wneuper/isa: src/HOL/Hyperreal/HyperNat.thy@83f1a514dcb4

     1 (*  Title       : HyperNat.thy

     2     Author      : Jacques D. Fleuriot

     3     Copyright   : 1998  University of Cambridge

     5 Converted to Isar and polished by lcp

     6 *)

     8 header{*Construction of Hypernaturals using Ultrafilters*}

    10 theory HyperNat = Star:

    12 constdefs

    13     hypnatrel :: "((nat=>nat)*(nat=>nat)) set"

    14     "hypnatrel == {p. \<exists>X Y. p = ((X::nat=>nat),Y) &

    15                        {n::nat. X(n) = Y(n)} \<in> FreeUltrafilterNat}"

    17 typedef hypnat = "UNIV//hypnatrel"

    18     by (auto simp add: quotient_def)

    20 instance hypnat :: "{ord, zero, one, plus, times, minus}" ..

    22 consts whn :: hypnat

    25 defs (overloaded)

    27   (** hypernatural arithmetic **)

    29   hypnat_zero_def:  "0 == Abs_hypnat(hypnatrel``{%n::nat. 0})"

    30   hypnat_one_def:   "1 == Abs_hypnat(hypnatrel``{%n::nat. 1})"

    32   (* omega is in fact an infinite hypernatural number = [<1,2,3,...>] *)

    33   hypnat_omega_def:  "whn == Abs_hypnat(hypnatrel``{%n::nat. n})"

    35   hypnat_add_def:

    36   "P + Q == Abs_hypnat(\<Union>X \<in> Rep_hypnat(P). \<Union>Y \<in> Rep_hypnat(Q).

    37                 hypnatrel``{%n::nat. X n + Y n})"

    39   hypnat_mult_def:

    40   "P * Q == Abs_hypnat(\<Union>X \<in> Rep_hypnat(P). \<Union>Y \<in> Rep_hypnat(Q).

    41                 hypnatrel``{%n::nat. X n * Y n})"

    43   hypnat_minus_def:

    44   "P - Q == Abs_hypnat(\<Union>X \<in> Rep_hypnat(P). \<Union>Y \<in> Rep_hypnat(Q).

    45                 hypnatrel``{%n::nat. X n - Y n})"

    47   hypnat_le_def:

    48   "P \<le> (Q::hypnat) == \<exists>X Y. X \<in> Rep_hypnat(P) & Y \<in> Rep_hypnat(Q) &

    49                             {n::nat. X n \<le> Y n} \<in> FreeUltrafilterNat"

    51   hypnat_less_def: "(x < (y::hypnat)) == (x \<le> y & x \<noteq> y)"

    55 subsection{*Properties of @{term hypnatrel}*}

    57 text{*Proving that @{term hypnatrel} is an equivalence relation*}

    59 lemma hypnatrel_iff:

    60      "((X,Y) \<in> hypnatrel) = ({n. X n = Y n}: FreeUltrafilterNat)"

    61 apply (simp add: hypnatrel_def)

    62 done

    64 lemma hypnatrel_refl: "(x,x) \<in> hypnatrel"

    65 by (simp add: hypnatrel_def)

    67 lemma hypnatrel_sym: "(x,y) \<in> hypnatrel ==> (y,x) \<in> hypnatrel"

    68 by (auto simp add: hypnatrel_def eq_commute)

    70 lemma hypnatrel_trans [rule_format (no_asm)]:

    71      "(x,y) \<in> hypnatrel --> (y,z) \<in> hypnatrel --> (x,z) \<in> hypnatrel"

    72 by (auto simp add: hypnatrel_def, ultra)

    74 lemma equiv_hypnatrel:

    75      "equiv UNIV hypnatrel"

    76 apply (simp add: equiv_def refl_def sym_def trans_def hypnatrel_refl)

    77 apply (blast intro: hypnatrel_sym hypnatrel_trans)

    78 done

    80 (* (hypnatrel `` {x} = hypnatrel `` {y}) = ((x,y) \<in> hypnatrel) *)

    81 lemmas equiv_hypnatrel_iff =

    82     eq_equiv_class_iff [OF equiv_hypnatrel UNIV_I UNIV_I, simp]

    84 lemma hypnatrel_in_hypnat [simp]: "hypnatrel``{x}:hypnat"

    85 by (simp add: hypnat_def hypnatrel_def quotient_def, blast)

    87 lemma inj_on_Abs_hypnat: "inj_on Abs_hypnat hypnat"

    88 apply (rule inj_on_inverseI)

    89 apply (erule Abs_hypnat_inverse)

    90 done

    92 declare inj_on_Abs_hypnat [THEN inj_on_iff, simp]

    93         Abs_hypnat_inverse [simp]

    95 declare equiv_hypnatrel [THEN eq_equiv_class_iff, simp]

    97 declare hypnatrel_iff [iff]

   100 lemma inj_Rep_hypnat: "inj(Rep_hypnat)"

   101 apply (rule inj_on_inverseI)

   102 apply (rule Rep_hypnat_inverse)

   103 done

   105 lemma lemma_hypnatrel_refl: "x \<in> hypnatrel `` {x}"

   106 by (simp add: hypnatrel_def)

   108 declare lemma_hypnatrel_refl [simp]

   110 lemma hypnat_empty_not_mem: "{} \<notin> hypnat"

   111 apply (simp add: hypnat_def)

   112 apply (auto elim!: quotientE equalityCE)

   113 done

   115 declare hypnat_empty_not_mem [simp]

   117 lemma Rep_hypnat_nonempty: "Rep_hypnat x \<noteq> {}"

   118 by (cut_tac x = x in Rep_hypnat, auto)

   120 declare Rep_hypnat_nonempty [simp]

   123 lemma eq_Abs_hypnat:

   124     "(!!x. z = Abs_hypnat(hypnatrel``{x}) ==> P) ==> P"

   125 apply (rule_tac x1=z in Rep_hypnat [unfolded hypnat_def, THEN quotientE])

   126 apply (drule_tac f = Abs_hypnat in arg_cong)

   127 apply (force simp add: Rep_hypnat_inverse)

   128 done

   130 theorem hypnat_cases [case_names Abs_hypnat, cases type: hypnat]:

   131     "(!!x. z = Abs_hypnat(hypnatrel``{x}) ==> P) ==> P"

   132 by (rule eq_Abs_hypnat [of z], blast)

   134 subsection{*Hypernat Addition*}

   136 lemma hypnat_add_congruent2:

   137      "congruent2 hypnatrel hypnatrel (%X Y. hypnatrel``{%n. X n + Y n})"

   138 by (simp add: congruent2_def, auto, ultra)

   140 lemma hypnat_add:

   141   "Abs_hypnat(hypnatrel``{%n. X n}) + Abs_hypnat(hypnatrel``{%n. Y n}) =

   142    Abs_hypnat(hypnatrel``{%n. X n + Y n})"

   143 by (simp add: hypnat_add_def

   144     UN_equiv_class2 [OF equiv_hypnatrel equiv_hypnatrel hypnat_add_congruent2])

   146 lemma hypnat_add_commute: "(z::hypnat) + w = w + z"

   147 apply (cases z, cases w)

   148 apply (simp add: add_ac hypnat_add)

   149 done

   151 lemma hypnat_add_assoc: "((z1::hypnat) + z2) + z3 = z1 + (z2 + z3)"

   152 apply (cases z1, cases z2, cases z3)

   153 apply (simp add: hypnat_add nat_add_assoc)

   154 done

   156 lemma hypnat_add_zero_left: "(0::hypnat) + z = z"

   157 apply (cases z)

   158 apply (simp add: hypnat_zero_def hypnat_add)

   159 done

   161 instance hypnat :: comm_monoid_add

   162   by intro_classes

   163     (assumption |

   164       rule hypnat_add_commute hypnat_add_assoc hypnat_add_zero_left)+

   167 subsection{*Subtraction inverse on @{typ hypreal}*}

   170 lemma hypnat_minus_congruent2:

   171     "congruent2 hypnatrel hypnatrel (%X Y. hypnatrel``{%n. X n - Y n})"

   172 by (simp add: congruent2_def, auto, ultra)

   174 lemma hypnat_minus:

   175   "Abs_hypnat(hypnatrel``{%n. X n}) - Abs_hypnat(hypnatrel``{%n. Y n}) =

   176    Abs_hypnat(hypnatrel``{%n. X n - Y n})"

   177 by (simp add: hypnat_minus_def

   178   UN_equiv_class2 [OF equiv_hypnatrel equiv_hypnatrel hypnat_minus_congruent2])

   180 lemma hypnat_minus_zero: "z - z = (0::hypnat)"

   181 apply (cases z)

   182 apply (simp add: hypnat_zero_def hypnat_minus)

   183 done

   185 lemma hypnat_diff_0_eq_0: "(0::hypnat) - n = 0"

   186 apply (cases n)

   187 apply (simp add: hypnat_minus hypnat_zero_def)

   188 done

   190 declare hypnat_minus_zero [simp] hypnat_diff_0_eq_0 [simp]

   192 lemma hypnat_add_is_0: "(m+n = (0::hypnat)) = (m=0 & n=0)"

   193 apply (cases m, cases n)

   194 apply (auto intro: FreeUltrafilterNat_subset dest: FreeUltrafilterNat_Int simp add: hypnat_zero_def hypnat_add)

   195 done

   197 declare hypnat_add_is_0 [iff]

   199 lemma hypnat_diff_diff_left: "(i::hypnat) - j - k = i - (j+k)"

   200 apply (cases i, cases j, cases k)

   201 apply (simp add: hypnat_minus hypnat_add diff_diff_left)

   202 done

   204 lemma hypnat_diff_commute: "(i::hypnat) - j - k = i-k-j"

   205 by (simp add: hypnat_diff_diff_left hypnat_add_commute)

   207 lemma hypnat_diff_add_inverse: "((n::hypnat) + m) - n = m"

   208 apply (cases m, cases n)

   209 apply (simp add: hypnat_minus hypnat_add)

   210 done

   211 declare hypnat_diff_add_inverse [simp]

   213 lemma hypnat_diff_add_inverse2:  "((m::hypnat) + n) - n = m"

   214 apply (cases m, cases n)

   215 apply (simp add: hypnat_minus hypnat_add)

   216 done

   217 declare hypnat_diff_add_inverse2 [simp]

   219 lemma hypnat_diff_cancel: "((k::hypnat) + m) - (k+n) = m - n"

   220 apply (cases k, cases m, cases n)

   221 apply (simp add: hypnat_minus hypnat_add)

   222 done

   223 declare hypnat_diff_cancel [simp]

   225 lemma hypnat_diff_cancel2: "((m::hypnat) + k) - (n+k) = m - n"

   226 by (simp add: hypnat_add_commute [of _ k])

   227 declare hypnat_diff_cancel2 [simp]

   229 lemma hypnat_diff_add_0: "(n::hypnat) - (n+m) = (0::hypnat)"

   230 apply (cases m, cases n)

   231 apply (simp add: hypnat_zero_def hypnat_minus hypnat_add)

   232 done

   233 declare hypnat_diff_add_0 [simp]

   236 subsection{*Hyperreal Multiplication*}

   238 lemma hypnat_mult_congruent2:

   239     "congruent2 hypnatrel hypnatrel (%X Y. hypnatrel``{%n. X n * Y n})"

   240 by (simp add: congruent2_def, auto, ultra)

   242 lemma hypnat_mult:

   243   "Abs_hypnat(hypnatrel``{%n. X n}) * Abs_hypnat(hypnatrel``{%n. Y n}) =

   244    Abs_hypnat(hypnatrel``{%n. X n * Y n})"

   245 by (simp add: hypnat_mult_def

   246    UN_equiv_class2 [OF equiv_hypnatrel equiv_hypnatrel hypnat_mult_congruent2])

   248 lemma hypnat_mult_commute: "(z::hypnat) * w = w * z"

   249 by (cases z, cases w, simp add: hypnat_mult mult_ac)

   251 lemma hypnat_mult_assoc: "((z1::hypnat) * z2) * z3 = z1 * (z2 * z3)"

   252 apply (cases z1, cases z2, cases z3)

   253 apply (simp add: hypnat_mult mult_assoc)

   254 done

   256 lemma hypnat_mult_1: "(1::hypnat) * z = z"

   257 apply (cases z)

   258 apply (simp add: hypnat_mult hypnat_one_def)

   259 done

   261 lemma hypnat_diff_mult_distrib: "((m::hypnat) - n) * k = (m * k) - (n * k)"

   262 apply (cases k, cases m, cases n)

   263 apply (simp add: hypnat_mult hypnat_minus diff_mult_distrib)

   264 done

   266 lemma hypnat_diff_mult_distrib2: "(k::hypnat) * (m - n) = (k * m) - (k * n)"

   267 by (simp add: hypnat_diff_mult_distrib hypnat_mult_commute [of k])

   269 lemma hypnat_add_mult_distrib: "((z1::hypnat) + z2) * w = (z1 * w) + (z2 * w)"

   270 apply (cases z1, cases z2, cases w)

   271 apply (simp add: hypnat_mult hypnat_add add_mult_distrib)

   272 done

   274 lemma hypnat_add_mult_distrib2: "(w::hypnat) * (z1 + z2) = (w * z1) + (w * z2)"

   275 by (simp add: hypnat_mult_commute [of w] hypnat_add_mult_distrib)

   277 text{*one and zero are distinct*}

   278 lemma hypnat_zero_not_eq_one: "(0::hypnat) \<noteq> (1::hypnat)"

   279 by (auto simp add: hypnat_zero_def hypnat_one_def)

   280 declare hypnat_zero_not_eq_one [THEN not_sym, simp]

   283 text{*The Hypernaturals Form A comm_semiring_1_cancel*}

   284 instance hypnat :: comm_semiring_1_cancel

   285 proof

   286   fix i j k :: hypnat

   287   show "(i * j) * k = i * (j * k)" by (rule hypnat_mult_assoc)

   288   show "i * j = j * i" by (rule hypnat_mult_commute)

   289   show "1 * i = i" by (rule hypnat_mult_1)

   290   show "(i + j) * k = i * k + j * k" by (simp add: hypnat_add_mult_distrib)

   291   show "0 \<noteq> (1::hypnat)" by (rule hypnat_zero_not_eq_one)

   292   assume "k+i = k+j"

   293   hence "(k+i) - k = (k+j) - k" by simp

   294   thus "i=j" by simp

   295 qed

   298 subsection{*Properties of The @{text "\<le>"} Relation*}

   300 lemma hypnat_le:

   301       "(Abs_hypnat(hypnatrel``{%n. X n}) \<le> Abs_hypnat(hypnatrel``{%n. Y n})) =

   302        ({n. X n \<le> Y n} \<in> FreeUltrafilterNat)"

   303 apply (simp add: hypnat_le_def)

   304 apply (auto intro!: lemma_hypnatrel_refl, ultra)

   305 done

   307 lemma hypnat_le_refl: "w \<le> (w::hypnat)"

   308 apply (cases w)

   309 apply (simp add: hypnat_le)

   310 done

   312 lemma hypnat_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::hypnat)"

   313 apply (cases i, cases j, cases k)

   314 apply (simp add: hypnat_le, ultra)

   315 done

   317 lemma hypnat_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::hypnat)"

   318 apply (cases z, cases w)

   319 apply (simp add: hypnat_le, ultra)

   320 done

   322 (* Axiom 'order_less_le' of class 'order': *)

   323 lemma hypnat_less_le: "((w::hypnat) < z) = (w \<le> z & w \<noteq> z)"

   324 by (simp add: hypnat_less_def)

   326 instance hypnat :: order

   327   by intro_classes

   328     (assumption |

   329       rule hypnat_le_refl hypnat_le_trans hypnat_le_anti_sym hypnat_less_le)+

   331 (* Axiom 'linorder_linear' of class 'linorder': *)

   332 lemma hypnat_le_linear: "(z::hypnat) \<le> w | w \<le> z"

   333 apply (cases z, cases w)

   334 apply (auto simp add: hypnat_le, ultra)

   335 done

   337 instance hypnat :: linorder

   338   by intro_classes (rule hypnat_le_linear)

   340 lemma hypnat_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::hypnat)"

   341 apply (cases x, cases y, cases z)

   342 apply (auto simp add: hypnat_le hypnat_add)

   343 done

   345 lemma hypnat_mult_less_mono2: "[| (0::hypnat)<z; x<y |] ==> z*x<z*y"

   346 apply (cases x, cases y, cases z)

   347 apply (simp add: hypnat_zero_def  hypnat_mult linorder_not_le [symmetric])

   348 apply (auto simp add: hypnat_le, ultra)

   349 done

   352 subsection{*The Hypernaturals Form an Ordered comm_semiring_1_cancel*}

   354 instance hypnat :: ordered_semidom

   355 proof

   356   fix x y z :: hypnat

   357   show "0 < (1::hypnat)"

   358     by (simp add: hypnat_zero_def hypnat_one_def linorder_not_le [symmetric],

   359         simp add: hypnat_le)

   360   show "x \<le> y ==> z + x \<le> z + y"

   361     by (rule hypnat_add_left_mono)

   362   show "x < y ==> 0 < z ==> z * x < z * y"

   363     by (simp add: hypnat_mult_less_mono2)

   364 qed

   366 lemma hypnat_le_zero_cancel [iff]: "(n \<le> (0::hypnat)) = (n = 0)"

   367 apply (cases n)

   368 apply (simp add: hypnat_zero_def hypnat_le)

   369 done

   371 lemma hypnat_mult_is_0 [simp]: "(m*n = (0::hypnat)) = (m=0 | n=0)"

   372 apply (cases m, cases n)

   373 apply (auto simp add: hypnat_zero_def hypnat_mult, ultra+)

   374 done

   376 lemma hypnat_diff_is_0_eq [simp]: "((m::hypnat) - n = 0) = (m \<le> n)"

   377 apply (cases m, cases n)

   378 apply (simp add: hypnat_le hypnat_minus hypnat_zero_def)

   379 done

   383 subsection{*Theorems for Ordering*}

   385 lemma hypnat_less:

   386       "(Abs_hypnat(hypnatrel``{%n. X n}) < Abs_hypnat(hypnatrel``{%n. Y n})) =

   387        ({n. X n < Y n} \<in> FreeUltrafilterNat)"

   388 apply (auto simp add: hypnat_le  linorder_not_le [symmetric])

   389 apply (ultra+)

   390 done

   392 lemma hypnat_not_less0 [iff]: "~ n < (0::hypnat)"

   393 apply (cases n)

   394 apply (auto simp add: hypnat_zero_def hypnat_less)

   395 done

   397 lemma hypnat_less_one [iff]:

   398       "(n < (1::hypnat)) = (n=0)"

   399 apply (cases n)

   400 apply (auto simp add: hypnat_zero_def hypnat_one_def hypnat_less)

   401 done

   403 lemma hypnat_add_diff_inverse: "~ m<n ==> n+(m-n) = (m::hypnat)"

   404 apply (cases m, cases n)

   405 apply (simp add: hypnat_minus hypnat_add hypnat_less split: nat_diff_split, ultra)

   406 done

   408 lemma hypnat_le_add_diff_inverse [simp]: "n \<le> m ==> n+(m-n) = (m::hypnat)"

   409 by (simp add: hypnat_add_diff_inverse linorder_not_less [symmetric])

   411 lemma hypnat_le_add_diff_inverse2 [simp]: "n\<le>m ==> (m-n)+n = (m::hypnat)"

   412 by (simp add: hypnat_le_add_diff_inverse hypnat_add_commute)

   414 declare hypnat_le_add_diff_inverse2 [OF order_less_imp_le]

   416 lemma hypnat_le0 [iff]: "(0::hypnat) \<le> n"

   417 by (simp add: linorder_not_less [symmetric])

   419 lemma hypnat_add_self_le [simp]: "(x::hypnat) \<le> n + x"

   420 by (insert add_right_mono [of 0 n x], simp)

   422 lemma hypnat_add_one_self_less [simp]: "(x::hypnat) < x + (1::hypnat)"

   423 by (insert add_strict_left_mono [OF zero_less_one], auto)

   425 lemma hypnat_neq0_conv [iff]: "(n \<noteq> 0) = (0 < (n::hypnat))"

   426 by (simp add: order_less_le)

   428 lemma hypnat_gt_zero_iff: "((0::hypnat) < n) = ((1::hypnat) \<le> n)"

   429 by (auto simp add: linorder_not_less [symmetric])

   431 lemma hypnat_gt_zero_iff2: "(0 < n) = (\<exists>m. n = m + (1::hypnat))"

   432 apply safe

   433  apply (rule_tac x = "n - (1::hypnat) " in exI)

   434  apply (simp add: hypnat_gt_zero_iff)

   435 apply (insert add_le_less_mono [OF _ zero_less_one, of 0], auto)

   436 done

   438 lemma hypnat_add_self_not_less: "~ (x + y < (x::hypnat))"

   439 by (simp add: linorder_not_le [symmetric] add_commute [of x])

   441 lemma hypnat_diff_split:

   442     "P(a - b::hypnat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"

   443     -- {* elimination of @{text -} on @{text hypnat} *}

   444 proof (cases "a<b" rule: case_split)

   445   case True

   446     thus ?thesis

   447       by (auto simp add: hypnat_add_self_not_less order_less_imp_le

   448                          hypnat_diff_is_0_eq [THEN iffD2])

   449 next

   450   case False

   451     thus ?thesis

   452       by (auto simp add: linorder_not_less dest: order_le_less_trans)

   453 qed

   456 subsection{*The Embedding @{term hypnat_of_nat} Preserves comm_ring_1 and

   457       Order Properties*}

   459 constdefs

   461   hypnat_of_nat   :: "nat => hypnat"

   462   "hypnat_of_nat m  == of_nat m"

   464   (* the set of infinite hypernatural numbers *)

   465   HNatInfinite :: "hypnat set"

   466   "HNatInfinite == {n. n \<notin> Nats}"

   469 lemma hypnat_of_nat_add:

   470       "hypnat_of_nat ((z::nat) + w) = hypnat_of_nat z + hypnat_of_nat w"

   471 by (simp add: hypnat_of_nat_def)

   473 lemma hypnat_of_nat_mult:

   474       "hypnat_of_nat (z * w) = hypnat_of_nat z * hypnat_of_nat w"

   475 by (simp add: hypnat_of_nat_def)

   477 lemma hypnat_of_nat_less_iff [simp]:

   478       "(hypnat_of_nat z < hypnat_of_nat w) = (z < w)"

   479 by (simp add: hypnat_of_nat_def)

   481 lemma hypnat_of_nat_le_iff [simp]:

   482       "(hypnat_of_nat z \<le> hypnat_of_nat w) = (z \<le> w)"

   483 by (simp add: hypnat_of_nat_def)

   485 lemma hypnat_of_nat_eq_iff [simp]:

   486       "(hypnat_of_nat z = hypnat_of_nat w) = (z = w)"

   487 by (simp add: hypnat_of_nat_def)

   489 lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = (1::hypnat)"

   490 by (simp add: hypnat_of_nat_def)

   492 lemma hypnat_of_nat_zero [simp]: "hypnat_of_nat 0 = 0"

   493 by (simp add: hypnat_of_nat_def)

   495 lemma hypnat_of_nat_zero_iff [simp]: "(hypnat_of_nat n = 0) = (n = 0)"

   496 by (simp add: hypnat_of_nat_def)

   498 lemma hypnat_of_nat_Suc [simp]:

   499      "hypnat_of_nat (Suc n) = hypnat_of_nat n + (1::hypnat)"

   500 by (simp add: hypnat_of_nat_def)

   502 lemma hypnat_of_nat_minus:

   503       "hypnat_of_nat ((j::nat) - k) = hypnat_of_nat j - hypnat_of_nat k"

   504 by (simp add: hypnat_of_nat_def split: nat_diff_split hypnat_diff_split)

   507 subsection{*Existence of an Infinite Hypernatural Number*}

   509 lemma hypnat_omega: "hypnatrel``{%n::nat. n} \<in> hypnat"

   510 by auto

   512 lemma Rep_hypnat_omega: "Rep_hypnat(whn) \<in> hypnat"

   513 by (simp add: hypnat_omega_def)

   515 text{*Existence of infinite number not corresponding to any natural number

   516 follows because member @{term FreeUltrafilterNat} is not finite.

   517 See @{text HyperDef.thy} for similar argument.*}

   520 subsection{*Properties of the set @{term Nats} of Embedded Natural Numbers*}

   522 lemma of_nat_eq_add [rule_format]:

   523      "\<forall>d::hypnat. of_nat m = of_nat n + d --> d \<in> range of_nat"

   524 apply (induct n)

   525 apply (auto simp add: add_assoc)

   526 apply (case_tac x)

   527 apply (auto simp add: add_commute [of 1])

   528 done

   530 lemma Nats_diff [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> (a-b :: hypnat) \<in> Nats"

   531 by (auto simp add: of_nat_eq_add Nats_def split: hypnat_diff_split)

   533 lemma lemma_unbounded_set [simp]: "{n::nat. m < n} \<in> FreeUltrafilterNat"

   534 apply (insert finite_atMost [of m])

   535 apply (simp add: atMost_def)

   536 apply (drule FreeUltrafilterNat_finite)

   537 apply (drule FreeUltrafilterNat_Compl_mem, ultra)

   538 done

   540 lemma Compl_Collect_le: "- {n::nat. N \<le> n} = {n. n < N}"

   541 by (simp add: Collect_neg_eq [symmetric] linorder_not_le)

   544 lemma hypnat_of_nat_eq:

   545      "hypnat_of_nat m  = Abs_hypnat(hypnatrel``{%n::nat. m})"

   546 apply (induct m)

   547 apply (simp_all add: hypnat_zero_def hypnat_one_def hypnat_add)

   548 done

   550 lemma SHNat_eq: "Nats = {n. \<exists>N. n = hypnat_of_nat N}"

   551 by (force simp add: hypnat_of_nat_def Nats_def)

   553 lemma hypnat_omega_gt_SHNat:

   554      "n \<in> Nats ==> n < whn"

   555 apply (auto simp add: hypnat_of_nat_eq hypnat_less_def hypnat_le_def

   556                       hypnat_omega_def SHNat_eq)

   557  prefer 2 apply (force dest: FreeUltrafilterNat_not_finite)

   558 apply (auto intro!: exI)

   559 apply (rule cofinite_mem_FreeUltrafilterNat)

   560 apply (simp add: Compl_Collect_le finite_nat_segment)

   561 done

   563 (* Infinite hypernatural not in embedded Nats *)

   564 lemma SHNAT_omega_not_mem [simp]: "whn \<notin> Nats"

   565 by (blast dest: hypnat_omega_gt_SHNat)

   567 lemma hypnat_of_nat_less_whn [simp]: "hypnat_of_nat n < whn"

   568 apply (insert hypnat_omega_gt_SHNat [of "hypnat_of_nat n"])

   569 apply (simp add: hypnat_of_nat_def)

   570 done

   572 lemma hypnat_of_nat_le_whn [simp]: "hypnat_of_nat n \<le> whn"

   573 by (rule hypnat_of_nat_less_whn [THEN order_less_imp_le])

   575 lemma hypnat_zero_less_hypnat_omega [simp]: "0 < whn"

   576 by (simp add: hypnat_omega_gt_SHNat)

   578 lemma hypnat_one_less_hypnat_omega [simp]: "(1::hypnat) < whn"

   579 by (simp add: hypnat_omega_gt_SHNat)

   582 subsection{*Infinite Hypernatural Numbers -- @{term HNatInfinite}*}

   584 lemma HNatInfinite_whn [simp]: "whn \<in> HNatInfinite"

   585 by (simp add: HNatInfinite_def)

   587 lemma Nats_not_HNatInfinite_iff: "(x \<in> Nats) = (x \<notin> HNatInfinite)"

   588 by (simp add: HNatInfinite_def)

   590 lemma HNatInfinite_not_Nats_iff: "(x \<in> HNatInfinite) = (x \<notin> Nats)"

   591 by (simp add: HNatInfinite_def)

   594 subsection{*Alternative Characterization of the Set of Infinite Hypernaturals:

   595 @{term "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"}*}

   597 (*??delete? similar reasoning in hypnat_omega_gt_SHNat above*)

   598 lemma HNatInfinite_FreeUltrafilterNat_lemma:

   599      "\<forall>N::nat. {n. f n \<noteq> N} \<in> FreeUltrafilterNat

   600       ==> {n. N < f n} \<in> FreeUltrafilterNat"

   601 apply (induct_tac "N")

   602 apply (drule_tac x = 0 in spec)

   603 apply (rule ccontr, drule FreeUltrafilterNat_Compl_mem, drule FreeUltrafilterNat_Int, assumption, simp)

   604 apply (drule_tac x = "Suc n" in spec, ultra)

   605 done

   607 lemma HNatInfinite_iff: "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"

   608 apply (auto simp add: HNatInfinite_def SHNat_eq hypnat_of_nat_eq)

   609 apply (rule_tac z = x in eq_Abs_hypnat)

   610 apply (auto elim: HNatInfinite_FreeUltrafilterNat_lemma

   611             simp add: hypnat_less FreeUltrafilterNat_Compl_iff1

   612                       Collect_neg_eq [symmetric])

   613 done

   616 subsection{*Alternative Characterization of @{term HNatInfinite} using

   617 Free Ultrafilter*}

   619 lemma HNatInfinite_FreeUltrafilterNat:

   620      "x \<in> HNatInfinite

   621       ==> \<exists>X \<in> Rep_hypnat x. \<forall>u. {n. u < X n}:  FreeUltrafilterNat"

   622 apply (cases x)

   623 apply (auto simp add: HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)

   624 apply (rule bexI [OF _ lemma_hypnatrel_refl], clarify)

   625 apply (auto simp add: hypnat_of_nat_def hypnat_less)

   626 done

   628 lemma FreeUltrafilterNat_HNatInfinite:

   629      "\<exists>X \<in> Rep_hypnat x. \<forall>u. {n. u < X n}:  FreeUltrafilterNat

   630       ==> x \<in> HNatInfinite"

   631 apply (cases x)

   632 apply (auto simp add: hypnat_less HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)

   633 apply (drule spec, ultra, auto)

   634 done

   636 lemma HNatInfinite_FreeUltrafilterNat_iff:

   637      "(x \<in> HNatInfinite) =

   638       (\<exists>X \<in> Rep_hypnat x. \<forall>u. {n. u < X n}:  FreeUltrafilterNat)"

   639 by (blast intro: HNatInfinite_FreeUltrafilterNat

   640                  FreeUltrafilterNat_HNatInfinite)

   642 lemma HNatInfinite_gt_one [simp]: "x \<in> HNatInfinite ==> (1::hypnat) < x"

   643 by (auto simp add: HNatInfinite_iff)

   645 lemma zero_not_mem_HNatInfinite [simp]: "0 \<notin> HNatInfinite"

   646 apply (auto simp add: HNatInfinite_iff)

   647 apply (drule_tac a = " (1::hypnat) " in equals0D)

   648 apply simp

   649 done

   651 lemma HNatInfinite_not_eq_zero: "x \<in> HNatInfinite ==> 0 < x"

   652 apply (drule HNatInfinite_gt_one)

   653 apply (auto simp add: order_less_trans [OF zero_less_one])

   654 done

   656 lemma HNatInfinite_ge_one [simp]: "x \<in> HNatInfinite ==> (1::hypnat) \<le> x"

   657 by (blast intro: order_less_imp_le HNatInfinite_gt_one)

   660 subsection{*Closure Rules*}

   662 lemma HNatInfinite_add:

   663      "[| x \<in> HNatInfinite; y \<in> HNatInfinite |] ==> x + y \<in> HNatInfinite"

   664 apply (auto simp add: HNatInfinite_iff)

   665 apply (drule bspec, assumption)

   666 apply (drule bspec [OF _ Nats_0])

   667 apply (drule add_strict_mono, assumption, simp)

   668 done

   670 lemma HNatInfinite_SHNat_add:

   671      "[| x \<in> HNatInfinite; y \<in> Nats |] ==> x + y \<in> HNatInfinite"

   672 apply (auto simp add: HNatInfinite_not_Nats_iff)

   673 apply (drule_tac a = "x + y" in Nats_diff, auto)

   674 done

   676 lemma HNatInfinite_Nats_imp_less: "[| x \<in> HNatInfinite; y \<in> Nats |] ==> y < x"

   677 by (simp add: HNatInfinite_iff)

   679 lemma HNatInfinite_SHNat_diff:

   680   assumes x: "x \<in> HNatInfinite" and y: "y \<in> Nats"

   681   shows "x - y \<in> HNatInfinite"

   682 proof -

   683   have "y < x" by (simp add: HNatInfinite_Nats_imp_less prems)

   684   hence "x - y + y = x" by (simp add: order_less_imp_le)

   685   with x show ?thesis

   686     by (force simp add: HNatInfinite_not_Nats_iff

   687               dest: Nats_add [of "x-y", OF _ y])

   688 qed

   690 lemma HNatInfinite_add_one:

   691      "x \<in> HNatInfinite ==> x + (1::hypnat) \<in> HNatInfinite"

   692 by (auto intro: HNatInfinite_SHNat_add)

   694 lemma HNatInfinite_is_Suc: "x \<in> HNatInfinite ==> \<exists>y. x = y + (1::hypnat)"

   695 apply (rule_tac x = "x - (1::hypnat) " in exI)

   696 apply auto

   697 done

   700 subsection{*Embedding of the Hypernaturals into the Hyperreals*}

   701 text{*Obtained using the nonstandard extension of the naturals*}

   703 constdefs

   704   hypreal_of_hypnat :: "hypnat => hypreal"

   705    "hypreal_of_hypnat N  ==

   706       Abs_hypreal(\<Union>X \<in> Rep_hypnat(N). hyprel``{%n::nat. real (X n)})"

   709 lemma HNat_hypreal_of_nat [simp]: "hypreal_of_nat N \<in> Nats"

   710 by (simp add: hypreal_of_nat_def)

   712 (*WARNING: FRAGILE!*)

   713 lemma lemma_hyprel_FUFN:

   714      "(Ya \<in> hyprel ``{%n. f(n)}) = ({n. f n = Ya n} \<in> FreeUltrafilterNat)"

   715 by force

   717 lemma hypreal_of_hypnat:

   718       "hypreal_of_hypnat (Abs_hypnat(hypnatrel``{%n. X n})) =

   719        Abs_hypreal(hyprel `` {%n. real (X n)})"

   720 apply (simp add: hypreal_of_hypnat_def)

   721 apply (rule_tac f = Abs_hypreal in arg_cong)

   722 apply (auto elim: FreeUltrafilterNat_Int [THEN FreeUltrafilterNat_subset]

   723        simp add: lemma_hyprel_FUFN)

   724 done

   726 lemma hypreal_of_hypnat_inject [simp]:

   727      "(hypreal_of_hypnat m = hypreal_of_hypnat n) = (m=n)"

   728 apply (cases m, cases n)

   729 apply (auto simp add: hypreal_of_hypnat)

   730 done

   732 lemma hypreal_of_hypnat_zero: "hypreal_of_hypnat 0 = 0"

   733 by (simp add: hypnat_zero_def hypreal_zero_def hypreal_of_hypnat)

   735 lemma hypreal_of_hypnat_one: "hypreal_of_hypnat (1::hypnat) = 1"

   736 by (simp add: hypnat_one_def hypreal_one_def hypreal_of_hypnat)

   738 lemma hypreal_of_hypnat_add [simp]:

   739      "hypreal_of_hypnat (m + n) = hypreal_of_hypnat m + hypreal_of_hypnat n"

   740 apply (cases m, cases n)

   741 apply (simp add: hypreal_of_hypnat hypreal_add hypnat_add real_of_nat_add)

   742 done

   744 lemma hypreal_of_hypnat_mult [simp]:

   745      "hypreal_of_hypnat (m * n) = hypreal_of_hypnat m * hypreal_of_hypnat n"

   746 apply (cases m, cases n)

   747 apply (simp add: hypreal_of_hypnat hypreal_mult hypnat_mult real_of_nat_mult)

   748 done

   750 lemma hypreal_of_hypnat_less_iff [simp]:

   751      "(hypreal_of_hypnat n < hypreal_of_hypnat m) = (n < m)"

   752 apply (cases m, cases n)

   753 apply (simp add: hypreal_of_hypnat hypreal_less hypnat_less)

   754 done

   756 lemma hypreal_of_hypnat_eq_zero_iff: "(hypreal_of_hypnat N = 0) = (N = 0)"

   757 by (simp add: hypreal_of_hypnat_zero [symmetric])

   758 declare hypreal_of_hypnat_eq_zero_iff [simp]

   760 lemma hypreal_of_hypnat_ge_zero [simp]: "0 \<le> hypreal_of_hypnat n"

   761 apply (cases n)

   762 apply (simp add: hypreal_of_hypnat hypreal_zero_num hypreal_le)

   763 done

   765 lemma HNatInfinite_inverse_Infinitesimal [simp]:

   766      "n \<in> HNatInfinite ==> inverse (hypreal_of_hypnat n) \<in> Infinitesimal"

   767 apply (cases n)

   768 apply (auto simp add: hypreal_of_hypnat hypreal_inverse

   769       HNatInfinite_FreeUltrafilterNat_iff Infinitesimal_FreeUltrafilterNat_iff2)

   770 apply (rule bexI, rule_tac [2] lemma_hyprel_refl, auto)

   771 apply (drule_tac x = "m + 1" in spec, ultra)

   772 done

   774 lemma HNatInfinite_hypreal_of_hypnat_gt_zero:

   775      "N \<in> HNatInfinite ==> 0 < hypreal_of_hypnat N"

   776 apply (rule ccontr)

   777 apply (simp add: hypreal_of_hypnat_zero [symmetric] linorder_not_less)

   778 done

   781 ML

   782 {*

   783 val hypnat_of_nat_def = thm"hypnat_of_nat_def";

   784 val HNatInfinite_def = thm"HNatInfinite_def";

   785 val hypreal_of_hypnat_def = thm"hypreal_of_hypnat_def";

   786 val hypnat_zero_def = thm"hypnat_zero_def";

   787 val hypnat_one_def = thm"hypnat_one_def";

   788 val hypnat_omega_def = thm"hypnat_omega_def";

   790 val hypnatrel_iff = thm "hypnatrel_iff";

   791 val hypnatrel_in_hypnat = thm "hypnatrel_in_hypnat";

   792 val inj_on_Abs_hypnat = thm "inj_on_Abs_hypnat";

   793 val inj_Rep_hypnat = thm "inj_Rep_hypnat";

   794 val lemma_hypnatrel_refl = thm "lemma_hypnatrel_refl";

   795 val hypnat_empty_not_mem = thm "hypnat_empty_not_mem";

   796 val Rep_hypnat_nonempty = thm "Rep_hypnat_nonempty";

   797 val eq_Abs_hypnat = thm "eq_Abs_hypnat";

   798 val hypnat_add = thm "hypnat_add";

   799 val hypnat_add_commute = thm "hypnat_add_commute";

   800 val hypnat_add_assoc = thm "hypnat_add_assoc";

   801 val hypnat_add_zero_left = thm "hypnat_add_zero_left";

   802 val hypnat_minus_congruent2 = thm "hypnat_minus_congruent2";

   803 val hypnat_minus = thm "hypnat_minus";

   804 val hypnat_minus_zero = thm "hypnat_minus_zero";

   805 val hypnat_diff_0_eq_0 = thm "hypnat_diff_0_eq_0";

   806 val hypnat_add_is_0 = thm "hypnat_add_is_0";

   807 val hypnat_diff_diff_left = thm "hypnat_diff_diff_left";

   808 val hypnat_diff_commute = thm "hypnat_diff_commute";

   809 val hypnat_diff_add_inverse = thm "hypnat_diff_add_inverse";

   810 val hypnat_diff_add_inverse2 = thm "hypnat_diff_add_inverse2";

   811 val hypnat_diff_cancel = thm "hypnat_diff_cancel";

   812 val hypnat_diff_cancel2 = thm "hypnat_diff_cancel2";

   813 val hypnat_diff_add_0 = thm "hypnat_diff_add_0";

   814 val hypnat_mult_congruent2 = thm "hypnat_mult_congruent2";

   815 val hypnat_mult = thm "hypnat_mult";

   816 val hypnat_mult_commute = thm "hypnat_mult_commute";

   817 val hypnat_mult_assoc = thm "hypnat_mult_assoc";

   818 val hypnat_mult_1 = thm "hypnat_mult_1";

   819 val hypnat_diff_mult_distrib = thm "hypnat_diff_mult_distrib";

   820 val hypnat_diff_mult_distrib2 = thm "hypnat_diff_mult_distrib2";

   821 val hypnat_add_mult_distrib = thm "hypnat_add_mult_distrib";

   822 val hypnat_add_mult_distrib2 = thm "hypnat_add_mult_distrib2";

   823 val hypnat_zero_not_eq_one = thm "hypnat_zero_not_eq_one";

   824 val hypnat_le = thm "hypnat_le";

   825 val hypnat_le_refl = thm "hypnat_le_refl";

   826 val hypnat_le_trans = thm "hypnat_le_trans";

   827 val hypnat_le_anti_sym = thm "hypnat_le_anti_sym";

   828 val hypnat_less_le = thm "hypnat_less_le";

   829 val hypnat_le_linear = thm "hypnat_le_linear";

   830 val hypnat_add_left_mono = thm "hypnat_add_left_mono";

   831 val hypnat_mult_less_mono2 = thm "hypnat_mult_less_mono2";

   832 val hypnat_mult_is_0 = thm "hypnat_mult_is_0";

   833 val hypnat_less = thm "hypnat_less";

   834 val hypnat_not_less0 = thm "hypnat_not_less0";

   835 val hypnat_less_one = thm "hypnat_less_one";

   836 val hypnat_add_diff_inverse = thm "hypnat_add_diff_inverse";

   837 val hypnat_le_add_diff_inverse = thm "hypnat_le_add_diff_inverse";

   838 val hypnat_le_add_diff_inverse2 = thm "hypnat_le_add_diff_inverse2";

   839 val hypnat_le0 = thm "hypnat_le0";

   840 val hypnat_add_self_le = thm "hypnat_add_self_le";

   841 val hypnat_add_one_self_less = thm "hypnat_add_one_self_less";

   842 val hypnat_neq0_conv = thm "hypnat_neq0_conv";

   843 val hypnat_gt_zero_iff = thm "hypnat_gt_zero_iff";

   844 val hypnat_gt_zero_iff2 = thm "hypnat_gt_zero_iff2";

   845 val hypnat_of_nat_add = thm "hypnat_of_nat_add";

   846 val hypnat_of_nat_minus = thm "hypnat_of_nat_minus";

   847 val hypnat_of_nat_mult = thm "hypnat_of_nat_mult";

   848 val hypnat_of_nat_less_iff = thm "hypnat_of_nat_less_iff";

   849 val hypnat_of_nat_le_iff = thm "hypnat_of_nat_le_iff";

   850 val hypnat_of_nat_eq = thm"hypnat_of_nat_eq"

   851 val SHNat_eq = thm"SHNat_eq"

   852 val hypnat_of_nat_one = thm "hypnat_of_nat_one";

   853 val hypnat_of_nat_zero = thm "hypnat_of_nat_zero";

   854 val hypnat_of_nat_zero_iff = thm "hypnat_of_nat_zero_iff";

   855 val hypnat_of_nat_Suc = thm "hypnat_of_nat_Suc";

   856 val hypnat_omega = thm "hypnat_omega";

   857 val Rep_hypnat_omega = thm "Rep_hypnat_omega";

   858 val SHNAT_omega_not_mem = thm "SHNAT_omega_not_mem";

   859 val cofinite_mem_FreeUltrafilterNat = thm "cofinite_mem_FreeUltrafilterNat";

   860 val hypnat_omega_gt_SHNat = thm "hypnat_omega_gt_SHNat";

   861 val hypnat_of_nat_less_whn = thm "hypnat_of_nat_less_whn";

   862 val hypnat_of_nat_le_whn = thm "hypnat_of_nat_le_whn";

   863 val hypnat_zero_less_hypnat_omega = thm "hypnat_zero_less_hypnat_omega";

   864 val hypnat_one_less_hypnat_omega = thm "hypnat_one_less_hypnat_omega";

   865 val HNatInfinite_whn = thm "HNatInfinite_whn";

   866 val HNatInfinite_iff = thm "HNatInfinite_iff";

   867 val HNatInfinite_FreeUltrafilterNat = thm "HNatInfinite_FreeUltrafilterNat";

   868 val FreeUltrafilterNat_HNatInfinite = thm "FreeUltrafilterNat_HNatInfinite";

   869 val HNatInfinite_FreeUltrafilterNat_iff = thm "HNatInfinite_FreeUltrafilterNat_iff";

   870 val HNatInfinite_gt_one = thm "HNatInfinite_gt_one";

   871 val zero_not_mem_HNatInfinite = thm "zero_not_mem_HNatInfinite";

   872 val HNatInfinite_not_eq_zero = thm "HNatInfinite_not_eq_zero";

   873 val HNatInfinite_ge_one = thm "HNatInfinite_ge_one";

   874 val HNatInfinite_add = thm "HNatInfinite_add";

   875 val HNatInfinite_SHNat_add = thm "HNatInfinite_SHNat_add";

   876 val HNatInfinite_SHNat_diff = thm "HNatInfinite_SHNat_diff";

   877 val HNatInfinite_add_one = thm "HNatInfinite_add_one";

   878 val HNatInfinite_is_Suc = thm "HNatInfinite_is_Suc";

   879 val HNat_hypreal_of_nat = thm "HNat_hypreal_of_nat";

   880 val hypreal_of_hypnat = thm "hypreal_of_hypnat";

   881 val hypreal_of_hypnat_zero = thm "hypreal_of_hypnat_zero";

   882 val hypreal_of_hypnat_one = thm "hypreal_of_hypnat_one";

   883 val hypreal_of_hypnat_add = thm "hypreal_of_hypnat_add";

   884 val hypreal_of_hypnat_mult = thm "hypreal_of_hypnat_mult";

   885 val hypreal_of_hypnat_less_iff = thm "hypreal_of_hypnat_less_iff";

   886 val hypreal_of_hypnat_ge_zero = thm "hypreal_of_hypnat_ge_zero";

   887 val HNatInfinite_inverse_Infinitesimal = thm "HNatInfinite_inverse_Infinitesimal";

   888 *}

   890 end

author	obua
	Tue, 11 May 2004 20:11:08 +0200
changeset 14738	83f1a514dcb4
parent 14691	e1eedc8cad37
child 15053	405be2b48f5b
permissions	-rw-r--r--