wneuper/isa: src/HOL/Nat.thy@83f1a514dcb4

     1 (*  Title:      HOL/Nat.thy

     2     ID:         $Id$

     3     Author:     Tobias Nipkow and Lawrence C Paulson

     5 Type "nat" is a linear order, and a datatype; arithmetic operators + -

     6 and * (for div, mod and dvd, see theory Divides).

     7 *)

     9 header {* Natural numbers *}

    11 theory Nat = Wellfounded_Recursion + Ring_and_Field:

    13 subsection {* Type @{text ind} *}

    15 typedecl ind

    17 consts

    18   Zero_Rep      :: ind

    19   Suc_Rep       :: "ind => ind"

    21 axioms

    22   -- {* the axiom of infinity in 2 parts *}

    23   inj_Suc_Rep:          "inj Suc_Rep"

    24   Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"

    27 subsection {* Type nat *}

    29 text {* Type definition *}

    31 consts

    32   Nat :: "ind set"

    34 inductive Nat

    35 intros

    36   Zero_RepI: "Zero_Rep : Nat"

    37   Suc_RepI: "i : Nat ==> Suc_Rep i : Nat"

    39 global

    41 typedef (open Nat)

    42   nat = Nat by (rule exI, rule Nat.Zero_RepI)

    44 instance nat :: "{ord, zero, one}" ..

    47 text {* Abstract constants and syntax *}

    49 consts

    50   Suc :: "nat => nat"

    51   pred_nat :: "(nat * nat) set"

    53 local

    55 defs

    56   Zero_nat_def: "0 == Abs_Nat Zero_Rep"

    57   Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"

    58   One_nat_def [simp]: "1 == Suc 0"

    60   -- {* nat operations *}

    61   pred_nat_def: "pred_nat == {(m, n). n = Suc m}"

    63   less_def: "m < n == (m, n) : trancl pred_nat"

    65   le_def: "m \<le> (n::nat) == ~ (n < m)"

    68 text {* Induction *}

    70 theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n"

    71   apply (unfold Zero_nat_def Suc_def)

    72   apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}

    73   apply (erule Rep_Nat [THEN Nat.induct])

    74   apply (rules elim: Abs_Nat_inverse [THEN subst])

    75   done

    78 text {* Isomorphisms: @{text Abs_Nat} and @{text Rep_Nat} *}

    80 lemma inj_Rep_Nat: "inj Rep_Nat"

    81   apply (rule inj_on_inverseI)

    82   apply (rule Rep_Nat_inverse)

    83   done

    85 lemma inj_on_Abs_Nat: "inj_on Abs_Nat Nat"

    86   apply (rule inj_on_inverseI)

    87   apply (erule Abs_Nat_inverse)

    88   done

    90 text {* Distinctness of constructors *}

    92 lemma Suc_not_Zero [iff]: "Suc m \<noteq> 0"

    93   apply (unfold Zero_nat_def Suc_def)

    94   apply (rule inj_on_Abs_Nat [THEN inj_on_contraD])

    95   apply (rule Suc_Rep_not_Zero_Rep)

    96   apply (rule Rep_Nat Nat.Suc_RepI Nat.Zero_RepI)+

    97   done

    99 lemma Zero_not_Suc [iff]: "0 \<noteq> Suc m"

   100   by (rule not_sym, rule Suc_not_Zero not_sym)

   102 lemma Suc_neq_Zero: "Suc m = 0 ==> R"

   103   by (rule notE, rule Suc_not_Zero)

   105 lemma Zero_neq_Suc: "0 = Suc m ==> R"

   106   by (rule Suc_neq_Zero, erule sym)

   108 text {* Injectiveness of @{term Suc} *}

   110 lemma inj_Suc: "inj Suc"

   111   apply (unfold Suc_def)

   112   apply (rule inj_onI)

   113   apply (drule inj_on_Abs_Nat [THEN inj_onD])

   114   apply (rule Rep_Nat Nat.Suc_RepI)+

   115   apply (drule inj_Suc_Rep [THEN injD])

   116   apply (erule inj_Rep_Nat [THEN injD])

   117   done

   119 lemma Suc_inject: "Suc x = Suc y ==> x = y"

   120   by (rule inj_Suc [THEN injD])

   122 lemma Suc_Suc_eq [iff]: "(Suc m = Suc n) = (m = n)"

   123   apply (rule iffI)

   124   apply (erule Suc_inject)

   125   apply (erule arg_cong)

   126   done

   128 lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False"

   129   by auto

   131 text {* @{typ nat} is a datatype *}

   133 rep_datatype nat

   134   distinct  Suc_not_Zero Zero_not_Suc

   135   inject    Suc_Suc_eq

   136   induction nat_induct

   138 lemma n_not_Suc_n: "n \<noteq> Suc n"

   139   by (induct n) simp_all

   141 lemma Suc_n_not_n: "Suc t \<noteq> t"

   142   by (rule not_sym, rule n_not_Suc_n)

   144 text {* A special form of induction for reasoning

   145   about @{term "m < n"} and @{term "m - n"} *}

   147 theorem diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>

   148     (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"

   149   apply (rule_tac x = m in spec)

   150   apply (induct_tac n)

   151   prefer 2

   152   apply (rule allI)

   153   apply (induct_tac x, rules+)

   154   done

   156 subsection {* Basic properties of "less than" *}

   158 lemma wf_pred_nat: "wf pred_nat"

   159   apply (unfold wf_def pred_nat_def, clarify)

   160   apply (induct_tac x, blast+)

   161   done

   163 lemma wf_less: "wf {(x, y::nat). x < y}"

   164   apply (unfold less_def)

   165   apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset], blast)

   166   done

   168 lemma less_eq: "((m, n) : pred_nat^+) = (m < n)"

   169   apply (unfold less_def)

   170   apply (rule refl)

   171   done

   173 subsubsection {* Introduction properties *}

   175 lemma less_trans: "i < j ==> j < k ==> i < (k::nat)"

   176   apply (unfold less_def)

   177   apply (rule trans_trancl [THEN transD], assumption+)

   178   done

   180 lemma lessI [iff]: "n < Suc n"

   181   apply (unfold less_def pred_nat_def)

   182   apply (simp add: r_into_trancl)

   183   done

   185 lemma less_SucI: "i < j ==> i < Suc j"

   186   apply (rule less_trans, assumption)

   187   apply (rule lessI)

   188   done

   190 lemma zero_less_Suc [iff]: "0 < Suc n"

   191   apply (induct n)

   192   apply (rule lessI)

   193   apply (erule less_trans)

   194   apply (rule lessI)

   195   done

   197 subsubsection {* Elimination properties *}

   199 lemma less_not_sym: "n < m ==> ~ m < (n::nat)"

   200   apply (unfold less_def)

   201   apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym])

   202   done

   204 lemma less_asym:

   205   assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P

   206   apply (rule contrapos_np)

   207   apply (rule less_not_sym)

   208   apply (rule h1)

   209   apply (erule h2)

   210   done

   212 lemma less_not_refl: "~ n < (n::nat)"

   213   apply (unfold less_def)

   214   apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl])

   215   done

   217 lemma less_irrefl [elim!]: "(n::nat) < n ==> R"

   218   by (rule notE, rule less_not_refl)

   220 lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)" by blast

   222 lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"

   223   by (rule not_sym, rule less_not_refl2)

   225 lemma lessE:

   226   assumes major: "i < k"

   227   and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"

   228   shows P

   229   apply (rule major [unfolded less_def pred_nat_def, THEN tranclE], simp_all)

   230   apply (erule p1)

   231   apply (rule p2)

   232   apply (simp add: less_def pred_nat_def, assumption)

   233   done

   235 lemma not_less0 [iff]: "~ n < (0::nat)"

   236   by (blast elim: lessE)

   238 lemma less_zeroE: "(n::nat) < 0 ==> R"

   239   by (rule notE, rule not_less0)

   241 lemma less_SucE: assumes major: "m < Suc n"

   242   and less: "m < n ==> P" and eq: "m = n ==> P" shows P

   243   apply (rule major [THEN lessE])

   244   apply (rule eq, blast)

   245   apply (rule less, blast)

   246   done

   248 lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"

   249   by (blast elim!: less_SucE intro: less_trans)

   251 lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"

   252   by (simp add: less_Suc_eq)

   254 lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"

   255   by (simp add: less_Suc_eq)

   257 lemma Suc_mono: "m < n ==> Suc m < Suc n"

   258   by (induct n) (fast elim: less_trans lessE)+

   260 text {* "Less than" is a linear ordering *}

   261 lemma less_linear: "m < n | m = n | n < (m::nat)"

   262   apply (induct_tac m)

   263   apply (induct_tac n)

   264   apply (rule refl [THEN disjI1, THEN disjI2])

   265   apply (rule zero_less_Suc [THEN disjI1])

   266   apply (blast intro: Suc_mono less_SucI elim: lessE)

   267   done

   269 text {* "Less than" is antisymmetric, sort of *}

   270 lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"

   271 apply(simp only:less_Suc_eq)

   272 apply blast

   273 done

   275 lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"

   276   using less_linear by blast

   278 lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"

   279   and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"

   280   shows "P n m"

   281   apply (rule less_linear [THEN disjE])

   282   apply (erule_tac [2] disjE)

   283   apply (erule lessCase)

   284   apply (erule sym [THEN eqCase])

   285   apply (erule major)

   286   done

   289 subsubsection {* Inductive (?) properties *}

   291 lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"

   292   apply (simp add: nat_neq_iff)

   293   apply (blast elim!: less_irrefl less_SucE elim: less_asym)

   294   done

   296 lemma Suc_lessD: "Suc m < n ==> m < n"

   297   apply (induct n)

   298   apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+

   299   done

   301 lemma Suc_lessE: assumes major: "Suc i < k"

   302   and minor: "!!j. i < j ==> k = Suc j ==> P" shows P

   303   apply (rule major [THEN lessE])

   304   apply (erule lessI [THEN minor])

   305   apply (erule Suc_lessD [THEN minor], assumption)

   306   done

   308 lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"

   309   by (blast elim: lessE dest: Suc_lessD)

   311 lemma Suc_less_eq [iff]: "(Suc m < Suc n) = (m < n)"

   312   apply (rule iffI)

   313   apply (erule Suc_less_SucD)

   314   apply (erule Suc_mono)

   315   done

   317 lemma less_trans_Suc:

   318   assumes le: "i < j" shows "j < k ==> Suc i < k"

   319   apply (induct k, simp_all)

   320   apply (insert le)

   321   apply (simp add: less_Suc_eq)

   322   apply (blast dest: Suc_lessD)

   323   done

   325 text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}

   326 lemma not_less_eq: "(~ m < n) = (n < Suc m)"

   327 by (rule_tac m = m and n = n in diff_induct, simp_all)

   329 text {* Complete induction, aka course-of-values induction *}

   330 lemma nat_less_induct:

   331   assumes prem: "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"

   332   apply (rule_tac a=n in wf_induct)

   333   apply (rule wf_pred_nat [THEN wf_trancl])

   334   apply (rule prem)

   335   apply (unfold less_def, assumption)

   336   done

   338 lemmas less_induct = nat_less_induct [rule_format, case_names less]

   340 subsection {* Properties of "less than or equal" *}

   342 text {* Was @{text le_eq_less_Suc}, but this orientation is more useful *}

   343 lemma less_Suc_eq_le: "(m < Suc n) = (m \<le> n)"

   344   by (unfold le_def, rule not_less_eq [symmetric])

   346 lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"

   347   by (rule less_Suc_eq_le [THEN iffD2])

   349 lemma le0 [iff]: "(0::nat) \<le> n"

   350   by (unfold le_def, rule not_less0)

   352 lemma Suc_n_not_le_n: "~ Suc n \<le> n"

   353   by (simp add: le_def)

   355 lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)"

   356   by (induct i) (simp_all add: le_def)

   358 lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"

   359   by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq)

   361 lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"

   362   by (drule le_Suc_eq [THEN iffD1], rules+)

   364 lemma leI: "~ n < m ==> m \<le> (n::nat)" by (simp add: le_def)

   366 lemma leD: "m \<le> n ==> ~ n < (m::nat)"

   367   by (simp add: le_def)

   369 lemmas leE = leD [elim_format]

   371 lemma not_less_iff_le: "(~ n < m) = (m \<le> (n::nat))"

   372   by (blast intro: leI elim: leE)

   374 lemma not_leE: "~ m \<le> n ==> n<(m::nat)"

   375   by (simp add: le_def)

   377 lemma not_le_iff_less: "(~ n \<le> m) = (m < (n::nat))"

   378   by (simp add: le_def)

   380 lemma Suc_leI: "m < n ==> Suc(m) \<le> n"

   381   apply (simp add: le_def less_Suc_eq)

   382   apply (blast elim!: less_irrefl less_asym)

   383   done -- {* formerly called lessD *}

   385 lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n"

   386   by (simp add: le_def less_Suc_eq)

   388 text {* Stronger version of @{text Suc_leD} *}

   389 lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"

   390   apply (simp add: le_def less_Suc_eq)

   391   using less_linear

   392   apply blast

   393   done

   395 lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)"

   396   by (blast intro: Suc_leI Suc_le_lessD)

   398 lemma le_SucI: "m \<le> n ==> m \<le> Suc n"

   399   by (unfold le_def) (blast dest: Suc_lessD)

   401 lemma less_imp_le: "m < n ==> m \<le> (n::nat)"

   402   by (unfold le_def) (blast elim: less_asym)

   404 text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}

   405 lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq

   408 text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}

   410 lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)"

   411   apply (unfold le_def)

   412   using less_linear

   413   apply (blast elim: less_irrefl less_asym)

   414   done

   416 lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"

   417   apply (unfold le_def)

   418   using less_linear

   419   apply (blast elim!: less_irrefl elim: less_asym)

   420   done

   422 lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"

   423   by (rules intro: less_or_eq_imp_le le_imp_less_or_eq)

   425 text {* Useful with @{text Blast}. *}

   426 lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"

   427   by (rule less_or_eq_imp_le, rule disjI2)

   429 lemma le_refl: "n \<le> (n::nat)"

   430   by (simp add: le_eq_less_or_eq)

   432 lemma le_less_trans: "[| i \<le> j; j < k |] ==> i < (k::nat)"

   433   by (blast dest!: le_imp_less_or_eq intro: less_trans)

   435 lemma less_le_trans: "[| i < j; j \<le> k |] ==> i < (k::nat)"

   436   by (blast dest!: le_imp_less_or_eq intro: less_trans)

   438 lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"

   439   by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans)

   441 lemma le_anti_sym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"

   442   by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym)

   444 lemma Suc_le_mono [iff]: "(Suc n \<le> Suc m) = (n \<le> m)"

   445   by (simp add: le_simps)

   447 text {* Axiom @{text order_less_le} of class @{text order}: *}

   448 lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"

   449   by (simp add: le_def nat_neq_iff) (blast elim!: less_asym)

   451 lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"

   452   by (rule iffD2, rule nat_less_le, rule conjI)

   454 text {* Axiom @{text linorder_linear} of class @{text linorder}: *}

   455 lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"

   456   apply (simp add: le_eq_less_or_eq)

   457   using less_linear

   458   apply blast

   459   done

   461 text {* Type {@typ nat} is a wellfounded linear order *}

   463 instance nat :: "{order, linorder, wellorder}"

   464   by intro_classes

   465     (assumption |

   466       rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+

   468 lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"

   469   by (blast elim!: less_SucE)

   471 text {*

   472   Rewrite @{term "n < Suc m"} to @{term "n = m"}

   473   if @{term "~ n < m"} or @{term "m \<le> n"} hold.

   474   Not suitable as default simprules because they often lead to looping

   475 *}

   476 lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"

   477   by (rule not_less_less_Suc_eq, rule leD)

   479 lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq

   482 text {*

   483   Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}.

   484   No longer added as simprules (they loop)

   485   but via @{text reorient_simproc} in Bin

   486 *}

   488 text {* Polymorphic, not just for @{typ nat} *}

   489 lemma zero_reorient: "(0 = x) = (x = 0)"

   490   by auto

   492 lemma one_reorient: "(1 = x) = (x = 1)"

   493   by auto

   495 subsection {* Arithmetic operators *}

   497 axclass power < type

   499 consts

   500   power :: "('a::power) => nat => 'a"            (infixr "^" 80)

   503 text {* arithmetic operators @{text "+ -"} and @{text "*"} *}

   505 instance nat :: "{plus, minus, times, power}" ..

   507 text {* size of a datatype value; overloaded *}

   508 consts size :: "'a => nat"

   510 primrec

   511   add_0:    "0 + n = n"

   512   add_Suc:  "Suc m + n = Suc (m + n)"

   514 primrec

   515   diff_0:   "m - 0 = m"

   516   diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"

   518 primrec

   519   mult_0:   "0 * n = 0"

   520   mult_Suc: "Suc m * n = n + (m * n)"

   522 text {* These two rules ease the use of primitive recursion.

   523 NOTE USE OF @{text "=="} *}

   524 lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"

   525   by simp

   527 lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"

   528   by simp

   530 lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"

   531   by (case_tac n) simp_all

   533 lemma gr_implies_not0: "!!n::nat. m<n ==> n \<noteq> 0"

   534   by (case_tac n) simp_all

   536 lemma neq0_conv [iff]: "!!n::nat. (n \<noteq> 0) = (0 < n)"

   537   by (case_tac n) simp_all

   539 text {* This theorem is useful with @{text blast} *}

   540 lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"

   541   by (rule iffD1, rule neq0_conv, rules)

   543 lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"

   544   by (fast intro: not0_implies_Suc)

   546 lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"

   547   apply (rule iffI)

   548   apply (rule ccontr, simp_all)

   549   done

   551 lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"

   552   by (induct m') simp_all

   554 text {* Useful in certain inductive arguments *}

   555 lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"

   556   by (case_tac m) simp_all

   558 lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"

   559   apply (rule nat_less_induct)

   560   apply (case_tac n)

   561   apply (case_tac [2] nat)

   562   apply (blast intro: less_trans)+

   563   done

   565 subsection {* @{text LEAST} theorems for type @{typ nat} by specialization *}

   567 lemmas LeastI = wellorder_LeastI

   568 lemmas Least_le = wellorder_Least_le

   569 lemmas not_less_Least = wellorder_not_less_Least

   571 lemma Least_Suc:

   572      "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"

   573   apply (case_tac "n", auto)

   574   apply (frule LeastI)

   575   apply (drule_tac P = "%x. P (Suc x) " in LeastI)

   576   apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")

   577   apply (erule_tac [2] Least_le)

   578   apply (case_tac "LEAST x. P x", auto)

   579   apply (drule_tac P = "%x. P (Suc x) " in Least_le)

   580   apply (blast intro: order_antisym)

   581   done

   583 lemma Least_Suc2:

   584      "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"

   585   by (erule (1) Least_Suc [THEN ssubst], simp)

   589 subsection {* @{term min} and @{term max} *}

   591 lemma min_0L [simp]: "min 0 n = (0::nat)"

   592   by (rule min_leastL) simp

   594 lemma min_0R [simp]: "min n 0 = (0::nat)"

   595   by (rule min_leastR) simp

   597 lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"

   598   by (simp add: min_of_mono)

   600 lemma max_0L [simp]: "max 0 n = (n::nat)"

   601   by (rule max_leastL) simp

   603 lemma max_0R [simp]: "max n 0 = (n::nat)"

   604   by (rule max_leastR) simp

   606 lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"

   607   by (simp add: max_of_mono)

   610 subsection {* Basic rewrite rules for the arithmetic operators *}

   612 text {* Difference *}

   614 lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"

   615   by (induct_tac n) simp_all

   617 lemma diff_Suc_Suc [simp, code]: "Suc(m) - Suc(n) = m - n"

   618   by (induct_tac n) simp_all

   621 text {*

   622   Could be (and is, below) generalized in various ways

   623   However, none of the generalizations are currently in the simpset,

   624   and I dread to think what happens if I put them in

   625 *}

   626 lemma Suc_pred [simp]: "0 < n ==> Suc (n - Suc 0) = n"

   627   by (simp split add: nat.split)

   629 declare diff_Suc [simp del, code del]

   632 subsection {* Addition *}

   634 lemma add_0_right [simp]: "m + 0 = (m::nat)"

   635   by (induct m) simp_all

   637 lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"

   638   by (induct m) simp_all

   640 lemma [code]: "Suc m + n = m + Suc n" by simp

   643 text {* Associative law for addition *}

   644 lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"

   645   by (induct m) simp_all

   647 text {* Commutative law for addition *}

   648 lemma nat_add_commute: "m + n = n + (m::nat)"

   649   by (induct m) simp_all

   651 lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"

   652   apply (rule mk_left_commute [of "op +"])

   653   apply (rule nat_add_assoc)

   654   apply (rule nat_add_commute)

   655   done

   657 lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"

   658   by (induct k) simp_all

   660 lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"

   661   by (induct k) simp_all

   663 lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"

   664   by (induct k) simp_all

   666 lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"

   667   by (induct k) simp_all

   669 text {* Reasoning about @{text "m + 0 = 0"}, etc. *}

   671 lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)"

   672   by (case_tac m) simp_all

   674 lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"

   675   by (case_tac m) simp_all

   677 lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"

   678   by (rule trans, rule eq_commute, rule add_is_1)

   680 lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m | 0 < n)"

   681   by (simp del: neq0_conv add: neq0_conv [symmetric])

   683 lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0"

   684   apply (drule add_0_right [THEN ssubst])

   685   apply (simp add: nat_add_assoc del: add_0_right)

   686   done

   689 subsection {* Multiplication *}

   691 text {* right annihilation in product *}

   692 lemma mult_0_right [simp]: "(m::nat) * 0 = 0"

   693   by (induct m) simp_all

   695 text {* right successor law for multiplication *}

   696 lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"

   697   by (induct m) (simp_all add: nat_add_left_commute)

   699 text {* Commutative law for multiplication *}

   700 lemma nat_mult_commute: "m * n = n * (m::nat)"

   701   by (induct m) simp_all

   703 text {* addition distributes over multiplication *}

   704 lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"

   705   by (induct m) (simp_all add: nat_add_assoc nat_add_left_commute)

   707 lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"

   708   by (induct m) (simp_all add: nat_add_assoc)

   710 text {* Associative law for multiplication *}

   711 lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"

   712   by (induct m) (simp_all add: add_mult_distrib)

   715 text{*The Naturals Form A comm_semiring_1_cancel*}

   716 instance nat :: comm_semiring_1_cancel

   717 proof

   718   fix i j k :: nat

   719   show "(i + j) + k = i + (j + k)" by (rule nat_add_assoc)

   720   show "i + j = j + i" by (rule nat_add_commute)

   721   show "0 + i = i" by simp

   722   show "(i * j) * k = i * (j * k)" by (rule nat_mult_assoc)

   723   show "i * j = j * i" by (rule nat_mult_commute)

   724   show "1 * i = i" by simp

   725   show "(i + j) * k = i * k + j * k" by (simp add: add_mult_distrib)

   726   show "0 \<noteq> (1::nat)" by simp

   727   assume "k+i = k+j" thus "i=j" by simp

   728 qed

   730 lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"

   731   apply (induct_tac m)

   732   apply (induct_tac [2] n, simp_all)

   733   done

   735 subsection {* Monotonicity of Addition *}

   737 text {* strict, in 1st argument *}

   738 lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"

   739   by (induct k) simp_all

   741 text {* strict, in both arguments *}

   742 lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"

   743   apply (rule add_less_mono1 [THEN less_trans], assumption+)

   744   apply (induct_tac j, simp_all)

   745   done

   747 text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}

   748 lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"

   749   apply (induct n)

   750   apply (simp_all add: order_le_less)

   751   apply (blast elim!: less_SucE

   752                intro!: add_0_right [symmetric] add_Suc_right [symmetric])

   753   done

   755 text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}

   756 lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j"

   757   apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp)

   758   apply (induct_tac x)

   759   apply (simp_all add: add_less_mono)

   760   done

   763 text{*The Naturals Form an Ordered comm_semiring_1_cancel*}

   764 instance nat :: ordered_semidom

   765 proof

   766   fix i j k :: nat

   767   show "0 < (1::nat)" by simp

   768   show "i \<le> j ==> k + i \<le> k + j" by simp

   769   show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)

   770 qed

   772 lemma nat_mult_1: "(1::nat) * n = n"

   773   by simp

   775 lemma nat_mult_1_right: "n * (1::nat) = n"

   776   by simp

   779 subsection {* Additional theorems about "less than" *}

   781 text {* A [clumsy] way of lifting @{text "<"}

   782   monotonicity to @{text "\<le>"} monotonicity *}

   783 lemma less_mono_imp_le_mono:

   784   assumes lt_mono: "!!i j::nat. i < j ==> f i < f j"

   785   and le: "i \<le> j" shows "f i \<le> ((f j)::nat)" using le

   786   apply (simp add: order_le_less)

   787   apply (blast intro!: lt_mono)

   788   done

   790 text {* non-strict, in 1st argument *}

   791 lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"

   792   by (rule add_right_mono)

   794 text {* non-strict, in both arguments *}

   795 lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"

   796   by (rule add_mono)

   798 lemma le_add2: "n \<le> ((m + n)::nat)"

   799   by (insert add_right_mono [of 0 m n], simp)

   801 lemma le_add1: "n \<le> ((n + m)::nat)"

   802   by (simp add: add_commute, rule le_add2)

   804 lemma less_add_Suc1: "i < Suc (i + m)"

   805   by (rule le_less_trans, rule le_add1, rule lessI)

   807 lemma less_add_Suc2: "i < Suc (m + i)"

   808   by (rule le_less_trans, rule le_add2, rule lessI)

   810 lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"

   811   by (rules intro!: less_add_Suc1 less_imp_Suc_add)

   813 lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"

   814   by (rule le_trans, assumption, rule le_add1)

   816 lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"

   817   by (rule le_trans, assumption, rule le_add2)

   819 lemma trans_less_add1: "(i::nat) < j ==> i < j + m"

   820   by (rule less_le_trans, assumption, rule le_add1)

   822 lemma trans_less_add2: "(i::nat) < j ==> i < m + j"

   823   by (rule less_le_trans, assumption, rule le_add2)

   825 lemma add_lessD1: "i + j < (k::nat) ==> i < k"

   826   apply (rule le_less_trans [of _ "i+j"])

   827   apply (simp_all add: le_add1)

   828   done

   830 lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"

   831   apply (rule notI)

   832   apply (erule add_lessD1 [THEN less_irrefl])

   833   done

   835 lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"

   836   by (simp add: add_commute not_add_less1)

   838 lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"

   839   apply (rule order_trans [of _ "m+k"])

   840   apply (simp_all add: le_add1)

   841   done

   843 lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"

   844   apply (simp add: add_commute)

   845   apply (erule add_leD1)

   846   done

   848 lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"

   849   by (blast dest: add_leD1 add_leD2)

   851 text {* needs @{text "!!k"} for @{text add_ac} to work *}

   852 lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"

   853   by (force simp del: add_Suc_right

   854     simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)

   858 subsection {* Difference *}

   860 lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"

   861   by (induct m) simp_all

   863 text {* Addition is the inverse of subtraction:

   864   if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}

   865 lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"

   866   by (induct m n rule: diff_induct) simp_all

   868 lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"

   869   by (simp add: add_diff_inverse not_less_iff_le)

   871 lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"

   872   by (simp add: le_add_diff_inverse add_commute)

   875 subsection {* More results about difference *}

   877 lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"

   878   by (induct m n rule: diff_induct) simp_all

   880 lemma diff_less_Suc: "m - n < Suc m"

   881   apply (induct m n rule: diff_induct)

   882   apply (erule_tac [3] less_SucE)

   883   apply (simp_all add: less_Suc_eq)

   884   done

   886 lemma diff_le_self [simp]: "m - n \<le> (m::nat)"

   887   by (induct m n rule: diff_induct) (simp_all add: le_SucI)

   889 lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"

   890   by (rule le_less_trans, rule diff_le_self)

   892 lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"

   893   by (induct i j rule: diff_induct) simp_all

   895 lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"

   896   by (simp add: diff_diff_left)

   898 lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"

   899   apply (case_tac "n", safe)

   900   apply (simp add: le_simps)

   901   done

   903 text {* This and the next few suggested by Florian Kammueller *}

   904 lemma diff_commute: "(i::nat) - j - k = i - k - j"

   905   by (simp add: diff_diff_left add_commute)

   907 lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"

   908   by (induct j k rule: diff_induct) simp_all

   910 lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"

   911   by (simp add: add_commute diff_add_assoc)

   913 lemma diff_add_inverse: "(n + m) - n = (m::nat)"

   914   by (induct n) simp_all

   916 lemma diff_add_inverse2: "(m + n) - n = (m::nat)"

   917   by (simp add: diff_add_assoc)

   919 lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"

   920   apply safe

   921   apply (simp_all add: diff_add_inverse2)

   922   done

   924 lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"

   925   by (induct m n rule: diff_induct) simp_all

   927 lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"

   928   by (rule iffD2, rule diff_is_0_eq)

   930 lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"

   931   by (induct m n rule: diff_induct) simp_all

   933 lemma less_imp_add_positive: "i < j  ==> \<exists>k::nat. 0 < k & i + k = j"

   934   apply (rule_tac x = "j - i" in exI)

   935   apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym)

   936   done

   938 lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"

   939   apply (induct k i rule: diff_induct)

   940   apply (simp_all (no_asm))

   941   apply rules

   942   done

   944 lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"

   945   apply (rule diff_self_eq_0 [THEN subst])

   946   apply (rule zero_induct_lemma, rules+)

   947   done

   949 lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"

   950   by (induct k) simp_all

   952 lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"

   953   by (simp add: diff_cancel add_commute)

   955 lemma diff_add_0: "n - (n + m) = (0::nat)"

   956   by (induct n) simp_all

   959 text {* Difference distributes over multiplication *}

   961 lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"

   962   by (induct m n rule: diff_induct) (simp_all add: diff_cancel)

   964 lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"

   965   by (simp add: diff_mult_distrib mult_commute [of k])

   966   -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}

   968 lemmas nat_distrib =

   969   add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2

   972 subsection {* Monotonicity of Multiplication *}

   974 lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"

   975   by (simp add: mult_right_mono)

   977 lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"

   978   by (simp add: mult_left_mono)

   980 text {* @{text "\<le>"} monotonicity, BOTH arguments *}

   981 lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"

   982   by (simp add: mult_mono)

   984 lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"

   985   by (simp add: mult_strict_right_mono)

   987 text{*Differs from the standard @{text zero_less_mult_iff} in that

   988       there are no negative numbers.*}

   989 lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"

   990   apply (induct m)

   991   apply (case_tac [2] n, simp_all)

   992   done

   994 lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"

   995   apply (induct m)

   996   apply (case_tac [2] n, simp_all)

   997   done

   999 lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"

  1000   apply (induct_tac m, simp)

  1001   apply (induct_tac n, simp, fastsimp)

  1002   done

  1004 lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"

  1005   apply (rule trans)

  1006   apply (rule_tac [2] mult_eq_1_iff, fastsimp)

  1007   done

  1009 lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"

  1010   apply (safe intro!: mult_less_mono1)

  1011   apply (case_tac k, auto)

  1012   apply (simp del: le_0_eq add: linorder_not_le [symmetric])

  1013   apply (blast intro: mult_le_mono1)

  1014   done

  1016 lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"

  1017   by (simp add: mult_commute [of k])

  1019 lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"

  1020 by (simp add: linorder_not_less [symmetric], auto)

  1022 lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"

  1023 by (simp add: linorder_not_less [symmetric], auto)

  1025 lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"

  1026   apply (cut_tac less_linear, safe, auto)

  1027   apply (drule mult_less_mono1, assumption, simp)+

  1028   done

  1030 lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"

  1031   by (simp add: mult_commute [of k])

  1033 lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"

  1034   by (subst mult_less_cancel1) simp

  1036 lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"

  1037   by (subst mult_le_cancel1) simp

  1039 lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"

  1040   by (subst mult_cancel1) simp

  1042 text {* Lemma for @{text gcd} *}

  1043 lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"

  1044   apply (drule sym)

  1045   apply (rule disjCI)

  1046   apply (rule nat_less_cases, erule_tac [2] _)

  1047   apply (fastsimp elim!: less_SucE)

  1048   apply (fastsimp dest: mult_less_mono2)

  1049   done

  1051 end

author	obua
	Tue, 11 May 2004 20:11:08 +0200
changeset 14738	83f1a514dcb4
parent 14691	e1eedc8cad37
child 14740	c8e1937110c2
permissions	-rw-r--r--