |
[hypnat_minus,hypnat_add]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
274 |
qed "hypnat_diff_add_0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
275 |
Addsimps [hypnat_diff_add_0]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
276 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
277 |
(*----------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
278 |
Multiplication for hyper naturals: hypnat_mult |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
279 |
-----------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
280 |
Goalw [congruent2_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
281 |
"congruent2 hypnatrel (%X Y. hypnatrel^^{%n. X n * Y n})"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
282 |
by (safe_tac (claset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
283 |
by (ALLGOALS(Fuf_tac)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
284 |
qed "hypnat_mult_congruent2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
285 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
286 |
Goalw [hypnat_mult_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
287 |
"Abs_hypnat(hypnatrel^^{%n. X n}) * Abs_hypnat(hypnatrel^^{%n. Y n}) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
288 |
\ Abs_hypnat(hypnatrel^^{%n. X n * Y n})"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
289 |
by (asm_simp_tac |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
290 |
(simpset() addsimps [[equiv_hypnatrel,hypnat_mult_congruent2] MRS |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
291 |
UN_equiv_class2]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
292 |
qed "hypnat_mult"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
293 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
294 |
Goal "(z::hypnat) * w = w * z"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
295 |
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
296 |
by (res_inst_tac [("z","w")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
297 |
by (asm_simp_tac (simpset() addsimps ([hypnat_mult] @ mult_ac)) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
298 |
qed "hypnat_mult_commute"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
299 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
300 |
Goal "((z1::hypnat) * z2) * z3 = z1 * (z2 * z3)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
301 |
by (res_inst_tac [("z","z1")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
302 |
by (res_inst_tac [("z","z2")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
303 |
by (res_inst_tac [("z","z3")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
304 |
by (asm_simp_tac (simpset() addsimps [hypnat_mult,mult_assoc]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
305 |
qed "hypnat_mult_assoc"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
306 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
307 |
qed_goal "hypnat_mult_left_commute" thy |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
308 |
"(z1::hypnat) * (z2 * z3) = z2 * (z1 * z3)" |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
309 |
(fn _ => [rtac (hypnat_mult_commute RS trans) 1, rtac (hypnat_mult_assoc RS trans) 1, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
310 |
rtac (hypnat_mult_commute RS arg_cong) 1]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
311 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
312 |
(* hypnat multiplication is an AC operator *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
313 |
val hypnat_mult_ac = [hypnat_mult_assoc, hypnat_mult_commute, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
314 |
hypnat_mult_left_commute]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
315 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
316 |
Goalw [hypnat_one_def] "1hn * z = z"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
317 |
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
318 |
by (asm_full_simp_tac (simpset() addsimps [hypnat_mult]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
319 |
qed "hypnat_mult_1"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
320 |
Addsimps [hypnat_mult_1]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
321 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
322 |
Goal "z * 1hn = z"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
323 |
by (simp_tac (simpset() addsimps [hypnat_mult_commute]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
324 |
qed "hypnat_mult_1_right"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
325 |
Addsimps [hypnat_mult_1_right]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
326 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
327 |
Goalw [hypnat_zero_def] "(0::hypnat) * z = 0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
328 |
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
329 |
by (asm_full_simp_tac (simpset() addsimps [hypnat_mult]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
330 |
qed "hypnat_mult_0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
331 |
Addsimps [hypnat_mult_0]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
332 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
333 |
Goal "z * (0::hypnat) = 0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
334 |
by (simp_tac (simpset() addsimps [hypnat_mult_commute]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
335 |
qed "hypnat_mult_0_right"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
336 |
Addsimps [hypnat_mult_0_right]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
337 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
338 |
Goal "!!m::hypnat. (m - n) * k = (m * k) - (n * k)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
339 |
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
340 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
341 |
by (res_inst_tac [("z","k")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
342 |
by (asm_simp_tac (simpset() addsimps [hypnat_mult, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
343 |
hypnat_minus,diff_mult_distrib]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
344 |
qed "hypnat_diff_mult_distrib" ; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
345 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
346 |
Goal "!!m::hypnat. k * (m - n) = (k * m) - (k * n)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
347 |
val hypnat_mult_commute_k = read_instantiate [("z","k")] hypnat_mult_commute; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
348 |
by (simp_tac (simpset() addsimps [hypnat_diff_mult_distrib, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
349 |
hypnat_mult_commute_k]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
350 |
qed "hypnat_diff_mult_distrib2" ; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
351 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
352 |
(*-------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
353 |
A Few more theorems |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
354 |
-------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
355 |
qed_goal "hypnat_add_assoc_cong" thy |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
356 |
"!!z. (z::hypnat) + v = z' + v' ==> z + (v + w) = z' + (v' + w)" |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
357 |
(fn _ => [(asm_simp_tac (simpset() addsimps [hypnat_add_assoc RS sym]) 1)]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
358 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
359 |
qed_goal "hypnat_add_assoc_swap" thy |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
360 |
"(z::hypnat) + (v + w) = v + (z + w)" |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
361 |
(fn _ => [(REPEAT (ares_tac [hypnat_add_commute RS hypnat_add_assoc_cong] 1))]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
362 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
363 |
Goal "((z1::hypnat) + z2) * w = (z1 * w) + (z2 * w)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
364 |
by (res_inst_tac [("z","z1")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
365 |
by (res_inst_tac [("z","z2")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
366 |
by (res_inst_tac [("z","w")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
367 |
by (asm_simp_tac (simpset() addsimps [hypnat_mult,hypnat_add, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
368 |
add_mult_distrib]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
369 |
qed "hypnat_add_mult_distrib"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
370 |
Addsimps [hypnat_add_mult_distrib]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
371 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
372 |
val hypnat_mult_commute'= read_instantiate [("z","w")] hypnat_mult_commute; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
373 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
374 |
Goal "(w::hypnat) * (z1 + z2) = (w * z1) + (w * z2)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
375 |
by (simp_tac (simpset() addsimps [hypnat_mult_commute']) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
376 |
qed "hypnat_add_mult_distrib2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
377 |
Addsimps [hypnat_add_mult_distrib2]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
378 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
379 |
(*** (hypnat) one and zero are distinct ***) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
380 |
Goalw [hypnat_zero_def,hypnat_one_def] "(0::hypnat) ~= 1hn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
381 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
382 |
qed "hypnat_zero_not_eq_one"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
383 |
Addsimps [hypnat_zero_not_eq_one]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
384 |
Addsimps [hypnat_zero_not_eq_one RS not_sym]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
385 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
386 |
Goalw [hypnat_zero_def] "(m*n = (0::hypnat)) = (m=0 | n=0)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
387 |
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
388 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
389 |
by (auto_tac (claset() addSDs [FreeUltrafilterNat_Compl_mem], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
390 |
simpset() addsimps [hypnat_mult])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
391 |
by (ALLGOALS(Fuf_tac)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
392 |
qed "hypnat_mult_is_0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
393 |
Addsimps [hypnat_mult_is_0]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
394 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
395 |
(*------------------------------------------------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
396 |
Theorems for ordering |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
397 |
------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
398 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
399 |
(* prove introduction and elimination rules for hypnat_less *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
400 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
401 |
Goalw [hypnat_less_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
402 |
"P < (Q::hypnat) = (EX X Y. X : Rep_hypnat(P) & \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
403 |
\ Y : Rep_hypnat(Q) & \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
404 |
\ {n. X n < Y n} : FreeUltrafilterNat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
405 |
by (Fast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
406 |
qed "hypnat_less_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
407 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
408 |
Goalw [hypnat_less_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
409 |
"!!P. [| {n. X n < Y n} : FreeUltrafilterNat; \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
410 |
\ X : Rep_hypnat(P); \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
411 |
\ Y : Rep_hypnat(Q) |] ==> P < (Q::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
412 |
by (Fast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
413 |
qed "hypnat_lessI"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
414 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
415 |
Goalw [hypnat_less_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
416 |
"!! R1. [| R1 < (R2::hypnat); \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
417 |
\ !!X Y. {n. X n < Y n} : FreeUltrafilterNat ==> P; \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
418 |
\ !!X. X : Rep_hypnat(R1) ==> P; \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
419 |
\ !!Y. Y : Rep_hypnat(R2) ==> P |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
420 |
\ ==> P"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
421 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
422 |
qed "hypnat_lessE"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
423 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
424 |
Goalw [hypnat_less_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
425 |
"!!R1. R1 < (R2::hypnat) ==> (EX X Y. {n. X n < Y n} : FreeUltrafilterNat & \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
426 |
\ X : Rep_hypnat(R1) & \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
427 |
\ Y : Rep_hypnat(R2))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
428 |
by (Fast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
429 |
qed "hypnat_lessD"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
430 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
431 |
Goal "~ (R::hypnat) < R"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
432 |
by (res_inst_tac [("z","R")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
433 |
by (auto_tac (claset(),simpset() addsimps [hypnat_less_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
434 |
by (Fuf_empty_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
435 |
qed "hypnat_less_not_refl"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
436 |
Addsimps [hypnat_less_not_refl]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
437 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
438 |
bind_thm("hypnat_less_irrefl",hypnat_less_not_refl RS notE); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
439 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
440 |
qed_goal "hypnat_not_refl2" thy |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
441 |
"!!(x::hypnat). x < y ==> x ~= y" (fn _ => [Auto_tac]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
442 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
443 |
Goalw [hypnat_less_def,hypnat_zero_def] "~ n<(0::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
444 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
445 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
446 |
by (Fuf_empty_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
447 |
qed "hypnat_not_less0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
448 |
AddIffs [hypnat_not_less0]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
449 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
450 |
(* n<(0::hypnat) ==> R *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
451 |
bind_thm ("hypnat_less_zeroE", hypnat_not_less0 RS notE); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
452 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
453 |
Goalw [hypnat_less_def,hypnat_zero_def,hypnat_one_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
454 |
"(n<1hn) = (n=0)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
455 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
456 |
by (auto_tac (claset() addSIs [exI] addEs |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
457 |
[FreeUltrafilterNat_subset],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
458 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
459 |
qed "hypnat_less_one"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
460 |
AddIffs [hypnat_less_one]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
461 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
462 |
Goal "!!(R1::hypnat). [| R1 < R2; R2 < R3 |] ==> R1 < R3"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
463 |
by (res_inst_tac [("z","R1")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
464 |
by (res_inst_tac [("z","R2")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
465 |
by (res_inst_tac [("z","R3")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
466 |
by (auto_tac (claset(),simpset() addsimps [hypnat_less_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
467 |
by (res_inst_tac [("x","X")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
468 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
469 |
by (res_inst_tac [("x","Ya")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
470 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
471 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
472 |
qed "hypnat_less_trans"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
473 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
474 |
Goal "!! (R1::hypnat). [| R1 < R2; R2 < R1 |] ==> P"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
475 |
by (dtac hypnat_less_trans 1 THEN assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
476 |
by (Asm_full_simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
477 |
qed "hypnat_less_asym"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
478 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
479 |
(*----- hypnat less iff less a.e -----*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
480 |
(* See comments in HYPER for corresponding thm *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
481 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
482 |
Goalw [hypnat_less_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
483 |
"(Abs_hypnat(hypnatrel^^{%n. X n}) < \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
484 |
\ Abs_hypnat(hypnatrel^^{%n. Y n})) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
485 |
\ ({n. X n < Y n} : FreeUltrafilterNat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
486 |
by (auto_tac (claset() addSIs [lemma_hypnatrel_refl],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
487 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
488 |
qed "hypnat_less"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
489 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
490 |
Goal "~ m<n --> n+(m-n) = (m::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
491 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
492 |
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
493 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
494 |
[hypnat_minus,hypnat_add,hypnat_less])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
495 |
by (dtac FreeUltrafilterNat_Compl_mem 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
496 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
497 |
qed_spec_mp "hypnat_add_diff_inverse"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
498 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
499 |
Goal "n<=m ==> n+(m-n) = (m::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
500 |
by (asm_full_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
501 |
[hypnat_add_diff_inverse, hypnat_le_def]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
502 |
qed "hypnat_le_add_diff_inverse"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
503 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
504 |
Goal "n<=m ==> (m-n)+n = (m::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
505 |
by (asm_simp_tac (simpset() addsimps [hypnat_le_add_diff_inverse, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
506 |
hypnat_add_commute]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
507 |
qed "hypnat_le_add_diff_inverse2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
508 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
509 |
(*--------------------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
510 |
Hyper naturals as a linearly ordered field |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
511 |
---------------------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
512 |
Goalw [hypnat_zero_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
513 |
"[| (0::hypnat) < x; 0 < y |] ==> 0 < x + y"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
514 |
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
515 |
by (res_inst_tac [("z","y")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
516 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
517 |
[hypnat_less_def,hypnat_add])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
518 |
by (REPEAT(Step_tac 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
519 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
520 |
qed "hypnat_add_order"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
521 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
522 |
Goalw [hypnat_zero_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
523 |
"!!(x::hypnat). [| (0::hypnat) < x; 0 < y |] ==> 0 < x * y"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
524 |
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
525 |
by (res_inst_tac [("z","y")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
526 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
527 |
[hypnat_less_def,hypnat_mult])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
528 |
by (REPEAT(Step_tac 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
529 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
530 |
qed "hypnat_mult_order"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
531 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
532 |
(*--------------------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
533 |
Trichotomy of the hyper naturals |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
534 |
--------------------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
535 |
Goalw [hypnatrel_def] "EX x. x: hypnatrel ^^ {%n. 0}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
536 |
by (res_inst_tac [("x","%n. 0")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
537 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
538 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
539 |
qed "lemma_hypnatrel_0_mem"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
540 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
541 |
(* linearity is actually proved further down! *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
542 |
Goalw [hypnat_zero_def, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
543 |
hypnat_less_def]"(0::hypnat) < x | x = 0 | x < 0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
544 |
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
545 |
by (Auto_tac ); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
546 |
by (REPEAT(Step_tac 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
547 |
by (REPEAT(dtac FreeUltrafilterNat_Compl_mem 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
548 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
549 |
qed "hypnat_trichotomy"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
550 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
551 |
Goal "!!x. [| (0::hypnat) < x ==> P; \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
552 |
\ x = 0 ==> P; \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
553 |
\ x < 0 ==> P |] ==> P"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
554 |
by (cut_inst_tac [("x","x")] hypnat_trichotomy 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
555 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
556 |
qed "hypnat_trichotomyE"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
557 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
558 |
(*------------------------------------------------------------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
559 |
More properties of < |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
560 |
------------------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
561 |
Goal "!!(A::hypnat). A < B ==> A + C < B + C"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
562 |
by (res_inst_tac [("z","A")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
563 |
by (res_inst_tac [("z","B")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
564 |
by (res_inst_tac [("z","C")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
565 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
566 |
[hypnat_less_def,hypnat_add])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
567 |
by (REPEAT(Step_tac 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
568 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
569 |
qed "hypnat_add_less_mono1"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
570 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
571 |
Goal "!!(A::hypnat). A < B ==> C + A < C + B"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
572 |
by (auto_tac (claset() addIs [hypnat_add_less_mono1], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
573 |
simpset() addsimps [hypnat_add_commute])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
574 |
qed "hypnat_add_less_mono2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
575 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
576 |
Goal "!!k l::hypnat. [|i<j; k<l |] ==> i + k < j + l"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
577 |
by (etac (hypnat_add_less_mono1 RS hypnat_less_trans) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
578 |
by (simp_tac (simpset() addsimps [hypnat_add_commute]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
579 |
(*j moves to the end because it is free while k, l are bound*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
580 |
by (etac hypnat_add_less_mono1 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
581 |
qed "hypnat_add_less_mono"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
582 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
583 |
(*--------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
584 |
hypnat_minus_less |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
585 |
---------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
586 |
Goalw [hypnat_less_def,hypnat_zero_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
587 |
"((x::hypnat) < y) = ((0::hypnat) < y - x)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
588 |
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
589 |
by (res_inst_tac [("z","y")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
590 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
591 |
[hypnat_minus])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
592 |
by (REPEAT(Step_tac 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
593 |
by (REPEAT(Step_tac 2)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
594 |
by (ALLGOALS(fuf_tac (claset() addDs [sym],simpset()))); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
595 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
596 |
(*** linearity ***) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
597 |
Goalw [hypnat_less_def] "(x::hypnat) < y | x = y | y < x"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
598 |
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
599 |
by (res_inst_tac [("z","y")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
600 |
by (Auto_tac ); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
601 |
by (REPEAT(Step_tac 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
602 |
by (REPEAT(dtac FreeUltrafilterNat_Compl_mem 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
603 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
604 |
qed "hypnat_linear"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
605 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
606 |
Goal |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
607 |
"!!(x::hypnat). [| x < y ==> P; x = y ==> P; \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
608 |
\ y < x ==> P |] ==> P"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
609 |
by (cut_inst_tac [("x","x"),("y","y")] hypnat_linear 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
610 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
611 |
qed "hypnat_linear_less2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
612 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
613 |
(*------------------------------------------------------------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
614 |
Properties of <= |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
615 |
------------------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
616 |
(*------ hypnat le iff nat le a.e ------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
617 |
Goalw [hypnat_le_def,le_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
618 |
"(Abs_hypnat(hypnatrel^^{%n. X n}) <= \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
619 |
\ Abs_hypnat(hypnatrel^^{%n. Y n})) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
620 |
\ ({n. X n <= Y n} : FreeUltrafilterNat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
621 |
by (auto_tac (claset() addSDs [FreeUltrafilterNat_Compl_mem], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
622 |
simpset() addsimps [hypnat_less])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
623 |
by (Fuf_tac 1 THEN Fuf_empty_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
624 |
qed "hypnat_le"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
625 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
626 |
(*---------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
627 |
(*---------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
628 |
Goalw [hypnat_le_def] "!!w. ~(w < z) ==> z <= (w::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
629 |
by (assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
630 |
qed "hypnat_leI"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
631 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
632 |
Goalw [hypnat_le_def] "!!w. z<=w ==> ~(w<(z::hypnat))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
633 |
by (assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
634 |
qed "hypnat_leD"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
635 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
636 |
val hypnat_leE = make_elim hypnat_leD; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
637 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
638 |
Goal "!!w. (~(w < z)) = (z <= (w::hypnat))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
639 |
by (fast_tac (claset() addSIs [hypnat_leI,hypnat_leD]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
640 |
qed "hypnat_less_le_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
641 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
642 |
Goalw [hypnat_le_def] "!!z. ~ z <= w ==> w<(z::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
643 |
by (Fast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
644 |
qed "not_hypnat_leE"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
645 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
646 |
Goalw [hypnat_le_def] "!!z. z < w ==> z <= (w::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
647 |
by (fast_tac (claset() addEs [hypnat_less_asym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
648 |
qed "hypnat_less_imp_le"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
649 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
650 |
Goalw [hypnat_le_def] "!!(x::hypnat). x <= y ==> x < y | x = y"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
651 |
by (cut_facts_tac [hypnat_linear] 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
652 |
by (fast_tac (claset() addEs [hypnat_less_irrefl,hypnat_less_asym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
653 |
qed "hypnat_le_imp_less_or_eq"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
654 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
655 |
Goalw [hypnat_le_def] "!!z. z<w | z=w ==> z <=(w::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
656 |
by (cut_facts_tac [hypnat_linear] 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
657 |
by (fast_tac (claset() addEs [hypnat_less_irrefl,hypnat_less_asym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
658 |
qed "hypnat_less_or_eq_imp_le"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
659 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
660 |
Goal "(x <= (y::hypnat)) = (x < y | x=y)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
661 |
by (REPEAT(ares_tac [iffI, hypnat_less_or_eq_imp_le, hypnat_le_imp_less_or_eq] 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
662 |
qed "hypnat_le_eq_less_or_eq"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
663 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
664 |
Goal "w <= (w::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
665 |
by (simp_tac (simpset() addsimps [hypnat_le_eq_less_or_eq]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
666 |
qed "hypnat_le_refl"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
667 |
Addsimps [hypnat_le_refl]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
668 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
669 |
val prems = goal thy "!!i. [| i <= j; j < k |] ==> i < (k::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
670 |
by (dtac hypnat_le_imp_less_or_eq 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
671 |
by (fast_tac (claset() addIs [hypnat_less_trans]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
672 |
qed "hypnat_le_less_trans"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
673 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
674 |
Goal "!! (i::hypnat). [| i < j; j <= k |] ==> i < k"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
675 |
by (dtac hypnat_le_imp_less_or_eq 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
676 |
by (fast_tac (claset() addIs [hypnat_less_trans]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
677 |
qed "hypnat_less_le_trans"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
678 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
679 |
Goal "!!i. [| i <= j; j <= k |] ==> i <= (k::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
680 |
by (EVERY1 [dtac hypnat_le_imp_less_or_eq, dtac hypnat_le_imp_less_or_eq, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
681 |
rtac hypnat_less_or_eq_imp_le, fast_tac (claset() addIs [hypnat_less_trans])]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
682 |
qed "hypnat_le_trans"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
683 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
684 |
Goal "!!z. [| z <= w; w <= z |] ==> z = (w::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
685 |
by (EVERY1 [dtac hypnat_le_imp_less_or_eq, dtac hypnat_le_imp_less_or_eq, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
686 |
fast_tac (claset() addEs [hypnat_less_irrefl,hypnat_less_asym])]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
687 |
qed "hypnat_le_anti_sym"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
688 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
689 |
Goal "!!x. [| ~ y < x; y ~= x |] ==> x < (y::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
690 |
by (rtac not_hypnat_leE 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
691 |
by (fast_tac (claset() addDs [hypnat_le_imp_less_or_eq]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
692 |
qed "not_less_not_eq_hypnat_less"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
693 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
694 |
Goal "!!x. [| (0::hypnat) <= x; 0 <= y |] ==> 0 <= x * y"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
695 |
by (REPEAT(dtac hypnat_le_imp_less_or_eq 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
696 |
by (auto_tac (claset() addIs [hypnat_mult_order, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
697 |
hypnat_less_imp_le],simpset() addsimps [hypnat_le_refl])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
698 |
qed "hypnat_le_mult_order"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
699 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
700 |
Goalw [hypnat_one_def,hypnat_zero_def,hypnat_less_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
701 |
"(0::hypnat) < 1hn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
702 |
by (res_inst_tac [("x","%n. 0")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
703 |
by (res_inst_tac [("x","%n. 1")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
704 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
705 |
qed "hypnat_zero_less_one"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
706 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
707 |
Goal "!!x. [| (0::hypnat) <= x; 0 <= y |] ==> 0 <= x + y"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
708 |
by (REPEAT(dtac hypnat_le_imp_less_or_eq 1)); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
709 |
by (auto_tac (claset() addIs [hypnat_add_order, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
710 |
hypnat_less_imp_le],simpset() addsimps [hypnat_le_refl])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
711 |
qed "hypnat_le_add_order"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
712 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
713 |
Goal "!!(q1::hypnat). q1 <= q2 ==> x + q1 <= x + q2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
714 |
by (dtac hypnat_le_imp_less_or_eq 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
715 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
716 |
by (auto_tac (claset() addSIs [hypnat_le_refl, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
717 |
hypnat_less_imp_le,hypnat_add_less_mono1], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
718 |
simpset() addsimps [hypnat_add_commute])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
719 |
qed "hypnat_add_le_mono2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
720 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
721 |
Goal "!!(q1::hypnat). q1 <= q2 ==> q1 + x <= q2 + x"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
722 |
by (auto_tac (claset() addDs [hypnat_add_le_mono2], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
723 |
simpset() addsimps [hypnat_add_commute])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
724 |
qed "hypnat_add_le_mono1"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
725 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
726 |
Goal "!!k l::hypnat. [|i<=j; k<=l |] ==> i + k <= j + l"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
727 |
by (etac (hypnat_add_le_mono1 RS hypnat_le_trans) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
728 |
by (simp_tac (simpset() addsimps [hypnat_add_commute]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
729 |
(*j moves to the end because it is free while k, l are bound*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
730 |
by (etac hypnat_add_le_mono1 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
731 |
qed "hypnat_add_le_mono"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
732 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
733 |
Goalw [hypnat_zero_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
734 |
"!!x::hypnat. [| (0::hypnat) < z; x < y |] ==> x * z < y * z"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
735 |
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
736 |
by (res_inst_tac [("z","y")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
737 |
by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
738 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
739 |
[hypnat_less,hypnat_mult])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
740 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
741 |
qed "hypnat_mult_less_mono1"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
742 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
743 |
Goal "!!x::hypnat. [| 0 < z; x < y |] ==> z * x < z * y"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
744 |
by (auto_tac (claset() addIs [hypnat_mult_less_mono1], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
745 |
simpset() addsimps [hypnat_mult_commute])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
746 |
qed "hypnat_mult_less_mono2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
747 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
748 |
Addsimps [hypnat_mult_less_mono2,hypnat_mult_less_mono1]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
749 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
750 |
Goal "(x::hypnat) <= n + x"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
751 |
by (res_inst_tac [("x","n")] hypnat_trichotomyE 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
752 |
by (auto_tac (claset() addDs [(hypnat_less_imp_le RS |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
753 |
hypnat_add_le_mono1)],simpset() addsimps [hypnat_le_refl])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
754 |
qed "hypnat_add_self_le"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
755 |
Addsimps [hypnat_add_self_le]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
756 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
757 |
Goal "(x::hypnat) < x + 1hn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
758 |
by (cut_facts_tac [hypnat_zero_less_one |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
759 |
RS hypnat_add_less_mono2] 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
760 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
761 |
qed "hypnat_add_one_self_less"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
762 |
Addsimps [hypnat_add_one_self_less]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
763 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
764 |
Goalw [hypnat_le_def] "~ x + 1hn <= x"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
765 |
by (Simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
766 |
qed "not_hypnat_add_one_le_self"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
767 |
Addsimps [not_hypnat_add_one_le_self]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
768 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
769 |
Goal "((0::hypnat) < n) = (1hn <= n)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
770 |
by (res_inst_tac [("x","n")] hypnat_trichotomyE 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
771 |
by (auto_tac (claset(),simpset() addsimps [hypnat_le_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
772 |
qed "hypnat_gt_zero_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
773 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
774 |
Addsimps [hypnat_le_add_diff_inverse, hypnat_le_add_diff_inverse2, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
775 |
hypnat_less_imp_le RS hypnat_le_add_diff_inverse2]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
776 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
777 |
Goal "(0 < n) = (EX m. n = m + 1hn)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
778 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
779 |
by (res_inst_tac [("x","m")] hypnat_trichotomyE 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
780 |
by (rtac hypnat_less_trans 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
781 |
by (res_inst_tac [("x","n - 1hn")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
782 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
783 |
[hypnat_gt_zero_iff,hypnat_add_commute])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
784 |
qed "hypnat_gt_zero_iff2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
785 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
786 |
Goalw [hypnat_zero_def] "(0::hypnat) <= n"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
787 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
788 |
by (asm_simp_tac (simpset() addsimps [hypnat_le]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
789 |
qed "hypnat_le_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
790 |
Addsimps [hypnat_le_zero]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
791 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
792 |
(*------------------------------------------------------------------ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
793 |
hypnat_of_nat: properties embedding of naturals in hypernaturals |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
794 |
-----------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
795 |
(** hypnat_of_nat preserves field and order properties **) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
796 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
797 |
Goalw [hypnat_of_nat_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
798 |
"hypnat_of_nat ((z1::nat) + z2) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
799 |
\ hypnat_of_nat z1 + hypnat_of_nat z2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
800 |
by (asm_simp_tac (simpset() addsimps [hypnat_add]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
801 |
qed "hypnat_of_nat_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
802 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
803 |
Goalw [hypnat_of_nat_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
804 |
"hypnat_of_nat ((z1::nat) - z2) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
805 |
\ hypnat_of_nat z1 - hypnat_of_nat z2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
806 |
by (asm_simp_tac (simpset() addsimps [hypnat_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
807 |
qed "hypnat_of_nat_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
808 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
809 |
Goalw [hypnat_of_nat_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
810 |
"hypnat_of_nat (z1 * z2) = hypnat_of_nat z1 * hypnat_of_nat z2"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
811 |
by (full_simp_tac (simpset() addsimps [hypnat_mult]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
812 |
qed "hypnat_of_nat_mult"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
813 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
814 |
Goalw [hypnat_less_def,hypnat_of_nat_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
815 |
"(z1 < z2) = (hypnat_of_nat z1 < hypnat_of_nat z2)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
816 |
by (auto_tac (claset() addSIs [exI] addIs |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
817 |
[FreeUltrafilterNat_all],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
818 |
by (rtac FreeUltrafilterNat_P 1 THEN Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
819 |
qed "hypnat_of_nat_less_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
820 |
Addsimps [hypnat_of_nat_less_iff RS sym]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
821 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
822 |
Goalw [hypnat_le_def,le_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
823 |
"(z1 <= z2) = (hypnat_of_nat z1 <= hypnat_of_nat z2)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
824 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
825 |
qed "hypnat_of_nat_le_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
826 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
827 |
Goalw [hypnat_of_nat_def,hypnat_one_def] "hypnat_of_nat 1 = 1hn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
828 |
by (Simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
829 |
qed "hypnat_of_nat_one"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
830 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
831 |
Goalw [hypnat_of_nat_def,hypnat_zero_def] "hypnat_of_nat 0 = 0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
832 |
by (Simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
833 |
qed "hypnat_of_nat_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
834 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
835 |
Goal "(hypnat_of_nat n = 0) = (n = 0)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
836 |
by (auto_tac (claset() addIs [FreeUltrafilterNat_P], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
837 |
simpset() addsimps [hypnat_of_nat_def, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
838 |
hypnat_zero_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
839 |
qed "hypnat_of_nat_zero_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
840 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
841 |
Goal "(hypnat_of_nat n ~= 0) = (n ~= 0)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
842 |
by (full_simp_tac (simpset() addsimps [hypnat_of_nat_zero_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
843 |
qed "hypnat_of_nat_not_zero_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
844 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
845 |
goalw HyperNat.thy [hypnat_of_nat_def,hypnat_one_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
846 |
"hypnat_of_nat (Suc n) = hypnat_of_nat n + 1hn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
847 |
by (auto_tac (claset(),simpset() addsimps [hypnat_add])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
848 |
qed "hypnat_of_nat_Suc"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
849 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
850 |
(*--------------------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
851 |
Existence of infinite hypernatural number |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
852 |
---------------------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
853 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
854 |
Goal "hypnatrel^^{%n::nat. n} : hypnat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
855 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
856 |
qed "hypnat_omega"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
857 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
858 |
Goalw [hypnat_omega_def] "Rep_hypnat(whn) : hypnat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
859 |
by (rtac Rep_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
860 |
qed "Rep_hypnat_omega"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
861 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
862 |
(* See Hyper.thy for similar argument*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
863 |
(* existence of infinite number not corresponding to any natural number *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
864 |
(* use assumption that member FreeUltrafilterNat is not finite *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
865 |
(* a few lemmas first *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
866 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
867 |
Goalw [hypnat_omega_def,hypnat_of_nat_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
868 |
"~ (EX x. hypnat_of_nat x = whn)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
869 |
by (auto_tac (claset() addDs [FreeUltrafilterNat_not_finite], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
870 |
simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
871 |
qed "not_ex_hypnat_of_nat_eq_omega"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
872 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
873 |
Goal "hypnat_of_nat x ~= whn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
874 |
by (cut_facts_tac [not_ex_hypnat_of_nat_eq_omega] 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
875 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
876 |
qed "hypnat_of_nat_not_eq_omega"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
877 |
Addsimps [hypnat_of_nat_not_eq_omega RS not_sym]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
878 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
879 |
(*----------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
880 |
Properties of the set SHNat of embedded natural numbers |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
881 |
(cf. set SReal in NSA.thy/NSA.ML) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
882 |
----------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
883 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
884 |
(* Infinite hypernatural not in embedded SHNat *) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
885 |
Goalw [SHNat_def] "whn ~: SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
886 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
887 |
qed "SHNAT_omega_not_mem"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
888 |
Addsimps [SHNAT_omega_not_mem]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
889 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
890 |
(*----------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
891 |
Closure laws for members of (embedded) set standard naturals SHNat |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
892 |
-----------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
893 |
Goalw [SHNat_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
894 |
"!!x. [| x: SHNat; y: SHNat |] ==> x + y: SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
895 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
896 |
by (res_inst_tac [("x","N + Na")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
897 |
by (simp_tac (simpset() addsimps [hypnat_of_nat_add]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
898 |
qed "SHNat_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
899 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
900 |
Goalw [SHNat_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
901 |
"!!x. [| x: SHNat; y: SHNat |] ==> x - y: SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
902 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
903 |
by (res_inst_tac [("x","N - Na")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
904 |
by (simp_tac (simpset() addsimps [hypnat_of_nat_minus]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
905 |
qed "SHNat_minus"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
906 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
907 |
Goalw [SHNat_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
908 |
"!!x. [| x: SHNat; y: SHNat |] ==> x * y: SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
909 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
910 |
by (res_inst_tac [("x","N * Na")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
911 |
by (simp_tac (simpset() addsimps [hypnat_of_nat_mult]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
912 |
qed "SHNat_mult"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
913 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
914 |
Goal "!!x. [| x + y : SHNat; y: SHNat |] ==> x: SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
915 |
by (dres_inst_tac [("x","x+y")] SHNat_minus 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
916 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
917 |
qed "SHNat_add_cancel"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
918 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
919 |
Goalw [SHNat_def] "hypnat_of_nat x : SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
920 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
921 |
qed "SHNat_hypnat_of_nat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
922 |
Addsimps [SHNat_hypnat_of_nat]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
923 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
924 |
Goal "hypnat_of_nat 1 : SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
925 |
by (Simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
926 |
qed "SHNat_hypnat_of_nat_one"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
927 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
928 |
Goal "hypnat_of_nat 0 : SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
929 |
by (Simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
930 |
qed "SHNat_hypnat_of_nat_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
931 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
932 |
Goal "1hn : SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
933 |
by (simp_tac (simpset() addsimps [SHNat_hypnat_of_nat_one, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
934 |
hypnat_of_nat_one RS sym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
935 |
qed "SHNat_one"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
936 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
937 |
Goal "0 : SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
938 |
by (simp_tac (simpset() addsimps [SHNat_hypnat_of_nat_zero, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
939 |
hypnat_of_nat_zero RS sym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
940 |
qed "SHNat_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
941 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
942 |
Addsimps [SHNat_hypnat_of_nat_one,SHNat_hypnat_of_nat_zero, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
943 |
SHNat_one,SHNat_zero]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
944 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
945 |
Goal "1hn + 1hn : SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
946 |
by (rtac ([SHNat_one,SHNat_one] MRS SHNat_add) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
947 |
qed "SHNat_two"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
948 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
949 |
Goalw [SHNat_def] "{x. hypnat_of_nat x : SHNat} = (UNIV::nat set)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
950 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
951 |
qed "SHNat_UNIV_nat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
952 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
953 |
Goalw [SHNat_def] "(x: SHNat) = (EX y. x = hypnat_of_nat y)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
954 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
955 |
qed "SHNat_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
956 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
957 |
Goalw [SHNat_def] "hypnat_of_nat ``(UNIV::nat set) = SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
958 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
959 |
qed "hypnat_of_nat_image"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
960 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
961 |
Goalw [SHNat_def] "inv hypnat_of_nat ``SHNat = (UNIV::nat set)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
962 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
963 |
by (rtac (inj_hypnat_of_nat RS inv_f_f RS subst) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
964 |
by (Blast_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
965 |
qed "inv_hypnat_of_nat_image"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
966 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
967 |
Goalw [SHNat_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
968 |
"!!P. [| EX x. x: P; P <= SHNat |] ==> \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
969 |
\ EX Q. P = hypnat_of_nat `` Q"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
970 |
by (Best_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
971 |
qed "SHNat_hypnat_of_nat_image"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
972 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
973 |
Goalw [SHNat_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
974 |
"SHNat = hypnat_of_nat `` (UNIV::nat set)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
975 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
976 |
qed "SHNat_hypnat_of_nat_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
977 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
978 |
Goalw [SHNat_def] "SHNat <= (UNIV::hypnat set)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
979 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
980 |
qed "SHNat_subset_UNIV"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
981 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
982 |
Goal "{n. n <= Suc m} = {n. n <= m} Un {n. n = Suc m}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
983 |
by (auto_tac (claset(),simpset() addsimps [le_Suc_eq])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
984 |
qed "leSuc_Un_eq"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
985 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
986 |
Goal "finite {n::nat. n <= m}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
987 |
by (nat_ind_tac "m" 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
988 |
by (auto_tac (claset(),simpset() addsimps [leSuc_Un_eq])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
989 |
qed "finite_nat_le_segment"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
990 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
991 |
Goal "{n::nat. m < n} : FreeUltrafilterNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
992 |
by (cut_inst_tac [("m2","m")] (finite_nat_le_segment RS |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
993 |
FreeUltrafilterNat_finite RS FreeUltrafilterNat_Compl_mem) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
994 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
995 |
qed "lemma_unbounded_set"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
996 |
Addsimps [lemma_unbounded_set]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
997 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
998 |
Goalw [SHNat_def,hypnat_of_nat_def, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
999 |
hypnat_less_def,hypnat_omega_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1000 |
"ALL n: SHNat. n < whn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1001 |
by (Clarify_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1002 |
by (auto_tac (claset() addSIs [exI],simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1003 |
qed "hypnat_omega_gt_SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1004 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1005 |
Goal "hypnat_of_nat n < whn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1006 |
by (cut_facts_tac [hypnat_omega_gt_SHNat] 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1007 |
by (dres_inst_tac [("x","hypnat_of_nat n")] bspec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1008 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1009 |
qed "hypnat_of_nat_less_whn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1010 |
Addsimps [hypnat_of_nat_less_whn]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1011 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1012 |
Goal "hypnat_of_nat n <= whn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1013 |
by (rtac (hypnat_of_nat_less_whn RS hypnat_less_imp_le) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1014 |
qed "hypnat_of_nat_le_whn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1015 |
Addsimps [hypnat_of_nat_le_whn]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1016 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1017 |
Goal "0 < whn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1018 |
by (rtac (hypnat_omega_gt_SHNat RS ballE) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1019 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1020 |
qed "hypnat_zero_less_hypnat_omega"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1021 |
Addsimps [hypnat_zero_less_hypnat_omega]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1022 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1023 |
Goal "1hn < whn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1024 |
by (rtac (hypnat_omega_gt_SHNat RS ballE) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1025 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1026 |
qed "hypnat_one_less_hypnat_omega"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1027 |
Addsimps [hypnat_one_less_hypnat_omega]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1028 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1029 |
(*-------------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1030 |
Theorems about infinite hypernatural numbers -- HNatInfinite |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1031 |
-------------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1032 |
Goalw [HNatInfinite_def,SHNat_def] "whn : HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1033 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1034 |
qed "HNatInfinite_whn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1035 |
Addsimps [HNatInfinite_whn]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1036 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1037 |
Goalw [HNatInfinite_def] "!!x. x: SHNat ==> x ~: HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1038 |
by (Simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1039 |
qed "SHNat_not_HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1040 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1041 |
Goalw [HNatInfinite_def] "!!x. x ~: HNatInfinite ==> x: SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1042 |
by (Asm_full_simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1043 |
qed "not_HNatInfinite_SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1044 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1045 |
Goalw [HNatInfinite_def] "!!x. x ~: SHNat ==> x: HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1046 |
by (Simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1047 |
qed "not_SHNat_HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1048 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1049 |
Goalw [HNatInfinite_def] "!!x. x: HNatInfinite ==> x ~: SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1050 |
by (Asm_full_simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1051 |
qed "HNatInfinite_not_SHNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1052 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1053 |
Goal "(x: SHNat) = (x ~: HNatInfinite)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1054 |
by (blast_tac (claset() addSIs [SHNat_not_HNatInfinite, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1055 |
not_HNatInfinite_SHNat]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1056 |
qed "SHNat_not_HNatInfinite_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1057 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1058 |
Goal "(x ~: SHNat) = (x: HNatInfinite)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1059 |
by (blast_tac (claset() addSIs [not_SHNat_HNatInfinite, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1060 |
HNatInfinite_not_SHNat]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1061 |
qed "not_SHNat_HNatInfinite_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1062 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1063 |
Goal "x : SHNat | x : HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1064 |
by (simp_tac (simpset() addsimps [SHNat_not_HNatInfinite_iff]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1065 |
qed "SHNat_HNatInfinite_disj"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1066 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1067 |
(*------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1068 |
Proof of alternative definition for set of Infinite hypernatural |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1069 |
numbers --- HNatInfinite = {N. ALL n: SHNat. n < N} |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1070 |
-------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1071 |
Goal "!!N (xa::nat=>nat). \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1072 |
\ (ALL N. {n. xa n ~= N} : FreeUltrafilterNat) ==> \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1073 |
\ ({n. N < xa n} : FreeUltrafilterNat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1074 |
by (nat_ind_tac "N" 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1075 |
by (dres_inst_tac [("x","0")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1076 |
by (rtac ccontr 1 THEN dtac FreeUltrafilterNat_Compl_mem 1 |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1077 |
THEN dtac FreeUltrafilterNat_Int 1 THEN assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1078 |
by (Asm_full_simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1079 |
by (dres_inst_tac [("x","Suc N")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1080 |
by (fuf_tac (claset() addSDs [Suc_leI],simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1081 |
[le_eq_less_or_eq]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1082 |
qed "HNatInfinite_FreeUltrafilterNat_lemma"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1083 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1084 |
(*** alternative definition ***) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1085 |
Goalw [HNatInfinite_def,SHNat_def,hypnat_of_nat_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1086 |
"HNatInfinite = {N. ALL n:SHNat. n < N}"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1087 |
by (Step_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1088 |
by (dres_inst_tac [("x","Abs_hypnat \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1089 |
\ (hypnatrel ^^ {%n. N})")] bspec 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1090 |
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1091 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1092 |
by (auto_tac (claset(),simpset() addsimps [hypnat_less_iff])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1093 |
by (auto_tac (claset() addSIs [exI] addEs |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1094 |
[HNatInfinite_FreeUltrafilterNat_lemma], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1095 |
simpset() addsimps [FreeUltrafilterNat_Compl_iff1, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1096 |
CLAIM "- {n. xa n = N} = {n. xa n ~= N}"])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1097 |
qed "HNatInfinite_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1098 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1099 |
(*-------------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1100 |
Alternative definition for HNatInfinite using Free ultrafilter |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1101 |
--------------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1102 |
Goal "!!x. x : HNatInfinite ==> EX X: Rep_hypnat x. \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1103 |
\ ALL u. {n. u < X n}: FreeUltrafilterNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1104 |
by (auto_tac (claset(),simpset() addsimps [hypnat_less_def, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1105 |
HNatInfinite_iff,SHNat_iff,hypnat_of_nat_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1106 |
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1107 |
by (EVERY[Auto_tac, rtac bexI 1, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1108 |
rtac lemma_hypnatrel_refl 2, Step_tac 1]); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1109 |
by (dres_inst_tac [("x","hypnat_of_nat u")] bspec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1110 |
by (Simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1111 |
by (auto_tac (claset(), |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1112 |
simpset() addsimps [hypnat_of_nat_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1113 |
by (Fuf_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1114 |
qed "HNatInfinite_FreeUltrafilterNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1115 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1116 |
Goal "!!x. EX X: Rep_hypnat x. \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1117 |
\ ALL u. {n. u < X n}: FreeUltrafilterNat \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1118 |
\ ==> x: HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1119 |
by (auto_tac (claset(),simpset() addsimps [hypnat_less_def, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1120 |
HNatInfinite_iff,SHNat_iff,hypnat_of_nat_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1121 |
by (rtac exI 1 THEN Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1122 |
qed "FreeUltrafilterNat_HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1123 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1124 |
Goal "!!x. (x : HNatInfinite) = (EX X: Rep_hypnat x. \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1125 |
\ ALL u. {n. u < X n}: FreeUltrafilterNat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1126 |
by (blast_tac (claset() addIs [HNatInfinite_FreeUltrafilterNat, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1127 |
FreeUltrafilterNat_HNatInfinite]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1128 |
qed "HNatInfinite_FreeUltrafilterNat_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1129 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1130 |
Goal "!!x. x : HNatInfinite ==> 1hn < x"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1131 |
by (auto_tac (claset(),simpset() addsimps [HNatInfinite_iff])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1132 |
qed "HNatInfinite_gt_one"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1133 |
Addsimps [HNatInfinite_gt_one]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1134 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1135 |
Goal "!!x. 0 ~: HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1136 |
by (auto_tac (claset(),simpset() |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1137 |
addsimps [HNatInfinite_iff])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1138 |
by (dres_inst_tac [("a","1hn")] equals0D 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1139 |
by (Asm_full_simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1140 |
qed "zero_not_mem_HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1141 |
Addsimps [zero_not_mem_HNatInfinite]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1142 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1143 |
Goal "!!x. x : HNatInfinite ==> x ~= 0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1144 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1145 |
qed "HNatInfinite_not_eq_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1146 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1147 |
Goal "!!x. x : HNatInfinite ==> 1hn <= x"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1148 |
by (blast_tac (claset() addIs [hypnat_less_imp_le, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1149 |
HNatInfinite_gt_one]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1150 |
qed "HNatInfinite_ge_one"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1151 |
Addsimps [HNatInfinite_ge_one]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1152 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1153 |
(*-------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1154 |
Closure Rules |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1155 |
--------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1156 |
Goal "!!x. [| x: HNatInfinite; y: HNatInfinite |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1157 |
\ ==> x + y: HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1158 |
by (auto_tac (claset(),simpset() addsimps [HNatInfinite_iff])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1159 |
by (dtac bspec 1 THEN assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1160 |
by (dtac (SHNat_zero RSN (2,bspec)) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1161 |
by (dtac hypnat_add_less_mono 1 THEN assume_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1162 |
by (Asm_full_simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1163 |
qed "HNatInfinite_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1164 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1165 |
Goal "!!x. [| x: HNatInfinite; y: SHNat |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1166 |
\ ==> x + y: HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1167 |
by (rtac ccontr 1 THEN dtac not_HNatInfinite_SHNat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1168 |
by (dres_inst_tac [("x","x + y")] SHNat_minus 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1169 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1170 |
[SHNat_not_HNatInfinite_iff])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1171 |
qed "HNatInfinite_SHNat_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1172 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1173 |
goal HyperNat.thy "!!x. [| x: HNatInfinite; y: SHNat |] \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1174 |
\ ==> x - y: HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1175 |
by (rtac ccontr 1 THEN dtac not_HNatInfinite_SHNat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1176 |
by (dres_inst_tac [("x","x - y")] SHNat_add 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1177 |
by (subgoal_tac "y <= x" 2); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1178 |
by (auto_tac (claset() addSDs [hypnat_le_add_diff_inverse2], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1179 |
simpset() addsimps [not_SHNat_HNatInfinite_iff RS sym])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1180 |
by (auto_tac (claset() addSIs [hypnat_less_imp_le], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1181 |
simpset() addsimps [not_SHNat_HNatInfinite_iff,HNatInfinite_iff])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1182 |
qed "HNatInfinite_SHNat_diff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1183 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1184 |
Goal |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1185 |
"!!x. x: HNatInfinite ==> x + 1hn: HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1186 |
by (auto_tac (claset() addIs [HNatInfinite_SHNat_add], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1187 |
simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1188 |
qed "HNatInfinite_add_one"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1189 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1190 |
Goal |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1191 |
"!!x. x: HNatInfinite ==> x - 1hn: HNatInfinite"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1192 |
by (rtac ccontr 1 THEN dtac not_HNatInfinite_SHNat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1193 |
by (dres_inst_tac [("x","x - 1hn"),("y","1hn")] SHNat_add 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1194 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1195 |
[not_SHNat_HNatInfinite_iff RS sym])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1196 |
qed "HNatInfinite_minus_one"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1197 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1198 |
Goal "!!x. x : HNatInfinite ==> EX y. x = y + 1hn"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1199 |
by (res_inst_tac [("x","x - 1hn")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1200 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1201 |
qed "HNatInfinite_is_Suc"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1202 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1203 |
(*--------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1204 |
HNat : the hypernaturals embedded in the hyperreals |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1205 |
Obtained using the NS extension of the naturals |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1206 |
--------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1207 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1208 |
Goalw [HNat_def,starset_def, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1209 |
hypreal_of_posnat_def,hypreal_of_real_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1210 |
"hypreal_of_posnat N : HNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1211 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1212 |
by (Ultra_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1213 |
by (res_inst_tac [("x","Suc N")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1214 |
by (dtac sym 1 THEN auto_tac (claset(),simpset() |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1215 |
addsimps [real_of_preal_real_of_nat_Suc])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1216 |
qed "HNat_hypreal_of_posnat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1217 |
Addsimps [HNat_hypreal_of_posnat]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1218 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1219 |
Goalw [HNat_def,starset_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1220 |
"[| x: HNat; y: HNat |] ==> x + y: HNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1221 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1222 |
by (res_inst_tac [("z","y")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1223 |
by (auto_tac (claset() addSDs [bspec] addIs [lemma_hyprel_refl], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1224 |
simpset() addsimps [hypreal_add])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1225 |
by (Ultra_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1226 |
by (dres_inst_tac [("t","Y xb")] sym 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1227 |
by (auto_tac (claset(),simpset() addsimps [real_of_nat_add RS sym])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1228 |
qed "HNat_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1229 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1230 |
Goalw [HNat_def,starset_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1231 |
"[| x: HNat; y: HNat |] ==> x * y: HNat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1232 |
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1233 |
by (res_inst_tac [("z","y")] eq_Abs_hypreal 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1234 |
by (auto_tac (claset() addSDs [bspec] addIs [lemma_hyprel_refl], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1235 |
simpset() addsimps [hypreal_mult])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1236 |
by (Ultra_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1237 |
by (dres_inst_tac [("t","Y xb")] sym 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1238 |
by (auto_tac (claset(),simpset() addsimps [real_of_nat_mult RS sym])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1239 |
qed "HNat_mult"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1240 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1241 |
(*--------------------------------------------------------------- |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1242 |
Embedding of the hypernaturals into the hyperreal |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1243 |
--------------------------------------------------------------*) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1244 |
(*** A lemma that should have been derived a long time ago! ***) |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1245 |
Goal "(Ya : hyprel ^^{%n. f(n)}) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1246 |
\ ({n. f n = Ya n} : FreeUltrafilterNat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1247 |
by (Auto_tac); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1248 |
qed "lemma_hyprel_FUFN"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1249 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1250 |
Goalw [hypreal_of_hypnat_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1251 |
"hypreal_of_hypnat (Abs_hypnat(hypnatrel^^{%n. X n})) = \ |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1252 |
\ Abs_hypreal(hyprel ^^ {%n. real_of_nat (X n)})"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1253 |
by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1254 |
by (auto_tac (claset() addEs [FreeUltrafilterNat_Int RS |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1255 |
FreeUltrafilterNat_subset],simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1256 |
[lemma_hyprel_FUFN])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1257 |
qed "hypreal_of_hypnat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1258 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1259 |
Goal "inj(hypreal_of_hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1260 |
by (rtac injI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1261 |
by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1262 |
by (res_inst_tac [("z","y")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1263 |
by (auto_tac (claset(),simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1264 |
[hypreal_of_hypnat,real_of_nat_eq_cancel])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1265 |
qed "inj_hypreal_of_hypnat"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1266 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1267 |
Goal "(hypreal_of_hypnat n = hypreal_of_hypnat m) = (n = m)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1268 |
by (auto_tac (claset(),simpset() addsimps [inj_hypreal_of_hypnat RS injD])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1269 |
qed "hypreal_of_hypnat_eq_cancel"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1270 |
Addsimps [hypreal_of_hypnat_eq_cancel]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1271 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1272 |
Goal "(hypnat_of_nat n = hypnat_of_nat m) = (n = m)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1273 |
by (auto_tac (claset() addDs [inj_hypnat_of_nat RS injD], |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1274 |
simpset())); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1275 |
qed "hypnat_of_nat_eq_cancel"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1276 |
Addsimps [hypnat_of_nat_eq_cancel]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1277 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1278 |
Goalw [hypreal_zero_def,hypnat_zero_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1279 |
"hypreal_of_hypnat 0 = 0"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1280 |
by (simp_tac (simpset() addsimps [hypreal_of_hypnat, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1281 |
real_of_nat_zero]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1282 |
qed "hypreal_of_hypnat_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1283 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1284 |
Goalw [hypreal_one_def,hypnat_one_def] |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1285 |
"hypreal_of_hypnat 1hn = 1hr"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1286 |
by (simp_tac (simpset() addsimps [hypreal_of_hypnat, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1287 |
real_of_nat_one]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1288 |
qed "hypreal_of_hypnat_one"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1289 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1290 |
Goal "hypreal_of_hypnat n1 + hypreal_of_hypnat n2 = hypreal_of_hypnat (n1 + n2)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1291 |
by (res_inst_tac [("z","n1")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1292 |
by (res_inst_tac [("z","n2")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1293 |
by (asm_simp_tac (simpset() addsimps [hypreal_of_hypnat, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1294 |
hypreal_add,hypnat_add,real_of_nat_add]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1295 |
qed "hypreal_of_hypnat_add"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1296 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1297 |
Goal "hypreal_of_hypnat n1 * hypreal_of_hypnat n2 = hypreal_of_hypnat (n1 * n2)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1298 |
by (res_inst_tac [("z","n1")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1299 |
by (res_inst_tac [("z","n2")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1300 |
by (asm_simp_tac (simpset() addsimps [hypreal_of_hypnat, |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1301 |
hypreal_mult,hypnat_mult,real_of_nat_mult]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1302 |
qed "hypreal_of_hypnat_mult"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1303 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1304 |
Goal "(hypreal_of_hypnat n < hypreal_of_hypnat m) = (n < m)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1305 |
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1306 |
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1307 |
by (asm_simp_tac (simpset() addsimps |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1308 |
[hypreal_of_hypnat,hypreal_less,hypnat_less]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1309 |
qed "hypreal_of_hypnat_less_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1310 |
Addsimps [hypreal_of_hypnat_less_iff]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1311 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1312 |
Goal "(hypreal_of_hypnat N = 0) = (N = 0)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1313 |
by (simp_tac (simpset() addsimps [hypreal_of_hypnat_zero RS sym]) 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1314 |
qed "hypreal_of_hypnat_eq_zero_iff"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1315 |
Addsimps [hypreal_of_hypnat_eq_zero_iff]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1316 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1317 |
Goal "ALL n. N <= n ==> N = (0::hypnat)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1318 |
by (dres_inst_tac [("x","0")] spec 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1319 |
by (res_inst_tac [("z","N")] eq_Abs_hypnat 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1320 |
by (auto_tac (claset(),simpset() addsimps [hypnat_le,hypnat_zero_def])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1321 |
qed "hypnat_eq_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1322 |
Addsimps [hypnat_eq_zero]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1323 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1324 |
Goal "~ (ALL n. n = (0::hypnat))"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1325 |
by Auto_tac; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1326 |
by (res_inst_tac [("x","1hn")] exI 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1327 |
by (Simp_tac 1); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1328 |
qed "hypnat_not_all_eq_zero"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1329 |
Addsimps [hypnat_not_all_eq_zero]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1330 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1331 |
Goal "n ~= 0 ==> (n <= 1hn) = (n = 1hn)"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1332 |
by (auto_tac (claset(),simpset() addsimps [hypnat_le_eq_less_or_eq])); |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1333 |
qed "hypnat_le_one_eq_one"; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1334 |
Addsimps [hypnat_le_one_eq_one]; |
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1335 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1336 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1337 |
|
c76b73e16711
New theories: construction of hypernaturals, nonstandard extensions,
fleuriot
parents:
diff
changeset
|
1338 |