src/HOL/SetInterval.thy
author nipkow
Tue Aug 28 15:34:15 2007 +0200 (2007-08-28)
changeset 24449 2f05cb7fed85
parent 24286 7619080e49f0
child 24691 e7f46ee04809
permissions -rw-r--r--
added (code) lemmas for setsum and foldl
     1 (*  Title:      HOL/SetInterval.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow and Clemens Ballarin
     4                 Additions by Jeremy Avigad in March 2004
     5     Copyright   2000  TU Muenchen
     6 
     7 lessThan, greaterThan, atLeast, atMost and two-sided intervals
     8 *)
     9 
    10 header {* Set intervals *}
    11 
    12 theory SetInterval
    13 imports IntArith
    14 begin
    15 
    16 constdefs
    17   lessThan    :: "('a::ord) => 'a set"	("(1{..<_})")
    18   "{..<u} == {x. x<u}"
    19 
    20   atMost      :: "('a::ord) => 'a set"	("(1{.._})")
    21   "{..u} == {x. x<=u}"
    22 
    23   greaterThan :: "('a::ord) => 'a set"	("(1{_<..})")
    24   "{l<..} == {x. l<x}"
    25 
    26   atLeast     :: "('a::ord) => 'a set"	("(1{_..})")
    27   "{l..} == {x. l<=x}"
    28 
    29   greaterThanLessThan :: "['a::ord, 'a] => 'a set"  ("(1{_<..<_})")
    30   "{l<..<u} == {l<..} Int {..<u}"
    31 
    32   atLeastLessThan :: "['a::ord, 'a] => 'a set"      ("(1{_..<_})")
    33   "{l..<u} == {l..} Int {..<u}"
    34 
    35   greaterThanAtMost :: "['a::ord, 'a] => 'a set"    ("(1{_<.._})")
    36   "{l<..u} == {l<..} Int {..u}"
    37 
    38   atLeastAtMost :: "['a::ord, 'a] => 'a set"        ("(1{_.._})")
    39   "{l..u} == {l..} Int {..u}"
    40 
    41 text{* A note of warning when using @{term"{..<n}"} on type @{typ
    42 nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
    43 @{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
    44 
    45 syntax
    46   "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3UN _<=_./ _)" 10)
    47   "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3UN _<_./ _)" 10)
    48   "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3INT _<=_./ _)" 10)
    49   "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3INT _<_./ _)" 10)
    50 
    51 syntax (input)
    52   "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" 10)
    53   "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _<_./ _)" 10)
    54   "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" 10)
    55   "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" 10)
    56 
    57 syntax (xsymbols)
    58   "@UNION_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
    59   "@UNION_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
    60   "@INTER_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
    61   "@INTER_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
    62 
    63 translations
    64   "UN i<=n. A"  == "UN i:{..n}. A"
    65   "UN i<n. A"   == "UN i:{..<n}. A"
    66   "INT i<=n. A" == "INT i:{..n}. A"
    67   "INT i<n. A"  == "INT i:{..<n}. A"
    68 
    69 
    70 subsection {* Various equivalences *}
    71 
    72 lemma lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
    73 by (simp add: lessThan_def)
    74 
    75 lemma Compl_lessThan [simp]:
    76     "!!k:: 'a::linorder. -lessThan k = atLeast k"
    77 apply (auto simp add: lessThan_def atLeast_def)
    78 done
    79 
    80 lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
    81 by auto
    82 
    83 lemma greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
    84 by (simp add: greaterThan_def)
    85 
    86 lemma Compl_greaterThan [simp]:
    87     "!!k:: 'a::linorder. -greaterThan k = atMost k"
    88 apply (simp add: greaterThan_def atMost_def le_def, auto)
    89 done
    90 
    91 lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
    92 apply (subst Compl_greaterThan [symmetric])
    93 apply (rule double_complement)
    94 done
    95 
    96 lemma atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
    97 by (simp add: atLeast_def)
    98 
    99 lemma Compl_atLeast [simp]:
   100     "!!k:: 'a::linorder. -atLeast k = lessThan k"
   101 apply (simp add: lessThan_def atLeast_def le_def, auto)
   102 done
   103 
   104 lemma atMost_iff [iff]: "(i: atMost k) = (i<=k)"
   105 by (simp add: atMost_def)
   106 
   107 lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
   108 by (blast intro: order_antisym)
   109 
   110 
   111 subsection {* Logical Equivalences for Set Inclusion and Equality *}
   112 
   113 lemma atLeast_subset_iff [iff]:
   114      "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
   115 by (blast intro: order_trans)
   116 
   117 lemma atLeast_eq_iff [iff]:
   118      "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
   119 by (blast intro: order_antisym order_trans)
   120 
   121 lemma greaterThan_subset_iff [iff]:
   122      "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
   123 apply (auto simp add: greaterThan_def)
   124  apply (subst linorder_not_less [symmetric], blast)
   125 done
   126 
   127 lemma greaterThan_eq_iff [iff]:
   128      "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
   129 apply (rule iffI)
   130  apply (erule equalityE)
   131  apply (simp_all add: greaterThan_subset_iff)
   132 done
   133 
   134 lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
   135 by (blast intro: order_trans)
   136 
   137 lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
   138 by (blast intro: order_antisym order_trans)
   139 
   140 lemma lessThan_subset_iff [iff]:
   141      "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
   142 apply (auto simp add: lessThan_def)
   143  apply (subst linorder_not_less [symmetric], blast)
   144 done
   145 
   146 lemma lessThan_eq_iff [iff]:
   147      "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
   148 apply (rule iffI)
   149  apply (erule equalityE)
   150  apply (simp_all add: lessThan_subset_iff)
   151 done
   152 
   153 
   154 subsection {*Two-sided intervals*}
   155 
   156 lemma greaterThanLessThan_iff [simp,noatp]:
   157   "(i : {l<..<u}) = (l < i & i < u)"
   158 by (simp add: greaterThanLessThan_def)
   159 
   160 lemma atLeastLessThan_iff [simp,noatp]:
   161   "(i : {l..<u}) = (l <= i & i < u)"
   162 by (simp add: atLeastLessThan_def)
   163 
   164 lemma greaterThanAtMost_iff [simp,noatp]:
   165   "(i : {l<..u}) = (l < i & i <= u)"
   166 by (simp add: greaterThanAtMost_def)
   167 
   168 lemma atLeastAtMost_iff [simp,noatp]:
   169   "(i : {l..u}) = (l <= i & i <= u)"
   170 by (simp add: atLeastAtMost_def)
   171 
   172 text {* The above four lemmas could be declared as iffs.
   173   If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
   174   seems to take forever (more than one hour). *}
   175 
   176 subsubsection{* Emptyness and singletons *}
   177 
   178 lemma atLeastAtMost_empty [simp]: "n < m ==> {m::'a::order..n} = {}";
   179   by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);
   180 
   181 lemma atLeastLessThan_empty[simp]: "n \<le> m ==> {m..<n::'a::order} = {}"
   182 by (auto simp add: atLeastLessThan_def)
   183 
   184 lemma greaterThanAtMost_empty[simp]:"l \<le> k ==> {k<..(l::'a::order)} = {}"
   185 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
   186 
   187 lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..(l::'a::order)} = {}"
   188 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
   189 
   190 lemma atLeastAtMost_singleton [simp]: "{a::'a::order..a} = {a}";
   191 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);
   192 
   193 subsection {* Intervals of natural numbers *}
   194 
   195 subsubsection {* The Constant @{term lessThan} *}
   196 
   197 lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
   198 by (simp add: lessThan_def)
   199 
   200 lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
   201 by (simp add: lessThan_def less_Suc_eq, blast)
   202 
   203 lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
   204 by (simp add: lessThan_def atMost_def less_Suc_eq_le)
   205 
   206 lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
   207 by blast
   208 
   209 subsubsection {* The Constant @{term greaterThan} *}
   210 
   211 lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
   212 apply (simp add: greaterThan_def)
   213 apply (blast dest: gr0_conv_Suc [THEN iffD1])
   214 done
   215 
   216 lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
   217 apply (simp add: greaterThan_def)
   218 apply (auto elim: linorder_neqE)
   219 done
   220 
   221 lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
   222 by blast
   223 
   224 subsubsection {* The Constant @{term atLeast} *}
   225 
   226 lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
   227 by (unfold atLeast_def UNIV_def, simp)
   228 
   229 lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
   230 apply (simp add: atLeast_def)
   231 apply (simp add: Suc_le_eq)
   232 apply (simp add: order_le_less, blast)
   233 done
   234 
   235 lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
   236   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
   237 
   238 lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
   239 by blast
   240 
   241 subsubsection {* The Constant @{term atMost} *}
   242 
   243 lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
   244 by (simp add: atMost_def)
   245 
   246 lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
   247 apply (simp add: atMost_def)
   248 apply (simp add: less_Suc_eq order_le_less, blast)
   249 done
   250 
   251 lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
   252 by blast
   253 
   254 subsubsection {* The Constant @{term atLeastLessThan} *}
   255 
   256 text{*The orientation of the following rule is tricky. The lhs is
   257 defined in terms of the rhs.  Hence the chosen orientation makes sense
   258 in this theory --- the reverse orientation complicates proofs (eg
   259 nontermination). But outside, when the definition of the lhs is rarely
   260 used, the opposite orientation seems preferable because it reduces a
   261 specific concept to a more general one. *}
   262 lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
   263 by(simp add:lessThan_def atLeastLessThan_def)
   264 
   265 declare atLeast0LessThan[symmetric, code unfold]
   266 
   267 lemma atLeastLessThan0: "{m..<0::nat} = {}"
   268 by (simp add: atLeastLessThan_def)
   269 
   270 subsubsection {* Intervals of nats with @{term Suc} *}
   271 
   272 text{*Not a simprule because the RHS is too messy.*}
   273 lemma atLeastLessThanSuc:
   274     "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
   275 by (auto simp add: atLeastLessThan_def)
   276 
   277 lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
   278 by (auto simp add: atLeastLessThan_def)
   279 (*
   280 lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
   281 by (induct k, simp_all add: atLeastLessThanSuc)
   282 
   283 lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
   284 by (auto simp add: atLeastLessThan_def)
   285 *)
   286 lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
   287   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
   288 
   289 lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
   290   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
   291     greaterThanAtMost_def)
   292 
   293 lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
   294   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
   295     greaterThanLessThan_def)
   296 
   297 lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
   298 by (auto simp add: atLeastAtMost_def)
   299 
   300 subsubsection {* Image *}
   301 
   302 lemma image_add_atLeastAtMost:
   303   "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
   304 proof
   305   show "?A \<subseteq> ?B" by auto
   306 next
   307   show "?B \<subseteq> ?A"
   308   proof
   309     fix n assume a: "n : ?B"
   310     hence "n - k : {i..j}" by auto
   311     moreover have "n = (n - k) + k" using a by auto
   312     ultimately show "n : ?A" by blast
   313   qed
   314 qed
   315 
   316 lemma image_add_atLeastLessThan:
   317   "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
   318 proof
   319   show "?A \<subseteq> ?B" by auto
   320 next
   321   show "?B \<subseteq> ?A"
   322   proof
   323     fix n assume a: "n : ?B"
   324     hence "n - k : {i..<j}" by auto
   325     moreover have "n = (n - k) + k" using a by auto
   326     ultimately show "n : ?A" by blast
   327   qed
   328 qed
   329 
   330 corollary image_Suc_atLeastAtMost[simp]:
   331   "Suc ` {i..j} = {Suc i..Suc j}"
   332 using image_add_atLeastAtMost[where k=1] by simp
   333 
   334 corollary image_Suc_atLeastLessThan[simp]:
   335   "Suc ` {i..<j} = {Suc i..<Suc j}"
   336 using image_add_atLeastLessThan[where k=1] by simp
   337 
   338 lemma image_add_int_atLeastLessThan:
   339     "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
   340   apply (auto simp add: image_def)
   341   apply (rule_tac x = "x - l" in bexI)
   342   apply auto
   343   done
   344 
   345 
   346 subsubsection {* Finiteness *}
   347 
   348 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
   349   by (induct k) (simp_all add: lessThan_Suc)
   350 
   351 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
   352   by (induct k) (simp_all add: atMost_Suc)
   353 
   354 lemma finite_greaterThanLessThan [iff]:
   355   fixes l :: nat shows "finite {l<..<u}"
   356 by (simp add: greaterThanLessThan_def)
   357 
   358 lemma finite_atLeastLessThan [iff]:
   359   fixes l :: nat shows "finite {l..<u}"
   360 by (simp add: atLeastLessThan_def)
   361 
   362 lemma finite_greaterThanAtMost [iff]:
   363   fixes l :: nat shows "finite {l<..u}"
   364 by (simp add: greaterThanAtMost_def)
   365 
   366 lemma finite_atLeastAtMost [iff]:
   367   fixes l :: nat shows "finite {l..u}"
   368 by (simp add: atLeastAtMost_def)
   369 
   370 lemma bounded_nat_set_is_finite:
   371     "(ALL i:N. i < (n::nat)) ==> finite N"
   372   -- {* A bounded set of natural numbers is finite. *}
   373   apply (rule finite_subset)
   374    apply (rule_tac [2] finite_lessThan, auto)
   375   done
   376 
   377 subsubsection {* Cardinality *}
   378 
   379 lemma card_lessThan [simp]: "card {..<u} = u"
   380   by (induct u, simp_all add: lessThan_Suc)
   381 
   382 lemma card_atMost [simp]: "card {..u} = Suc u"
   383   by (simp add: lessThan_Suc_atMost [THEN sym])
   384 
   385 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
   386   apply (subgoal_tac "card {l..<u} = card {..<u-l}")
   387   apply (erule ssubst, rule card_lessThan)
   388   apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
   389   apply (erule subst)
   390   apply (rule card_image)
   391   apply (simp add: inj_on_def)
   392   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
   393   apply (rule_tac x = "x - l" in exI)
   394   apply arith
   395   done
   396 
   397 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
   398   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
   399 
   400 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
   401   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
   402 
   403 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
   404   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
   405 
   406 subsection {* Intervals of integers *}
   407 
   408 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
   409   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
   410 
   411 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
   412   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
   413 
   414 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
   415     "{l+1..<u} = {l<..<u::int}"
   416   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
   417 
   418 subsubsection {* Finiteness *}
   419 
   420 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
   421     {(0::int)..<u} = int ` {..<nat u}"
   422   apply (unfold image_def lessThan_def)
   423   apply auto
   424   apply (rule_tac x = "nat x" in exI)
   425   apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
   426   done
   427 
   428 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
   429   apply (case_tac "0 \<le> u")
   430   apply (subst image_atLeastZeroLessThan_int, assumption)
   431   apply (rule finite_imageI)
   432   apply auto
   433   done
   434 
   435 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
   436   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
   437   apply (erule subst)
   438   apply (rule finite_imageI)
   439   apply (rule finite_atLeastZeroLessThan_int)
   440   apply (rule image_add_int_atLeastLessThan)
   441   done
   442 
   443 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
   444   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
   445 
   446 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
   447   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
   448 
   449 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
   450   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
   451 
   452 subsubsection {* Cardinality *}
   453 
   454 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
   455   apply (case_tac "0 \<le> u")
   456   apply (subst image_atLeastZeroLessThan_int, assumption)
   457   apply (subst card_image)
   458   apply (auto simp add: inj_on_def)
   459   done
   460 
   461 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
   462   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
   463   apply (erule ssubst, rule card_atLeastZeroLessThan_int)
   464   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
   465   apply (erule subst)
   466   apply (rule card_image)
   467   apply (simp add: inj_on_def)
   468   apply (rule image_add_int_atLeastLessThan)
   469   done
   470 
   471 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
   472   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
   473   apply (auto simp add: compare_rls)
   474   done
   475 
   476 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
   477   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
   478 
   479 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
   480   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
   481 
   482 
   483 subsection {*Lemmas useful with the summation operator setsum*}
   484 
   485 text {* For examples, see Algebra/poly/UnivPoly2.thy *}
   486 
   487 subsubsection {* Disjoint Unions *}
   488 
   489 text {* Singletons and open intervals *}
   490 
   491 lemma ivl_disj_un_singleton:
   492   "{l::'a::linorder} Un {l<..} = {l..}"
   493   "{..<u} Un {u::'a::linorder} = {..u}"
   494   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
   495   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
   496   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
   497   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
   498 by auto
   499 
   500 text {* One- and two-sided intervals *}
   501 
   502 lemma ivl_disj_un_one:
   503   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
   504   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
   505   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
   506   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
   507   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
   508   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
   509   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
   510   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
   511 by auto
   512 
   513 text {* Two- and two-sided intervals *}
   514 
   515 lemma ivl_disj_un_two:
   516   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
   517   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
   518   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
   519   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
   520   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
   521   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
   522   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
   523   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
   524 by auto
   525 
   526 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
   527 
   528 subsubsection {* Disjoint Intersections *}
   529 
   530 text {* Singletons and open intervals *}
   531 
   532 lemma ivl_disj_int_singleton:
   533   "{l::'a::order} Int {l<..} = {}"
   534   "{..<u} Int {u} = {}"
   535   "{l} Int {l<..<u} = {}"
   536   "{l<..<u} Int {u} = {}"
   537   "{l} Int {l<..u} = {}"
   538   "{l..<u} Int {u} = {}"
   539   by simp+
   540 
   541 text {* One- and two-sided intervals *}
   542 
   543 lemma ivl_disj_int_one:
   544   "{..l::'a::order} Int {l<..<u} = {}"
   545   "{..<l} Int {l..<u} = {}"
   546   "{..l} Int {l<..u} = {}"
   547   "{..<l} Int {l..u} = {}"
   548   "{l<..u} Int {u<..} = {}"
   549   "{l<..<u} Int {u..} = {}"
   550   "{l..u} Int {u<..} = {}"
   551   "{l..<u} Int {u..} = {}"
   552   by auto
   553 
   554 text {* Two- and two-sided intervals *}
   555 
   556 lemma ivl_disj_int_two:
   557   "{l::'a::order<..<m} Int {m..<u} = {}"
   558   "{l<..m} Int {m<..<u} = {}"
   559   "{l..<m} Int {m..<u} = {}"
   560   "{l..m} Int {m<..<u} = {}"
   561   "{l<..<m} Int {m..u} = {}"
   562   "{l<..m} Int {m<..u} = {}"
   563   "{l..<m} Int {m..u} = {}"
   564   "{l..m} Int {m<..u} = {}"
   565   by auto
   566 
   567 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
   568 
   569 subsubsection {* Some Differences *}
   570 
   571 lemma ivl_diff[simp]:
   572  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
   573 by(auto)
   574 
   575 
   576 subsubsection {* Some Subset Conditions *}
   577 
   578 lemma ivl_subset [simp,noatp]:
   579  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
   580 apply(auto simp:linorder_not_le)
   581 apply(rule ccontr)
   582 apply(insert linorder_le_less_linear[of i n])
   583 apply(clarsimp simp:linorder_not_le)
   584 apply(fastsimp)
   585 done
   586 
   587 
   588 subsection {* Summation indexed over intervals *}
   589 
   590 syntax
   591   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
   592   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
   593   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
   594   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
   595 syntax (xsymbols)
   596   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
   597   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
   598   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
   599   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
   600 syntax (HTML output)
   601   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
   602   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
   603   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
   604   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
   605 syntax (latex_sum output)
   606   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   607  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
   608   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   609  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
   610   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   611  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
   612   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   613  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
   614 
   615 translations
   616   "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
   617   "\<Sum>x=a..<b. t" == "setsum (%x. t) {a..<b}"
   618   "\<Sum>i\<le>n. t" == "setsum (\<lambda>i. t) {..n}"
   619   "\<Sum>i<n. t" == "setsum (\<lambda>i. t) {..<n}"
   620 
   621 text{* The above introduces some pretty alternative syntaxes for
   622 summation over intervals:
   623 \begin{center}
   624 \begin{tabular}{lll}
   625 Old & New & \LaTeX\\
   626 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
   627 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
   628 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
   629 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
   630 \end{tabular}
   631 \end{center}
   632 The left column shows the term before introduction of the new syntax,
   633 the middle column shows the new (default) syntax, and the right column
   634 shows a special syntax. The latter is only meaningful for latex output
   635 and has to be activated explicitly by setting the print mode to
   636 @{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
   637 antiquotations). It is not the default \LaTeX\ output because it only
   638 works well with italic-style formulae, not tt-style.
   639 
   640 Note that for uniformity on @{typ nat} it is better to use
   641 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
   642 not provide all lemmas available for @{term"{m..<n}"} also in the
   643 special form for @{term"{..<n}"}. *}
   644 
   645 text{* This congruence rule should be used for sums over intervals as
   646 the standard theorem @{text[source]setsum_cong} does not work well
   647 with the simplifier who adds the unsimplified premise @{term"x:B"} to
   648 the context. *}
   649 
   650 lemma setsum_ivl_cong:
   651  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
   652  setsum f {a..<b} = setsum g {c..<d}"
   653 by(rule setsum_cong, simp_all)
   654 
   655 (* FIXME why are the following simp rules but the corresponding eqns
   656 on intervals are not? *)
   657 
   658 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
   659 by (simp add:atMost_Suc add_ac)
   660 
   661 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
   662 by (simp add:lessThan_Suc add_ac)
   663 
   664 lemma setsum_cl_ivl_Suc[simp]:
   665   "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
   666 by (auto simp:add_ac atLeastAtMostSuc_conv)
   667 
   668 lemma setsum_op_ivl_Suc[simp]:
   669   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
   670 by (auto simp:add_ac atLeastLessThanSuc)
   671 (*
   672 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
   673     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
   674 by (auto simp:add_ac atLeastAtMostSuc_conv)
   675 *)
   676 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
   677   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
   678 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
   679 
   680 lemma setsum_diff_nat_ivl:
   681 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
   682 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
   683   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
   684 using setsum_add_nat_ivl [of m n p f,symmetric]
   685 apply (simp add: add_ac)
   686 done
   687 
   688 subsection{* Shifting bounds *}
   689 
   690 lemma setsum_shift_bounds_nat_ivl:
   691   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
   692 by (induct "n", auto simp:atLeastLessThanSuc)
   693 
   694 lemma setsum_shift_bounds_cl_nat_ivl:
   695   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
   696 apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
   697 apply (simp add:image_add_atLeastAtMost o_def)
   698 done
   699 
   700 corollary setsum_shift_bounds_cl_Suc_ivl:
   701   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
   702 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k=1,simplified])
   703 
   704 corollary setsum_shift_bounds_Suc_ivl:
   705   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
   706 by (simp add:setsum_shift_bounds_nat_ivl[where k=1,simplified])
   707 
   708 lemma setsum_head:
   709   fixes n :: nat
   710   assumes mn: "m <= n" 
   711   shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
   712 proof -
   713   from mn
   714   have "{m..n} = {m} \<union> {m<..n}"
   715     by (auto intro: ivl_disj_un_singleton)
   716   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
   717     by (simp add: atLeast0LessThan)
   718   also have "\<dots> = ?rhs" by simp
   719   finally show ?thesis .
   720 qed
   721 
   722 lemma setsum_head_upt:
   723   fixes m::nat
   724   assumes m: "0 < m"
   725   shows "(\<Sum>x<m. P x) = P 0 + (\<Sum>x\<in>{1..<m}. P x)"
   726 proof -
   727   have "(\<Sum>x<m. P x) = (\<Sum>x\<in>{0..<m}. P x)" 
   728     by (simp add: atLeast0LessThan)
   729   also 
   730   from m 
   731   have "\<dots> = (\<Sum>x\<in>{0..m - 1}. P x)"
   732     by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)
   733   also
   734   have "\<dots> = P 0 + (\<Sum>x\<in>{0<..m - 1}. P x)"
   735     by (simp add: setsum_head)
   736   also 
   737   from m 
   738   have "{0<..m - 1} = {1..<m}" 
   739     by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)
   740   finally show ?thesis .
   741 qed
   742 
   743 subsection {* The formula for geometric sums *}
   744 
   745 lemma geometric_sum:
   746   "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
   747   (x ^ n - 1) / (x - 1::'a::{field, recpower})"
   748 by (induct "n") (simp_all add:field_simps power_Suc)
   749 
   750 subsection {* The formula for arithmetic sums *}
   751 
   752 lemma gauss_sum:
   753   "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
   754    of_nat n*((of_nat n)+1)"
   755 proof (induct n)
   756   case 0
   757   show ?case by simp
   758 next
   759   case (Suc n)
   760   then show ?case by (simp add: ring_simps)
   761 qed
   762 
   763 theorem arith_series_general:
   764   "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
   765   of_nat n * (a + (a + of_nat(n - 1)*d))"
   766 proof cases
   767   assume ngt1: "n > 1"
   768   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
   769   have
   770     "(\<Sum>i\<in>{..<n}. a+?I i*d) =
   771      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
   772     by (rule setsum_addf)
   773   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
   774   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
   775     by (simp add: setsum_right_distrib setsum_head_upt mult_ac)
   776   also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
   777     by (simp add: left_distrib right_distrib)
   778   also from ngt1 have "{1..<n} = {1..n - 1}"
   779     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)    
   780   also from ngt1 
   781   have "(1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..n - 1}. ?I i) = ((1+1)*?n*a + d*?I (n - 1)*?I n)"
   782     by (simp only: mult_ac gauss_sum [of "n - 1"])
   783        (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
   784   finally show ?thesis by (simp add: mult_ac add_ac right_distrib)
   785 next
   786   assume "\<not>(n > 1)"
   787   hence "n = 1 \<or> n = 0" by auto
   788   thus ?thesis by (auto simp: mult_ac right_distrib)
   789 qed
   790 
   791 lemma arith_series_nat:
   792   "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
   793 proof -
   794   have
   795     "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
   796     of_nat(n) * (a + (a + of_nat(n - 1)*d))"
   797     by (rule arith_series_general)
   798   thus ?thesis by (auto simp add: of_nat_id)
   799 qed
   800 
   801 lemma arith_series_int:
   802   "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
   803   of_nat n * (a + (a + of_nat(n - 1)*d))"
   804 proof -
   805   have
   806     "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
   807     of_nat(n) * (a + (a + of_nat(n - 1)*d))"
   808     by (rule arith_series_general)
   809   thus ?thesis by simp
   810 qed
   811 
   812 lemma sum_diff_distrib:
   813   fixes P::"nat\<Rightarrow>nat"
   814   shows
   815   "\<forall>x. Q x \<le> P x  \<Longrightarrow>
   816   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
   817 proof (induct n)
   818   case 0 show ?case by simp
   819 next
   820   case (Suc n)
   821 
   822   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
   823   let ?rhs = "\<Sum>x<n. P x - Q x"
   824 
   825   from Suc have "?lhs = ?rhs" by simp
   826   moreover
   827   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
   828   moreover
   829   from Suc have
   830     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
   831     by (subst diff_diff_left[symmetric],
   832         subst diff_add_assoc2)
   833        (auto simp: diff_add_assoc2 intro: setsum_mono)
   834   ultimately
   835   show ?case by simp
   836 qed
   837 
   838 
   839 ML
   840 {*
   841 val Compl_atLeast = thm "Compl_atLeast";
   842 val Compl_atMost = thm "Compl_atMost";
   843 val Compl_greaterThan = thm "Compl_greaterThan";
   844 val Compl_lessThan = thm "Compl_lessThan";
   845 val INT_greaterThan_UNIV = thm "INT_greaterThan_UNIV";
   846 val UN_atLeast_UNIV = thm "UN_atLeast_UNIV";
   847 val UN_atMost_UNIV = thm "UN_atMost_UNIV";
   848 val UN_lessThan_UNIV = thm "UN_lessThan_UNIV";
   849 val atLeastAtMost_def = thm "atLeastAtMost_def";
   850 val atLeastAtMost_iff = thm "atLeastAtMost_iff";
   851 val atLeastLessThan_def  = thm "atLeastLessThan_def";
   852 val atLeastLessThan_iff = thm "atLeastLessThan_iff";
   853 val atLeast_0 = thm "atLeast_0";
   854 val atLeast_Suc = thm "atLeast_Suc";
   855 val atLeast_def      = thm "atLeast_def";
   856 val atLeast_iff = thm "atLeast_iff";
   857 val atMost_0 = thm "atMost_0";
   858 val atMost_Int_atLeast = thm "atMost_Int_atLeast";
   859 val atMost_Suc = thm "atMost_Suc";
   860 val atMost_def       = thm "atMost_def";
   861 val atMost_iff = thm "atMost_iff";
   862 val greaterThanAtMost_def  = thm "greaterThanAtMost_def";
   863 val greaterThanAtMost_iff = thm "greaterThanAtMost_iff";
   864 val greaterThanLessThan_def  = thm "greaterThanLessThan_def";
   865 val greaterThanLessThan_iff = thm "greaterThanLessThan_iff";
   866 val greaterThan_0 = thm "greaterThan_0";
   867 val greaterThan_Suc = thm "greaterThan_Suc";
   868 val greaterThan_def  = thm "greaterThan_def";
   869 val greaterThan_iff = thm "greaterThan_iff";
   870 val ivl_disj_int = thms "ivl_disj_int";
   871 val ivl_disj_int_one = thms "ivl_disj_int_one";
   872 val ivl_disj_int_singleton = thms "ivl_disj_int_singleton";
   873 val ivl_disj_int_two = thms "ivl_disj_int_two";
   874 val ivl_disj_un = thms "ivl_disj_un";
   875 val ivl_disj_un_one = thms "ivl_disj_un_one";
   876 val ivl_disj_un_singleton = thms "ivl_disj_un_singleton";
   877 val ivl_disj_un_two = thms "ivl_disj_un_two";
   878 val lessThan_0 = thm "lessThan_0";
   879 val lessThan_Suc = thm "lessThan_Suc";
   880 val lessThan_Suc_atMost = thm "lessThan_Suc_atMost";
   881 val lessThan_def     = thm "lessThan_def";
   882 val lessThan_iff = thm "lessThan_iff";
   883 val single_Diff_lessThan = thm "single_Diff_lessThan";
   884 
   885 val bounded_nat_set_is_finite = thm "bounded_nat_set_is_finite";
   886 val finite_atMost = thm "finite_atMost";
   887 val finite_lessThan = thm "finite_lessThan";
   888 *}
   889 
   890 end