(*  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]: "(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]: "(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]: "!!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))"