wneuper/isa: src/HOL/Nat.thy@7dc7dcd63224

     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

    12 imports Wellfounded_Recursion Ring_and_Field

    13 begin

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

    17 typedecl ind

    19 consts

    20   Zero_Rep      :: ind

    21   Suc_Rep       :: "ind => ind"

    23 axioms

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

    25   inj_Suc_Rep:          "inj Suc_Rep"

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

    27 finalconsts

    28   Zero_Rep

    29   Suc_Rep

    31 subsection {* Type nat *}

    33 text {* Type definition *}

    35 consts

    36   Nat :: "ind set"

    38 inductive Nat

    39 intros

    40   Zero_RepI: "Zero_Rep : Nat"

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

    43 global

    45 typedef (open Nat)

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

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

    51 text {* Abstract constants and syntax *}

    53 consts

    54   Suc :: "nat => nat"

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

    57 local

    59 defs

    60   Zero_nat_def: "0 == Abs_Nat Zero_Rep"

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

    62   One_nat_def:  "1 == Suc 0"

    64   -- {* nat operations *}

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

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

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

    71 declare One_nat_def [simp]

    74 text {* Induction *}

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

    77   apply (unfold Zero_nat_def Suc_def)

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

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

    80   apply (iprover elim: Abs_Nat_inverse [THEN subst])

    81   done

    83 text {* Distinctness of constructors *}

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

    86   by (simp add: Zero_nat_def Suc_def Abs_Nat_inject Rep_Nat Suc_RepI Zero_RepI

    87                 Suc_Rep_not_Zero_Rep)

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

    90   by (rule not_sym, rule Suc_not_Zero not_sym)

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

    93   by (rule notE, rule Suc_not_Zero)

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

    96   by (rule Suc_neq_Zero, erule sym)

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

   100 lemma inj_Suc[simp]: "inj_on Suc N"

   101   by (simp add: Suc_def inj_on_def Abs_Nat_inject Rep_Nat Suc_RepI

   102                 inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject)

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

   105   by (rule inj_Suc [THEN injD])

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

   108   by (rule inj_Suc [THEN inj_eq])

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

   111   by auto

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

   115 rep_datatype nat

   116   distinct  Suc_not_Zero Zero_not_Suc

   117   inject    Suc_Suc_eq

   118   induction nat_induct

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

   121   by (induct n) simp_all

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

   124   by (rule not_sym, rule n_not_Suc_n)

   126 text {* A special form of induction for reasoning

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

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

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

   131   apply (rule_tac x = m in spec)

   132   apply (induct n)

   133   prefer 2

   134   apply (rule allI)

   135   apply (induct_tac x, iprover+)

   136   done

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

   140 lemma wf_pred_nat: "wf pred_nat"

   141   apply (unfold wf_def pred_nat_def, clarify)

   142   apply (induct_tac x, blast+)

   143   done

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

   146   apply (unfold less_def)

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

   148   done

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

   151   apply (unfold less_def)

   152   apply (rule refl)

   153   done

   155 subsubsection {* Introduction properties *}

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

   158   apply (unfold less_def)

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

   160   done

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

   163   apply (unfold less_def pred_nat_def)

   164   apply (simp add: r_into_trancl)

   165   done

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

   168   apply (rule less_trans, assumption)

   169   apply (rule lessI)

   170   done

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

   173   apply (induct n)

   174   apply (rule lessI)

   175   apply (erule less_trans)

   176   apply (rule lessI)

   177   done

   179 subsubsection {* Elimination properties *}

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

   182   apply (unfold less_def)

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

   184   done

   186 lemma less_asym:

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

   188   apply (rule contrapos_np)

   189   apply (rule less_not_sym)

   190   apply (rule h1)

   191   apply (erule h2)

   192   done

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

   195   apply (unfold less_def)

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

   197   done

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

   200   by (rule notE, rule less_not_refl)

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

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

   205   by (rule not_sym, rule less_not_refl2)

   207 lemma lessE:

   208   assumes major: "i < k"

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

   210   shows P

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

   212   apply (erule p1)

   213   apply (rule p2)

   214   apply (simp add: less_def pred_nat_def, assumption)

   215   done

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

   218   by (blast elim: lessE)

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

   221   by (rule notE, rule not_less0)

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

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

   225   apply (rule major [THEN lessE])

   226   apply (rule eq, blast)

   227   apply (rule less, blast)

   228   done

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

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

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

   234   by (simp add: less_Suc_eq)

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

   237   by (simp add: less_Suc_eq)

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

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

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

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

   244   apply (induct m)

   245   apply (induct n)

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

   247   apply (rule zero_less_Suc [THEN disjI1])

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

   249   done

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

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

   253 apply(simp only:less_Suc_eq)

   254 apply blast

   255 done

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

   258   using less_linear by blast

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

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

   262   shows "P n m"

   263   apply (rule less_linear [THEN disjE])

   264   apply (erule_tac [2] disjE)

   265   apply (erule lessCase)

   266   apply (erule sym [THEN eqCase])

   267   apply (erule major)

   268   done

   271 subsubsection {* Inductive (?) properties *}

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

   274   apply (simp add: nat_neq_iff)

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

   276   done

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

   279   apply (induct n)

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

   281   done

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

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

   285   apply (rule major [THEN lessE])

   286   apply (erule lessI [THEN minor])

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

   288   done

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

   291   by (blast elim: lessE dest: Suc_lessD)

   293 lemma Suc_less_eq [iff, code]: "(Suc m < Suc n) = (m < n)"

   294   apply (rule iffI)

   295   apply (erule Suc_less_SucD)

   296   apply (erule Suc_mono)

   297   done

   299 lemma less_trans_Suc:

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

   301   apply (induct k, simp_all)

   302   apply (insert le)

   303   apply (simp add: less_Suc_eq)

   304   apply (blast dest: Suc_lessD)

   305   done

   307 lemma [code]: "((n::nat) < 0) = False" by simp

   308 lemma [code]: "(0 < Suc n) = True" by simp

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

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

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

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

   315 lemma nat_less_induct:

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

   317   apply (rule_tac a=n in wf_induct)

   318   apply (rule wf_pred_nat [THEN wf_trancl])

   319   apply (rule prem)

   320   apply (unfold less_def, assumption)

   321   done

   323 lemmas less_induct = nat_less_induct [rule_format, case_names less]

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

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

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

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

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

   332   by (rule less_Suc_eq_le [THEN iffD2])

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

   335   by (unfold le_def, rule not_less0)

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

   338   by (simp add: le_def)

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

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

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

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

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

   347   by (drule le_Suc_eq [THEN iffD1], iprover+)

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

   350   apply (simp add: le_def less_Suc_eq)

   351   apply (blast elim!: less_irrefl less_asym)

   352   done -- {* formerly called lessD *}

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

   355   by (simp add: le_def less_Suc_eq)

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

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

   359   apply (simp add: le_def less_Suc_eq)

   360   using less_linear

   361   apply blast

   362   done

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

   365   by (blast intro: Suc_leI Suc_le_lessD)

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

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

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

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

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

   374 lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq

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

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

   380   apply (unfold le_def)

   381   using less_linear

   382   apply (blast elim: less_irrefl less_asym)

   383   done

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

   386   apply (unfold le_def)

   387   using less_linear

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

   389   done

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

   392   by (iprover intro: less_or_eq_imp_le le_imp_less_or_eq)

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

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

   396   by (rule less_or_eq_imp_le, rule disjI2)

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

   399   by (simp add: le_eq_less_or_eq)

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

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

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

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

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

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

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

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

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

   414   by (simp add: le_simps)

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

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

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

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

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

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

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

   425   apply (simp add: le_eq_less_or_eq)

   426   using less_linear

   427   apply blast

   428   done

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

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

   433   by intro_classes

   434     (assumption |

   435       rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+

   437 lemmas linorder_neqE_nat = linorder_neqE[where 'a = nat]

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

   440   by (blast elim!: less_SucE)

   442 text {*

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

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

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

   446 *}

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

   448   by (rule not_less_less_Suc_eq, rule leD)

   450 lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq

   453 text {*

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

   455   No longer added as simprules (they loop)

   456   but via @{text reorient_simproc} in Bin

   457 *}

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

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

   461   by auto

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

   464   by auto

   466 subsection {* Arithmetic operators *}

   468 axclass power < type

   470 consts

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

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

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

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

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

   481 primrec

   482   add_0:    "0 + n = n"

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

   485 primrec

   486   diff_0:   "m - 0 = m"

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

   489 primrec

   490   mult_0:   "0 * n = 0"

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

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

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

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

   496   by simp

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

   499   by simp

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

   502   by (case_tac n) simp_all

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

   505   by (case_tac n) simp_all

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

   508   by (case_tac n) simp_all

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

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

   512   by (rule iffD1, rule neq0_conv, iprover)

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

   515   by (fast intro: not0_implies_Suc)

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

   518   apply (rule iffI)

   519   apply (rule ccontr, simp_all)

   520   done

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

   523   by (induct m') simp_all

   525 text {* Useful in certain inductive arguments *}

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

   527   by (case_tac m) simp_all

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

   530   apply (rule nat_less_induct)

   531   apply (case_tac n)

   532   apply (case_tac [2] nat)

   533   apply (blast intro: less_trans)+

   534   done

   536 subsection {* @{text LEAST} theorems for type @{typ nat}*}

   538 lemma Least_Suc:

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

   540   apply (case_tac "n", auto)

   541   apply (frule LeastI)

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

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

   544   apply (erule_tac [2] Least_le)

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

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

   547   apply (blast intro: order_antisym)

   548   done

   550 lemma Least_Suc2:

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

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

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

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

   559   by (rule min_leastL) simp

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

   562   by (rule min_leastR) simp

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

   565   by (simp add: min_of_mono)

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

   568   by (rule max_leastL) simp

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

   571   by (rule max_leastR) simp

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

   574   by (simp add: max_of_mono)

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

   579 text {* Difference *}

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

   582   by (induct n) simp_all

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

   585   by (induct n) simp_all

   588 text {*

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

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

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

   592 *}

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

   594   by (simp split add: nat.split)

   596 declare diff_Suc [simp del, code del]

   599 subsection {* Addition *}

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

   602   by (induct m) simp_all

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

   605   by (induct m) simp_all

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

   610 text {* Associative law for addition *}

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

   612   by (induct m) simp_all

   614 text {* Commutative law for addition *}

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

   616   by (induct m) simp_all

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

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

   620   apply (rule nat_add_assoc)

   621   apply (rule nat_add_commute)

   622   done

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

   625   by (induct k) simp_all

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

   628   by (induct k) simp_all

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

   631   by (induct k) simp_all

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

   634   by (induct k) simp_all

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

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

   639   by (case_tac m) simp_all

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

   642   by (case_tac m) simp_all

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

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

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

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

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

   651   apply (drule add_0_right [THEN ssubst])

   652   apply (simp add: nat_add_assoc del: add_0_right)

   653   done

   656 lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"

   657 apply(induct k)

   658  apply simp

   659 apply(drule comp_inj_on[OF _ inj_Suc])

   660 apply (simp add:o_def)

   661 done

   664 subsection {* Multiplication *}

   666 text {* right annihilation in product *}

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

   668   by (induct m) simp_all

   670 text {* right successor law for multiplication *}

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

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

   674 text {* Commutative law for multiplication *}

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

   676   by (induct m) simp_all

   678 text {* addition distributes over multiplication *}

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

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

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

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

   685 text {* Associative law for multiplication *}

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

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

   690 text{*The naturals form a @{text comm_semiring_1_cancel}*}

   691 instance nat :: comm_semiring_1_cancel

   692 proof

   693   fix i j k :: nat

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

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

   696   show "0 + i = i" by simp

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

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

   699   show "1 * i = i" by simp

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

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

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

   703 qed

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

   706   apply (induct m)

   707   apply (induct_tac [2] n, simp_all)

   708   done

   710 subsection {* Monotonicity of Addition *}

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

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

   714   by (induct k) simp_all

   716 text {* strict, in both arguments *}

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

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

   719   apply (induct j, simp_all)

   720   done

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

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

   724   apply (induct n)

   725   apply (simp_all add: order_le_less)

   726   apply (blast elim!: less_SucE

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

   728   done

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

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

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

   733   apply (induct_tac x)

   734   apply (simp_all add: add_less_mono)

   735   done

   738 text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}

   739 instance nat :: ordered_semidom

   740 proof

   741   fix i j k :: nat

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

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

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

   745 qed

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

   748   by simp

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

   751   by simp

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

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

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

   758 lemma less_mono_imp_le_mono:

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

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

   761   apply (simp add: order_le_less)

   762   apply (blast intro!: lt_mono)

   763   done

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

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

   767   by (rule add_right_mono)

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

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

   771   by (rule add_mono)

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

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

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

   777   by (simp add: add_commute, rule le_add2)

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

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

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

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

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

   786   by (iprover intro!: less_add_Suc1 less_imp_Suc_add)

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

   789   by (rule le_trans, assumption, rule le_add1)

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

   792   by (rule le_trans, assumption, rule le_add2)

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

   795   by (rule less_le_trans, assumption, rule le_add1)

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

   798   by (rule less_le_trans, assumption, rule le_add2)

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

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

   802   apply (simp_all add: le_add1)

   803   done

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

   806   apply (rule notI)

   807   apply (erule add_lessD1 [THEN less_irrefl])

   808   done

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

   811   by (simp add: add_commute not_add_less1)

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

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

   815   apply (simp_all add: le_add1)

   816   done

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

   819   apply (simp add: add_commute)

   820   apply (erule add_leD1)

   821   done

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

   824   by (blast dest: add_leD1 add_leD2)

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

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

   828   by (force simp del: add_Suc_right

   829     simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)

   833 subsection {* Difference *}

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

   836   by (induct m) simp_all

   838 text {* Addition is the inverse of subtraction:

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

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

   841   by (induct m n rule: diff_induct) simp_all

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

   844   by (simp add: add_diff_inverse linorder_not_less)

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

   847   by (simp add: le_add_diff_inverse add_commute)

   850 subsection {* More results about difference *}

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

   853   by (induct m n rule: diff_induct) simp_all

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

   856   apply (induct m n rule: diff_induct)

   857   apply (erule_tac [3] less_SucE)

   858   apply (simp_all add: less_Suc_eq)

   859   done

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

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

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

   865   by (rule le_less_trans, rule diff_le_self)

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

   868   by (induct i j rule: diff_induct) simp_all

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

   871   by (simp add: diff_diff_left)

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

   874   apply (case_tac "n", safe)

   875   apply (simp add: le_simps)

   876   done

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

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

   880   by (simp add: diff_diff_left add_commute)

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

   883   by (induct j k rule: diff_induct) simp_all

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

   886   by (simp add: add_commute diff_add_assoc)

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

   889   by (induct n) simp_all

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

   892   by (simp add: diff_add_assoc)

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

   895   apply safe

   896   apply (simp_all add: diff_add_inverse2)

   897   done

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

   900   by (induct m n rule: diff_induct) simp_all

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

   903   by (rule iffD2, rule diff_is_0_eq)

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

   906   by (induct m n rule: diff_induct) simp_all

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

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

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

   911   done

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

   914   apply (induct k i rule: diff_induct)

   915   apply (simp_all (no_asm))

   916   apply iprover

   917   done

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

   920   apply (rule diff_self_eq_0 [THEN subst])

   921   apply (rule zero_induct_lemma, iprover+)

   922   done

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

   925   by (induct k) simp_all

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

   928   by (simp add: diff_cancel add_commute)

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

   931   by (induct n) simp_all

   934 text {* Difference distributes over multiplication *}

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

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

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

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

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

   943 lemmas nat_distrib =

   944   add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2

   947 subsection {* Monotonicity of Multiplication *}

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

   950   by (simp add: mult_right_mono)

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

   953   by (simp add: mult_left_mono)

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

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

   957   by (simp add: mult_mono)

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

   960   by (simp add: mult_strict_right_mono)

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

   963       there are no negative numbers.*}

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

   965   apply (induct m)

   966   apply (case_tac [2] n, simp_all)

   967   done

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

   970   apply (induct m)

   971   apply (case_tac [2] n, simp_all)

   972   done

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

   975   apply (induct m, simp)

   976   apply (induct n, simp, fastsimp)

   977   done

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

   980   apply (rule trans)

   981   apply (rule_tac [2] mult_eq_1_iff, fastsimp)

   982   done

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

   985   apply (safe intro!: mult_less_mono1)

   986   apply (case_tac k, auto)

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

   988   apply (blast intro: mult_le_mono1)

   989   done

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

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

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

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

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

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

  1000 lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"

  1001   apply (cut_tac less_linear, safe, auto)

  1002   apply (drule mult_less_mono1, assumption, simp)+

  1003   done

  1005 lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"

  1006   by (simp add: mult_commute [of k])

  1008 lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"

  1009   by (subst mult_less_cancel1) simp

  1011 lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"

  1012   by (subst mult_le_cancel1) simp

  1014 lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"

  1015   by (subst mult_cancel1) simp

  1017 text {* Lemma for @{text gcd} *}

  1018 lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"

  1019   apply (drule sym)

  1020   apply (rule disjCI)

  1021   apply (rule nat_less_cases, erule_tac [2] _)

  1022   apply (fastsimp elim!: less_SucE)

  1023   apply (fastsimp dest: mult_less_mono2)

  1024   done

  1026 subsection {* Code generator setup *}

  1028 code_alias

  1029   "nat" "Nat.nat"

  1030   "0" "Nat.Zero"

  1031   "1" "Nat.One"

  1032   "Suc" "Nat.Suc"

  1034 end

author	haftmann
	Tue, 17 Jan 2006 16:36:57 +0100
changeset 18702	7dc7dcd63224
parent 18648	22f96cd085d5
child 19573	340c466c9605
permissions	-rw-r--r--