src/HOL/Nat.thy

 
 Type "nat" is a linear order, and a datatype; arithmetic operators + -
 and * (for div, mod and dvd, see theory Divides).
 *)
 
-Nat = NatDef + 
-
-(* type "nat" is a wellfounded linear order, and a datatype *)
+header {* Natural numbers *}
+
+theory Nat = Wellfounded_Recursion:
+
+subsection {* Type @{text ind} *}
+
+typedecl ind
+
+consts
+  Zero_Rep      :: ind
+  Suc_Rep       :: "ind => ind"
+
+axioms
+  -- {* the axiom of infinity in 2 parts *}
+  inj_Suc_Rep:          "inj Suc_Rep"
+  Suc_Rep_not_Zero_Rep: "Suc_Rep x ~= Zero_Rep"
+
+
+subsection {* Type nat *}
+
+text {* Type definition *}
+
+consts
+  Nat :: "ind set"
+
+inductive Nat
+intros
+  Zero_RepI: "Zero_Rep : Nat"
+  Suc_RepI: "i : Nat ==> Suc_Rep i : Nat"
+
+global
+
+typedef (open Nat)
+  nat = "Nat" by (rule exI, rule Nat.Zero_RepI)
+
+instance nat :: ord ..
+instance nat :: zero ..
+instance nat :: one ..
+
+
+text {* Abstract constants and syntax *}
+
+consts
+  Suc :: "nat => nat"
+  pred_nat :: "(nat * nat) set"
+
+local
+
+defs
+  Zero_nat_def: "0 == Abs_Nat Zero_Rep"
+  Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"
+  One_nat_def [simp]: "1 == Suc 0"
+
+  -- {* nat operations *}
+  pred_nat_def: "pred_nat == {(m, n). n = Suc m}"
+
+  less_def: "m < n == (m, n) : trancl pred_nat"
+
+  le_def: "m <= (n::nat) == ~ (n < m)"
+
+
+text {* Induction *}
+
+theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n"
+  apply (unfold Zero_nat_def Suc_def)
+  apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}
+  apply (erule Rep_Nat [THEN Nat.induct])
+  apply (rules elim: Abs_Nat_inverse [THEN subst])
+  done
+
+
+text {* Isomorphisms: @{text Abs_Nat} and @{text Rep_Nat} *}
+
+lemma inj_Rep_Nat: "inj Rep_Nat"
+  apply (rule inj_inverseI)
+  apply (rule Rep_Nat_inverse)
+  done
+
+lemma inj_on_Abs_Nat: "inj_on Abs_Nat Nat"
+  apply (rule inj_on_inverseI)
+  apply (erule Abs_Nat_inverse)
+  done
+
+text {* Distinctness of constructors *}
+
+lemma Suc_not_Zero [iff]: "Suc m ~= 0"
+  apply (unfold Zero_nat_def Suc_def)
+  apply (rule inj_on_Abs_Nat [THEN inj_on_contraD])
+  apply (rule Suc_Rep_not_Zero_Rep)
+  apply (rule Rep_Nat Nat.Suc_RepI Nat.Zero_RepI)+
+  done
+
+lemma Zero_not_Suc [iff]: "0 ~= Suc m"
+  by (rule not_sym, rule Suc_not_Zero not_sym)
+
+lemma Suc_neq_Zero: "Suc m = 0 ==> R"
+  by (rule notE, rule Suc_not_Zero)
+
+lemma Zero_neq_Suc: "0 = Suc m ==> R"
+  by (rule Suc_neq_Zero, erule sym)
+
+text {* Injectiveness of @{term Suc} *}
+
+lemma inj_Suc: "inj Suc"
+  apply (unfold Suc_def)
+  apply (rule injI)
+  apply (drule inj_on_Abs_Nat [THEN inj_onD])
+  apply (rule Rep_Nat Nat.Suc_RepI)+
+  apply (drule inj_Suc_Rep [THEN injD])
+  apply (erule inj_Rep_Nat [THEN injD])
+  done
+
+lemma Suc_inject: "Suc x = Suc y ==> x = y"
+  by (rule inj_Suc [THEN injD])
+
+lemma Suc_Suc_eq [iff]: "(Suc m = Suc n) = (m = n)"
+  apply (rule iffI)
+  apply (erule Suc_inject)
+  apply (erule arg_cong)
+  done
+
+lemma nat_not_singleton: "(ALL x. x = (0::nat)) = False"
+  by auto
+
+text {* @{typ nat} is a datatype *}
 
 rep_datatype nat
-  distinct Suc_not_Zero, Zero_not_Suc
-  inject   Suc_Suc_eq
-  induct   nat_induct
-
-instance nat :: order (le_refl,le_trans,le_anti_sym,nat_less_le)
-instance nat :: linorder (nat_le_linear)
-instance nat :: wellorder (wf_less)
+  distinct  Suc_not_Zero Zero_not_Suc
+  inject    Suc_Suc_eq
+  induction nat_induct
+
+lemma n_not_Suc_n: "n ~= Suc n"
+  by (induct n) simp_all
+
+lemma Suc_n_not_n: "Suc t ~= t"
+  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"} *}
+
+theorem 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_tac n)
+  prefer 2
+  apply (rule allI)
+  apply (induct_tac x)
+  apply rules+
+  done
+
+subsection {* Basic properties of "less than" *}
+
+lemma wf_pred_nat: "wf pred_nat"
+  apply (unfold wf_def pred_nat_def)
+  apply clarify
+  apply (induct_tac x)
+  apply blast+
+  done
+
+lemma wf_less: "wf {(x, y::nat). x < y}"
+  apply (unfold less_def)
+  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset])
+  apply blast
+  done
+
+lemma less_eq: "((m, n) : pred_nat^+) = (m < n)"
+  apply (unfold less_def)
+  apply (rule refl)
+  done
+
+subsubsection {* Introduction properties *}
+
+lemma less_trans: "i < j ==> j < k ==> i < (k::nat)"
+  apply (unfold less_def)
+  apply (rule trans_trancl [THEN transD])
+  apply assumption+
+  done
+
+lemma lessI [iff]: "n < Suc n"
+  apply (unfold less_def pred_nat_def)
+  apply (simp add: r_into_trancl)
+  done
+
+lemma less_SucI: "i < j ==> i < Suc j"
+  apply (rule less_trans)
+  apply assumption
+  apply (rule lessI)
+  done
+
+lemma zero_less_Suc [iff]: "0 < Suc n"
+  apply (induct n)
+  apply (rule lessI)
+  apply (erule less_trans)
+  apply (rule lessI)
+  done
+
+subsubsection {* Elimination properties *}
+
+lemma less_not_sym: "n < m ==> ~ m < (n::nat)"
+  apply (unfold less_def)
+  apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym])
+  done
+
+lemma less_asym:
+  assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P
+  apply (rule contrapos_np)
+  apply (rule less_not_sym)
+  apply (rule h1)
+  apply (erule h2)
+  done
+
+lemma less_not_refl: "~ n < (n::nat)"
+  apply (unfold less_def)
+  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl])
+  done
+
+lemma less_irrefl [elim!]: "(n::nat) < n ==> R"
+  by (rule notE, rule less_not_refl)
+
+lemma less_not_refl2: "n < m ==> m ~= (n::nat)" by blast
+
+lemma less_not_refl3: "(s::nat) < t ==> s ~= t"
+  by (rule not_sym, rule less_not_refl2)
+
+lemma lessE:
+  assumes major: "i < k"
+  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
+  shows P
+  apply (rule major [unfolded less_def pred_nat_def, THEN tranclE])
+  apply simp_all
+  apply (erule p1)
+  apply (rule p2)
+  apply (simp add: less_def pred_nat_def)
+  apply assumption
+  done
+
+lemma not_less0 [iff]: "~ n < (0::nat)"
+  by (blast elim: lessE)
+
+lemma less_zeroE: "(n::nat) < 0 ==> R"
+  by (rule notE, rule not_less0)
+
+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)
+  apply blast
+  apply (rule less)
+  apply blast
+  done
+
+lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
+  by (blast elim!: less_SucE intro: less_trans)
+
+lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
+  by (simp add: less_Suc_eq)
+
+lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
+  by (simp add: less_Suc_eq)
+
+lemma Suc_mono: "m < n ==> Suc m < Suc n"
+  by (induct n) (fast elim: less_trans lessE)+
+
+text {* "Less than" is a linear ordering *}
+lemma less_linear: "m < n | m = n | n < (m::nat)"
+  apply (induct_tac m)
+  apply (induct_tac n)
+  apply (rule refl [THEN disjI1, THEN disjI2])
+  apply (rule zero_less_Suc [THEN disjI1])
+  apply (blast intro: Suc_mono less_SucI elim: lessE)
+  done
+
+lemma nat_neq_iff: "((m::nat) ~= n) = (m < n | n < m)"
+  using less_linear by blast
+
+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 ~= n ==> Suc m < n"
+  apply (simp add: nat_neq_iff)
+  apply (blast elim!: less_irrefl less_SucE elim: less_asym)
+  done
+
+lemma Suc_lessD: "Suc m < n ==> m < n"
+  apply (induct n)
+  apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+
+  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])
+  apply assumption
+  done
+
+lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
+  by (blast elim: lessE dest: Suc_lessD)
+
+lemma Suc_less_eq [iff]: "(Suc m < Suc n) = (m < n)"
+  apply (rule iffI)
+  apply (erule Suc_less_SucD)
+  apply (erule Suc_mono)
+  done
+
+lemma less_trans_Suc:
+  assumes le: "i < j" shows "j < k ==> Suc i < k"
+  apply (induct k)
+  apply 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: "(~ m < n) = (n < Suc m)"
+  apply (rule_tac m = "m" and n = "n" in diff_induct)
+  apply simp_all
+  done
+
+text {* Complete induction, aka course-of-values induction *}
+lemma nat_less_induct:
+  assumes prem: "!!n. ALL m::nat. m < n --> P m ==> P n" shows "P n"
+  apply (rule_tac a=n in wf_induct)
+  apply (rule wf_pred_nat [THEN wf_trancl])
+  apply (rule prem)
+  apply (unfold less_def)
+  apply assumption
+  done
+
+subsection {* Properties of "less or equal than" *}
+
+text {* Was @{text le_eq_less_Suc}, but this orientation is more useful *}
+lemma less_Suc_eq_le: "(m < Suc n) = (m <= n)"
+  by (unfold le_def, rule not_less_eq [symmetric])
+
+lemma le_imp_less_Suc: "m <= n ==> m < Suc n"
+  by (rule less_Suc_eq_le [THEN iffD2])
+
+lemma le0 [iff]: "(0::nat) <= n"
+  by (unfold le_def, rule not_less0)
+
+lemma Suc_n_not_le_n: "~ Suc n <= n"
+  by (simp add: le_def)
+
+lemma le_0_eq [iff]: "((i::nat) <= 0) = (i = 0)"
+  by (induct i) (simp_all add: le_def)
+
+lemma le_Suc_eq: "(m <= Suc n) = (m <= n | m = Suc n)"
+  by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq)
+
+lemma le_SucE: "m <= Suc n ==> (m <= n ==> R) ==> (m = Suc n ==> R) ==> R"
+  by (drule le_Suc_eq [THEN iffD1], rules+)
+
+lemma leI: "~ n < m ==> m <= (n::nat)" by (simp add: le_def)
+
+lemma leD: "m <= n ==> ~ n < (m::nat)"
+  by (simp add: le_def)
+
+lemmas leE = leD [elim_format]
+
+lemma not_less_iff_le: "(~ n < m) = (m <= (n::nat))"
+  by (blast intro: leI elim: leE)
+
+lemma not_leE: "~ m <= n ==> n<(m::nat)"
+  by (simp add: le_def)
+
+lemma not_le_iff_less: "(~ n <= m) = (m < (n::nat))"
+  by (simp add: le_def)
+
+lemma Suc_leI: "m < n ==> Suc(m) <= n"
+  apply (simp add: le_def less_Suc_eq)
+  apply (blast elim!: less_irrefl less_asym)
+  done -- {* formerly called lessD *}
+
+lemma Suc_leD: "Suc(m) <= n ==> m <= n"
+  by (simp add: le_def less_Suc_eq)
+
+text {* Stronger version of @{text Suc_leD} *}
+lemma Suc_le_lessD: "Suc m <= n ==> m < n"
+  apply (simp add: le_def less_Suc_eq)
+  using less_linear
+  apply blast
+  done
+
+lemma Suc_le_eq: "(Suc m <= n) = (m < n)"
+  by (blast intro: Suc_leI Suc_le_lessD)
+
+lemma le_SucI: "m <= n ==> m <= Suc n"
+  by (unfold le_def) (blast dest: Suc_lessD)
+
+lemma less_imp_le: "m < n ==> m <= (n::nat)"
+  by (unfold le_def) (blast elim: less_asym)
+
+text {* For instance, @{text "(Suc m < Suc n) = (Suc m <= n) = (m < n)"} *}
+lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq
+
+
+text {* Equivalence of @{term "m <= n"} and @{term "m < n | m = n"} *}
+
+lemma le_imp_less_or_eq: "m <= n ==> m < n | m = (n::nat)"
+  apply (unfold le_def)
+  using less_linear
+  apply (blast elim: less_irrefl less_asym)
+  done
+
+lemma less_or_eq_imp_le: "m < n | m = n ==> m <= (n::nat)"
+  apply (unfold le_def)
+  using less_linear
+  apply (blast elim!: less_irrefl elim: less_asym)
+  done
+
+lemma le_eq_less_or_eq: "(m <= (n::nat)) = (m < n | m=n)"
+  by (rules intro: less_or_eq_imp_le le_imp_less_or_eq)
+
+text {* Useful with @{text Blast}. *}
+lemma eq_imp_le: "(m::nat) = n ==> m <= n"
+  by (rule less_or_eq_imp_le, rule disjI2)
+
+lemma le_refl: "n <= (n::nat)"
+  by (simp add: le_eq_less_or_eq)
+
+lemma le_less_trans: "[| i <= j; j < k |] ==> i < (k::nat)"
+  by (blast dest!: le_imp_less_or_eq intro: less_trans)
+
+lemma less_le_trans: "[| i < j; j <= k |] ==> i < (k::nat)"
+  by (blast dest!: le_imp_less_or_eq intro: less_trans)
+
+lemma le_trans: "[| i <= j; j <= k |] ==> i <= (k::nat)"
+  by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans)
+
+lemma le_anti_sym: "[| m <= n; n <= m |] ==> m = (n::nat)"
+  -- {* @{text order_less_irrefl} could make this proof fail *}
+  by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym)
+
+lemma Suc_le_mono [iff]: "(Suc n <= Suc m) = (n <= m)"
+  by (simp add: le_simps)
+
+text {* Axiom @{text order_less_le} of class @{text order}: *}
+lemma nat_less_le: "((m::nat) < n) = (m <= n & m ~= n)"
+  by (simp add: le_def nat_neq_iff) (blast elim!: less_asym)
+
+lemma le_neq_implies_less: "(m::nat) <= n ==> m ~= n ==> m < n"
+  by (rule iffD2, rule nat_less_le, rule conjI)
+
+text {* Axiom @{text linorder_linear} of class @{text linorder}: *}
+lemma nat_le_linear: "(m::nat) <= n | n <= m"
+  apply (simp add: le_eq_less_or_eq)
+  using less_linear
+  apply blast
+  done
+
+lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
+  by (blast elim!: less_SucE)
+
+
+text {*
+  Rewrite @{term "n < Suc m"} to @{term "n = m"}
+  if @{term "~ n < m"} or @{term "m <= n"} hold.
+  Not suitable as default simprules because they often lead to looping
+*}
+lemma le_less_Suc_eq: "m <= n ==> (n < Suc m) = (n = m)"
+  by (rule not_less_less_Suc_eq, rule leD)
+
+lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
+
+
+text {*
+  Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}. 
+  No longer added as simprules (they loop) 
+  but via @{text reorient_simproc} in Bin
+*}
+
+text {* Polymorphic, not just for @{typ nat} *}
+lemma zero_reorient: "(0 = x) = (x = 0)"
+  by auto
+
+lemma one_reorient: "(1 = x) = (x = 1)"
+  by auto
+
+text {* Type {@typ nat} is a wellfounded linear order *}
+
+instance nat :: order by (intro_classes,
+  (assumption | rule le_refl le_trans le_anti_sym nat_less_le)+)
+instance nat :: linorder by (intro_classes, rule nat_le_linear)
+instance nat :: wellorder by (intro_classes, rule wf_less)
+
+subsection {* Arithmetic operators *}
 
 axclass power < type
 
 consts
-  power :: ['a::power, nat] => 'a            (infixr "^" 80)
-
-
-(* arithmetic operators + - and * *)
-
-instance
-  nat :: {plus, minus, times, power}
-
-(* size of a datatype value; overloaded *)
-consts size :: 'a => nat
+  power :: "('a::power) => nat => 'a"            (infixr "^" 80)
+
+
+text {* arithmetic operators @{text "+ -"} and @{text "*"} *}
+
+instance nat :: plus ..
+instance nat :: minus ..
+instance nat :: times ..
+instance nat :: power ..
+
+text {* size of a datatype value; overloaded *}
+consts size :: "'a => nat"
 
 primrec
-  add_0    "0 + n = n"
-  add_Suc  "Suc m + n = Suc(m + n)"
+  add_0:    "0 + n = n"
+  add_Suc:  "Suc m + n = Suc (m + n)"
 
 primrec
-  diff_0   "m - 0 = m"
-  diff_Suc "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
+  diff_0:   "m - 0 = m"
+  diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
 
 primrec
-  mult_0   "0 * n = 0"
-  mult_Suc "Suc m * n = n + (m * n)"
+  mult_0:   "0 * n = 0"
+  mult_Suc: "Suc m * n = n + (m * n)"
+
+text {* These 2 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 ~= 0 ==> EX m. n = Suc m"
+  by (case_tac n) simp_all
+
+lemma gr_implies_not0: "!!n::nat. m<n ==> n ~= 0"
+  by (case_tac n) simp_all
+
+lemma neq0_conv [iff]: "!!n::nat. (n ~= 0) = (0 < n)"
+  by (case_tac n) simp_all
+
+text {* This theorem is useful with @{text blast} *}
+lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
+  by (rule iffD1, rule neq0_conv, rules)
+
+lemma gr0_conv_Suc: "(0 < n) = (EX m. n = Suc m)"
+  by (fast intro: not0_implies_Suc)
+
+lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
+  apply (rule iffI)
+  apply (rule ccontr)
+  apply simp_all
+  done
+
+lemma Suc_le_D: "(Suc n <= 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 | (EX j. m = Suc j & j < n))"
+  by (case_tac m) simp_all
+
+lemma nat_induct2: "P 0 ==> P (Suc 0) ==> (!!k. P k ==> P (Suc (Suc k))) ==> P n"
+  apply (rule nat_less_induct)
+  apply (case_tac n)
+  apply (case_tac [2] nat)
+  apply (blast intro: less_trans)+
+  done
+
+subsection {* @{text LEAST} theorems for type @{typ nat} by specialization *}
+
+lemmas LeastI = wellorder_LeastI
+lemmas Least_le = wellorder_Least_le
+lemmas not_less_Least = wellorder_not_less_Least
+
+lemma Least_Suc: "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
+  apply (case_tac "n")
+  apply auto
+  apply (frule LeastI)
+  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
+  apply (subgoal_tac " (LEAST x. P x) <= Suc (LEAST x. P (Suc x))")
+  apply (erule_tac [2] Least_le)
+  apply (case_tac "LEAST x. P x")
+  apply 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
+
+
+subsection {* @{term min} and @{term max} *}
+
+lemma min_0L [simp]: "min 0 n = (0::nat)"
+  by (rule min_leastL) simp
+
+lemma min_0R [simp]: "min n 0 = (0::nat)"
+  by (rule min_leastR) simp
+
+lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
+  by (simp add: min_of_mono)
+
+lemma max_0L [simp]: "max 0 n = (n::nat)"
+  by (rule max_leastL) simp
+
+lemma max_0R [simp]: "max n 0 = (n::nat)"
+  by (rule max_leastR) simp
+
+lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
+  by (simp add: max_of_mono)
+
+
+subsection {* Basic rewrite rules for the arithmetic operators *}
+
+text {* Difference *}
+
+lemma diff_0_eq_0 [simp]: "0 - n = (0::nat)"
+  by (induct_tac n) simp_all
+
+lemma diff_Suc_Suc [simp]: "Suc(m) - Suc(n) = m - n"
+  by (induct_tac n) simp_all
+
+
+text {*
+  Could be (and is, below) generalized in various ways
+  However, none of the generalizations are currently in the simpset,
+  and I dread to think what happens if I put them in
+*}
+lemma Suc_pred [simp]: "0 < n ==> Suc (n - Suc 0) = n"
+  by (simp split add: nat.split)
+
+declare diff_Suc [simp del]
+
+
+subsection {* Addition *}
+
+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
+
+
+text {* Associative law for addition *}
+lemma add_assoc: "(m + n) + k = m + ((n + k)::nat)"
+  by (induct m) simp_all
+
+text {* Commutative law for addition *}
+lemma add_commute: "m + n = n + (m::nat)"
+  by (induct m) simp_all
+
+lemma add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
+  apply (rule mk_left_commute [of "op +"])
+  apply (rule add_assoc)
+  apply (rule add_commute)
+  done
+
+text {* Addition is an AC-operator *}
+lemmas add_ac = add_assoc add_commute add_left_commute
+
+lemma add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
+  by (induct k) simp_all
+
+lemma add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
+  by (induct k) simp_all
+
+lemma add_left_cancel_le [simp]: "(k + m <= k + n) = (m<=(n::nat))"
+  by (induct k) simp_all
+
+lemma add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
+  by (induct k) simp_all
+
+text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
+
+lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)"
+  by (case_tac m) simp_all
+
+lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
+  by (case_tac 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_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m | 0 < n)"
+  by (simp del: neq0_conv add: neq0_conv [symmetric])
+
+lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0"
+  apply (drule add_0_right [THEN ssubst])
+  apply (simp add: add_assoc del: add_0_right)
+  done
+
+subsection {* Additional theorems about "less than" *}
+
+text {* Deleted @{text less_natE}; instead use @{text "less_imp_Suc_add RS exE"} *}
+lemma less_imp_Suc_add: "m < n ==> (EX k. n = Suc (m + k))"
+  apply (induct n)
+  apply (simp_all add: order_le_less)
+  apply (blast elim!: less_SucE intro!: add_0_right [symmetric] add_Suc_right [symmetric])
+  done
+
+lemma le_add2: "n <= ((m + n)::nat)"
+  apply (induct m)
+  apply simp_all
+  apply (erule le_SucI)
+  done
+
+lemma le_add1: "n <= ((n + m)::nat)"
+  apply (simp add: add_ac)
+  apply (rule le_add2)
+  done
+
+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) = (EX k. n = Suc (m + k))"
+  by (rules intro!: less_add_Suc1 less_imp_Suc_add)
+
+
+lemma trans_le_add1: "(i::nat) <= j ==> i <= j + m"
+  by (rule le_trans, assumption, rule le_add1)
+
+lemma trans_le_add2: "(i::nat) <= j ==> i <= 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 (induct j)
+  apply simp_all
+  apply (blast dest: Suc_lessD)
+  done
+
+lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
+  apply (rule notI)
+  apply (erule add_lessD1 [THEN less_irrefl])
+  done
+
+lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
+  by (simp add: add_commute not_add_less1)
+
+lemma add_leD1: "m + k <= n ==> m <= (n::nat)"
+  by (induct k) (simp_all add: le_simps)
+
+lemma add_leD2: "m + k <= n ==> k <= (n::nat)"
+  apply (simp add: add_commute)
+  apply (erule add_leD1)
+  done
+
+lemma add_leE: "(m::nat) + k <= n ==> (m <= n ==> k <= 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)
+
+
+subsection {* Monotonicity of Addition *}
+
+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])
+  apply assumption+
+  apply (induct_tac j)
+  apply simp_all
+  done
+
+text {* A [clumsy] way of lifting @{text "<"}
+  monotonicity to @{text "<="} monotonicity *}
+lemma less_mono_imp_le_mono:
+  assumes lt_mono: "!!i j::nat. i < j ==> f i < f j"
+  and le: "i <= j" shows "f i <= ((f j)::nat)" using le
+  apply (simp add: order_le_less)
+  apply (blast intro!: lt_mono)
+  done
+
+text {* non-strict, in 1st argument *}
+lemma add_le_mono1: "i <= j ==> i + k <= j + (k::nat)"
+  apply (rule_tac f = "%j. j + k" in less_mono_imp_le_mono)
+  apply (erule add_less_mono1)
+  apply assumption
+  done
+
+text {* non-strict, in both arguments *}
+lemma add_le_mono: "[| i <= j;  k <= l |] ==> i + k <= j + (l::nat)"
+  apply (erule add_le_mono1 [THEN le_trans])
+  apply (simp add: add_commute)
+  done
+
+
+subsection {* Multiplication *}
+
+text {* right annihilation in product *}
+lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
+  by (induct m) simp_all
+
+text {* right successor law for multiplication *}
+lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
+  by (induct m) (simp_all add: add_ac)
+
+lemma mult_1: "(1::nat) * n = n" by simp
+
+lemma mult_1_right: "n * (1::nat) = n" by simp
+
+text {* Commutative law for multiplication *}
+lemma mult_commute: "m * n = n * (m::nat)"
+  by (induct m) simp_all
+
+text {* addition distributes over multiplication *}
+lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
+  by (induct m) (simp_all add: add_ac)
+
+lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
+  by (induct m) (simp_all add: add_ac)
+
+text {* Associative law for multiplication *}
+lemma mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
+  by (induct m) (simp_all add: add_mult_distrib)
+
+lemma mult_left_commute: "x * (y * z) = y * ((x * z)::nat)"
+  apply (rule mk_left_commute [of "op *"])
+  apply (rule mult_assoc)
+  apply (rule mult_commute)
+  done
+
+lemmas mult_ac = mult_assoc mult_commute mult_left_commute
+
+lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
+  apply (induct_tac m)
+  apply (induct_tac [2] n)
+  apply simp_all
+  done
+
+
+subsection {* Difference *}
+
+lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
+  by (induct m) simp_all
+
+text {* Addition is the inverse of subtraction:
+  if @{term "n <= 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 <= m ==> n + (m - n) = (m::nat)"
+  by (simp add: add_diff_inverse not_less_iff_le)
+
+lemma le_add_diff_inverse2 [simp]: "n <= m ==> (m - n) + n = (m::nat)"
+  by (simp add: le_add_diff_inverse add_commute)
+
+
+subsection {* More results about difference *}
+
+lemma Suc_diff_le: "n <= 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 <= (m::nat)"
+  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
+
+lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
+  by (rule le_less_trans, rule diff_le_self)
+
+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_Suc_less [simp]: "0<n ==> n - Suc i < n"
+  apply (case_tac "n")
+  apply safe
+  apply (simp add: le_simps)
+  done
+
+text {* This and the next few suggested by Florian Kammueller *}
+lemma diff_commute: "(i::nat) - j - k = i - k - j"
+  by (simp add: diff_diff_left add_commute)
+
+lemma diff_add_assoc: "k <= (j::nat) ==> (i + j) - k = i + (j - k)"
+  by (induct j k rule: diff_induct) simp_all
+
+lemma diff_add_assoc2: "k <= (j::nat) ==> (j + i) - k = (j - k) + i"
+  by (simp add: add_commute diff_add_assoc)
+
+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_assoc)
+
+lemma le_imp_diff_is_add: "i <= (j::nat) ==> (j - i = k) = (j = k + i)"
+  apply safe
+  apply (simp_all add: diff_add_inverse2)
+  done
+
+lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m <= n)"
+  by (induct m n rule: diff_induct) simp_all
+
+lemma diff_is_0_eq' [simp]: "m <= 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: "i < j  ==> EX k::nat. 0 < k & i + k = j"
+  apply (rule_tac x = "j - i" in exI)
+  apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym)
+  done
+
+lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
+  apply (induct k i rule: diff_induct)
+  apply (simp_all (no_asm))
+  apply rules
+  done
+
+lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
+  apply (rule diff_self_eq_0 [THEN subst])
+  apply (rule zero_induct_lemma)
+  apply rules+
+  done
+
+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
+
+
+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 *}
+
+lemmas nat_distrib =
+  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
+
+
+subsection {* Monotonicity of Multiplication *}
+
+lemma mult_le_mono1: "i <= (j::nat) ==> i * k <= j * k"
+  by (induct k) (simp_all add: add_le_mono)
+
+lemma mult_le_mono2: "i <= (j::nat) ==> k * i <= k * j"
+  apply (drule mult_le_mono1)
+  apply (simp add: mult_commute)
+  done
+
+text {* @{text "<="} monotonicity, BOTH arguments *}
+lemma mult_le_mono: "i <= (j::nat) ==> k <= l ==> i * k <= j * l"
+  apply (erule mult_le_mono1 [THEN le_trans])
+  apply (erule mult_le_mono2)
+  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 (erule_tac m1 = "0" in less_imp_Suc_add [THEN exE])
+  apply simp
+  apply (induct_tac x)
+  apply (simp_all add: add_less_mono)
+  done
+
+lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
+  by (drule mult_less_mono2) (simp_all add: mult_commute)
+
+lemma zero_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
+  apply (induct m)
+  apply (case_tac [2] n)
+  apply simp_all
+  done
+
+lemma one_le_mult_iff [simp]: "(Suc 0 <= m * n) = (1 <= m & 1 <= n)"
+  apply (induct m)
+  apply (case_tac [2] n)
+  apply simp_all
+  done
+
+lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
+  apply (induct_tac m)
+  apply simp
+  apply (induct_tac n)
+  apply simp
+  apply fastsimp
+  done
+
+lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
+  apply (rule trans)
+  apply (rule_tac [2] mult_eq_1_iff)
+  apply fastsimp
+  done
+
+lemma mult_less_cancel2: "((m::nat) * k < n * k) = (0 < k & m < n)"
+  apply (safe intro!: mult_less_mono1)
+  apply (case_tac k)
+  apply 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_less_cancel2 mult_commute [of k])
+
+declare mult_less_cancel2 [simp]
+
+lemma mult_le_cancel1 [simp]: "(k * (m::nat) <= k * n) = (0 < k --> m <= n)"
+  apply (simp add: linorder_not_less [symmetric])
+  apply auto
+  done
+
+lemma mult_le_cancel2 [simp]: "((m::nat) * k <= n * k) = (0 < k --> m <= n)"
+  apply (simp add: linorder_not_less [symmetric])
+  apply auto
+  done
+
+lemma mult_cancel2: "(m * k = n * k) = (m = n | (k = (0::nat)))"
+  apply (cut_tac less_linear)
+  apply safe
+  apply auto
+  apply (drule mult_less_mono1, assumption, simp)+
+  done
+
+lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
+  by (simp add: mult_cancel2 mult_commute [of k])
+
+declare mult_cancel2 [simp]
+
+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 <= Suc k * n) = (m <= n)"
+  by (subst mult_le_cancel1) simp
+
+lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
+  by (subst mult_cancel1) simp
+
+
+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 (fastsimp elim!: less_SucE)
+  apply (fastsimp dest: mult_less_mono2)
+  done
 
 end