(*  Title:      HOL/Nat.thy

     Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel

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

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

 *)

 header {* Natural numbers *}

 theory Nat

 imports Inductive Typedef Fun Fields

 begin

 ML_file "~~/src/Tools/rat.ML"

 ML_file "Tools/arith_data.ML"

 ML_file "~~/src/Provers/Arith/fast_lin_arith.ML"

 subsection {* Type @{text ind} *}

 typedecl ind

 axiomatization Zero_Rep :: ind and Suc_Rep :: "ind => ind" where

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

   Suc_Rep_inject:       "Suc_Rep x = Suc_Rep y ==> x = y" and

   Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"

 subsection {* Type nat *}

 text {* Type definition *}

 inductive Nat :: "ind \<Rightarrow> bool" where

   Zero_RepI: "Nat Zero_Rep"

 | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"

 typedef nat = "{n. Nat n}"

   morphisms Rep_Nat Abs_Nat

   using Nat.Zero_RepI by auto

 lemma Nat_Rep_Nat:

   "Nat (Rep_Nat n)"

   using Rep_Nat by simp

 lemma Nat_Abs_Nat_inverse:

   "Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n"

   using Abs_Nat_inverse by simp

 lemma Nat_Abs_Nat_inject:

   "Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m"

   using Abs_Nat_inject by simp

 instantiation nat :: zero

 begin

 definition Zero_nat_def:

   "0 = Abs_Nat Zero_Rep"

 instance ..

 end

 definition Suc :: "nat \<Rightarrow> nat" where

   "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"

 lemma Suc_not_Zero: "Suc m \<noteq> 0"

   by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat)

 lemma Zero_not_Suc: "0 \<noteq> Suc m"

   by (rule not_sym, rule Suc_not_Zero not_sym)

 lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y"

   by (rule iffI, rule Suc_Rep_inject) simp_all

 rep_datatype "0 \<Colon> nat" Suc

   apply (unfold Zero_nat_def Suc_def)

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

    apply (erule Nat_Rep_Nat [THEN Nat.induct])

    apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst])

     apply (simp_all add: Nat_Abs_Nat_inject Nat_Rep_Nat

       Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep

       Suc_Rep_not_Zero_Rep [symmetric]

       Suc_Rep_inject' Rep_Nat_inject)

   done

 lemma nat_induct [case_names 0 Suc, induct type: nat]:

   -- {* for backward compatibility -- names of variables differ *}

   fixes n

   assumes "P 0"

     and "\<And>n. P n \<Longrightarrow> P (Suc n)"

   shows "P n"

   using assms by (rule nat.induct)

 declare nat.exhaust [case_names 0 Suc, cases type: nat]

 lemmas nat_rec_0 = nat.recs(1)

   and nat_rec_Suc = nat.recs(2)

 lemmas nat_case_0 = nat.cases(1)

   and nat_case_Suc = nat.cases(2)

 text {* Injectiveness and distinctness lemmas *}

 lemma inj_Suc[simp]: "inj_on Suc N"

   by (simp add: inj_on_def)

 lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"

 by (rule notE, rule Suc_not_Zero)

 lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"

 by (rule Suc_neq_Zero, erule sym)

 lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"

 by (rule inj_Suc [THEN injD])

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

 by (induct n) simp_all

 lemma Suc_n_not_n: "Suc n \<noteq> n"

 by (rule not_sym, rule n_not_Suc_n)

 text {* A special form of induction for reasoning

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

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

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

   apply (rule_tac x = m in spec)

   apply (induct n)

   prefer 2

   apply (rule allI)

   apply (induct_tac x, iprover+)

   done

 subsection {* Arithmetic operators *}

 instantiation nat :: comm_monoid_diff

 begin

 primrec plus_nat where

   add_0:      "0 + n = (n\<Colon>nat)"

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

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

   by (induct m) simp_all

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

   by (induct m) simp_all

 declare add_0 [code]

 lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"

   by simp

 primrec minus_nat where

   diff_0 [code]: "m - 0 = (m\<Colon>nat)"

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

 declare diff_Suc [simp del]

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

   by (induct n) (simp_all add: diff_Suc)

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

   by (induct n) (simp_all add: diff_Suc)

 instance proof

   fix n m q :: nat

   show "(n + m) + q = n + (m + q)" by (induct n) simp_all

   show "n + m = m + n" by (induct n) simp_all

   show "0 + n = n" by simp

   show "n - 0 = n" by simp

   show "0 - n = 0" by simp

   show "(q + n) - (q + m) = n - m" by (induct q) simp_all

   show "n - m - q = n - (m + q)" by (induct q) (simp_all add: diff_Suc)

 qed

 end

 hide_fact (open) add_0 add_0_right diff_0

 instantiation nat :: comm_semiring_1_cancel

 begin

 definition

   One_nat_def [simp]: "1 = Suc 0"

 primrec times_nat where

   mult_0:     "0 * n = (0\<Colon>nat)"

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

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

   by (induct m) simp_all

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

   by (induct m) (simp_all add: add_left_commute)

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

   by (induct m) (simp_all add: add_assoc)

 instance proof

   fix n m q :: nat

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

   show "1 * n = n" unfolding One_nat_def by simp

   show "n * m = m * n" by (induct n) simp_all

   show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib)

   show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib)

   assume "n + m = n + q" thus "m = q" by (induct n) simp_all

 qed

 end

 subsubsection {* Addition *}

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

   by (rule add_assoc)

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

   by (rule add_commute)

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

   by (rule add_left_commute)

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

   by (rule add_left_cancel)

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

   by (rule add_right_cancel)

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

 lemma add_is_0 [iff]:

   fixes m n :: nat

   shows "(m + n = 0) = (m = 0 & n = 0)"

   by (cases m) simp_all

 lemma add_is_1:

   "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"

   by (cases m) simp_all

 lemma one_is_add:

   "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"

   by (rule trans, rule eq_commute, rule add_is_1)

 lemma add_eq_self_zero:

   fixes m n :: nat

   shows "m + n = m \<Longrightarrow> n = 0"

   by (induct m) simp_all

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

   apply (induct k)

    apply simp

   apply(drule comp_inj_on[OF _ inj_Suc])

   apply (simp add:o_def)

   done

 lemma Suc_eq_plus1: "Suc n = n + 1"

   unfolding One_nat_def by simp

 lemma Suc_eq_plus1_left: "Suc n = 1 + n"

   unfolding One_nat_def by simp

 subsubsection {* Difference *}

 lemma diff_self_eq_0 [simp]: "(m\<Colon>nat) - m = 0"

   by (induct m) simp_all

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

   by (induct i j rule: diff_induct) simp_all

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

   by (simp add: diff_diff_left)

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

   by (simp add: diff_diff_left add_commute)

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

   by (induct n) simp_all

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

   by (simp add: diff_add_inverse add_commute [of m n])

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

   by (induct k) simp_all

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

   by (simp add: diff_cancel add_commute)

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

   by (induct n) simp_all

 lemma diff_Suc_1 [simp]: "Suc n - 1 = n"

   unfolding One_nat_def by simp

 text {* Difference distributes over multiplication *}

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

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

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

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

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

 subsubsection {* Multiplication *}

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

   by (rule mult_assoc)

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

   by (rule mult_commute)

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

   by (rule distrib_left)

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

   by (induct m) auto

 lemmas nat_distrib =

   add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2

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

   apply (induct m)

    apply simp

   apply (induct n)

    apply auto

   done

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

   apply (rule trans)

   apply (rule_tac [2] mult_eq_1_iff, fastforce)

   done

 lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"

   unfolding One_nat_def by (rule mult_eq_1_iff)

 lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"

   unfolding One_nat_def by (rule one_eq_mult_iff)

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

 proof -

   have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"

   proof (induct n arbitrary: m)

     case 0 then show "m = 0" by simp

   next

     case (Suc n) then show "m = Suc n"

       by (cases m) (simp_all add: eq_commute [of "0"])

   qed

   then show ?thesis by auto

 qed

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

   by (simp add: mult_commute)

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

   by (subst mult_cancel1) simp

 subsection {* Orders on @{typ nat} *}

 subsubsection {* Operation definition *}

 instantiation nat :: linorder

 begin

 primrec less_eq_nat where

   "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"

 | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"

 declare less_eq_nat.simps [simp del]

 lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps)

 lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)

 definition less_nat where

   less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"

 lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"

   by (simp add: less_eq_nat.simps(2))

 lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"

   unfolding less_eq_Suc_le ..

 lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"

   by (induct n) (simp_all add: less_eq_nat.simps(2))

 lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"

   by (simp add: less_eq_Suc_le)

 lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"

   by simp

 lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"

   by (simp add: less_eq_Suc_le)

 lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"

   by (simp add: less_eq_Suc_le)

 lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"

   by (induct m arbitrary: n)

     (simp_all add: less_eq_nat.simps(2) split: nat.splits)

 lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"

   by (cases n) (auto intro: le_SucI)

 lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"

   by (simp add: less_eq_Suc_le) (erule Suc_leD)

 lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"

   by (simp add: less_eq_Suc_le) (erule Suc_leD)

 instance

 proof

   fix n m :: nat

   show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" 

   proof (induct n arbitrary: m)

     case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)

   next

     case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)

   qed

 next

   fix n :: nat show "n \<le> n" by (induct n) simp_all

 next

   fix n m :: nat assume "n \<le> m" and "m \<le> n"

   then show "n = m"

     by (induct n arbitrary: m)

       (simp_all add: less_eq_nat.simps(2) split: nat.splits)

 next

   fix n m q :: nat assume "n \<le> m" and "m \<le> q"

   then show "n \<le> q"

   proof (induct n arbitrary: m q)

     case 0 show ?case by simp

   next

     case (Suc n) then show ?case

       by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,

         simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,

         simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)

   qed

 next

   fix n m :: nat show "n \<le> m \<or> m \<le> n"

     by (induct n arbitrary: m)

       (simp_all add: less_eq_nat.simps(2) split: nat.splits)

 qed

 end

 instantiation nat :: order_bot

 begin

 definition bot_nat :: nat where

   "bot_nat = 0"

 instance proof

 qed (simp add: bot_nat_def)

 end

 instance nat :: no_top

   by default (auto intro: less_Suc_eq_le [THEN iffD2])

 subsubsection {* Introduction properties *}

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

   by (simp add: less_Suc_eq_le)

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

   by (simp add: less_Suc_eq_le)

 subsubsection {* Elimination properties *}

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

   by (rule order_less_irrefl)

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

   by (rule not_sym) (rule less_imp_neq) 

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

   by (rule less_imp_neq)

 lemma less_irrefl_nat: "(n::nat) < n ==> R"

   by (rule notE, rule less_not_refl)

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

   by (rule notE) (rule not_less0)

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

   unfolding less_Suc_eq_le le_less ..

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

   by (simp add: less_Suc_eq)

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

   unfolding One_nat_def by (rule less_Suc0)

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

   by simp

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

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

   unfolding not_less less_Suc_eq_le by (rule antisym)

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

   by (rule linorder_neq_iff)

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

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

   shows "P n m"

   apply (rule less_linear [THEN disjE])

   apply (erule_tac [2] disjE)

   apply (erule lessCase)

   apply (erule sym [THEN eqCase])

   apply (erule major)

   done

 subsubsection {* Inductive (?) properties *}

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

   unfolding less_eq_Suc_le [of m] le_less by simp 

 lemma lessE:

   assumes major: "i < k"

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

   shows P

 proof -

   from major have "\<exists>j. i \<le> j \<and> k = Suc j"

     unfolding less_eq_Suc_le by (induct k) simp_all

   then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"

     by (clarsimp simp add: less_le)

   with p1 p2 show P by auto

 qed

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

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

   apply (rule major [THEN lessE])

   apply (rule eq, blast)

   apply (rule less, blast)

   done

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

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

   apply (rule major [THEN lessE])

   apply (erule lessI [THEN minor])

   apply (erule Suc_lessD [THEN minor], assumption)

   done

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

   by simp

 lemma less_trans_Suc:

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

   apply (induct k, simp_all)

   apply (insert le)

   apply (simp add: less_Suc_eq)

   apply (blast dest: Suc_lessD)

   done

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

 lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"

   unfolding not_less less_Suc_eq_le ..

 lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"

   unfolding not_le Suc_le_eq ..

 text {* Properties of "less than or equal" *}

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

   unfolding less_Suc_eq_le .

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

   unfolding not_le less_Suc_eq_le ..

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

   by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)

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

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

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

   unfolding Suc_le_eq .

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

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

   unfolding Suc_le_eq .

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

   unfolding less_eq_Suc_le by (rule Suc_leD)

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

 lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq

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

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

   unfolding le_less .

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

   by (rule le_less)

 text {* Useful with @{text blast}. *}

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

   by auto

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

   by simp

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

   by (rule order_trans)

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

   by (rule antisym)

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

   by (rule less_le)

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

   unfolding less_le ..

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

   by (rule linear)

 lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]

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

   unfolding less_Suc_eq_le by auto

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

   unfolding not_less by (rule le_less_Suc_eq)

 lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq

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

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

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

 by simp

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

 by simp

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

 by (cases n) simp_all

 lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"

 by (cases n) simp_all

 lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"

 by (cases n) simp_all

 lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"

 by (cases n) simp_all

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

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

 by (rule neq0_conv[THEN iffD1], iprover)

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

 by (fast intro: not0_implies_Suc)

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

 using neq0_conv by blast

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

 by (induct m') simp_all

 text {* Useful in certain inductive arguments *}

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

 by (cases m) simp_all

 subsubsection {* Monotonicity of Addition *}

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

 by (simp add: diff_Suc split: nat.split)

 lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"

 unfolding One_nat_def by (rule Suc_pred)

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

 by (induct k) simp_all

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

 by (induct k) simp_all

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

 by(auto dest:gr0_implies_Suc)

 text {* strict, in 1st argument *}

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

 by (induct k) simp_all

 text {* strict, in both arguments *}

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

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

   apply (induct j, simp_all)

   done

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

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

   apply (induct n)

   apply (simp_all add: order_le_less)

   apply (blast elim!: less_SucE

                intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])

   done

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

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

 apply(auto simp: gr0_conv_Suc)

 apply (induct_tac m)

 apply (simp_all add: add_less_mono)

 done

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

 instance nat :: linordered_semidom

 proof

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

   show "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp

   show "\<And>m n q :: nat. m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2)

 qed

 instance nat :: no_zero_divisors

 proof

   fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto

 qed

 subsubsection {* @{term min} and @{term max} *}

 lemma mono_Suc: "mono Suc"

 by (rule monoI) simp

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

 by (rule min_absorb1) simp

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

 by (rule min_absorb2) simp

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

 by (simp add: mono_Suc min_of_mono)

 lemma min_Suc1:

    "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"

 by (simp split: nat.split)

 lemma min_Suc2:

    "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"

 by (simp split: nat.split)

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

 by (rule max_absorb2) simp

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

 by (rule max_absorb1) simp

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

 by (simp add: mono_Suc max_of_mono)

 lemma max_Suc1:

    "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"

 by (simp split: nat.split)

 lemma max_Suc2:

    "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"

 by (simp split: nat.split)

 lemma nat_mult_min_left:

   fixes m n q :: nat

   shows "min m n * q = min (m * q) (n * q)"

   by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)

 lemma nat_mult_min_right:

   fixes m n q :: nat

   shows "m * min n q = min (m * n) (m * q)"

   by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)

 lemma nat_add_max_left:

   fixes m n q :: nat

   shows "max m n + q = max (m + q) (n + q)"

   by (simp add: max_def)

 lemma nat_add_max_right:

   fixes m n q :: nat

   shows "m + max n q = max (m + n) (m + q)"

   by (simp add: max_def)

 lemma nat_mult_max_left:

   fixes m n q :: nat

   shows "max m n * q = max (m * q) (n * q)"

   by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)

 lemma nat_mult_max_right:

   fixes m n q :: nat

   shows "m * max n q = max (m * n) (m * q)"

   by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)

 subsubsection {* Additional theorems about @{term "op \<le>"} *}

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

 instance nat :: wellorder proof

   fix P and n :: nat

   assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"

   have "\<And>q. q \<le> n \<Longrightarrow> P q"

   proof (induct n)

     case (0 n)

     have "P 0" by (rule step) auto

     thus ?case using 0 by auto

   next

     case (Suc m n)

     then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)

     thus ?case

     proof

       assume "n \<le> m" thus "P n" by (rule Suc(1))

     next

       assume n: "n = Suc m"

       show "P n"

         by (rule step) (rule Suc(1), simp add: n le_simps)

     qed

   qed

   then show "P n" by auto

 qed

 lemma Least_Suc:

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

   apply (cases n, auto)

   apply (frule LeastI)

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

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

   apply (erule_tac [2] Least_le)

   apply (cases "LEAST x. P x", auto)

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

   apply (blast intro: order_antisym)

   done

 lemma Least_Suc2:

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

   apply (erule (1) Least_Suc [THEN ssubst])

   apply simp

   done

 lemma ex_least_nat_le: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not>P i) & P(k)"

   apply (cases n)

    apply blast

   apply (rule_tac x="LEAST k. P(k)" in exI)

   apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)

   done

 lemma ex_least_nat_less: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not>P i) & P(k+1)"

   unfolding One_nat_def

   apply (cases n)

    apply blast

   apply (frule (1) ex_least_nat_le)

   apply (erule exE)

   apply (case_tac k)

    apply simp

   apply (rename_tac k1)

   apply (rule_tac x=k1 in exI)

   apply (auto simp add: less_eq_Suc_le)

   done

 lemma nat_less_induct:

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

   using assms less_induct by blast

 lemma measure_induct_rule [case_names less]:

   fixes f :: "'a \<Rightarrow> nat"

   assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"

   shows "P a"

 by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)

 text {* old style induction rules: *}

 lemma measure_induct:

   fixes f :: "'a \<Rightarrow> nat"

   shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"

   by (rule measure_induct_rule [of f P a]) iprover

 lemma full_nat_induct:

   assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"

   shows "P n"

   by (rule less_induct) (auto intro: step simp:le_simps)

 text{*An induction rule for estabilishing binary relations*}

 lemma less_Suc_induct:

   assumes less:  "i < j"

      and  step:  "!!i. P i (Suc i)"

      and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"

   shows "P i j"

 proof -

   from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)

   have "P i (Suc (i + k))"

   proof (induct k)

     case 0

     show ?case by (simp add: step)

   next

     case (Suc k)

     have "0 + i < Suc k + i" by (rule add_less_mono1) simp

     hence "i < Suc (i + k)" by (simp add: add_commute)

     from trans[OF this lessI Suc step]

     show ?case by simp

   qed

   thus "P i j" by (simp add: j)

 qed

 text {* The method of infinite descent, frequently used in number theory.

 Provided by Roelof Oosterhuis.

 $P(n)$ is true for all $n\in\mathbb{N}$ if

 \begin{itemize}

   \item case ``0'': given $n=0$ prove $P(n)$,

   \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists

         a smaller integer $m$ such that $\neg P(m)$.

 \end{itemize} *}

 text{* A compact version without explicit base case: *}

 lemma infinite_descent:

   "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"

 by (induct n rule: less_induct) auto

 lemma infinite_descent0[case_names 0 smaller]: 

   "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"

 by (rule infinite_descent) (case_tac "n>0", auto)

 text {*

 Infinite descent using a mapping to $\mathbb{N}$:

 $P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and

 \begin{itemize}

 \item case ``0'': given $V(x)=0$ prove $P(x)$,

 \item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$.

 \end{itemize}

 NB: the proof also shows how to use the previous lemma. *}

 corollary infinite_descent0_measure [case_names 0 smaller]:

   assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"

     and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"

   shows "P x"

 proof -

   obtain n where "n = V x" by auto

   moreover have "\<And>x. V x = n \<Longrightarrow> P x"

   proof (induct n rule: infinite_descent0)

     case 0 -- "i.e. $V(x) = 0$"

     with A0 show "P x" by auto

   next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"

     case (smaller n)

     then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto

     with A1 obtain y where "V y < V x \<and> \<not> P y" by auto

     with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto

     then show ?case by auto

   qed

   ultimately show "P x" by auto

 qed

 text{* Again, without explicit base case: *}

 lemma infinite_descent_measure:

 assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"

 proof -

   from assms obtain n where "n = V x" by auto

   moreover have "!!x. V x = n \<Longrightarrow> P x"

   proof (induct n rule: infinite_descent, auto)

     fix x assume "\<not> P x"

     with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto

   qed

   ultimately show "P x" by auto

 qed

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

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

 lemma less_mono_imp_le_mono:

   "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"

 by (simp add: order_le_less) (blast)

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

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

 by (rule add_right_mono)

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

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

 by (rule add_mono)

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

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

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

 by (simp add: add_commute, rule le_add2)

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

 by (rule le_less_trans, rule le_add1, rule lessI)

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

 by (rule le_less_trans, rule le_add2, rule lessI)

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

 by (iprover intro!: less_add_Suc1 less_imp_Suc_add)

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

 by (rule le_trans, assumption, rule le_add1)

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

 by (rule le_trans, assumption, rule le_add2)

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

 by (rule less_le_trans, assumption, rule le_add1)

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

 by (rule less_le_trans, assumption, rule le_add2)

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

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

 apply (simp_all add: le_add1)

 done

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

 apply (rule notI)

 apply (drule add_lessD1)

 apply (erule less_irrefl [THEN notE])

 done

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

 by (simp add: add_commute)

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

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

 apply (simp_all add: le_add1)

 done

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

 apply (simp add: add_commute)

 apply (erule add_leD1)

 done

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

 by (blast dest: add_leD1 add_leD2)

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

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

 by (force simp del: add_Suc_right

     simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)

 subsubsection {* More results about difference *}

 text {* Addition is the inverse of subtraction:

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

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

 by (induct m n rule: diff_induct) simp_all

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

 by (simp add: add_diff_inverse linorder_not_less)

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

 by (simp add: add_commute)

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

 by (induct m n rule: diff_induct) simp_all

 lemma diff_less_Suc: "m - n < Suc m"

 apply (induct m n rule: diff_induct)

 apply (erule_tac [3] less_SucE)

 apply (simp_all add: less_Suc_eq)

 done

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

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

 lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"

   by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])

 instance nat :: ordered_cancel_comm_monoid_diff

 proof

   show "\<And>m n :: nat. m \<le> n \<longleftrightarrow> (\<exists>q. n = m + q)" by (fact le_iff_add)

 qed

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

 by (rule le_less_trans, rule diff_le_self)

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

 by (cases n) (auto simp add: le_simps)

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

 by (induct j k rule: diff_induct) simp_all

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

 by (simp add: add_commute diff_add_assoc)

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

 by (auto simp add: diff_add_inverse2)

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

 by (induct m n rule: diff_induct) simp_all

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

 by (rule iffD2, rule diff_is_0_eq)

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

 by (induct m n rule: diff_induct) simp_all

 lemma less_imp_add_positive:

   assumes "i < j"

   shows "\<exists>k::nat. 0 < k & i + k = j"

 proof

   from assms show "0 < j - i & i + (j - i) = j"

     by (simp add: order_less_imp_le)

 qed

 text {* a nice rewrite for bounded subtraction *}

 lemma nat_minus_add_max:

   fixes n m :: nat

   shows "n - m + m = max n m"

     by (simp add: max_def not_le order_less_imp_le)

 lemma nat_diff_split:

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

     -- {* elimination of @{text -} on @{text nat} *}

 by (cases "a < b")

   (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse

     not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)

 lemma nat_diff_split_asm:

   "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"

     -- {* elimination of @{text -} on @{text nat} in assumptions *}

 by (auto split: nat_diff_split)

 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"

   by simp

 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"

   unfolding One_nat_def by (cases m) simp_all

 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"

   unfolding One_nat_def by (cases m) simp_all

 lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - 1)"

   unfolding One_nat_def by (cases n) simp_all

 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"

   unfolding One_nat_def by (cases m) simp_all

 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"

   by (fact Let_def)

 subsubsection {* Monotonicity of Multiplication *}

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

 by (simp add: mult_right_mono)

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

 by (simp add: mult_left_mono)

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

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

 by (simp add: mult_mono)

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

 by (simp add: mult_strict_right_mono)

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

       there are no negative numbers.*}

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

   apply (induct m)

    apply simp

   apply (case_tac n)

    apply simp_all

   done

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

   apply (induct m)

    apply simp

   apply (case_tac n)

    apply simp_all

   done

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

   apply (safe intro!: mult_less_mono1)

   apply (cases k, auto)

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

   apply (blast intro: mult_le_mono1)

   done

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

 by (simp add: mult_commute [of k])

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

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

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

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

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

 by (subst mult_less_cancel1) simp

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

 by (subst mult_le_cancel1) simp

 lemma le_square: "m \<le> m * (m::nat)"

   by (cases m) (auto intro: le_add1)

 lemma le_cube: "(m::nat) \<le> m * (m * m)"

   by (cases m) (auto intro: le_add1)

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

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

   apply (drule sym)

   apply (rule disjCI)

   apply (rule nat_less_cases, erule_tac [2] _)

    apply (drule_tac [2] mult_less_mono2)

     apply (auto)

   done

 lemma mono_times_nat:

   fixes n :: nat

   assumes "n > 0"

   shows "mono (times n)"

 proof

   fix m q :: nat

   assume "m \<le> q"

   with assms show "n * m \<le> n * q" by simp

 qed

 text {* the lattice order on @{typ nat} *}

 instantiation nat :: distrib_lattice

 begin

 definition

   "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"

 definition

   "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"

 instance by intro_classes

   (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def

     intro: order_less_imp_le antisym elim!: order_trans order_less_trans)

 end

 subsection {* Natural operation of natural numbers on functions *}

 text {*

   We use the same logical constant for the power operations on

   functions and relations, in order to share the same syntax.

 *}

 consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"

 abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where

   "f ^^ n \<equiv> compow n f"

 notation (latex output)

   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)

 notation (HTML output)

   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)

 text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}

 overloading

   funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"

 begin

 primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where

   "funpow 0 f = id"

 | "funpow (Suc n) f = f o funpow n f"

 end

 lemma funpow_Suc_right:

   "f ^^ Suc n = f ^^ n \<circ> f"

 proof (induct n)

   case 0 then show ?case by simp

 next

   fix n

   assume "f ^^ Suc n = f ^^ n \<circ> f"

   then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"

     by (simp add: o_assoc)

 qed

 lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right

 text {* for code generation *}

 definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where

   funpow_code_def [code_abbrev]: "funpow = compow"

 lemma [code]:

   "funpow (Suc n) f = f o funpow n f"

   "funpow 0 f = id"

   by (simp_all add: funpow_code_def)

 hide_const (open) funpow

 lemma funpow_add:

   "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"

   by (induct m) simp_all

 lemma funpow_mult:

   fixes f :: "'a \<Rightarrow> 'a"

   shows "(f ^^ m) ^^ n = f ^^ (m * n)"

   by (induct n) (simp_all add: funpow_add)

 lemma funpow_swap1:

   "f ((f ^^ n) x) = (f ^^ n) (f x)"

 proof -

   have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp

   also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)

   also have "\<dots> = (f ^^ n) (f x)" by simp

   finally show ?thesis .

 qed

 lemma comp_funpow:

   fixes f :: "'a \<Rightarrow> 'a"

   shows "comp f ^^ n = comp (f ^^ n)"

   by (induct n) simp_all

 subsection {* Kleene iteration *}

 lemma Kleene_iter_lpfp:

 assumes "mono f" and "f p \<le> p" shows "(f^^k) (bot::'a::order_bot) \<le> p"

 proof(induction k)

   case 0 show ?case by simp

 next

   case Suc

   from monoD[OF assms(1) Suc] assms(2)

   show ?case by simp

 qed

 lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"

 shows "lfp f = (f^^k) bot"

 proof(rule antisym)

   show "lfp f \<le> (f^^k) bot"

   proof(rule lfp_lowerbound)

     show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp

   qed

 next

   show "(f^^k) bot \<le> lfp f"

     using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp

 qed

 subsection {* Embedding of the Naturals into any @{text semiring_1}: @{term of_nat} *}

 context semiring_1

 begin

 definition of_nat :: "nat \<Rightarrow> 'a" where

   "of_nat n = (plus 1 ^^ n) 0"

 lemma of_nat_simps [simp]:

   shows of_nat_0: "of_nat 0 = 0"

     and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"

   by (simp_all add: of_nat_def)

 lemma of_nat_1 [simp]: "of_nat 1 = 1"

   by (simp add: of_nat_def)

 lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"

   by (induct m) (simp_all add: add_ac)

 lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"

   by (induct m) (simp_all add: add_ac distrib_right)

 primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where

   "of_nat_aux inc 0 i = i"

 | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}

 lemma of_nat_code:

   "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"

 proof (induct n)

   case 0 then show ?case by simp

 next

   case (Suc n)

   have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"

     by (induct n) simp_all

   from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"

     by simp

   with Suc show ?case by (simp add: add_commute)

 qed

 end

 declare of_nat_code [code]

 text{*Class for unital semirings with characteristic zero.

  Includes non-ordered rings like the complex numbers.*}

 class semiring_char_0 = semiring_1 +

   assumes inj_of_nat: "inj of_nat"

 begin

 lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"

   by (auto intro: inj_of_nat injD)

 text{*Special cases where either operand is zero*}

 lemma of_nat_0_eq_iff [simp, no_atp]: "0 = of_nat n \<longleftrightarrow> 0 = n"

   by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])

 lemma of_nat_eq_0_iff [simp, no_atp]: "of_nat m = 0 \<longleftrightarrow> m = 0"

   by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])

 end

 context linordered_semidom

 begin

 lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"

   by (induct n) simp_all

 lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"

   by (simp add: not_less)

 lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"

   by (induct m n rule: diff_induct, simp_all add: add_pos_nonneg)

 lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"

   by (simp add: not_less [symmetric] linorder_not_less [symmetric])

 lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"

   by simp

 lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"

   by simp

 text{*Every @{text linordered_semidom} has characteristic zero.*}

 subclass semiring_char_0 proof

 qed (auto intro!: injI simp add: eq_iff)

 text{*Special cases where either operand is zero*}

 lemma of_nat_le_0_iff [simp, no_atp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"

   by (rule of_nat_le_iff [of _ 0, simplified])

 lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"

   by (rule of_nat_less_iff [of 0, simplified])

 end

 context ring_1

 begin

 lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"

 by (simp add: algebra_simps of_nat_add [symmetric])

 end

 context linordered_idom

 begin

 lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"

   unfolding abs_if by auto

 end

 lemma of_nat_id [simp]: "of_nat n = n"

   by (induct n) simp_all

 lemma of_nat_eq_id [simp]: "of_nat = id"

   by (auto simp add: fun_eq_iff)

 subsection {* The Set of Natural Numbers *}

 context semiring_1

 begin

 definition Nats  :: "'a set" where

   "Nats = range of_nat"

 notation (xsymbols)

   Nats  ("\<nat>")

 lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"

   by (simp add: Nats_def)

 lemma Nats_0 [simp]: "0 \<in> \<nat>"

 apply (simp add: Nats_def)

 apply (rule range_eqI)

 apply (rule of_nat_0 [symmetric])

 done

 lemma Nats_1 [simp]: "1 \<in> \<nat>"

 apply (simp add: Nats_def)

 apply (rule range_eqI)

 apply (rule of_nat_1 [symmetric])

 done

 lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"

 apply (auto simp add: Nats_def)

 apply (rule range_eqI)

 apply (rule of_nat_add [symmetric])

 done

 lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"

 apply (auto simp add: Nats_def)

 apply (rule range_eqI)

 apply (rule of_nat_mult [symmetric])

 done

 lemma Nats_cases [cases set: Nats]:

   assumes "x \<in> \<nat>"

   obtains (of_nat) n where "x = of_nat n"

   unfolding Nats_def

 proof -

   from `x \<in> \<nat>` have "x \<in> range of_nat" unfolding Nats_def .

   then obtain n where "x = of_nat n" ..

   then show thesis ..

 qed

 lemma Nats_induct [case_names of_nat, induct set: Nats]:

   "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"

   by (rule Nats_cases) auto

 end

 subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}

 lemma subst_equals:

   assumes 1: "t = s" and 2: "u = t"

   shows "u = s"

   using 2 1 by (rule trans)

 setup Arith_Data.setup

 ML_file "Tools/nat_arith.ML"

 simproc_setup nateq_cancel_sums

   ("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =

   {* fn phi => fn ss => try Nat_Arith.cancel_eq_conv *}

 simproc_setup natless_cancel_sums

   ("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =

   {* fn phi => fn ss => try Nat_Arith.cancel_less_conv *}

 simproc_setup natle_cancel_sums

   ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =

   {* fn phi => fn ss => try Nat_Arith.cancel_le_conv *}

 simproc_setup natdiff_cancel_sums

   ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =

   {* fn phi => fn ss => try Nat_Arith.cancel_diff_conv *}

 ML_file "Tools/lin_arith.ML"

 setup {* Lin_Arith.global_setup *}

 declaration {* K Lin_Arith.setup *}

 simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) <= n" | "(m::nat) = n") =

   {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}

 (* Because of this simproc, the arithmetic solver is really only

 useful to detect inconsistencies among the premises for subgoals which are

 *not* themselves (in)equalities, because the latter activate

 fast_nat_arith_simproc anyway. However, it seems cheaper to activate the

 solver all the time rather than add the additional check. *)

 lemmas [arith_split] = nat_diff_split split_min split_max

 context order

 begin

 lemma lift_Suc_mono_le:

   assumes mono: "!!n. f n \<le> f(Suc n)" and "n\<le>n'"

   shows "f n \<le> f n'"

 proof (cases "n < n'")

   case True

   thus ?thesis

     by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)

 qed (insert `n \<le> n'`, auto) -- {*trivial for @{prop "n = n'"} *}

 lemma lift_Suc_mono_less:

   assumes mono: "!!n. f n < f(Suc n)" and "n < n'"

   shows "f n < f n'"

 using `n < n'`

 by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)

 lemma lift_Suc_mono_less_iff:

   "(!!n. f n < f(Suc n)) \<Longrightarrow> f(n) < f(m) \<longleftrightarrow> n<m"

 by(blast intro: less_asym' lift_Suc_mono_less[of f]

          dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq[THEN iffD1])

 end

 lemma mono_iff_le_Suc: "mono f = (\<forall>n. f n \<le> f (Suc n))"

   unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])

 lemma mono_nat_linear_lb:

   "(!!m n::nat. m<n \<Longrightarrow> f m < f n) \<Longrightarrow> f(m)+k \<le> f(m+k)"

 apply(induct_tac k)

  apply simp

 apply(erule_tac x="m+n" in meta_allE)

 apply(erule_tac x="Suc(m+n)" in meta_allE)

 apply simp

 done

 text{*Subtraction laws, mostly by Clemens Ballarin*}

 lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"

 by arith

 lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"

 by arith

 lemma less_diff_conv2:

   fixes j k i :: nat

   assumes "k \<le> j"

   shows "j - k < i \<longleftrightarrow> j < i + k"

   using assms by arith

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

 by arith

 lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"

 by arith

 lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"

 by arith

 lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"

 by arith

 (*Replaces the previous diff_less and le_diff_less, which had the stronger

   second premise n\<le>m*)

 lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"

 by arith

 text {* Simplification of relational expressions involving subtraction *}

 lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"

 by (simp split add: nat_diff_split)

 hide_fact (open) diff_diff_eq

 lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"

 by (auto split add: nat_diff_split)

 lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"

 by (auto split add: nat_diff_split)

 lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"

 by (auto split add: nat_diff_split)

 text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}

 (* Monotonicity of subtraction in first argument *)

 lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"

 by (simp split add: nat_diff_split)

 lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"

 by (simp split add: nat_diff_split)

 lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"

 by (simp split add: nat_diff_split)

 lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"

 by (simp split add: nat_diff_split)

 lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"

 by auto

 lemma inj_on_diff_nat: 

   assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"

   shows "inj_on (\<lambda>n. n - k) N"

 proof (rule inj_onI)

   fix x y

   assume a: "x \<in> N" "y \<in> N" "x - k = y - k"

   with k_le_n have "x - k + k = y - k + k" by auto

   with a k_le_n show "x = y" by auto

 qed

 text{*Rewriting to pull differences out*}

 lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"

 by arith

 lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"

 by arith

 lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"

 by arith

 lemma Suc_diff_Suc: "n < m \<Longrightarrow> Suc (m - Suc n) = m - n"

 by simp

 (*The others are

       i - j - k = i - (j + k),

       k \<le> j ==> j - k + i = j + i - k,

       k \<le> j ==> i + (j - k) = i + j - k *)

 lemmas add_diff_assoc = diff_add_assoc [symmetric]

 lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]

 declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]

 text{*At present we prove no analogue of @{text not_less_Least} or @{text

 Least_Suc}, since there appears to be no need.*}

 text{*Lemmas for ex/Factorization*}

 lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"

 by (cases m) auto

 lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"

 by (cases m) auto

 lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"

 by (cases m) auto

 text {* Specialized induction principles that work "backwards": *}

 lemma inc_induct[consumes 1, case_names base step]:

   assumes less: "i <= j"

   assumes base: "P j"

   assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"

   shows "P i"

   using less

 proof (induct d=="j - i" arbitrary: i)

   case (0 i)

   hence "i = j" by simp

   with base show ?case by simp

 next

   case (Suc d i)

   hence "i < j" "P (Suc i)"

     by simp_all

   thus "P i" by (rule step)

 qed

 lemma strict_inc_induct[consumes 1, case_names base step]:

   assumes less: "i < j"

   assumes base: "!!i. j = Suc i ==> P i"

   assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"

   shows "P i"

   using less

 proof (induct d=="j - i - 1" arbitrary: i)

   case (0 i)

   with `i < j` have "j = Suc i" by simp

   with base show ?case by simp

 next

   case (Suc d i)

   hence "i < j" "P (Suc i)"

     by simp_all

   thus "P i" by (rule step)

 qed

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

   using inc_induct[of "k - i" k P, simplified] by blast

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

   using inc_induct[of 0 k P] by blast

 text {* Further induction rule similar to @{thm inc_induct} *}

 lemma dec_induct[consumes 1, case_names base step]:

   "i \<le> j \<Longrightarrow> P i \<Longrightarrow> (\<And>n. i \<le> n \<Longrightarrow> P n \<Longrightarrow> P (Suc n)) \<Longrightarrow> P j"

   by (induct j arbitrary: i) (auto simp: le_Suc_eq)

 subsection {* The divides relation on @{typ nat} *}

 lemma dvd_1_left [iff]: "Suc 0 dvd k"

 unfolding dvd_def by simp

 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"

 by (simp add: dvd_def)

 lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"

 by (simp add: dvd_def)

 lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"

   unfolding dvd_def

   by (force dest: mult_eq_self_implies_10 simp add: mult_assoc)

 text {* @{term "op dvd"} is a partial order *}

 interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"

   proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)

 lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"

 unfolding dvd_def

 by (blast intro: diff_mult_distrib2 [symmetric])

 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"

   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])

   apply (blast intro: dvd_add)

   done

 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"

 by (drule_tac m = m in dvd_diff_nat, auto)

 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"

   apply (rule iffI)

    apply (erule_tac [2] dvd_add)

    apply (rule_tac [2] dvd_refl)

   apply (subgoal_tac "n = (n+k) -k")

    prefer 2 apply simp

   apply (erule ssubst)

   apply (erule dvd_diff_nat)

   apply (rule dvd_refl)

   done

 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"

   unfolding dvd_def

   apply (erule exE)

   apply (simp add: mult_ac)

   done

 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"

   apply auto

    apply (subgoal_tac "m*n dvd m*1")

    apply (drule dvd_mult_cancel, auto)

   done

 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"

   apply (subst mult_commute)

   apply (erule dvd_mult_cancel1)

   done

 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"

 by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)

 lemma nat_dvd_not_less:

   fixes m n :: nat

   shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"

 by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)

 lemma dvd_plusE:

   fixes m n q :: nat

   assumes "m dvd n + q" "m dvd n"

   obtains "m dvd q"

 proof (cases "m = 0")

   case True with assms that show thesis by simp

 next

   case False then have "m > 0" by simp

   from assms obtain r s where "n = m * r" and "n + q = m * s" by (blast elim: dvdE)

   then have *: "m * r + q = m * s" by simp

   show thesis proof (cases "r \<le> s")

     case False then have "s < r" by (simp add: not_le)

     with * have "m * r + q - m * s = m * s - m * s" by simp

     then have "m * r + q - m * s = 0" by simp

     with `m > 0` `s < r` have "m * r - m * s + q = 0" by simp

     then have "m * (r - s) + q = 0" by auto

     then have "m * (r - s) = 0" by simp

     then have "m = 0 \<or> r - s = 0" by simp

     with `s < r` have "m = 0" by arith

     with `m > 0` show thesis by auto

   next

     case True with * have "m * r + q - m * r = m * s - m * r" by simp

     with `m > 0` `r \<le> s` have "m * r - m * r + q = m * s - m * r" by simp

     then have "q = m * (s - r)" by (simp add: diff_mult_distrib2)

     with assms that show thesis by (auto intro: dvdI)

   qed

 qed

 lemma dvd_plus_eq_right:

   fixes m n q :: nat

   assumes "m dvd n"

   shows "m dvd n + q \<longleftrightarrow> m dvd q"

   using assms by (auto elim: dvd_plusE)

 lemma dvd_plus_eq_left:

   fixes m n q :: nat

   assumes "m dvd q"

   shows "m dvd n + q \<longleftrightarrow> m dvd n"

   using assms by (simp add: dvd_plus_eq_right add_commute [of n])

 lemma less_dvd_minus:

   fixes m n :: nat

   assumes "m < n"

   shows "m dvd n \<longleftrightarrow> m dvd (n - m)"

 proof -

   from assms have "n = m + (n - m)" by arith

   then obtain q where "n = m + q" ..

   then show ?thesis by (simp add: dvd_reduce add_commute [of m])

 qed

 lemma dvd_minus_self:

   fixes m n :: nat

   shows "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"

   by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add)

 lemma dvd_minus_add:

   fixes m n q r :: nat

   assumes "q \<le> n" "q \<le> r * m"

   shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)"

 proof -

   have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)"

     by (auto elim: dvd_plusE)

   also with assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp

   also with assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp

   also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add_commute)

   finally show ?thesis .

 qed

 subsection {* aliasses *}

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

   by simp

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

   by simp

 subsection {* size of a datatype value *}

 class size =

   fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}

 subsection {* code module namespace *}

 code_identifier

   code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

 hide_const (open) of_nat_aux

 end