src/HOL/Nat.thy
author wenzelm
Wed Nov 08 13:48:29 2006 +0100 (2006-11-08)
changeset 21243 afffe1f72143
parent 21191 c00161fbf990
child 21252 9bffcdfd7553
permissions -rw-r--r--
removed theory NatArith (now part of Nat);
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(*  Title:      HOL/Nat.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
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Type "nat" is a linear order, and a datatype; arithmetic operators + -
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and * (for div, mod and dvd, see theory Divides).
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*)
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header {* Natural numbers *}
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theory Nat
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imports Wellfounded_Recursion Ring_and_Field
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uses ("arith_data.ML")
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begin
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subsection {* Type @{text ind} *}
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typedecl ind
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axiomatization
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  Zero_Rep :: ind and
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  Suc_Rep :: "ind => ind"
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where
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  -- {* the axiom of infinity in 2 parts *}
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  inj_Suc_Rep:          "inj Suc_Rep" and
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  Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
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subsection {* Type nat *}
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text {* Type definition *}
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consts
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  Nat :: "ind set"
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inductive Nat
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intros
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  Zero_RepI: "Zero_Rep : Nat"
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  Suc_RepI: "i : Nat ==> Suc_Rep i : Nat"
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global
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typedef (open Nat)
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  nat = Nat
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proof
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  show "Zero_Rep : Nat" by (rule Nat.Zero_RepI)
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qed
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instance nat :: "{ord, zero, one}" ..
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text {* Abstract constants and syntax *}
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consts
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  Suc :: "nat => nat"
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  pred_nat :: "(nat * nat) set"
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local
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defs
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  Zero_nat_def: "0 == Abs_Nat Zero_Rep"
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  Suc_def:      "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"
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  One_nat_def:  "1 == Suc 0"
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  -- {* nat operations *}
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  pred_nat_def: "pred_nat == {(m, n). n = Suc m}"
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  less_def: "m < n == (m, n) : trancl pred_nat"
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  le_def: "m \<le> (n::nat) == ~ (n < m)"
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declare One_nat_def [simp]
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text {* Induction *}
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theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n"
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  apply (unfold Zero_nat_def Suc_def)
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  apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}
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  apply (erule Rep_Nat [THEN Nat.induct])
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  apply (iprover elim: Abs_Nat_inverse [THEN subst])
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  done
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text {* Distinctness of constructors *}
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lemma Suc_not_Zero [iff]: "Suc m \<noteq> 0"
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  by (simp add: Zero_nat_def Suc_def Abs_Nat_inject Rep_Nat Suc_RepI Zero_RepI
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                Suc_Rep_not_Zero_Rep) 
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lemma Zero_not_Suc [iff]: "0 \<noteq> Suc m"
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  by (rule not_sym, rule Suc_not_Zero not_sym)
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lemma Suc_neq_Zero: "Suc m = 0 ==> R"
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  by (rule notE, rule Suc_not_Zero)
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lemma Zero_neq_Suc: "0 = Suc m ==> R"
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  by (rule Suc_neq_Zero, erule sym)
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text {* Injectiveness of @{term Suc} *}
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lemma inj_Suc[simp]: "inj_on Suc N"
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  by (simp add: Suc_def inj_on_def Abs_Nat_inject Rep_Nat Suc_RepI 
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                inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject) 
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lemma Suc_inject: "Suc x = Suc y ==> x = y"
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  by (rule inj_Suc [THEN injD])
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lemma Suc_Suc_eq [iff]: "(Suc m = Suc n) = (m = n)"
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  by (rule inj_Suc [THEN inj_eq])
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lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False"
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  by auto
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text {* size of a datatype value; overloaded *}
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consts size :: "'a => nat"
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text {* @{typ nat} is a datatype *}
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rep_datatype nat
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  distinct  Suc_not_Zero Zero_not_Suc
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  inject    Suc_Suc_eq
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  induction nat_induct [case_names 0 Suc]
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text {* fix syntax translation for nat case *}
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setup {*
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let
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  val thy = the_context ();
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  val info = DatatypePackage.the_datatype thy "nat";
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  val constrs = (#3 o snd o hd o #descr) info;
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  val constrs' = ["0", "Suc"];
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  val case_name = Sign.extern_const thy (#case_name info);
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  fun nat_case_tr' context ts =
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    if length ts <> length constrs + 1 then raise Match else
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    let
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      val (fs, x) = split_last ts;
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      fun strip_abs 0 t = ([], t)
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        | strip_abs i (Abs p) =
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          let val (x, u) = Syntax.atomic_abs_tr' p
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          in apfst (cons x) (strip_abs (i-1) u) end
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        | strip_abs i (Const ("split", _) $ t) = (case strip_abs (i+1) t of
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            (v :: v' :: vs, u) => (Syntax.const "Pair" $ v $ v' :: vs, u));
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      fun is_dependent i t =
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        let val k = length (strip_abs_vars t) - i
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        in k < 0 orelse exists (fn j => j >= k)
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          (loose_bnos (strip_abs_body t))
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        end;
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      val cases = map (fn (((cname, dts), cname'), t) =>
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        (cname', strip_abs (length dts) t, is_dependent (length dts) t))
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        (constrs ~~ constrs' ~~ fs);
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      fun count_cases (_, _, true) = I
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        | count_cases (cname, (_, body), false) =
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            AList.map_default (op = : term * term -> bool)
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              (body, []) (cons cname)
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      val cases' = sort (int_ord o swap o pairself (length o snd))
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        (fold_rev count_cases cases []);
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      fun mk_case1 (cname, (vs, body), _) = Syntax.const "_case1" $
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        list_comb (Syntax.const cname, vs) $ body;
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      fun is_undefined (Const ("undefined", _)) = true
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        | is_undefined _ = false;
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    in
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      Syntax.const "_case_syntax" $ x $
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        foldr1 (fn (t, u) => Syntax.const "_case2" $ t $ u) (map mk_case1
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          (case find_first (is_undefined o fst) cases' of
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             SOME (_, cnames) =>
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             if length cnames = length constrs then [hd cases]
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             else filter_out (fn (_, (_, body), _) => is_undefined body) cases
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           | NONE => case cases' of
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             [] => cases
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           | (default, cnames) :: _ =>
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             if length cnames = 1 then cases
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             else if length cnames = length constrs then
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               [hd cases, ("dummy_pattern", ([], default), false)]
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             else
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               filter_out (fn (cname, _, _) => cname mem cnames) cases @
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               [("dummy_pattern", ([], default), false)]))
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    end
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in
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  Theory.add_advanced_trfuns ([], [], [(case_name, nat_case_tr')], [])
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end
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*}
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lemma n_not_Suc_n: "n \<noteq> Suc n"
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  by (induct n) simp_all
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lemma Suc_n_not_n: "Suc t \<noteq> t"
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  by (rule not_sym, rule n_not_Suc_n)
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text {* A special form of induction for reasoning
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  about @{term "m < n"} and @{term "m - n"} *}
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theorem diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
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    (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
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  apply (rule_tac x = m in spec)
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  apply (induct n)
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  prefer 2
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  apply (rule allI)
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  apply (induct_tac x, iprover+)
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  done
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subsection {* Basic properties of "less than" *}
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lemma wf_pred_nat: "wf pred_nat"
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  apply (unfold wf_def pred_nat_def, clarify)
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  apply (induct_tac x, blast+)
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  done
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lemma wf_less: "wf {(x, y::nat). x < y}"
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  apply (unfold less_def)
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  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset], blast)
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  done
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lemma less_eq: "((m, n) : pred_nat^+) = (m < n)"
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  apply (unfold less_def)
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  apply (rule refl)
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  done
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subsubsection {* Introduction properties *}
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lemma less_trans: "i < j ==> j < k ==> i < (k::nat)"
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  apply (unfold less_def)
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  apply (rule trans_trancl [THEN transD], assumption+)
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  done
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lemma lessI [iff]: "n < Suc n"
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  apply (unfold less_def pred_nat_def)
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  apply (simp add: r_into_trancl)
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  done
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lemma less_SucI: "i < j ==> i < Suc j"
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  apply (rule less_trans, assumption)
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  apply (rule lessI)
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  done
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lemma zero_less_Suc [iff]: "0 < Suc n"
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  apply (induct n)
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  apply (rule lessI)
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  apply (erule less_trans)
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  apply (rule lessI)
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  done
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subsubsection {* Elimination properties *}
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lemma less_not_sym: "n < m ==> ~ m < (n::nat)"
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  apply (unfold less_def)
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  apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym])
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  done
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lemma less_asym:
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  assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P
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  apply (rule contrapos_np)
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  apply (rule less_not_sym)
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  apply (rule h1)
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  apply (erule h2)
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  done
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lemma less_not_refl: "~ n < (n::nat)"
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  apply (unfold less_def)
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  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl])
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  done
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lemma less_irrefl [elim!]: "(n::nat) < n ==> R"
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  by (rule notE, rule less_not_refl)
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lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)" by blast
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lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
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  by (rule not_sym, rule less_not_refl2)
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lemma lessE:
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  assumes major: "i < k"
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  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
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  shows P
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  apply (rule major [unfolded less_def pred_nat_def, THEN tranclE], simp_all)
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  apply (erule p1)
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  apply (rule p2)
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  apply (simp add: less_def pred_nat_def, assumption)
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  done
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lemma not_less0 [iff]: "~ n < (0::nat)"
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  by (blast elim: lessE)
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lemma less_zeroE: "(n::nat) < 0 ==> R"
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  by (rule notE, rule not_less0)
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lemma less_SucE: assumes major: "m < Suc n"
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  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
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  apply (rule major [THEN lessE])
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  apply (rule eq, blast)
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  apply (rule less, blast)
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  done
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lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
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  by (blast elim!: less_SucE intro: less_trans)
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lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
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  by (simp add: less_Suc_eq)
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lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
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  by (simp add: less_Suc_eq)
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lemma Suc_mono: "m < n ==> Suc m < Suc n"
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  by (induct n) (fast elim: less_trans lessE)+
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text {* "Less than" is a linear ordering *}
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lemma less_linear: "m < n | m = n | n < (m::nat)"
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  apply (induct m)
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  apply (induct n)
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  apply (rule refl [THEN disjI1, THEN disjI2])
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  apply (rule zero_less_Suc [THEN disjI1])
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  apply (blast intro: Suc_mono less_SucI elim: lessE)
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  done
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text {* "Less than" is antisymmetric, sort of *}
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lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
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apply(simp only:less_Suc_eq)
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apply blast
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done
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lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
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  using less_linear by blast
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lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
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  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
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  shows "P n m"
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  apply (rule less_linear [THEN disjE])
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  apply (erule_tac [2] disjE)
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  apply (erule lessCase)
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  apply (erule sym [THEN eqCase])
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  apply (erule major)
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  done
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subsubsection {* Inductive (?) properties *}
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lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
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  apply (simp add: nat_neq_iff)
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  apply (blast elim!: less_irrefl less_SucE elim: less_asym)
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  done
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   342
lemma Suc_lessD: "Suc m < n ==> m < n"
berghofe@13449
   343
  apply (induct n)
berghofe@13449
   344
  apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+
berghofe@13449
   345
  done
berghofe@13449
   346
berghofe@13449
   347
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   348
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   349
  apply (rule major [THEN lessE])
berghofe@13449
   350
  apply (erule lessI [THEN minor])
paulson@14208
   351
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   352
  done
berghofe@13449
   353
berghofe@13449
   354
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
berghofe@13449
   355
  by (blast elim: lessE dest: Suc_lessD)
wenzelm@4104
   356
berghofe@16635
   357
lemma Suc_less_eq [iff, code]: "(Suc m < Suc n) = (m < n)"
berghofe@13449
   358
  apply (rule iffI)
berghofe@13449
   359
  apply (erule Suc_less_SucD)
berghofe@13449
   360
  apply (erule Suc_mono)
berghofe@13449
   361
  done
berghofe@13449
   362
berghofe@13449
   363
lemma less_trans_Suc:
berghofe@13449
   364
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   365
  apply (induct k, simp_all)
berghofe@13449
   366
  apply (insert le)
berghofe@13449
   367
  apply (simp add: less_Suc_eq)
berghofe@13449
   368
  apply (blast dest: Suc_lessD)
berghofe@13449
   369
  done
berghofe@13449
   370
berghofe@16635
   371
lemma [code]: "((n::nat) < 0) = False" by simp
berghofe@16635
   372
lemma [code]: "(0 < Suc n) = True" by simp
berghofe@16635
   373
berghofe@13449
   374
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
berghofe@13449
   375
lemma not_less_eq: "(~ m < n) = (n < Suc m)"
paulson@14208
   376
by (rule_tac m = m and n = n in diff_induct, simp_all)
berghofe@13449
   377
berghofe@13449
   378
text {* Complete induction, aka course-of-values induction *}
berghofe@13449
   379
lemma nat_less_induct:
paulson@14267
   380
  assumes prem: "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
berghofe@13449
   381
  apply (rule_tac a=n in wf_induct)
berghofe@13449
   382
  apply (rule wf_pred_nat [THEN wf_trancl])
berghofe@13449
   383
  apply (rule prem)
paulson@14208
   384
  apply (unfold less_def, assumption)
berghofe@13449
   385
  done
berghofe@13449
   386
paulson@14131
   387
lemmas less_induct = nat_less_induct [rule_format, case_names less]
paulson@14131
   388
wenzelm@21243
   389
paulson@14131
   390
subsection {* Properties of "less than or equal" *}
berghofe@13449
   391
berghofe@13449
   392
text {* Was @{text le_eq_less_Suc}, but this orientation is more useful *}
paulson@14267
   393
lemma less_Suc_eq_le: "(m < Suc n) = (m \<le> n)"
berghofe@13449
   394
  by (unfold le_def, rule not_less_eq [symmetric])
berghofe@13449
   395
paulson@14267
   396
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
berghofe@13449
   397
  by (rule less_Suc_eq_le [THEN iffD2])
berghofe@13449
   398
paulson@14267
   399
lemma le0 [iff]: "(0::nat) \<le> n"
berghofe@13449
   400
  by (unfold le_def, rule not_less0)
berghofe@13449
   401
paulson@14267
   402
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
berghofe@13449
   403
  by (simp add: le_def)
berghofe@13449
   404
paulson@14267
   405
lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)"
berghofe@13449
   406
  by (induct i) (simp_all add: le_def)
berghofe@13449
   407
paulson@14267
   408
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
berghofe@13449
   409
  by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   410
paulson@14267
   411
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
nipkow@17589
   412
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   413
paulson@14267
   414
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
berghofe@13449
   415
  apply (simp add: le_def less_Suc_eq)
berghofe@13449
   416
  apply (blast elim!: less_irrefl less_asym)
berghofe@13449
   417
  done -- {* formerly called lessD *}
berghofe@13449
   418
paulson@14267
   419
lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n"
berghofe@13449
   420
  by (simp add: le_def less_Suc_eq)
berghofe@13449
   421
berghofe@13449
   422
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   423
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
berghofe@13449
   424
  apply (simp add: le_def less_Suc_eq)
berghofe@13449
   425
  using less_linear
berghofe@13449
   426
  apply blast
berghofe@13449
   427
  done
berghofe@13449
   428
paulson@14267
   429
lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)"
berghofe@13449
   430
  by (blast intro: Suc_leI Suc_le_lessD)
berghofe@13449
   431
paulson@14267
   432
lemma le_SucI: "m \<le> n ==> m \<le> Suc n"
berghofe@13449
   433
  by (unfold le_def) (blast dest: Suc_lessD)
berghofe@13449
   434
paulson@14267
   435
lemma less_imp_le: "m < n ==> m \<le> (n::nat)"
berghofe@13449
   436
  by (unfold le_def) (blast elim: less_asym)
berghofe@13449
   437
paulson@14267
   438
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
berghofe@13449
   439
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq
berghofe@13449
   440
berghofe@13449
   441
paulson@14267
   442
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   443
paulson@14267
   444
lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)"
berghofe@13449
   445
  apply (unfold le_def)
berghofe@13449
   446
  using less_linear
berghofe@13449
   447
  apply (blast elim: less_irrefl less_asym)
berghofe@13449
   448
  done
berghofe@13449
   449
paulson@14267
   450
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
berghofe@13449
   451
  apply (unfold le_def)
berghofe@13449
   452
  using less_linear
berghofe@13449
   453
  apply (blast elim!: less_irrefl elim: less_asym)
berghofe@13449
   454
  done
berghofe@13449
   455
paulson@14267
   456
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
nipkow@17589
   457
  by (iprover intro: less_or_eq_imp_le le_imp_less_or_eq)
berghofe@13449
   458
berghofe@13449
   459
text {* Useful with @{text Blast}. *}
paulson@14267
   460
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
berghofe@13449
   461
  by (rule less_or_eq_imp_le, rule disjI2)
berghofe@13449
   462
paulson@14267
   463
lemma le_refl: "n \<le> (n::nat)"
berghofe@13449
   464
  by (simp add: le_eq_less_or_eq)
berghofe@13449
   465
paulson@14267
   466
lemma le_less_trans: "[| i \<le> j; j < k |] ==> i < (k::nat)"
berghofe@13449
   467
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
berghofe@13449
   468
paulson@14267
   469
lemma less_le_trans: "[| i < j; j \<le> k |] ==> i < (k::nat)"
berghofe@13449
   470
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
berghofe@13449
   471
paulson@14267
   472
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
berghofe@13449
   473
  by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans)
berghofe@13449
   474
paulson@14267
   475
lemma le_anti_sym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
berghofe@13449
   476
  by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym)
berghofe@13449
   477
paulson@14267
   478
lemma Suc_le_mono [iff]: "(Suc n \<le> Suc m) = (n \<le> m)"
berghofe@13449
   479
  by (simp add: le_simps)
berghofe@13449
   480
berghofe@13449
   481
text {* Axiom @{text order_less_le} of class @{text order}: *}
paulson@14267
   482
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
berghofe@13449
   483
  by (simp add: le_def nat_neq_iff) (blast elim!: less_asym)
berghofe@13449
   484
paulson@14267
   485
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
berghofe@13449
   486
  by (rule iffD2, rule nat_less_le, rule conjI)
berghofe@13449
   487
berghofe@13449
   488
text {* Axiom @{text linorder_linear} of class @{text linorder}: *}
paulson@14267
   489
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
berghofe@13449
   490
  apply (simp add: le_eq_less_or_eq)
berghofe@13449
   491
  using less_linear
berghofe@13449
   492
  apply blast
berghofe@13449
   493
  done
berghofe@13449
   494
paulson@14341
   495
text {* Type {@typ nat} is a wellfounded linear order *}
paulson@14341
   496
wenzelm@14691
   497
instance nat :: "{order, linorder, wellorder}"
wenzelm@14691
   498
  by intro_classes
wenzelm@14691
   499
    (assumption |
wenzelm@14691
   500
      rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+
paulson@14341
   501
nipkow@15921
   502
lemmas linorder_neqE_nat = linorder_neqE[where 'a = nat]
nipkow@15921
   503
berghofe@13449
   504
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
berghofe@13449
   505
  by (blast elim!: less_SucE)
berghofe@13449
   506
berghofe@13449
   507
text {*
berghofe@13449
   508
  Rewrite @{term "n < Suc m"} to @{term "n = m"}
paulson@14267
   509
  if @{term "~ n < m"} or @{term "m \<le> n"} hold.
berghofe@13449
   510
  Not suitable as default simprules because they often lead to looping
berghofe@13449
   511
*}
paulson@14267
   512
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
berghofe@13449
   513
  by (rule not_less_less_Suc_eq, rule leD)
berghofe@13449
   514
berghofe@13449
   515
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   516
berghofe@13449
   517
berghofe@13449
   518
text {*
berghofe@13449
   519
  Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}. 
berghofe@13449
   520
  No longer added as simprules (they loop) 
berghofe@13449
   521
  but via @{text reorient_simproc} in Bin
berghofe@13449
   522
*}
berghofe@13449
   523
berghofe@13449
   524
text {* Polymorphic, not just for @{typ nat} *}
berghofe@13449
   525
lemma zero_reorient: "(0 = x) = (x = 0)"
berghofe@13449
   526
  by auto
berghofe@13449
   527
berghofe@13449
   528
lemma one_reorient: "(1 = x) = (x = 1)"
berghofe@13449
   529
  by auto
berghofe@13449
   530
wenzelm@21243
   531
berghofe@13449
   532
subsection {* Arithmetic operators *}
oheimb@1660
   533
wenzelm@12338
   534
axclass power < type
wenzelm@10435
   535
paulson@3370
   536
consts
berghofe@13449
   537
  power :: "('a::power) => nat => 'a"            (infixr "^" 80)
paulson@3370
   538
wenzelm@9436
   539
berghofe@13449
   540
text {* arithmetic operators @{text "+ -"} and @{text "*"} *}
berghofe@13449
   541
wenzelm@14691
   542
instance nat :: "{plus, minus, times, power}" ..
wenzelm@9436
   543
berghofe@13449
   544
primrec
berghofe@13449
   545
  add_0:    "0 + n = n"
berghofe@13449
   546
  add_Suc:  "Suc m + n = Suc (m + n)"
berghofe@13449
   547
berghofe@13449
   548
primrec
berghofe@13449
   549
  diff_0:   "m - 0 = m"
berghofe@13449
   550
  diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
wenzelm@9436
   551
wenzelm@9436
   552
primrec
berghofe@13449
   553
  mult_0:   "0 * n = 0"
berghofe@13449
   554
  mult_Suc: "Suc m * n = n + (m * n)"
berghofe@13449
   555
paulson@14341
   556
text {* These two rules ease the use of primitive recursion. 
paulson@14341
   557
NOTE USE OF @{text "=="} *}
berghofe@13449
   558
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
berghofe@13449
   559
  by simp
berghofe@13449
   560
berghofe@13449
   561
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
berghofe@13449
   562
  by simp
berghofe@13449
   563
paulson@14267
   564
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
berghofe@13449
   565
  by (case_tac n) simp_all
berghofe@13449
   566
paulson@14267
   567
lemma gr_implies_not0: "!!n::nat. m<n ==> n \<noteq> 0"
berghofe@13449
   568
  by (case_tac n) simp_all
berghofe@13449
   569
paulson@14267
   570
lemma neq0_conv [iff]: "!!n::nat. (n \<noteq> 0) = (0 < n)"
berghofe@13449
   571
  by (case_tac n) simp_all
berghofe@13449
   572
berghofe@13449
   573
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   574
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@17589
   575
  by (rule iffD1, rule neq0_conv, iprover)
berghofe@13449
   576
paulson@14267
   577
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
berghofe@13449
   578
  by (fast intro: not0_implies_Suc)
berghofe@13449
   579
berghofe@13449
   580
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
berghofe@13449
   581
  apply (rule iffI)
paulson@14208
   582
  apply (rule ccontr, simp_all)
berghofe@13449
   583
  done
berghofe@13449
   584
paulson@14267
   585
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
berghofe@13449
   586
  by (induct m') simp_all
berghofe@13449
   587
berghofe@13449
   588
text {* Useful in certain inductive arguments *}
paulson@14267
   589
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
berghofe@13449
   590
  by (case_tac m) simp_all
berghofe@13449
   591
paulson@14341
   592
lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"
berghofe@13449
   593
  apply (rule nat_less_induct)
berghofe@13449
   594
  apply (case_tac n)
berghofe@13449
   595
  apply (case_tac [2] nat)
berghofe@13449
   596
  apply (blast intro: less_trans)+
berghofe@13449
   597
  done
berghofe@13449
   598
wenzelm@21243
   599
paulson@15341
   600
subsection {* @{text LEAST} theorems for type @{typ nat}*}
berghofe@13449
   601
paulson@14267
   602
lemma Least_Suc:
paulson@14267
   603
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
paulson@14208
   604
  apply (case_tac "n", auto)
berghofe@13449
   605
  apply (frule LeastI)
berghofe@13449
   606
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
paulson@14267
   607
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
berghofe@13449
   608
  apply (erule_tac [2] Least_le)
paulson@14208
   609
  apply (case_tac "LEAST x. P x", auto)
berghofe@13449
   610
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
berghofe@13449
   611
  apply (blast intro: order_antisym)
berghofe@13449
   612
  done
berghofe@13449
   613
paulson@14267
   614
lemma Least_Suc2:
paulson@14267
   615
     "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
paulson@14267
   616
  by (erule (1) Least_Suc [THEN ssubst], simp)
berghofe@13449
   617
berghofe@13449
   618
berghofe@13449
   619
subsection {* @{term min} and @{term max} *}
berghofe@13449
   620
berghofe@13449
   621
lemma min_0L [simp]: "min 0 n = (0::nat)"
berghofe@13449
   622
  by (rule min_leastL) simp
berghofe@13449
   623
berghofe@13449
   624
lemma min_0R [simp]: "min n 0 = (0::nat)"
berghofe@13449
   625
  by (rule min_leastR) simp
berghofe@13449
   626
berghofe@13449
   627
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
berghofe@13449
   628
  by (simp add: min_of_mono)
berghofe@13449
   629
berghofe@13449
   630
lemma max_0L [simp]: "max 0 n = (n::nat)"
berghofe@13449
   631
  by (rule max_leastL) simp
berghofe@13449
   632
berghofe@13449
   633
lemma max_0R [simp]: "max n 0 = (n::nat)"
berghofe@13449
   634
  by (rule max_leastR) simp
berghofe@13449
   635
berghofe@13449
   636
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
berghofe@13449
   637
  by (simp add: max_of_mono)
berghofe@13449
   638
berghofe@13449
   639
berghofe@13449
   640
subsection {* Basic rewrite rules for the arithmetic operators *}
berghofe@13449
   641
berghofe@13449
   642
text {* Difference *}
berghofe@13449
   643
berghofe@14193
   644
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
paulson@15251
   645
  by (induct n) simp_all
berghofe@13449
   646
berghofe@14193
   647
lemma diff_Suc_Suc [simp, code]: "Suc(m) - Suc(n) = m - n"
paulson@15251
   648
  by (induct n) simp_all
berghofe@13449
   649
berghofe@13449
   650
berghofe@13449
   651
text {*
berghofe@13449
   652
  Could be (and is, below) generalized in various ways
berghofe@13449
   653
  However, none of the generalizations are currently in the simpset,
berghofe@13449
   654
  and I dread to think what happens if I put them in
berghofe@13449
   655
*}
berghofe@13449
   656
lemma Suc_pred [simp]: "0 < n ==> Suc (n - Suc 0) = n"
berghofe@13449
   657
  by (simp split add: nat.split)
berghofe@13449
   658
berghofe@14193
   659
declare diff_Suc [simp del, code del]
berghofe@13449
   660
berghofe@13449
   661
berghofe@13449
   662
subsection {* Addition *}
berghofe@13449
   663
berghofe@13449
   664
lemma add_0_right [simp]: "m + 0 = (m::nat)"
berghofe@13449
   665
  by (induct m) simp_all
berghofe@13449
   666
berghofe@13449
   667
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
berghofe@13449
   668
  by (induct m) simp_all
berghofe@13449
   669
haftmann@19890
   670
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
haftmann@19890
   671
  by simp
berghofe@14193
   672
berghofe@13449
   673
berghofe@13449
   674
text {* Associative law for addition *}
paulson@14267
   675
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
berghofe@13449
   676
  by (induct m) simp_all
berghofe@13449
   677
berghofe@13449
   678
text {* Commutative law for addition *}
paulson@14267
   679
lemma nat_add_commute: "m + n = n + (m::nat)"
berghofe@13449
   680
  by (induct m) simp_all
berghofe@13449
   681
paulson@14267
   682
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
berghofe@13449
   683
  apply (rule mk_left_commute [of "op +"])
paulson@14267
   684
  apply (rule nat_add_assoc)
paulson@14267
   685
  apply (rule nat_add_commute)
berghofe@13449
   686
  done
berghofe@13449
   687
paulson@14331
   688
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
berghofe@13449
   689
  by (induct k) simp_all
berghofe@13449
   690
paulson@14331
   691
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
berghofe@13449
   692
  by (induct k) simp_all
berghofe@13449
   693
paulson@14331
   694
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
berghofe@13449
   695
  by (induct k) simp_all
berghofe@13449
   696
paulson@14331
   697
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
berghofe@13449
   698
  by (induct k) simp_all
berghofe@13449
   699
berghofe@13449
   700
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
berghofe@13449
   701
berghofe@13449
   702
lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)"
berghofe@13449
   703
  by (case_tac m) simp_all
berghofe@13449
   704
berghofe@13449
   705
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
berghofe@13449
   706
  by (case_tac m) simp_all
berghofe@13449
   707
berghofe@13449
   708
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
berghofe@13449
   709
  by (rule trans, rule eq_commute, rule add_is_1)
berghofe@13449
   710
berghofe@13449
   711
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m | 0 < n)"
berghofe@13449
   712
  by (simp del: neq0_conv add: neq0_conv [symmetric])
berghofe@13449
   713
berghofe@13449
   714
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0"
berghofe@13449
   715
  apply (drule add_0_right [THEN ssubst])
paulson@14267
   716
  apply (simp add: nat_add_assoc del: add_0_right)
berghofe@13449
   717
  done
berghofe@13449
   718
paulson@14267
   719
nipkow@16733
   720
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
nipkow@16733
   721
apply(induct k)
nipkow@16733
   722
 apply simp
nipkow@16733
   723
apply(drule comp_inj_on[OF _ inj_Suc])
nipkow@16733
   724
apply (simp add:o_def)
nipkow@16733
   725
done
nipkow@16733
   726
nipkow@16733
   727
paulson@14267
   728
subsection {* Multiplication *}
paulson@14267
   729
paulson@14267
   730
text {* right annihilation in product *}
paulson@14267
   731
lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
paulson@14267
   732
  by (induct m) simp_all
paulson@14267
   733
paulson@14267
   734
text {* right successor law for multiplication *}
paulson@14267
   735
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
paulson@14267
   736
  by (induct m) (simp_all add: nat_add_left_commute)
paulson@14267
   737
paulson@14267
   738
text {* Commutative law for multiplication *}
paulson@14267
   739
lemma nat_mult_commute: "m * n = n * (m::nat)"
paulson@14267
   740
  by (induct m) simp_all
paulson@14267
   741
paulson@14267
   742
text {* addition distributes over multiplication *}
paulson@14267
   743
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
paulson@14267
   744
  by (induct m) (simp_all add: nat_add_assoc nat_add_left_commute)
paulson@14267
   745
paulson@14267
   746
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
paulson@14267
   747
  by (induct m) (simp_all add: nat_add_assoc)
paulson@14267
   748
paulson@14267
   749
text {* Associative law for multiplication *}
paulson@14267
   750
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
paulson@14267
   751
  by (induct m) (simp_all add: add_mult_distrib)
paulson@14267
   752
paulson@14267
   753
nipkow@14740
   754
text{*The naturals form a @{text comm_semiring_1_cancel}*}
obua@14738
   755
instance nat :: comm_semiring_1_cancel
paulson@14267
   756
proof
paulson@14267
   757
  fix i j k :: nat
paulson@14267
   758
  show "(i + j) + k = i + (j + k)" by (rule nat_add_assoc)
paulson@14267
   759
  show "i + j = j + i" by (rule nat_add_commute)
paulson@14267
   760
  show "0 + i = i" by simp
paulson@14267
   761
  show "(i * j) * k = i * (j * k)" by (rule nat_mult_assoc)
paulson@14267
   762
  show "i * j = j * i" by (rule nat_mult_commute)
paulson@14267
   763
  show "1 * i = i" by simp
paulson@14267
   764
  show "(i + j) * k = i * k + j * k" by (simp add: add_mult_distrib)
paulson@14267
   765
  show "0 \<noteq> (1::nat)" by simp
paulson@14341
   766
  assume "k+i = k+j" thus "i=j" by simp
paulson@14341
   767
qed
paulson@14341
   768
paulson@14341
   769
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
paulson@15251
   770
  apply (induct m)
paulson@14341
   771
  apply (induct_tac [2] n, simp_all)
paulson@14341
   772
  done
paulson@14341
   773
wenzelm@21243
   774
paulson@14341
   775
subsection {* Monotonicity of Addition *}
paulson@14341
   776
paulson@14341
   777
text {* strict, in 1st argument *}
paulson@14341
   778
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
paulson@14341
   779
  by (induct k) simp_all
paulson@14341
   780
paulson@14341
   781
text {* strict, in both arguments *}
paulson@14341
   782
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   783
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   784
  apply (induct j, simp_all)
paulson@14341
   785
  done
paulson@14341
   786
paulson@14341
   787
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   788
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   789
  apply (induct n)
paulson@14341
   790
  apply (simp_all add: order_le_less)
paulson@14341
   791
  apply (blast elim!: less_SucE 
paulson@14341
   792
               intro!: add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   793
  done
paulson@14341
   794
paulson@14341
   795
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
paulson@14341
   796
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j"
paulson@14341
   797
  apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp)
paulson@14341
   798
  apply (induct_tac x) 
paulson@14341
   799
  apply (simp_all add: add_less_mono)
paulson@14341
   800
  done
paulson@14341
   801
paulson@14341
   802
nipkow@14740
   803
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
obua@14738
   804
instance nat :: ordered_semidom
paulson@14341
   805
proof
paulson@14341
   806
  fix i j k :: nat
paulson@14348
   807
  show "0 < (1::nat)" by simp
paulson@14267
   808
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   809
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   810
qed
paulson@14267
   811
paulson@14267
   812
lemma nat_mult_1: "(1::nat) * n = n"
paulson@14267
   813
  by simp
paulson@14267
   814
paulson@14267
   815
lemma nat_mult_1_right: "n * (1::nat) = n"
paulson@14267
   816
  by simp
paulson@14267
   817
paulson@14267
   818
paulson@14267
   819
subsection {* Additional theorems about "less than" *}
paulson@14267
   820
paulson@19870
   821
text{*An induction rule for estabilishing binary relations*}
paulson@19870
   822
lemma less_Suc_induct: 
paulson@19870
   823
  assumes less:  "i < j"
paulson@19870
   824
     and  step:  "!!i. P i (Suc i)"
paulson@19870
   825
     and  trans: "!!i j k. P i j ==> P j k ==> P i k"
paulson@19870
   826
  shows "P i j"
paulson@19870
   827
proof -
paulson@19870
   828
  from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add) 
paulson@19870
   829
  have "P i (Suc(i+k))"
paulson@19870
   830
  proof (induct k)
paulson@19870
   831
    case 0 
paulson@19870
   832
    show ?case by (simp add: step) 
paulson@19870
   833
  next
paulson@19870
   834
    case (Suc k)
paulson@19870
   835
    thus ?case by (auto intro: prems)
paulson@19870
   836
  qed
paulson@19870
   837
  thus "P i j" by (simp add: j) 
paulson@19870
   838
qed
paulson@19870
   839
paulson@19870
   840
paulson@14267
   841
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   842
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   843
lemma less_mono_imp_le_mono:
paulson@14267
   844
  assumes lt_mono: "!!i j::nat. i < j ==> f i < f j"
paulson@14267
   845
  and le: "i \<le> j" shows "f i \<le> ((f j)::nat)" using le
paulson@14267
   846
  apply (simp add: order_le_less)
paulson@14267
   847
  apply (blast intro!: lt_mono)
paulson@14267
   848
  done
paulson@14267
   849
paulson@14267
   850
text {* non-strict, in 1st argument *}
paulson@14267
   851
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
paulson@14267
   852
  by (rule add_right_mono)
paulson@14267
   853
paulson@14267
   854
text {* non-strict, in both arguments *}
paulson@14267
   855
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
paulson@14267
   856
  by (rule add_mono)
paulson@14267
   857
paulson@14267
   858
lemma le_add2: "n \<le> ((m + n)::nat)"
paulson@14341
   859
  by (insert add_right_mono [of 0 m n], simp) 
berghofe@13449
   860
paulson@14267
   861
lemma le_add1: "n \<le> ((n + m)::nat)"
paulson@14341
   862
  by (simp add: add_commute, rule le_add2)
berghofe@13449
   863
berghofe@13449
   864
lemma less_add_Suc1: "i < Suc (i + m)"
berghofe@13449
   865
  by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   866
berghofe@13449
   867
lemma less_add_Suc2: "i < Suc (m + i)"
berghofe@13449
   868
  by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   869
paulson@14267
   870
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@17589
   871
  by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   872
paulson@14267
   873
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
berghofe@13449
   874
  by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   875
paulson@14267
   876
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
berghofe@13449
   877
  by (rule le_trans, assumption, rule le_add2)
berghofe@13449
   878
berghofe@13449
   879
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
berghofe@13449
   880
  by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
   881
berghofe@13449
   882
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
berghofe@13449
   883
  by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
   884
berghofe@13449
   885
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
paulson@14341
   886
  apply (rule le_less_trans [of _ "i+j"]) 
paulson@14341
   887
  apply (simp_all add: le_add1)
berghofe@13449
   888
  done
berghofe@13449
   889
berghofe@13449
   890
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
berghofe@13449
   891
  apply (rule notI)
berghofe@13449
   892
  apply (erule add_lessD1 [THEN less_irrefl])
berghofe@13449
   893
  done
berghofe@13449
   894
berghofe@13449
   895
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
berghofe@13449
   896
  by (simp add: add_commute not_add_less1)
berghofe@13449
   897
paulson@14267
   898
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
paulson@14341
   899
  apply (rule order_trans [of _ "m+k"]) 
paulson@14341
   900
  apply (simp_all add: le_add1)
paulson@14341
   901
  done
berghofe@13449
   902
paulson@14267
   903
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
berghofe@13449
   904
  apply (simp add: add_commute)
berghofe@13449
   905
  apply (erule add_leD1)
berghofe@13449
   906
  done
berghofe@13449
   907
paulson@14267
   908
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
berghofe@13449
   909
  by (blast dest: add_leD1 add_leD2)
berghofe@13449
   910
berghofe@13449
   911
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
   912
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
berghofe@13449
   913
  by (force simp del: add_Suc_right
berghofe@13449
   914
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
   915
berghofe@13449
   916
berghofe@13449
   917
subsection {* Difference *}
berghofe@13449
   918
berghofe@13449
   919
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
berghofe@13449
   920
  by (induct m) simp_all
berghofe@13449
   921
berghofe@13449
   922
text {* Addition is the inverse of subtraction:
paulson@14267
   923
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
   924
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
berghofe@13449
   925
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   926
paulson@14267
   927
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
paulson@16796
   928
  by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
   929
paulson@14267
   930
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
berghofe@13449
   931
  by (simp add: le_add_diff_inverse add_commute)
berghofe@13449
   932
berghofe@13449
   933
berghofe@13449
   934
subsection {* More results about difference *}
berghofe@13449
   935
paulson@14267
   936
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
berghofe@13449
   937
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   938
berghofe@13449
   939
lemma diff_less_Suc: "m - n < Suc m"
berghofe@13449
   940
  apply (induct m n rule: diff_induct)
berghofe@13449
   941
  apply (erule_tac [3] less_SucE)
berghofe@13449
   942
  apply (simp_all add: less_Suc_eq)
berghofe@13449
   943
  done
berghofe@13449
   944
paulson@14267
   945
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
berghofe@13449
   946
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
   947
berghofe@13449
   948
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
berghofe@13449
   949
  by (rule le_less_trans, rule diff_le_self)
berghofe@13449
   950
berghofe@13449
   951
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
berghofe@13449
   952
  by (induct i j rule: diff_induct) simp_all
berghofe@13449
   953
berghofe@13449
   954
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
berghofe@13449
   955
  by (simp add: diff_diff_left)
berghofe@13449
   956
berghofe@13449
   957
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
paulson@14208
   958
  apply (case_tac "n", safe)
berghofe@13449
   959
  apply (simp add: le_simps)
berghofe@13449
   960
  done
berghofe@13449
   961
berghofe@13449
   962
text {* This and the next few suggested by Florian Kammueller *}
berghofe@13449
   963
lemma diff_commute: "(i::nat) - j - k = i - k - j"
berghofe@13449
   964
  by (simp add: diff_diff_left add_commute)
berghofe@13449
   965
paulson@14267
   966
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
berghofe@13449
   967
  by (induct j k rule: diff_induct) simp_all
berghofe@13449
   968
paulson@14267
   969
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
berghofe@13449
   970
  by (simp add: add_commute diff_add_assoc)
berghofe@13449
   971
berghofe@13449
   972
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
berghofe@13449
   973
  by (induct n) simp_all
berghofe@13449
   974
berghofe@13449
   975
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
berghofe@13449
   976
  by (simp add: diff_add_assoc)
berghofe@13449
   977
paulson@14267
   978
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
berghofe@13449
   979
  apply safe
berghofe@13449
   980
  apply (simp_all add: diff_add_inverse2)
berghofe@13449
   981
  done
berghofe@13449
   982
paulson@14267
   983
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
berghofe@13449
   984
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   985
paulson@14267
   986
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
berghofe@13449
   987
  by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
   988
berghofe@13449
   989
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
berghofe@13449
   990
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   991
paulson@14267
   992
lemma less_imp_add_positive: "i < j  ==> \<exists>k::nat. 0 < k & i + k = j"
berghofe@13449
   993
  apply (rule_tac x = "j - i" in exI)
berghofe@13449
   994
  apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym)
berghofe@13449
   995
  done
wenzelm@9436
   996
berghofe@13449
   997
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
berghofe@13449
   998
  apply (induct k i rule: diff_induct)
berghofe@13449
   999
  apply (simp_all (no_asm))
nipkow@17589
  1000
  apply iprover
berghofe@13449
  1001
  done
berghofe@13449
  1002
berghofe@13449
  1003
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
berghofe@13449
  1004
  apply (rule diff_self_eq_0 [THEN subst])
nipkow@17589
  1005
  apply (rule zero_induct_lemma, iprover+)
berghofe@13449
  1006
  done
berghofe@13449
  1007
berghofe@13449
  1008
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
berghofe@13449
  1009
  by (induct k) simp_all
berghofe@13449
  1010
berghofe@13449
  1011
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
berghofe@13449
  1012
  by (simp add: diff_cancel add_commute)
berghofe@13449
  1013
berghofe@13449
  1014
lemma diff_add_0: "n - (n + m) = (0::nat)"
berghofe@13449
  1015
  by (induct n) simp_all
berghofe@13449
  1016
berghofe@13449
  1017
berghofe@13449
  1018
text {* Difference distributes over multiplication *}
berghofe@13449
  1019
berghofe@13449
  1020
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
berghofe@13449
  1021
  by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
berghofe@13449
  1022
berghofe@13449
  1023
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
berghofe@13449
  1024
  by (simp add: diff_mult_distrib mult_commute [of k])
berghofe@13449
  1025
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
berghofe@13449
  1026
berghofe@13449
  1027
lemmas nat_distrib =
berghofe@13449
  1028
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
berghofe@13449
  1029
berghofe@13449
  1030
berghofe@13449
  1031
subsection {* Monotonicity of Multiplication *}
berghofe@13449
  1032
paulson@14267
  1033
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
paulson@14341
  1034
  by (simp add: mult_right_mono) 
berghofe@13449
  1035
paulson@14267
  1036
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
paulson@14341
  1037
  by (simp add: mult_left_mono) 
berghofe@13449
  1038
paulson@14267
  1039
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
  1040
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
paulson@14341
  1041
  by (simp add: mult_mono) 
berghofe@13449
  1042
berghofe@13449
  1043
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
paulson@14341
  1044
  by (simp add: mult_strict_right_mono) 
berghofe@13449
  1045
paulson@14266
  1046
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
  1047
      there are no negative numbers.*}
paulson@14266
  1048
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
  1049
  apply (induct m)
paulson@14208
  1050
  apply (case_tac [2] n, simp_all)
berghofe@13449
  1051
  done
berghofe@13449
  1052
paulson@14267
  1053
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"
berghofe@13449
  1054
  apply (induct m)
paulson@14208
  1055
  apply (case_tac [2] n, simp_all)
berghofe@13449
  1056
  done
berghofe@13449
  1057
berghofe@13449
  1058
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
paulson@15251
  1059
  apply (induct m, simp)
paulson@15251
  1060
  apply (induct n, simp, fastsimp)
berghofe@13449
  1061
  done
berghofe@13449
  1062
berghofe@13449
  1063
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
berghofe@13449
  1064
  apply (rule trans)
paulson@14208
  1065
  apply (rule_tac [2] mult_eq_1_iff, fastsimp)
berghofe@13449
  1066
  done
berghofe@13449
  1067
paulson@14341
  1068
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1069
  apply (safe intro!: mult_less_mono1)
paulson@14208
  1070
  apply (case_tac k, auto)
berghofe@13449
  1071
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1072
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1073
  done
berghofe@13449
  1074
berghofe@13449
  1075
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
paulson@14341
  1076
  by (simp add: mult_commute [of k])
berghofe@13449
  1077
paulson@14267
  1078
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
paulson@14208
  1079
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1080
paulson@14267
  1081
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
paulson@14208
  1082
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1083
paulson@14341
  1084
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
paulson@14208
  1085
  apply (cut_tac less_linear, safe, auto)
berghofe@13449
  1086
  apply (drule mult_less_mono1, assumption, simp)+
berghofe@13449
  1087
  done
berghofe@13449
  1088
berghofe@13449
  1089
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
paulson@14341
  1090
  by (simp add: mult_commute [of k])
berghofe@13449
  1091
berghofe@13449
  1092
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
berghofe@13449
  1093
  by (subst mult_less_cancel1) simp
berghofe@13449
  1094
paulson@14267
  1095
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
berghofe@13449
  1096
  by (subst mult_le_cancel1) simp
berghofe@13449
  1097
berghofe@13449
  1098
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
berghofe@13449
  1099
  by (subst mult_cancel1) simp
berghofe@13449
  1100
berghofe@13449
  1101
text {* Lemma for @{text gcd} *}
berghofe@13449
  1102
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1103
  apply (drule sym)
berghofe@13449
  1104
  apply (rule disjCI)
berghofe@13449
  1105
  apply (rule nat_less_cases, erule_tac [2] _)
berghofe@13449
  1106
  apply (fastsimp elim!: less_SucE)
berghofe@13449
  1107
  apply (fastsimp dest: mult_less_mono2)
berghofe@13449
  1108
  done
wenzelm@9436
  1109
haftmann@20588
  1110
haftmann@18702
  1111
subsection {* Code generator setup *}
haftmann@18702
  1112
haftmann@20355
  1113
lemma one_is_suc_zero [code inline]:
haftmann@20355
  1114
  "1 = Suc 0"
haftmann@20355
  1115
  by simp
haftmann@20355
  1116
haftmann@20588
  1117
instance nat :: eq ..
haftmann@20588
  1118
haftmann@20588
  1119
lemma [code func]:
haftmann@21043
  1120
  "Code_Generator.eq (0\<Colon>nat) 0 = True" unfolding eq_def by auto
haftmann@20588
  1121
haftmann@20588
  1122
lemma [code func]:
haftmann@21043
  1123
  "Code_Generator.eq (Suc n) (Suc m) = Code_Generator.eq n m" unfolding eq_def by auto
haftmann@20588
  1124
haftmann@20588
  1125
lemma [code func]:
haftmann@21043
  1126
  "Code_Generator.eq (Suc n) 0 = False" unfolding eq_def by auto
haftmann@20588
  1127
haftmann@20588
  1128
lemma [code func]:
haftmann@21043
  1129
  "Code_Generator.eq 0 (Suc m) = False" unfolding eq_def by auto
haftmann@20588
  1130
wenzelm@21243
  1131
wenzelm@21243
  1132
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
wenzelm@21243
  1133
wenzelm@21243
  1134
use "arith_data.ML"
wenzelm@21243
  1135
setup arith_setup
wenzelm@21243
  1136
wenzelm@21243
  1137
text{*The following proofs may rely on the arithmetic proof procedures.*}
wenzelm@21243
  1138
wenzelm@21243
  1139
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
wenzelm@21243
  1140
  by (auto simp: le_eq_less_or_eq dest: less_imp_Suc_add)
wenzelm@21243
  1141
wenzelm@21243
  1142
lemma pred_nat_trancl_eq_le: "((m, n) : pred_nat^*) = (m \<le> n)"
wenzelm@21243
  1143
by (simp add: less_eq reflcl_trancl [symmetric]
wenzelm@21243
  1144
            del: reflcl_trancl, arith)
wenzelm@21243
  1145
wenzelm@21243
  1146
lemma nat_diff_split:
wenzelm@21243
  1147
    "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
wenzelm@21243
  1148
    -- {* elimination of @{text -} on @{text nat} *}
wenzelm@21243
  1149
  by (cases "a<b" rule: case_split)
wenzelm@21243
  1150
     (auto simp add: diff_is_0_eq [THEN iffD2])
wenzelm@21243
  1151
wenzelm@21243
  1152
lemma nat_diff_split_asm:
wenzelm@21243
  1153
    "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
wenzelm@21243
  1154
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
wenzelm@21243
  1155
  by (simp split: nat_diff_split)
wenzelm@21243
  1156
wenzelm@21243
  1157
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1158
wenzelm@21243
  1159
wenzelm@21243
  1160
wenzelm@21243
  1161
lemma le_square: "m \<le> m * (m::nat)"
wenzelm@21243
  1162
  by (induct m) auto
wenzelm@21243
  1163
wenzelm@21243
  1164
lemma le_cube: "(m::nat) \<le> m * (m * m)"
wenzelm@21243
  1165
  by (induct m) auto
wenzelm@21243
  1166
wenzelm@21243
  1167
wenzelm@21243
  1168
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1169
wenzelm@21243
  1170
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
wenzelm@21243
  1171
by arith
wenzelm@21243
  1172
wenzelm@21243
  1173
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
wenzelm@21243
  1174
by arith
wenzelm@21243
  1175
wenzelm@21243
  1176
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
wenzelm@21243
  1177
by arith
wenzelm@21243
  1178
wenzelm@21243
  1179
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
wenzelm@21243
  1180
by arith
wenzelm@21243
  1181
wenzelm@21243
  1182
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
wenzelm@21243
  1183
by arith
wenzelm@21243
  1184
wenzelm@21243
  1185
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
wenzelm@21243
  1186
by arith
wenzelm@21243
  1187
wenzelm@21243
  1188
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1189
  second premise n\<le>m*)
wenzelm@21243
  1190
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
wenzelm@21243
  1191
by arith
wenzelm@21243
  1192
wenzelm@21243
  1193
wenzelm@21243
  1194
(** Simplification of relational expressions involving subtraction **)
wenzelm@21243
  1195
wenzelm@21243
  1196
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
wenzelm@21243
  1197
by (simp split add: nat_diff_split)
wenzelm@21243
  1198
wenzelm@21243
  1199
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
wenzelm@21243
  1200
by (auto split add: nat_diff_split)
wenzelm@21243
  1201
wenzelm@21243
  1202
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
wenzelm@21243
  1203
by (auto split add: nat_diff_split)
wenzelm@21243
  1204
wenzelm@21243
  1205
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
wenzelm@21243
  1206
by (auto split add: nat_diff_split)
wenzelm@21243
  1207
wenzelm@21243
  1208
wenzelm@21243
  1209
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1210
wenzelm@21243
  1211
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1212
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
wenzelm@21243
  1213
by (simp split add: nat_diff_split)
wenzelm@21243
  1214
wenzelm@21243
  1215
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
wenzelm@21243
  1216
by (simp split add: nat_diff_split)
wenzelm@21243
  1217
wenzelm@21243
  1218
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
wenzelm@21243
  1219
by (simp split add: nat_diff_split)
wenzelm@21243
  1220
wenzelm@21243
  1221
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
wenzelm@21243
  1222
by (simp split add: nat_diff_split)
wenzelm@21243
  1223
wenzelm@21243
  1224
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1225
wenzelm@21243
  1226
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
wenzelm@21243
  1227
by (case_tac "m", auto)
wenzelm@21243
  1228
wenzelm@21243
  1229
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
wenzelm@21243
  1230
by (case_tac "m", auto)
wenzelm@21243
  1231
wenzelm@21243
  1232
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
wenzelm@21243
  1233
by (case_tac "m", auto)
wenzelm@21243
  1234
wenzelm@21243
  1235
wenzelm@21243
  1236
text{*Rewriting to pull differences out*}
wenzelm@21243
  1237
wenzelm@21243
  1238
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
wenzelm@21243
  1239
by arith
wenzelm@21243
  1240
wenzelm@21243
  1241
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
wenzelm@21243
  1242
by arith
wenzelm@21243
  1243
wenzelm@21243
  1244
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
wenzelm@21243
  1245
by arith
wenzelm@21243
  1246
wenzelm@21243
  1247
(*The others are
wenzelm@21243
  1248
      i - j - k = i - (j + k),
wenzelm@21243
  1249
      k \<le> j ==> j - k + i = j + i - k,
wenzelm@21243
  1250
      k \<le> j ==> i + (j - k) = i + j - k *)
wenzelm@21243
  1251
lemmas add_diff_assoc = diff_add_assoc [symmetric]
wenzelm@21243
  1252
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
wenzelm@21243
  1253
declare diff_diff_left [simp]  add_diff_assoc [simp]  add_diff_assoc2[simp]
wenzelm@21243
  1254
wenzelm@21243
  1255
text{*At present we prove no analogue of @{text not_less_Least} or @{text
wenzelm@21243
  1256
Least_Suc}, since there appears to be no need.*}
wenzelm@21243
  1257
wenzelm@21243
  1258
ML
wenzelm@21243
  1259
{*
wenzelm@21243
  1260
val pred_nat_trancl_eq_le = thm "pred_nat_trancl_eq_le";
wenzelm@21243
  1261
val nat_diff_split = thm "nat_diff_split";
wenzelm@21243
  1262
val nat_diff_split_asm = thm "nat_diff_split_asm";
wenzelm@21243
  1263
val le_square = thm "le_square";
wenzelm@21243
  1264
val le_cube = thm "le_cube";
wenzelm@21243
  1265
val diff_less_mono = thm "diff_less_mono";
wenzelm@21243
  1266
val less_diff_conv = thm "less_diff_conv";
wenzelm@21243
  1267
val le_diff_conv = thm "le_diff_conv";
wenzelm@21243
  1268
val le_diff_conv2 = thm "le_diff_conv2";
wenzelm@21243
  1269
val diff_diff_cancel = thm "diff_diff_cancel";
wenzelm@21243
  1270
val le_add_diff = thm "le_add_diff";
wenzelm@21243
  1271
val diff_less = thm "diff_less";
wenzelm@21243
  1272
val diff_diff_eq = thm "diff_diff_eq";
wenzelm@21243
  1273
val eq_diff_iff = thm "eq_diff_iff";
wenzelm@21243
  1274
val less_diff_iff = thm "less_diff_iff";
wenzelm@21243
  1275
val le_diff_iff = thm "le_diff_iff";
wenzelm@21243
  1276
val diff_le_mono = thm "diff_le_mono";
wenzelm@21243
  1277
val diff_le_mono2 = thm "diff_le_mono2";
wenzelm@21243
  1278
val diff_less_mono2 = thm "diff_less_mono2";
wenzelm@21243
  1279
val diffs0_imp_equal = thm "diffs0_imp_equal";
wenzelm@21243
  1280
val one_less_mult = thm "one_less_mult";
wenzelm@21243
  1281
val n_less_m_mult_n = thm "n_less_m_mult_n";
wenzelm@21243
  1282
val n_less_n_mult_m = thm "n_less_n_mult_m";
wenzelm@21243
  1283
val diff_diff_right = thm "diff_diff_right";
wenzelm@21243
  1284
val diff_Suc_diff_eq1 = thm "diff_Suc_diff_eq1";
wenzelm@21243
  1285
val diff_Suc_diff_eq2 = thm "diff_Suc_diff_eq2";
wenzelm@21243
  1286
*}
wenzelm@21243
  1287
wenzelm@21243
  1288
subsection{*Embedding of the Naturals into any @{text
wenzelm@21243
  1289
semiring_1_cancel}: @{term of_nat}*}
wenzelm@21243
  1290
wenzelm@21243
  1291
consts of_nat :: "nat => 'a::semiring_1_cancel"
wenzelm@21243
  1292
wenzelm@21243
  1293
primrec
wenzelm@21243
  1294
  of_nat_0:   "of_nat 0 = 0"
wenzelm@21243
  1295
  of_nat_Suc: "of_nat (Suc m) = of_nat m + 1"
wenzelm@21243
  1296
wenzelm@21243
  1297
lemma of_nat_1 [simp]: "of_nat 1 = 1"
wenzelm@21243
  1298
by simp
wenzelm@21243
  1299
wenzelm@21243
  1300
lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"
wenzelm@21243
  1301
apply (induct m)
wenzelm@21243
  1302
apply (simp_all add: add_ac)
wenzelm@21243
  1303
done
wenzelm@21243
  1304
wenzelm@21243
  1305
lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n"
wenzelm@21243
  1306
apply (induct m)
wenzelm@21243
  1307
apply (simp_all add: add_ac left_distrib)
wenzelm@21243
  1308
done
wenzelm@21243
  1309
wenzelm@21243
  1310
lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)"
wenzelm@21243
  1311
apply (induct m, simp_all)
wenzelm@21243
  1312
apply (erule order_trans)
wenzelm@21243
  1313
apply (rule less_add_one [THEN order_less_imp_le])
wenzelm@21243
  1314
done
wenzelm@21243
  1315
wenzelm@21243
  1316
lemma less_imp_of_nat_less:
wenzelm@21243
  1317
     "m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)"
wenzelm@21243
  1318
apply (induct m n rule: diff_induct, simp_all)
wenzelm@21243
  1319
apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)
wenzelm@21243
  1320
done
wenzelm@21243
  1321
wenzelm@21243
  1322
lemma of_nat_less_imp_less:
wenzelm@21243
  1323
     "of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n"
wenzelm@21243
  1324
apply (induct m n rule: diff_induct, simp_all)
wenzelm@21243
  1325
apply (insert zero_le_imp_of_nat)
wenzelm@21243
  1326
apply (force simp add: linorder_not_less [symmetric])
wenzelm@21243
  1327
done
wenzelm@21243
  1328
wenzelm@21243
  1329
lemma of_nat_less_iff [simp]:
wenzelm@21243
  1330
     "(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)"
wenzelm@21243
  1331
by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
wenzelm@21243
  1332
wenzelm@21243
  1333
text{*Special cases where either operand is zero*}
wenzelm@21243
  1334
lemmas of_nat_0_less_iff = of_nat_less_iff [of 0, simplified]
wenzelm@21243
  1335
lemmas of_nat_less_0_iff = of_nat_less_iff [of _ 0, simplified]
wenzelm@21243
  1336
declare of_nat_0_less_iff [simp]
wenzelm@21243
  1337
declare of_nat_less_0_iff [simp]
wenzelm@21243
  1338
wenzelm@21243
  1339
lemma of_nat_le_iff [simp]:
wenzelm@21243
  1340
     "(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)"
wenzelm@21243
  1341
by (simp add: linorder_not_less [symmetric])
wenzelm@21243
  1342
wenzelm@21243
  1343
text{*Special cases where either operand is zero*}
wenzelm@21243
  1344
lemmas of_nat_0_le_iff = of_nat_le_iff [of 0, simplified]
wenzelm@21243
  1345
lemmas of_nat_le_0_iff = of_nat_le_iff [of _ 0, simplified]
wenzelm@21243
  1346
declare of_nat_0_le_iff [simp]
wenzelm@21243
  1347
declare of_nat_le_0_iff [simp]
wenzelm@21243
  1348
wenzelm@21243
  1349
text{*The ordering on the @{text semiring_1_cancel} is necessary
wenzelm@21243
  1350
to exclude the possibility of a finite field, which indeed wraps back to
wenzelm@21243
  1351
zero.*}
wenzelm@21243
  1352
lemma of_nat_eq_iff [simp]:
wenzelm@21243
  1353
     "(of_nat m = (of_nat n::'a::ordered_semidom)) = (m = n)"
wenzelm@21243
  1354
by (simp add: order_eq_iff)
wenzelm@21243
  1355
wenzelm@21243
  1356
text{*Special cases where either operand is zero*}
wenzelm@21243
  1357
lemmas of_nat_0_eq_iff = of_nat_eq_iff [of 0, simplified]
wenzelm@21243
  1358
lemmas of_nat_eq_0_iff = of_nat_eq_iff [of _ 0, simplified]
wenzelm@21243
  1359
declare of_nat_0_eq_iff [simp]
wenzelm@21243
  1360
declare of_nat_eq_0_iff [simp]
wenzelm@21243
  1361
wenzelm@21243
  1362
lemma of_nat_diff [simp]:
wenzelm@21243
  1363
     "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::ring_1)"
wenzelm@21243
  1364
by (simp del: of_nat_add
wenzelm@21243
  1365
	 add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
wenzelm@21243
  1366
clasohm@923
  1367
end