src/HOL/Nat.thy
author haftmann
Thu Jul 19 21:47:37 2007 +0200 (2007-07-19)
changeset 23852 3736cdf9398b
parent 23740 d7f18c837ce7
child 24075 366d4d234814
permissions -rw-r--r--
moved set Nats to Nat.thy
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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
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  "~~/src/Tools/rat.ML"
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  "~~/src/Provers/Arith/fast_lin_arith.ML"
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  "~~/src/Provers/Arith/cancel_sums.ML"
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  ("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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inductive_set Nat :: "ind set"
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where
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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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text {* Abstract constants and syntax *}
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consts
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  Suc :: "nat => nat"
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local
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defs
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  Suc_def:      "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"
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definition
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  pred_nat :: "(nat * nat) set" where
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  "pred_nat = {(m, n). n = Suc m}"
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instance nat :: "{ord, zero, one}"
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  Zero_nat_def: "0 == Abs_Nat Zero_Rep"
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  One_nat_def [simp]: "1 == Suc 0"
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  less_def: "m < n == (m, n) : pred_nat^+"
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  le_def:   "m \<le> (n::nat) == ~ (n < m)" ..
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lemmas [code func del] = less_def le_def
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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 *}
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class size = type +
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  fixes size :: "'a \<Rightarrow> 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
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declare nat.induct [case_names 0 Suc, induct type: nat]
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declare nat.exhaust [case_names 0 Suc, cases type: nat]
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lemmas nat_rec_0 = nat.recs(1)
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  and nat_rec_Suc = nat.recs(2)
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lemmas nat_case_0 = nat.cases(1)
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  and nat_case_Suc = nat.cases(2)
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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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lemma Suc_lessD: "Suc m < n ==> m < n"
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  apply (induct n)
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  apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+
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  done
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lemma Suc_lessE: assumes major: "Suc i < k"
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  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
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  apply (rule major [THEN lessE])
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  apply (erule lessI [THEN minor])
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  apply (erule Suc_lessD [THEN minor], assumption)
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  done
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lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
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  by (blast elim: lessE dest: Suc_lessD)
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lemma Suc_less_eq [iff, code]: "(Suc m < Suc n) = (m < n)"
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  apply (rule iffI)
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  apply (erule Suc_less_SucD)
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  apply (erule Suc_mono)
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  done
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lemma less_trans_Suc:
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  assumes le: "i < j" shows "j < k ==> Suc i < k"
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  apply (induct k, simp_all)
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  apply (insert le)
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  apply (simp add: less_Suc_eq)
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  apply (blast dest: Suc_lessD)
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  done
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lemma [code]: "((n::nat) < 0) = False" by simp
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lemma [code]: "(0 < Suc n) = True" by simp
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text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
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lemma not_less_eq: "(~ m < n) = (n < Suc m)"
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  by (induct m n rule: diff_induct) simp_all
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text {* Complete induction, aka course-of-values induction *}
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lemma nat_less_induct:
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  assumes prem: "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
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  apply (induct n rule: wf_induct [OF wf_pred_nat [THEN wf_trancl]])
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  apply (rule prem)
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  apply (unfold less_def, assumption)
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  done
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lemmas less_induct = nat_less_induct [rule_format, case_names less]
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subsection {* Properties of "less than or equal" *}
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text {* Was @{text le_eq_less_Suc}, but this orientation is more useful *}
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lemma less_Suc_eq_le: "(m < Suc n) = (m \<le> n)"
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  unfolding le_def by (rule not_less_eq [symmetric])
berghofe@13449
   343
paulson@14267
   344
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
berghofe@13449
   345
  by (rule less_Suc_eq_le [THEN iffD2])
berghofe@13449
   346
paulson@14267
   347
lemma le0 [iff]: "(0::nat) \<le> n"
wenzelm@22718
   348
  unfolding le_def by (rule not_less0)
berghofe@13449
   349
paulson@14267
   350
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
berghofe@13449
   351
  by (simp add: le_def)
berghofe@13449
   352
paulson@14267
   353
lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)"
berghofe@13449
   354
  by (induct i) (simp_all add: le_def)
berghofe@13449
   355
paulson@14267
   356
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
berghofe@13449
   357
  by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   358
paulson@14267
   359
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
nipkow@17589
   360
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   361
paulson@14267
   362
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
berghofe@13449
   363
  apply (simp add: le_def less_Suc_eq)
berghofe@13449
   364
  apply (blast elim!: less_irrefl less_asym)
berghofe@13449
   365
  done -- {* formerly called lessD *}
berghofe@13449
   366
paulson@14267
   367
lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n"
berghofe@13449
   368
  by (simp add: le_def less_Suc_eq)
berghofe@13449
   369
berghofe@13449
   370
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   371
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
berghofe@13449
   372
  apply (simp add: le_def less_Suc_eq)
berghofe@13449
   373
  using less_linear
berghofe@13449
   374
  apply blast
berghofe@13449
   375
  done
berghofe@13449
   376
paulson@14267
   377
lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)"
berghofe@13449
   378
  by (blast intro: Suc_leI Suc_le_lessD)
berghofe@13449
   379
paulson@14267
   380
lemma le_SucI: "m \<le> n ==> m \<le> Suc n"
berghofe@13449
   381
  by (unfold le_def) (blast dest: Suc_lessD)
berghofe@13449
   382
paulson@14267
   383
lemma less_imp_le: "m < n ==> m \<le> (n::nat)"
berghofe@13449
   384
  by (unfold le_def) (blast elim: less_asym)
berghofe@13449
   385
paulson@14267
   386
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
berghofe@13449
   387
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq
berghofe@13449
   388
berghofe@13449
   389
paulson@14267
   390
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   391
paulson@14267
   392
lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)"
wenzelm@22718
   393
  unfolding le_def
berghofe@13449
   394
  using less_linear
wenzelm@22718
   395
  by (blast elim: less_irrefl less_asym)
berghofe@13449
   396
paulson@14267
   397
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
wenzelm@22718
   398
  unfolding le_def
berghofe@13449
   399
  using less_linear
wenzelm@22718
   400
  by (blast elim!: less_irrefl elim: less_asym)
berghofe@13449
   401
paulson@14267
   402
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
nipkow@17589
   403
  by (iprover intro: less_or_eq_imp_le le_imp_less_or_eq)
berghofe@13449
   404
wenzelm@22718
   405
text {* Useful with @{text blast}. *}
paulson@14267
   406
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
wenzelm@22718
   407
  by (rule less_or_eq_imp_le) (rule disjI2)
berghofe@13449
   408
paulson@14267
   409
lemma le_refl: "n \<le> (n::nat)"
berghofe@13449
   410
  by (simp add: le_eq_less_or_eq)
berghofe@13449
   411
paulson@14267
   412
lemma le_less_trans: "[| i \<le> j; j < k |] ==> i < (k::nat)"
berghofe@13449
   413
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
berghofe@13449
   414
paulson@14267
   415
lemma less_le_trans: "[| i < j; j \<le> k |] ==> i < (k::nat)"
berghofe@13449
   416
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
berghofe@13449
   417
paulson@14267
   418
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
berghofe@13449
   419
  by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans)
berghofe@13449
   420
paulson@14267
   421
lemma le_anti_sym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
berghofe@13449
   422
  by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym)
berghofe@13449
   423
paulson@14267
   424
lemma Suc_le_mono [iff]: "(Suc n \<le> Suc m) = (n \<le> m)"
berghofe@13449
   425
  by (simp add: le_simps)
berghofe@13449
   426
berghofe@13449
   427
text {* Axiom @{text order_less_le} of class @{text order}: *}
paulson@14267
   428
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
berghofe@13449
   429
  by (simp add: le_def nat_neq_iff) (blast elim!: less_asym)
berghofe@13449
   430
paulson@14267
   431
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
berghofe@13449
   432
  by (rule iffD2, rule nat_less_le, rule conjI)
berghofe@13449
   433
berghofe@13449
   434
text {* Axiom @{text linorder_linear} of class @{text linorder}: *}
paulson@14267
   435
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
berghofe@13449
   436
  apply (simp add: le_eq_less_or_eq)
wenzelm@22718
   437
  using less_linear by blast
berghofe@13449
   438
paulson@14341
   439
text {* Type {@typ nat} is a wellfounded linear order *}
paulson@14341
   440
haftmann@22318
   441
instance nat :: wellorder
wenzelm@14691
   442
  by intro_classes
wenzelm@14691
   443
    (assumption |
wenzelm@14691
   444
      rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+
paulson@14341
   445
wenzelm@22718
   446
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   447
berghofe@13449
   448
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
berghofe@13449
   449
  by (blast elim!: less_SucE)
berghofe@13449
   450
berghofe@13449
   451
text {*
berghofe@13449
   452
  Rewrite @{term "n < Suc m"} to @{term "n = m"}
paulson@14267
   453
  if @{term "~ n < m"} or @{term "m \<le> n"} hold.
berghofe@13449
   454
  Not suitable as default simprules because they often lead to looping
berghofe@13449
   455
*}
paulson@14267
   456
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
berghofe@13449
   457
  by (rule not_less_less_Suc_eq, rule leD)
berghofe@13449
   458
berghofe@13449
   459
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   460
berghofe@13449
   461
berghofe@13449
   462
text {*
wenzelm@22718
   463
  Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}.
wenzelm@22718
   464
  No longer added as simprules (they loop)
berghofe@13449
   465
  but via @{text reorient_simproc} in Bin
berghofe@13449
   466
*}
berghofe@13449
   467
berghofe@13449
   468
text {* Polymorphic, not just for @{typ nat} *}
berghofe@13449
   469
lemma zero_reorient: "(0 = x) = (x = 0)"
berghofe@13449
   470
  by auto
berghofe@13449
   471
berghofe@13449
   472
lemma one_reorient: "(1 = x) = (x = 1)"
berghofe@13449
   473
  by auto
berghofe@13449
   474
wenzelm@21243
   475
berghofe@13449
   476
subsection {* Arithmetic operators *}
oheimb@1660
   477
haftmann@22473
   478
class power = type +
haftmann@21411
   479
  fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"            (infixr "\<^loc>^" 80)
wenzelm@9436
   480
berghofe@13449
   481
text {* arithmetic operators @{text "+ -"} and @{text "*"} *}
berghofe@13449
   482
haftmann@21456
   483
instance nat :: "{plus, minus, times}" ..
wenzelm@9436
   484
berghofe@13449
   485
primrec
berghofe@13449
   486
  add_0:    "0 + n = n"
berghofe@13449
   487
  add_Suc:  "Suc m + n = Suc (m + n)"
berghofe@13449
   488
berghofe@13449
   489
primrec
berghofe@13449
   490
  diff_0:   "m - 0 = m"
berghofe@13449
   491
  diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
wenzelm@9436
   492
wenzelm@9436
   493
primrec
berghofe@13449
   494
  mult_0:   "0 * n = 0"
berghofe@13449
   495
  mult_Suc: "Suc m * n = n + (m * n)"
berghofe@13449
   496
wenzelm@22718
   497
text {* These two rules ease the use of primitive recursion.
paulson@14341
   498
NOTE USE OF @{text "=="} *}
berghofe@13449
   499
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
berghofe@13449
   500
  by simp
berghofe@13449
   501
berghofe@13449
   502
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
berghofe@13449
   503
  by simp
berghofe@13449
   504
paulson@14267
   505
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
wenzelm@22718
   506
  by (cases n) simp_all
berghofe@13449
   507
wenzelm@22718
   508
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
wenzelm@22718
   509
  by (cases n) simp_all
berghofe@13449
   510
wenzelm@22718
   511
lemma neq0_conv [iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
wenzelm@22718
   512
  by (cases n) simp_all
berghofe@13449
   513
berghofe@13449
   514
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   515
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@17589
   516
  by (rule iffD1, rule neq0_conv, iprover)
berghofe@13449
   517
paulson@14267
   518
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
berghofe@13449
   519
  by (fast intro: not0_implies_Suc)
berghofe@13449
   520
berghofe@13449
   521
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
berghofe@13449
   522
  apply (rule iffI)
wenzelm@22718
   523
  apply (rule ccontr)
wenzelm@22718
   524
  apply simp_all
berghofe@13449
   525
  done
berghofe@13449
   526
paulson@14267
   527
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
berghofe@13449
   528
  by (induct m') simp_all
berghofe@13449
   529
berghofe@13449
   530
text {* Useful in certain inductive arguments *}
paulson@14267
   531
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
wenzelm@22718
   532
  by (cases m) simp_all
berghofe@13449
   533
paulson@14341
   534
lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"
berghofe@13449
   535
  apply (rule nat_less_induct)
berghofe@13449
   536
  apply (case_tac n)
berghofe@13449
   537
  apply (case_tac [2] nat)
berghofe@13449
   538
  apply (blast intro: less_trans)+
berghofe@13449
   539
  done
berghofe@13449
   540
wenzelm@21243
   541
paulson@15341
   542
subsection {* @{text LEAST} theorems for type @{typ nat}*}
berghofe@13449
   543
paulson@14267
   544
lemma Least_Suc:
paulson@14267
   545
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
paulson@14208
   546
  apply (case_tac "n", auto)
berghofe@13449
   547
  apply (frule LeastI)
berghofe@13449
   548
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
paulson@14267
   549
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
berghofe@13449
   550
  apply (erule_tac [2] Least_le)
paulson@14208
   551
  apply (case_tac "LEAST x. P x", auto)
berghofe@13449
   552
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
berghofe@13449
   553
  apply (blast intro: order_antisym)
berghofe@13449
   554
  done
berghofe@13449
   555
paulson@14267
   556
lemma Least_Suc2:
paulson@14267
   557
     "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
paulson@14267
   558
  by (erule (1) Least_Suc [THEN ssubst], simp)
berghofe@13449
   559
berghofe@13449
   560
berghofe@13449
   561
subsection {* @{term min} and @{term max} *}
berghofe@13449
   562
berghofe@13449
   563
lemma min_0L [simp]: "min 0 n = (0::nat)"
berghofe@13449
   564
  by (rule min_leastL) simp
berghofe@13449
   565
berghofe@13449
   566
lemma min_0R [simp]: "min n 0 = (0::nat)"
berghofe@13449
   567
  by (rule min_leastR) simp
berghofe@13449
   568
berghofe@13449
   569
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
berghofe@13449
   570
  by (simp add: min_of_mono)
berghofe@13449
   571
paulson@22191
   572
lemma min_Suc1:
paulson@22191
   573
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
wenzelm@22718
   574
  by (simp split: nat.split)
paulson@22191
   575
paulson@22191
   576
lemma min_Suc2:
paulson@22191
   577
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
paulson@22191
   578
  by (simp split: nat.split)
paulson@22191
   579
berghofe@13449
   580
lemma max_0L [simp]: "max 0 n = (n::nat)"
berghofe@13449
   581
  by (rule max_leastL) simp
berghofe@13449
   582
berghofe@13449
   583
lemma max_0R [simp]: "max n 0 = (n::nat)"
berghofe@13449
   584
  by (rule max_leastR) simp
berghofe@13449
   585
berghofe@13449
   586
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
berghofe@13449
   587
  by (simp add: max_of_mono)
berghofe@13449
   588
paulson@22191
   589
lemma max_Suc1:
paulson@22191
   590
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
wenzelm@22718
   591
  by (simp split: nat.split)
paulson@22191
   592
paulson@22191
   593
lemma max_Suc2:
paulson@22191
   594
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
paulson@22191
   595
  by (simp split: nat.split)
paulson@22191
   596
berghofe@13449
   597
berghofe@13449
   598
subsection {* Basic rewrite rules for the arithmetic operators *}
berghofe@13449
   599
berghofe@13449
   600
text {* Difference *}
berghofe@13449
   601
berghofe@14193
   602
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
paulson@15251
   603
  by (induct n) simp_all
berghofe@13449
   604
berghofe@14193
   605
lemma diff_Suc_Suc [simp, code]: "Suc(m) - Suc(n) = m - n"
paulson@15251
   606
  by (induct n) simp_all
berghofe@13449
   607
berghofe@13449
   608
berghofe@13449
   609
text {*
berghofe@13449
   610
  Could be (and is, below) generalized in various ways
berghofe@13449
   611
  However, none of the generalizations are currently in the simpset,
berghofe@13449
   612
  and I dread to think what happens if I put them in
berghofe@13449
   613
*}
berghofe@13449
   614
lemma Suc_pred [simp]: "0 < n ==> Suc (n - Suc 0) = n"
berghofe@13449
   615
  by (simp split add: nat.split)
berghofe@13449
   616
berghofe@14193
   617
declare diff_Suc [simp del, code del]
berghofe@13449
   618
berghofe@13449
   619
berghofe@13449
   620
subsection {* Addition *}
berghofe@13449
   621
berghofe@13449
   622
lemma add_0_right [simp]: "m + 0 = (m::nat)"
berghofe@13449
   623
  by (induct m) simp_all
berghofe@13449
   624
berghofe@13449
   625
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
berghofe@13449
   626
  by (induct m) simp_all
berghofe@13449
   627
haftmann@19890
   628
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
haftmann@19890
   629
  by simp
berghofe@14193
   630
berghofe@13449
   631
berghofe@13449
   632
text {* Associative law for addition *}
paulson@14267
   633
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
berghofe@13449
   634
  by (induct m) simp_all
berghofe@13449
   635
berghofe@13449
   636
text {* Commutative law for addition *}
paulson@14267
   637
lemma nat_add_commute: "m + n = n + (m::nat)"
berghofe@13449
   638
  by (induct m) simp_all
berghofe@13449
   639
paulson@14267
   640
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
berghofe@13449
   641
  apply (rule mk_left_commute [of "op +"])
paulson@14267
   642
  apply (rule nat_add_assoc)
paulson@14267
   643
  apply (rule nat_add_commute)
berghofe@13449
   644
  done
berghofe@13449
   645
paulson@14331
   646
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
berghofe@13449
   647
  by (induct k) simp_all
berghofe@13449
   648
paulson@14331
   649
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
berghofe@13449
   650
  by (induct k) simp_all
berghofe@13449
   651
paulson@14331
   652
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
berghofe@13449
   653
  by (induct k) simp_all
berghofe@13449
   654
paulson@14331
   655
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
berghofe@13449
   656
  by (induct k) simp_all
berghofe@13449
   657
berghofe@13449
   658
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
berghofe@13449
   659
wenzelm@22718
   660
lemma add_is_0 [iff]: fixes m :: nat shows "(m + n = 0) = (m = 0 & n = 0)"
wenzelm@22718
   661
  by (cases m) simp_all
berghofe@13449
   662
berghofe@13449
   663
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
wenzelm@22718
   664
  by (cases m) simp_all
berghofe@13449
   665
berghofe@13449
   666
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
berghofe@13449
   667
  by (rule trans, rule eq_commute, rule add_is_1)
berghofe@13449
   668
berghofe@13449
   669
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m | 0 < n)"
berghofe@13449
   670
  by (simp del: neq0_conv add: neq0_conv [symmetric])
berghofe@13449
   671
berghofe@13449
   672
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0"
berghofe@13449
   673
  apply (drule add_0_right [THEN ssubst])
paulson@14267
   674
  apply (simp add: nat_add_assoc del: add_0_right)
berghofe@13449
   675
  done
berghofe@13449
   676
nipkow@16733
   677
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
wenzelm@22718
   678
  apply (induct k)
wenzelm@22718
   679
   apply simp
wenzelm@22718
   680
  apply(drule comp_inj_on[OF _ inj_Suc])
wenzelm@22718
   681
  apply (simp add:o_def)
wenzelm@22718
   682
  done
nipkow@16733
   683
nipkow@16733
   684
paulson@14267
   685
subsection {* Multiplication *}
paulson@14267
   686
paulson@14267
   687
text {* right annihilation in product *}
paulson@14267
   688
lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
paulson@14267
   689
  by (induct m) simp_all
paulson@14267
   690
paulson@14267
   691
text {* right successor law for multiplication *}
paulson@14267
   692
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
paulson@14267
   693
  by (induct m) (simp_all add: nat_add_left_commute)
paulson@14267
   694
paulson@14267
   695
text {* Commutative law for multiplication *}
paulson@14267
   696
lemma nat_mult_commute: "m * n = n * (m::nat)"
paulson@14267
   697
  by (induct m) simp_all
paulson@14267
   698
paulson@14267
   699
text {* addition distributes over multiplication *}
paulson@14267
   700
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
paulson@14267
   701
  by (induct m) (simp_all add: nat_add_assoc nat_add_left_commute)
paulson@14267
   702
paulson@14267
   703
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
paulson@14267
   704
  by (induct m) (simp_all add: nat_add_assoc)
paulson@14267
   705
paulson@14267
   706
text {* Associative law for multiplication *}
paulson@14267
   707
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
paulson@14267
   708
  by (induct m) (simp_all add: add_mult_distrib)
paulson@14267
   709
paulson@14267
   710
nipkow@14740
   711
text{*The naturals form a @{text comm_semiring_1_cancel}*}
obua@14738
   712
instance nat :: comm_semiring_1_cancel
paulson@14267
   713
proof
paulson@14267
   714
  fix i j k :: nat
paulson@14267
   715
  show "(i + j) + k = i + (j + k)" by (rule nat_add_assoc)
paulson@14267
   716
  show "i + j = j + i" by (rule nat_add_commute)
paulson@14267
   717
  show "0 + i = i" by simp
paulson@14267
   718
  show "(i * j) * k = i * (j * k)" by (rule nat_mult_assoc)
paulson@14267
   719
  show "i * j = j * i" by (rule nat_mult_commute)
paulson@14267
   720
  show "1 * i = i" by simp
paulson@14267
   721
  show "(i + j) * k = i * k + j * k" by (simp add: add_mult_distrib)
paulson@14267
   722
  show "0 \<noteq> (1::nat)" by simp
paulson@14341
   723
  assume "k+i = k+j" thus "i=j" by simp
paulson@14341
   724
qed
paulson@14341
   725
paulson@14341
   726
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
paulson@15251
   727
  apply (induct m)
wenzelm@22718
   728
   apply (induct_tac [2] n)
wenzelm@22718
   729
    apply simp_all
paulson@14341
   730
  done
paulson@14341
   731
wenzelm@21243
   732
paulson@14341
   733
subsection {* Monotonicity of Addition *}
paulson@14341
   734
paulson@14341
   735
text {* strict, in 1st argument *}
paulson@14341
   736
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
paulson@14341
   737
  by (induct k) simp_all
paulson@14341
   738
paulson@14341
   739
text {* strict, in both arguments *}
paulson@14341
   740
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   741
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   742
  apply (induct j, simp_all)
paulson@14341
   743
  done
paulson@14341
   744
paulson@14341
   745
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   746
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   747
  apply (induct n)
paulson@14341
   748
  apply (simp_all add: order_le_less)
wenzelm@22718
   749
  apply (blast elim!: less_SucE
paulson@14341
   750
               intro!: add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   751
  done
paulson@14341
   752
paulson@14341
   753
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
paulson@14341
   754
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j"
paulson@14341
   755
  apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp)
wenzelm@22718
   756
  apply (induct_tac x)
paulson@14341
   757
  apply (simp_all add: add_less_mono)
paulson@14341
   758
  done
paulson@14341
   759
paulson@14341
   760
nipkow@14740
   761
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
obua@14738
   762
instance nat :: ordered_semidom
paulson@14341
   763
proof
paulson@14341
   764
  fix i j k :: nat
paulson@14348
   765
  show "0 < (1::nat)" by simp
paulson@14267
   766
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   767
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   768
qed
paulson@14267
   769
paulson@14267
   770
lemma nat_mult_1: "(1::nat) * n = n"
paulson@14267
   771
  by simp
paulson@14267
   772
paulson@14267
   773
lemma nat_mult_1_right: "n * (1::nat) = n"
paulson@14267
   774
  by simp
paulson@14267
   775
paulson@14267
   776
paulson@14267
   777
subsection {* Additional theorems about "less than" *}
paulson@14267
   778
paulson@19870
   779
text{*An induction rule for estabilishing binary relations*}
wenzelm@22718
   780
lemma less_Suc_induct:
paulson@19870
   781
  assumes less:  "i < j"
paulson@19870
   782
     and  step:  "!!i. P i (Suc i)"
paulson@19870
   783
     and  trans: "!!i j k. P i j ==> P j k ==> P i k"
paulson@19870
   784
  shows "P i j"
paulson@19870
   785
proof -
wenzelm@22718
   786
  from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   787
  have "P i (Suc (i + k))"
paulson@19870
   788
  proof (induct k)
wenzelm@22718
   789
    case 0
wenzelm@22718
   790
    show ?case by (simp add: step)
paulson@19870
   791
  next
paulson@19870
   792
    case (Suc k)
wenzelm@22718
   793
    thus ?case by (auto intro: assms)
paulson@19870
   794
  qed
wenzelm@22718
   795
  thus "P i j" by (simp add: j)
paulson@19870
   796
qed
paulson@19870
   797
paulson@19870
   798
paulson@14267
   799
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   800
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   801
lemma less_mono_imp_le_mono:
paulson@14267
   802
  assumes lt_mono: "!!i j::nat. i < j ==> f i < f j"
wenzelm@22718
   803
    and le: "i \<le> j"
wenzelm@22718
   804
  shows "f i \<le> ((f j)::nat)"
wenzelm@22718
   805
  using le
paulson@14267
   806
  apply (simp add: order_le_less)
paulson@14267
   807
  apply (blast intro!: lt_mono)
paulson@14267
   808
  done
paulson@14267
   809
paulson@14267
   810
text {* non-strict, in 1st argument *}
paulson@14267
   811
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
paulson@14267
   812
  by (rule add_right_mono)
paulson@14267
   813
paulson@14267
   814
text {* non-strict, in both arguments *}
paulson@14267
   815
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
paulson@14267
   816
  by (rule add_mono)
paulson@14267
   817
paulson@14267
   818
lemma le_add2: "n \<le> ((m + n)::nat)"
wenzelm@22718
   819
  by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
   820
paulson@14267
   821
lemma le_add1: "n \<le> ((n + m)::nat)"
paulson@14341
   822
  by (simp add: add_commute, rule le_add2)
berghofe@13449
   823
berghofe@13449
   824
lemma less_add_Suc1: "i < Suc (i + m)"
berghofe@13449
   825
  by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   826
berghofe@13449
   827
lemma less_add_Suc2: "i < Suc (m + i)"
berghofe@13449
   828
  by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   829
paulson@14267
   830
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@17589
   831
  by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   832
paulson@14267
   833
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
berghofe@13449
   834
  by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   835
paulson@14267
   836
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
berghofe@13449
   837
  by (rule le_trans, assumption, rule le_add2)
berghofe@13449
   838
berghofe@13449
   839
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
berghofe@13449
   840
  by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
   841
berghofe@13449
   842
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
berghofe@13449
   843
  by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
   844
berghofe@13449
   845
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
wenzelm@22718
   846
  apply (rule le_less_trans [of _ "i+j"])
paulson@14341
   847
  apply (simp_all add: le_add1)
berghofe@13449
   848
  done
berghofe@13449
   849
berghofe@13449
   850
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
berghofe@13449
   851
  apply (rule notI)
berghofe@13449
   852
  apply (erule add_lessD1 [THEN less_irrefl])
berghofe@13449
   853
  done
berghofe@13449
   854
berghofe@13449
   855
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
berghofe@13449
   856
  by (simp add: add_commute not_add_less1)
berghofe@13449
   857
paulson@14267
   858
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
wenzelm@22718
   859
  apply (rule order_trans [of _ "m+k"])
paulson@14341
   860
  apply (simp_all add: le_add1)
paulson@14341
   861
  done
berghofe@13449
   862
paulson@14267
   863
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
berghofe@13449
   864
  apply (simp add: add_commute)
berghofe@13449
   865
  apply (erule add_leD1)
berghofe@13449
   866
  done
berghofe@13449
   867
paulson@14267
   868
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
berghofe@13449
   869
  by (blast dest: add_leD1 add_leD2)
berghofe@13449
   870
berghofe@13449
   871
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
   872
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
berghofe@13449
   873
  by (force simp del: add_Suc_right
berghofe@13449
   874
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
   875
berghofe@13449
   876
berghofe@13449
   877
subsection {* Difference *}
berghofe@13449
   878
berghofe@13449
   879
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
berghofe@13449
   880
  by (induct m) simp_all
berghofe@13449
   881
berghofe@13449
   882
text {* Addition is the inverse of subtraction:
paulson@14267
   883
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
   884
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
berghofe@13449
   885
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   886
paulson@14267
   887
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
paulson@16796
   888
  by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
   889
paulson@14267
   890
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
berghofe@13449
   891
  by (simp add: le_add_diff_inverse add_commute)
berghofe@13449
   892
berghofe@13449
   893
berghofe@13449
   894
subsection {* More results about difference *}
berghofe@13449
   895
paulson@14267
   896
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
berghofe@13449
   897
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   898
berghofe@13449
   899
lemma diff_less_Suc: "m - n < Suc m"
berghofe@13449
   900
  apply (induct m n rule: diff_induct)
berghofe@13449
   901
  apply (erule_tac [3] less_SucE)
berghofe@13449
   902
  apply (simp_all add: less_Suc_eq)
berghofe@13449
   903
  done
berghofe@13449
   904
paulson@14267
   905
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
berghofe@13449
   906
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
   907
berghofe@13449
   908
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
berghofe@13449
   909
  by (rule le_less_trans, rule diff_le_self)
berghofe@13449
   910
berghofe@13449
   911
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
berghofe@13449
   912
  by (induct i j rule: diff_induct) simp_all
berghofe@13449
   913
berghofe@13449
   914
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
berghofe@13449
   915
  by (simp add: diff_diff_left)
berghofe@13449
   916
berghofe@13449
   917
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
wenzelm@22718
   918
  by (cases n) (auto simp add: le_simps)
berghofe@13449
   919
berghofe@13449
   920
text {* This and the next few suggested by Florian Kammueller *}
berghofe@13449
   921
lemma diff_commute: "(i::nat) - j - k = i - k - j"
berghofe@13449
   922
  by (simp add: diff_diff_left add_commute)
berghofe@13449
   923
paulson@14267
   924
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
berghofe@13449
   925
  by (induct j k rule: diff_induct) simp_all
berghofe@13449
   926
paulson@14267
   927
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
berghofe@13449
   928
  by (simp add: add_commute diff_add_assoc)
berghofe@13449
   929
berghofe@13449
   930
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
berghofe@13449
   931
  by (induct n) simp_all
berghofe@13449
   932
berghofe@13449
   933
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
berghofe@13449
   934
  by (simp add: diff_add_assoc)
berghofe@13449
   935
paulson@14267
   936
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
wenzelm@22718
   937
  by (auto simp add: diff_add_inverse2)
berghofe@13449
   938
paulson@14267
   939
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
berghofe@13449
   940
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   941
paulson@14267
   942
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
berghofe@13449
   943
  by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
   944
berghofe@13449
   945
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
berghofe@13449
   946
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   947
wenzelm@22718
   948
lemma less_imp_add_positive:
wenzelm@22718
   949
  assumes "i < j"
wenzelm@22718
   950
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
   951
proof
wenzelm@22718
   952
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
   953
    by (simp add: order_less_imp_le)
wenzelm@22718
   954
qed
wenzelm@9436
   955
berghofe@13449
   956
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
berghofe@13449
   957
  by (induct k) simp_all
berghofe@13449
   958
berghofe@13449
   959
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
berghofe@13449
   960
  by (simp add: diff_cancel add_commute)
berghofe@13449
   961
berghofe@13449
   962
lemma diff_add_0: "n - (n + m) = (0::nat)"
berghofe@13449
   963
  by (induct n) simp_all
berghofe@13449
   964
berghofe@13449
   965
berghofe@13449
   966
text {* Difference distributes over multiplication *}
berghofe@13449
   967
berghofe@13449
   968
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
berghofe@13449
   969
  by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
berghofe@13449
   970
berghofe@13449
   971
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
berghofe@13449
   972
  by (simp add: diff_mult_distrib mult_commute [of k])
berghofe@13449
   973
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
berghofe@13449
   974
berghofe@13449
   975
lemmas nat_distrib =
berghofe@13449
   976
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
berghofe@13449
   977
berghofe@13449
   978
berghofe@13449
   979
subsection {* Monotonicity of Multiplication *}
berghofe@13449
   980
paulson@14267
   981
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
wenzelm@22718
   982
  by (simp add: mult_right_mono)
berghofe@13449
   983
paulson@14267
   984
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
wenzelm@22718
   985
  by (simp add: mult_left_mono)
berghofe@13449
   986
paulson@14267
   987
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
   988
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
wenzelm@22718
   989
  by (simp add: mult_mono)
berghofe@13449
   990
berghofe@13449
   991
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
wenzelm@22718
   992
  by (simp add: mult_strict_right_mono)
berghofe@13449
   993
paulson@14266
   994
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
   995
      there are no negative numbers.*}
paulson@14266
   996
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
   997
  apply (induct m)
wenzelm@22718
   998
   apply simp
wenzelm@22718
   999
  apply (case_tac n)
wenzelm@22718
  1000
   apply simp_all
berghofe@13449
  1001
  done
berghofe@13449
  1002
paulson@14267
  1003
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"
berghofe@13449
  1004
  apply (induct m)
wenzelm@22718
  1005
   apply simp
wenzelm@22718
  1006
  apply (case_tac n)
wenzelm@22718
  1007
   apply simp_all
berghofe@13449
  1008
  done
berghofe@13449
  1009
berghofe@13449
  1010
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
wenzelm@22718
  1011
  apply (induct m)
wenzelm@22718
  1012
   apply simp
wenzelm@22718
  1013
  apply (induct n)
wenzelm@22718
  1014
   apply auto
berghofe@13449
  1015
  done
berghofe@13449
  1016
berghofe@13449
  1017
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
berghofe@13449
  1018
  apply (rule trans)
paulson@14208
  1019
  apply (rule_tac [2] mult_eq_1_iff, fastsimp)
berghofe@13449
  1020
  done
berghofe@13449
  1021
paulson@14341
  1022
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1023
  apply (safe intro!: mult_less_mono1)
paulson@14208
  1024
  apply (case_tac k, auto)
berghofe@13449
  1025
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1026
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1027
  done
berghofe@13449
  1028
berghofe@13449
  1029
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
paulson@14341
  1030
  by (simp add: mult_commute [of k])
berghofe@13449
  1031
paulson@14267
  1032
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
wenzelm@22718
  1033
  by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1034
paulson@14267
  1035
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
wenzelm@22718
  1036
  by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1037
paulson@14341
  1038
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
paulson@14208
  1039
  apply (cut_tac less_linear, safe, auto)
berghofe@13449
  1040
  apply (drule mult_less_mono1, assumption, simp)+
berghofe@13449
  1041
  done
berghofe@13449
  1042
berghofe@13449
  1043
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
paulson@14341
  1044
  by (simp add: mult_commute [of k])
berghofe@13449
  1045
berghofe@13449
  1046
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
berghofe@13449
  1047
  by (subst mult_less_cancel1) simp
berghofe@13449
  1048
paulson@14267
  1049
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
berghofe@13449
  1050
  by (subst mult_le_cancel1) simp
berghofe@13449
  1051
berghofe@13449
  1052
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
berghofe@13449
  1053
  by (subst mult_cancel1) simp
berghofe@13449
  1054
berghofe@13449
  1055
text {* Lemma for @{text gcd} *}
berghofe@13449
  1056
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1057
  apply (drule sym)
berghofe@13449
  1058
  apply (rule disjCI)
berghofe@13449
  1059
  apply (rule nat_less_cases, erule_tac [2] _)
berghofe@13449
  1060
  apply (fastsimp elim!: less_SucE)
berghofe@13449
  1061
  apply (fastsimp dest: mult_less_mono2)
berghofe@13449
  1062
  done
wenzelm@9436
  1063
haftmann@20588
  1064
haftmann@18702
  1065
subsection {* Code generator setup *}
haftmann@18702
  1066
wenzelm@22718
  1067
lemma one_is_Suc_zero [code inline]: "1 = Suc 0"
haftmann@20355
  1068
  by simp
haftmann@20355
  1069
haftmann@20588
  1070
instance nat :: eq ..
haftmann@20588
  1071
haftmann@20588
  1072
lemma [code func]:
wenzelm@22718
  1073
    "(0\<Colon>nat) = 0 \<longleftrightarrow> True"
wenzelm@22718
  1074
    "Suc n = Suc m \<longleftrightarrow> n = m"
wenzelm@22718
  1075
    "Suc n = 0 \<longleftrightarrow> False"
wenzelm@22718
  1076
    "0 = Suc m \<longleftrightarrow> False"
haftmann@22348
  1077
  by auto
haftmann@20588
  1078
haftmann@20588
  1079
lemma [code func]:
wenzelm@22718
  1080
    "(0\<Colon>nat) \<le> m \<longleftrightarrow> True"
wenzelm@22718
  1081
    "Suc (n\<Colon>nat) \<le> m \<longleftrightarrow> n < m"
wenzelm@22718
  1082
    "(n\<Colon>nat) < 0 \<longleftrightarrow> False"
wenzelm@22718
  1083
    "(n\<Colon>nat) < Suc m \<longleftrightarrow> n \<le> m"
haftmann@22348
  1084
  using Suc_le_eq less_Suc_eq_le by simp_all
haftmann@20588
  1085
wenzelm@21243
  1086
wenzelm@21243
  1087
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
wenzelm@21243
  1088
haftmann@22845
  1089
lemma subst_equals:
haftmann@22845
  1090
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1091
  shows "u = s"
haftmann@22845
  1092
  using 2 1 by (rule trans)
haftmann@22845
  1093
wenzelm@21243
  1094
use "arith_data.ML"
wenzelm@21243
  1095
setup arith_setup
wenzelm@21243
  1096
wenzelm@21243
  1097
text{*The following proofs may rely on the arithmetic proof procedures.*}
wenzelm@21243
  1098
wenzelm@21243
  1099
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
wenzelm@21243
  1100
  by (auto simp: le_eq_less_or_eq dest: less_imp_Suc_add)
wenzelm@21243
  1101
wenzelm@21243
  1102
lemma pred_nat_trancl_eq_le: "((m, n) : pred_nat^*) = (m \<le> n)"
wenzelm@22718
  1103
  by (simp add: less_eq reflcl_trancl [symmetric] del: reflcl_trancl, arith)
wenzelm@21243
  1104
wenzelm@21243
  1105
lemma nat_diff_split:
wenzelm@22718
  1106
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
wenzelm@21243
  1107
    -- {* elimination of @{text -} on @{text nat} *}
wenzelm@22718
  1108
  by (cases "a<b" rule: case_split) (auto simp add: diff_is_0_eq [THEN iffD2])
wenzelm@21243
  1109
wenzelm@21243
  1110
lemma nat_diff_split_asm:
wenzelm@21243
  1111
    "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
wenzelm@21243
  1112
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
wenzelm@21243
  1113
  by (simp split: nat_diff_split)
wenzelm@21243
  1114
wenzelm@21243
  1115
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1116
wenzelm@21243
  1117
wenzelm@21243
  1118
lemma le_square: "m \<le> m * (m::nat)"
wenzelm@21243
  1119
  by (induct m) auto
wenzelm@21243
  1120
wenzelm@21243
  1121
lemma le_cube: "(m::nat) \<le> m * (m * m)"
wenzelm@21243
  1122
  by (induct m) auto
wenzelm@21243
  1123
wenzelm@21243
  1124
wenzelm@21243
  1125
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1126
wenzelm@21243
  1127
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
wenzelm@22718
  1128
  by arith
wenzelm@21243
  1129
wenzelm@21243
  1130
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
wenzelm@22718
  1131
  by arith
wenzelm@21243
  1132
wenzelm@21243
  1133
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
wenzelm@22718
  1134
  by arith
wenzelm@21243
  1135
wenzelm@21243
  1136
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
wenzelm@22718
  1137
  by arith
wenzelm@21243
  1138
wenzelm@21243
  1139
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
wenzelm@22718
  1140
  by arith
wenzelm@21243
  1141
wenzelm@21243
  1142
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
wenzelm@22718
  1143
  by arith
wenzelm@21243
  1144
wenzelm@21243
  1145
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1146
  second premise n\<le>m*)
wenzelm@21243
  1147
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
wenzelm@22718
  1148
  by arith
wenzelm@21243
  1149
wenzelm@21243
  1150
wenzelm@21243
  1151
(** Simplification of relational expressions involving subtraction **)
wenzelm@21243
  1152
wenzelm@21243
  1153
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
wenzelm@22718
  1154
  by (simp split add: nat_diff_split)
wenzelm@21243
  1155
wenzelm@21243
  1156
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
wenzelm@22718
  1157
  by (auto split add: nat_diff_split)
wenzelm@21243
  1158
wenzelm@21243
  1159
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
wenzelm@22718
  1160
  by (auto split add: nat_diff_split)
wenzelm@21243
  1161
wenzelm@21243
  1162
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
wenzelm@22718
  1163
  by (auto split add: nat_diff_split)
wenzelm@21243
  1164
wenzelm@21243
  1165
wenzelm@21243
  1166
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1167
wenzelm@21243
  1168
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1169
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
wenzelm@22718
  1170
  by (simp split add: nat_diff_split)
wenzelm@21243
  1171
wenzelm@21243
  1172
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
wenzelm@22718
  1173
  by (simp split add: nat_diff_split)
wenzelm@21243
  1174
wenzelm@21243
  1175
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
wenzelm@22718
  1176
  by (simp split add: nat_diff_split)
wenzelm@21243
  1177
wenzelm@21243
  1178
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
wenzelm@22718
  1179
  by (simp split add: nat_diff_split)
wenzelm@21243
  1180
wenzelm@21243
  1181
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1182
wenzelm@21243
  1183
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
wenzelm@22718
  1184
  by (cases m) auto
wenzelm@21243
  1185
wenzelm@21243
  1186
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
wenzelm@22718
  1187
  by (cases m) auto
wenzelm@21243
  1188
wenzelm@21243
  1189
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
wenzelm@22718
  1190
  by (cases m) auto
wenzelm@21243
  1191
krauss@23001
  1192
text {* Specialized induction principles that work "backwards": *}
krauss@23001
  1193
krauss@23001
  1194
lemma inc_induct[consumes 1, case_names base step]:
krauss@23001
  1195
  assumes less: "i <= j"
krauss@23001
  1196
  assumes base: "P j"
krauss@23001
  1197
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1198
  shows "P i"
krauss@23001
  1199
  using less
krauss@23001
  1200
proof (induct d=="j - i" arbitrary: i)
krauss@23001
  1201
  case (0 i)
krauss@23001
  1202
  hence "i = j" by simp
krauss@23001
  1203
  with base show ?case by simp
krauss@23001
  1204
next
krauss@23001
  1205
  case (Suc d i)
krauss@23001
  1206
  hence "i < j" "P (Suc i)"
krauss@23001
  1207
    by simp_all
krauss@23001
  1208
  thus "P i" by (rule step)
krauss@23001
  1209
qed
krauss@23001
  1210
krauss@23001
  1211
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1212
  assumes less: "i < j"
krauss@23001
  1213
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1214
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1215
  shows "P i"
krauss@23001
  1216
  using less
krauss@23001
  1217
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1218
  case (0 i)
krauss@23001
  1219
  with `i < j` have "j = Suc i" by simp
krauss@23001
  1220
  with base show ?case by simp
krauss@23001
  1221
next
krauss@23001
  1222
  case (Suc d i)
krauss@23001
  1223
  hence "i < j" "P (Suc i)"
krauss@23001
  1224
    by simp_all
krauss@23001
  1225
  thus "P i" by (rule step)
krauss@23001
  1226
qed
krauss@23001
  1227
krauss@23001
  1228
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1229
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1230
krauss@23001
  1231
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1232
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1233
wenzelm@21243
  1234
text{*Rewriting to pull differences out*}
wenzelm@21243
  1235
wenzelm@21243
  1236
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
wenzelm@22718
  1237
  by arith
wenzelm@21243
  1238
wenzelm@21243
  1239
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
wenzelm@22718
  1240
  by arith
wenzelm@21243
  1241
wenzelm@21243
  1242
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
wenzelm@22718
  1243
  by arith
wenzelm@21243
  1244
wenzelm@21243
  1245
(*The others are
wenzelm@21243
  1246
      i - j - k = i - (j + k),
wenzelm@21243
  1247
      k \<le> j ==> j - k + i = j + i - k,
wenzelm@21243
  1248
      k \<le> j ==> i + (j - k) = i + j - k *)
wenzelm@21243
  1249
lemmas add_diff_assoc = diff_add_assoc [symmetric]
wenzelm@21243
  1250
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
wenzelm@21243
  1251
declare diff_diff_left [simp]  add_diff_assoc [simp]  add_diff_assoc2[simp]
wenzelm@21243
  1252
wenzelm@21243
  1253
text{*At present we prove no analogue of @{text not_less_Least} or @{text
wenzelm@21243
  1254
Least_Suc}, since there appears to be no need.*}
wenzelm@21243
  1255
wenzelm@21243
  1256
ML
wenzelm@21243
  1257
{*
wenzelm@21243
  1258
val pred_nat_trancl_eq_le = thm "pred_nat_trancl_eq_le";
wenzelm@21243
  1259
val nat_diff_split = thm "nat_diff_split";
wenzelm@21243
  1260
val nat_diff_split_asm = thm "nat_diff_split_asm";
wenzelm@21243
  1261
val le_square = thm "le_square";
wenzelm@21243
  1262
val le_cube = thm "le_cube";
wenzelm@21243
  1263
val diff_less_mono = thm "diff_less_mono";
wenzelm@21243
  1264
val less_diff_conv = thm "less_diff_conv";
wenzelm@21243
  1265
val le_diff_conv = thm "le_diff_conv";
wenzelm@21243
  1266
val le_diff_conv2 = thm "le_diff_conv2";
wenzelm@21243
  1267
val diff_diff_cancel = thm "diff_diff_cancel";
wenzelm@21243
  1268
val le_add_diff = thm "le_add_diff";
wenzelm@21243
  1269
val diff_less = thm "diff_less";
wenzelm@21243
  1270
val diff_diff_eq = thm "diff_diff_eq";
wenzelm@21243
  1271
val eq_diff_iff = thm "eq_diff_iff";
wenzelm@21243
  1272
val less_diff_iff = thm "less_diff_iff";
wenzelm@21243
  1273
val le_diff_iff = thm "le_diff_iff";
wenzelm@21243
  1274
val diff_le_mono = thm "diff_le_mono";
wenzelm@21243
  1275
val diff_le_mono2 = thm "diff_le_mono2";
wenzelm@21243
  1276
val diff_less_mono2 = thm "diff_less_mono2";
wenzelm@21243
  1277
val diffs0_imp_equal = thm "diffs0_imp_equal";
wenzelm@21243
  1278
val one_less_mult = thm "one_less_mult";
wenzelm@21243
  1279
val n_less_m_mult_n = thm "n_less_m_mult_n";
wenzelm@21243
  1280
val n_less_n_mult_m = thm "n_less_n_mult_m";
wenzelm@21243
  1281
val diff_diff_right = thm "diff_diff_right";
wenzelm@21243
  1282
val diff_Suc_diff_eq1 = thm "diff_Suc_diff_eq1";
wenzelm@21243
  1283
val diff_Suc_diff_eq2 = thm "diff_Suc_diff_eq2";
wenzelm@21243
  1284
*}
wenzelm@21243
  1285
wenzelm@22718
  1286
wenzelm@22718
  1287
subsection{*Embedding of the Naturals into any
huffman@23276
  1288
  @{text semiring_1}: @{term of_nat}*}
wenzelm@21243
  1289
huffman@23276
  1290
consts of_nat :: "nat => 'a::semiring_1"
wenzelm@21243
  1291
wenzelm@21243
  1292
primrec
wenzelm@21243
  1293
  of_nat_0:   "of_nat 0 = 0"
huffman@23431
  1294
  of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
wenzelm@21243
  1295
haftmann@22920
  1296
lemma of_nat_id [simp]: "(of_nat n \<Colon> nat) = n"
haftmann@22920
  1297
  by (induct n) auto
haftmann@22920
  1298
wenzelm@21243
  1299
lemma of_nat_1 [simp]: "of_nat 1 = 1"
wenzelm@22718
  1300
  by simp
wenzelm@21243
  1301
wenzelm@21243
  1302
lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"
wenzelm@22718
  1303
  by (induct m) (simp_all add: add_ac)
wenzelm@21243
  1304
huffman@23431
  1305
lemma of_nat_mult: "of_nat (m*n) = of_nat m * of_nat n"
wenzelm@22718
  1306
  by (induct m) (simp_all add: add_ac left_distrib)
wenzelm@21243
  1307
wenzelm@21243
  1308
lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)"
wenzelm@22718
  1309
  apply (induct m, simp_all)
wenzelm@22718
  1310
  apply (erule order_trans)
huffman@23431
  1311
  apply (rule ord_le_eq_trans [OF _ add_commute])
wenzelm@22718
  1312
  apply (rule less_add_one [THEN order_less_imp_le])
wenzelm@22718
  1313
  done
wenzelm@21243
  1314
wenzelm@21243
  1315
lemma less_imp_of_nat_less:
wenzelm@22718
  1316
    "m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)"
wenzelm@22718
  1317
  apply (induct m n rule: diff_induct, simp_all)
huffman@23431
  1318
  apply (insert add_less_le_mono [OF zero_less_one zero_le_imp_of_nat], force)
wenzelm@22718
  1319
  done
wenzelm@21243
  1320
wenzelm@21243
  1321
lemma of_nat_less_imp_less:
wenzelm@22718
  1322
    "of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n"
wenzelm@22718
  1323
  apply (induct m n rule: diff_induct, simp_all)
wenzelm@22718
  1324
  apply (insert zero_le_imp_of_nat)
wenzelm@22718
  1325
  apply (force simp add: linorder_not_less [symmetric])
wenzelm@22718
  1326
  done
wenzelm@21243
  1327
wenzelm@21243
  1328
lemma of_nat_less_iff [simp]:
wenzelm@22718
  1329
    "(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)"
wenzelm@22718
  1330
  by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
wenzelm@21243
  1331
wenzelm@21243
  1332
text{*Special cases where either operand is zero*}
wenzelm@22718
  1333
wenzelm@22718
  1334
lemma of_nat_0_less_iff [simp]: "((0::'a::ordered_semidom) < of_nat n) = (0 < n)"
wenzelm@22718
  1335
  by (rule of_nat_less_iff [of 0, simplified])
wenzelm@22718
  1336
wenzelm@22718
  1337
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < (0::'a::ordered_semidom)"
wenzelm@22718
  1338
  by (rule of_nat_less_iff [of _ 0, simplified])
wenzelm@21243
  1339
wenzelm@21243
  1340
lemma of_nat_le_iff [simp]:
wenzelm@22718
  1341
    "(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)"
wenzelm@22718
  1342
  by (simp add: linorder_not_less [symmetric])
wenzelm@21243
  1343
wenzelm@21243
  1344
text{*Special cases where either operand is zero*}
wenzelm@22718
  1345
lemma of_nat_0_le_iff [simp]: "(0::'a::ordered_semidom) \<le> of_nat n"
wenzelm@22718
  1346
  by (rule of_nat_le_iff [of 0, simplified])
wenzelm@22718
  1347
lemma of_nat_le_0_iff [simp]: "(of_nat m \<le> (0::'a::ordered_semidom)) = (m = 0)"
wenzelm@22718
  1348
  by (rule of_nat_le_iff [of _ 0, simplified])
wenzelm@21243
  1349
huffman@23282
  1350
text{*Class for unital semirings with characteristic zero.
huffman@23282
  1351
 Includes non-ordered rings like the complex numbers.*}
huffman@23282
  1352
axclass semiring_char_0 < semiring_1
huffman@23282
  1353
  of_nat_eq_iff [simp]: "(of_nat m = of_nat n) = (m = n)"
huffman@23282
  1354
huffman@23282
  1355
text{*Every @{text ordered_semidom} has characteristic zero.*}
huffman@23282
  1356
instance ordered_semidom < semiring_char_0
huffman@23282
  1357
by intro_classes (simp add: order_eq_iff)
wenzelm@21243
  1358
wenzelm@21243
  1359
text{*Special cases where either operand is zero*}
huffman@23282
  1360
lemma of_nat_0_eq_iff [simp]: "((0::'a::semiring_char_0) = of_nat n) = (0 = n)"
wenzelm@22718
  1361
  by (rule of_nat_eq_iff [of 0, simplified])
huffman@23282
  1362
lemma of_nat_eq_0_iff [simp]: "(of_nat m = (0::'a::semiring_char_0)) = (m = 0)"
wenzelm@22718
  1363
  by (rule of_nat_eq_iff [of _ 0, simplified])
wenzelm@21243
  1364
huffman@23347
  1365
lemma inj_of_nat: "inj (of_nat :: nat \<Rightarrow> 'a::semiring_char_0)"
huffman@23347
  1366
  by (simp add: inj_on_def)
huffman@23347
  1367
huffman@23438
  1368
lemma of_nat_diff:
wenzelm@22718
  1369
    "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::ring_1)"
wenzelm@22718
  1370
  by (simp del: of_nat_add
wenzelm@22718
  1371
    add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
wenzelm@21243
  1372
haftmann@23852
  1373
haftmann@23852
  1374
subsection {*The Set of Natural Numbers*}
haftmann@23852
  1375
haftmann@23852
  1376
definition
haftmann@23852
  1377
  Nats  :: "'a::semiring_1 set"
haftmann@23852
  1378
where
haftmann@23852
  1379
  "Nats = range of_nat"
haftmann@23852
  1380
haftmann@23852
  1381
notation (xsymbols)
haftmann@23852
  1382
  Nats  ("\<nat>")
haftmann@23852
  1383
haftmann@23852
  1384
lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"
haftmann@23852
  1385
  by (simp add: Nats_def)
haftmann@23852
  1386
haftmann@23852
  1387
lemma Nats_0 [simp]: "0 \<in> Nats"
haftmann@23852
  1388
apply (simp add: Nats_def)
haftmann@23852
  1389
apply (rule range_eqI)
haftmann@23852
  1390
apply (rule of_nat_0 [symmetric])
haftmann@23852
  1391
done
haftmann@23852
  1392
haftmann@23852
  1393
lemma Nats_1 [simp]: "1 \<in> Nats"
haftmann@23852
  1394
apply (simp add: Nats_def)
haftmann@23852
  1395
apply (rule range_eqI)
haftmann@23852
  1396
apply (rule of_nat_1 [symmetric])
haftmann@23852
  1397
done
haftmann@23852
  1398
haftmann@23852
  1399
lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"
haftmann@23852
  1400
apply (auto simp add: Nats_def)
haftmann@23852
  1401
apply (rule range_eqI)
haftmann@23852
  1402
apply (rule of_nat_add [symmetric])
haftmann@23852
  1403
done
haftmann@23852
  1404
haftmann@23852
  1405
lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"
haftmann@23852
  1406
apply (auto simp add: Nats_def)
haftmann@23852
  1407
apply (rule range_eqI)
haftmann@23852
  1408
apply (rule of_nat_mult [symmetric])
haftmann@23852
  1409
done
haftmann@23852
  1410
haftmann@23852
  1411
lemma of_nat_eq_id [simp]: "of_nat = (id :: nat => nat)"
haftmann@23852
  1412
  by (auto simp add: expand_fun_eq)
haftmann@23852
  1413
haftmann@23852
  1414
haftmann@22483
  1415
instance nat :: distrib_lattice
haftmann@22483
  1416
  "inf \<equiv> min"
haftmann@22483
  1417
  "sup \<equiv> max"
haftmann@22483
  1418
  by intro_classes (auto simp add: inf_nat_def sup_nat_def)
haftmann@22483
  1419
krauss@22157
  1420
krauss@22157
  1421
subsection {* Size function *}
krauss@22157
  1422
haftmann@22920
  1423
lemma nat_size [simp, code func]: "size (n\<Colon>nat) = n"
krauss@22157
  1424
  by (induct n) simp_all
krauss@22157
  1425
clasohm@923
  1426
end