src/HOL/Nat.thy
author nipkow
Mon Aug 27 14:19:38 2007 +0200 (2007-08-27)
changeset 24438 2d8058804a76
parent 24286 7619080e49f0
child 24523 cd723b2209ea
permissions -rw-r--r--
Added infinite_descent
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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/cancel_sums.ML"
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  ("arith_data.ML")
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  "~~/src/Provers/Arith/fast_lin_arith.ML"
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  ("Tools/lin_arith.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] = Zero_nat_def One_nat_def 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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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,noatp]: "(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])
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paulson@14267
   342
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
berghofe@13449
   343
  by (rule less_Suc_eq_le [THEN iffD2])
berghofe@13449
   344
paulson@14267
   345
lemma le0 [iff]: "(0::nat) \<le> n"
wenzelm@22718
   346
  unfolding le_def by (rule not_less0)
berghofe@13449
   347
paulson@14267
   348
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
berghofe@13449
   349
  by (simp add: le_def)
berghofe@13449
   350
paulson@14267
   351
lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)"
berghofe@13449
   352
  by (induct i) (simp_all add: le_def)
berghofe@13449
   353
paulson@14267
   354
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
berghofe@13449
   355
  by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   356
paulson@14267
   357
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
nipkow@17589
   358
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   359
paulson@14267
   360
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
berghofe@13449
   361
  apply (simp add: le_def less_Suc_eq)
berghofe@13449
   362
  apply (blast elim!: less_irrefl less_asym)
berghofe@13449
   363
  done -- {* formerly called lessD *}
berghofe@13449
   364
paulson@14267
   365
lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n"
berghofe@13449
   366
  by (simp add: le_def less_Suc_eq)
berghofe@13449
   367
berghofe@13449
   368
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   369
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
berghofe@13449
   370
  apply (simp add: le_def less_Suc_eq)
berghofe@13449
   371
  using less_linear
berghofe@13449
   372
  apply blast
berghofe@13449
   373
  done
berghofe@13449
   374
paulson@14267
   375
lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)"
berghofe@13449
   376
  by (blast intro: Suc_leI Suc_le_lessD)
berghofe@13449
   377
paulson@14267
   378
lemma le_SucI: "m \<le> n ==> m \<le> Suc n"
berghofe@13449
   379
  by (unfold le_def) (blast dest: Suc_lessD)
berghofe@13449
   380
paulson@14267
   381
lemma less_imp_le: "m < n ==> m \<le> (n::nat)"
berghofe@13449
   382
  by (unfold le_def) (blast elim: less_asym)
berghofe@13449
   383
paulson@14267
   384
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
berghofe@13449
   385
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq
berghofe@13449
   386
berghofe@13449
   387
paulson@14267
   388
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   389
paulson@14267
   390
lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)"
wenzelm@22718
   391
  unfolding le_def
berghofe@13449
   392
  using less_linear
wenzelm@22718
   393
  by (blast elim: less_irrefl less_asym)
berghofe@13449
   394
paulson@14267
   395
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
wenzelm@22718
   396
  unfolding le_def
berghofe@13449
   397
  using less_linear
wenzelm@22718
   398
  by (blast elim!: less_irrefl elim: less_asym)
berghofe@13449
   399
paulson@14267
   400
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
nipkow@17589
   401
  by (iprover intro: less_or_eq_imp_le le_imp_less_or_eq)
berghofe@13449
   402
wenzelm@22718
   403
text {* Useful with @{text blast}. *}
paulson@14267
   404
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
wenzelm@22718
   405
  by (rule less_or_eq_imp_le) (rule disjI2)
berghofe@13449
   406
paulson@14267
   407
lemma le_refl: "n \<le> (n::nat)"
berghofe@13449
   408
  by (simp add: le_eq_less_or_eq)
berghofe@13449
   409
paulson@14267
   410
lemma le_less_trans: "[| i \<le> j; j < k |] ==> i < (k::nat)"
berghofe@13449
   411
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
berghofe@13449
   412
paulson@14267
   413
lemma less_le_trans: "[| i < j; j \<le> k |] ==> i < (k::nat)"
berghofe@13449
   414
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
berghofe@13449
   415
paulson@14267
   416
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
berghofe@13449
   417
  by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans)
berghofe@13449
   418
paulson@14267
   419
lemma le_anti_sym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
berghofe@13449
   420
  by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym)
berghofe@13449
   421
paulson@14267
   422
lemma Suc_le_mono [iff]: "(Suc n \<le> Suc m) = (n \<le> m)"
berghofe@13449
   423
  by (simp add: le_simps)
berghofe@13449
   424
berghofe@13449
   425
text {* Axiom @{text order_less_le} of class @{text order}: *}
paulson@14267
   426
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
berghofe@13449
   427
  by (simp add: le_def nat_neq_iff) (blast elim!: less_asym)
berghofe@13449
   428
paulson@14267
   429
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
berghofe@13449
   430
  by (rule iffD2, rule nat_less_le, rule conjI)
berghofe@13449
   431
berghofe@13449
   432
text {* Axiom @{text linorder_linear} of class @{text linorder}: *}
paulson@14267
   433
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
berghofe@13449
   434
  apply (simp add: le_eq_less_or_eq)
wenzelm@22718
   435
  using less_linear by blast
berghofe@13449
   436
paulson@14341
   437
text {* Type {@typ nat} is a wellfounded linear order *}
paulson@14341
   438
haftmann@22318
   439
instance nat :: wellorder
wenzelm@14691
   440
  by intro_classes
wenzelm@14691
   441
    (assumption |
wenzelm@14691
   442
      rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+
paulson@14341
   443
wenzelm@22718
   444
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   445
berghofe@13449
   446
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
berghofe@13449
   447
  by (blast elim!: less_SucE)
berghofe@13449
   448
berghofe@13449
   449
text {*
berghofe@13449
   450
  Rewrite @{term "n < Suc m"} to @{term "n = m"}
paulson@14267
   451
  if @{term "~ n < m"} or @{term "m \<le> n"} hold.
berghofe@13449
   452
  Not suitable as default simprules because they often lead to looping
berghofe@13449
   453
*}
paulson@14267
   454
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
berghofe@13449
   455
  by (rule not_less_less_Suc_eq, rule leD)
berghofe@13449
   456
berghofe@13449
   457
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   458
berghofe@13449
   459
berghofe@13449
   460
text {*
wenzelm@22718
   461
  Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}.
wenzelm@22718
   462
  No longer added as simprules (they loop)
berghofe@13449
   463
  but via @{text reorient_simproc} in Bin
berghofe@13449
   464
*}
berghofe@13449
   465
berghofe@13449
   466
text {* Polymorphic, not just for @{typ nat} *}
berghofe@13449
   467
lemma zero_reorient: "(0 = x) = (x = 0)"
berghofe@13449
   468
  by auto
berghofe@13449
   469
berghofe@13449
   470
lemma one_reorient: "(1 = x) = (x = 1)"
berghofe@13449
   471
  by auto
berghofe@13449
   472
wenzelm@21243
   473
berghofe@13449
   474
subsection {* Arithmetic operators *}
oheimb@1660
   475
haftmann@22473
   476
class power = type +
haftmann@21411
   477
  fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"            (infixr "\<^loc>^" 80)
wenzelm@9436
   478
berghofe@13449
   479
text {* arithmetic operators @{text "+ -"} and @{text "*"} *}
berghofe@13449
   480
haftmann@21456
   481
instance nat :: "{plus, minus, times}" ..
wenzelm@9436
   482
berghofe@13449
   483
primrec
berghofe@13449
   484
  add_0:    "0 + n = n"
berghofe@13449
   485
  add_Suc:  "Suc m + n = Suc (m + n)"
berghofe@13449
   486
berghofe@13449
   487
primrec
berghofe@13449
   488
  diff_0:   "m - 0 = m"
berghofe@13449
   489
  diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
wenzelm@9436
   490
wenzelm@9436
   491
primrec
berghofe@13449
   492
  mult_0:   "0 * n = 0"
berghofe@13449
   493
  mult_Suc: "Suc m * n = n + (m * n)"
berghofe@13449
   494
wenzelm@22718
   495
text {* These two rules ease the use of primitive recursion.
paulson@14341
   496
NOTE USE OF @{text "=="} *}
berghofe@13449
   497
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
berghofe@13449
   498
  by simp
berghofe@13449
   499
berghofe@13449
   500
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
berghofe@13449
   501
  by simp
berghofe@13449
   502
paulson@14267
   503
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
wenzelm@22718
   504
  by (cases n) simp_all
berghofe@13449
   505
wenzelm@22718
   506
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
wenzelm@22718
   507
  by (cases n) simp_all
berghofe@13449
   508
wenzelm@22718
   509
lemma neq0_conv [iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
wenzelm@22718
   510
  by (cases n) simp_all
berghofe@13449
   511
berghofe@13449
   512
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   513
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@17589
   514
  by (rule iffD1, rule neq0_conv, iprover)
berghofe@13449
   515
paulson@14267
   516
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
berghofe@13449
   517
  by (fast intro: not0_implies_Suc)
berghofe@13449
   518
paulson@24286
   519
lemma not_gr0 [iff,noatp]: "!!n::nat. (~ (0 < n)) = (n = 0)"
berghofe@13449
   520
  apply (rule iffI)
wenzelm@22718
   521
  apply (rule ccontr)
wenzelm@22718
   522
  apply simp_all
berghofe@13449
   523
  done
berghofe@13449
   524
paulson@14267
   525
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
berghofe@13449
   526
  by (induct m') simp_all
berghofe@13449
   527
berghofe@13449
   528
text {* Useful in certain inductive arguments *}
paulson@14267
   529
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
wenzelm@22718
   530
  by (cases m) simp_all
berghofe@13449
   531
paulson@14341
   532
lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"
berghofe@13449
   533
  apply (rule nat_less_induct)
berghofe@13449
   534
  apply (case_tac n)
berghofe@13449
   535
  apply (case_tac [2] nat)
berghofe@13449
   536
  apply (blast intro: less_trans)+
berghofe@13449
   537
  done
berghofe@13449
   538
wenzelm@21243
   539
paulson@15341
   540
subsection {* @{text LEAST} theorems for type @{typ nat}*}
berghofe@13449
   541
paulson@14267
   542
lemma Least_Suc:
paulson@14267
   543
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
paulson@14208
   544
  apply (case_tac "n", auto)
berghofe@13449
   545
  apply (frule LeastI)
berghofe@13449
   546
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
paulson@14267
   547
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
berghofe@13449
   548
  apply (erule_tac [2] Least_le)
paulson@14208
   549
  apply (case_tac "LEAST x. P x", auto)
berghofe@13449
   550
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
berghofe@13449
   551
  apply (blast intro: order_antisym)
berghofe@13449
   552
  done
berghofe@13449
   553
paulson@14267
   554
lemma Least_Suc2:
paulson@14267
   555
     "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
paulson@14267
   556
  by (erule (1) Least_Suc [THEN ssubst], simp)
berghofe@13449
   557
berghofe@13449
   558
berghofe@13449
   559
subsection {* @{term min} and @{term max} *}
berghofe@13449
   560
berghofe@13449
   561
lemma min_0L [simp]: "min 0 n = (0::nat)"
berghofe@13449
   562
  by (rule min_leastL) simp
berghofe@13449
   563
berghofe@13449
   564
lemma min_0R [simp]: "min n 0 = (0::nat)"
berghofe@13449
   565
  by (rule min_leastR) simp
berghofe@13449
   566
berghofe@13449
   567
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
berghofe@13449
   568
  by (simp add: min_of_mono)
berghofe@13449
   569
paulson@22191
   570
lemma min_Suc1:
paulson@22191
   571
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
wenzelm@22718
   572
  by (simp split: nat.split)
paulson@22191
   573
paulson@22191
   574
lemma min_Suc2:
paulson@22191
   575
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
paulson@22191
   576
  by (simp split: nat.split)
paulson@22191
   577
berghofe@13449
   578
lemma max_0L [simp]: "max 0 n = (n::nat)"
berghofe@13449
   579
  by (rule max_leastL) simp
berghofe@13449
   580
berghofe@13449
   581
lemma max_0R [simp]: "max n 0 = (n::nat)"
berghofe@13449
   582
  by (rule max_leastR) simp
berghofe@13449
   583
berghofe@13449
   584
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
berghofe@13449
   585
  by (simp add: max_of_mono)
berghofe@13449
   586
paulson@22191
   587
lemma max_Suc1:
paulson@22191
   588
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
wenzelm@22718
   589
  by (simp split: nat.split)
paulson@22191
   590
paulson@22191
   591
lemma max_Suc2:
paulson@22191
   592
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
paulson@22191
   593
  by (simp split: nat.split)
paulson@22191
   594
berghofe@13449
   595
berghofe@13449
   596
subsection {* Basic rewrite rules for the arithmetic operators *}
berghofe@13449
   597
berghofe@13449
   598
text {* Difference *}
berghofe@13449
   599
berghofe@14193
   600
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
paulson@15251
   601
  by (induct n) simp_all
berghofe@13449
   602
berghofe@14193
   603
lemma diff_Suc_Suc [simp, code]: "Suc(m) - Suc(n) = m - n"
paulson@15251
   604
  by (induct n) simp_all
berghofe@13449
   605
berghofe@13449
   606
berghofe@13449
   607
text {*
berghofe@13449
   608
  Could be (and is, below) generalized in various ways
berghofe@13449
   609
  However, none of the generalizations are currently in the simpset,
berghofe@13449
   610
  and I dread to think what happens if I put them in
berghofe@13449
   611
*}
berghofe@13449
   612
lemma Suc_pred [simp]: "0 < n ==> Suc (n - Suc 0) = n"
berghofe@13449
   613
  by (simp split add: nat.split)
berghofe@13449
   614
berghofe@14193
   615
declare diff_Suc [simp del, code del]
berghofe@13449
   616
berghofe@13449
   617
berghofe@13449
   618
subsection {* Addition *}
berghofe@13449
   619
berghofe@13449
   620
lemma add_0_right [simp]: "m + 0 = (m::nat)"
berghofe@13449
   621
  by (induct m) simp_all
berghofe@13449
   622
berghofe@13449
   623
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
berghofe@13449
   624
  by (induct m) simp_all
berghofe@13449
   625
haftmann@19890
   626
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
haftmann@19890
   627
  by simp
berghofe@14193
   628
berghofe@13449
   629
berghofe@13449
   630
text {* Associative law for addition *}
paulson@14267
   631
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
berghofe@13449
   632
  by (induct m) simp_all
berghofe@13449
   633
berghofe@13449
   634
text {* Commutative law for addition *}
paulson@14267
   635
lemma nat_add_commute: "m + n = n + (m::nat)"
berghofe@13449
   636
  by (induct m) simp_all
berghofe@13449
   637
paulson@14267
   638
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
berghofe@13449
   639
  apply (rule mk_left_commute [of "op +"])
paulson@14267
   640
  apply (rule nat_add_assoc)
paulson@14267
   641
  apply (rule nat_add_commute)
berghofe@13449
   642
  done
berghofe@13449
   643
paulson@14331
   644
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
berghofe@13449
   645
  by (induct k) simp_all
berghofe@13449
   646
paulson@14331
   647
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
berghofe@13449
   648
  by (induct k) simp_all
berghofe@13449
   649
paulson@14331
   650
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
berghofe@13449
   651
  by (induct k) simp_all
berghofe@13449
   652
paulson@14331
   653
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
berghofe@13449
   654
  by (induct k) simp_all
berghofe@13449
   655
berghofe@13449
   656
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
berghofe@13449
   657
wenzelm@22718
   658
lemma add_is_0 [iff]: fixes m :: nat shows "(m + n = 0) = (m = 0 & n = 0)"
wenzelm@22718
   659
  by (cases m) simp_all
berghofe@13449
   660
berghofe@13449
   661
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
wenzelm@22718
   662
  by (cases m) simp_all
berghofe@13449
   663
berghofe@13449
   664
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
berghofe@13449
   665
  by (rule trans, rule eq_commute, rule add_is_1)
berghofe@13449
   666
berghofe@13449
   667
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m | 0 < n)"
berghofe@13449
   668
  by (simp del: neq0_conv add: neq0_conv [symmetric])
berghofe@13449
   669
berghofe@13449
   670
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0"
berghofe@13449
   671
  apply (drule add_0_right [THEN ssubst])
paulson@14267
   672
  apply (simp add: nat_add_assoc del: add_0_right)
berghofe@13449
   673
  done
berghofe@13449
   674
nipkow@16733
   675
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
wenzelm@22718
   676
  apply (induct k)
wenzelm@22718
   677
   apply simp
wenzelm@22718
   678
  apply(drule comp_inj_on[OF _ inj_Suc])
wenzelm@22718
   679
  apply (simp add:o_def)
wenzelm@22718
   680
  done
nipkow@16733
   681
nipkow@16733
   682
paulson@14267
   683
subsection {* Multiplication *}
paulson@14267
   684
paulson@14267
   685
text {* right annihilation in product *}
paulson@14267
   686
lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
paulson@14267
   687
  by (induct m) simp_all
paulson@14267
   688
paulson@14267
   689
text {* right successor law for multiplication *}
paulson@14267
   690
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
paulson@14267
   691
  by (induct m) (simp_all add: nat_add_left_commute)
paulson@14267
   692
paulson@14267
   693
text {* Commutative law for multiplication *}
paulson@14267
   694
lemma nat_mult_commute: "m * n = n * (m::nat)"
paulson@14267
   695
  by (induct m) simp_all
paulson@14267
   696
paulson@14267
   697
text {* addition distributes over multiplication *}
paulson@14267
   698
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
paulson@14267
   699
  by (induct m) (simp_all add: nat_add_assoc nat_add_left_commute)
paulson@14267
   700
paulson@14267
   701
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
paulson@14267
   702
  by (induct m) (simp_all add: nat_add_assoc)
paulson@14267
   703
paulson@14267
   704
text {* Associative law for multiplication *}
paulson@14267
   705
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
paulson@14267
   706
  by (induct m) (simp_all add: add_mult_distrib)
paulson@14267
   707
paulson@14267
   708
nipkow@14740
   709
text{*The naturals form a @{text comm_semiring_1_cancel}*}
obua@14738
   710
instance nat :: comm_semiring_1_cancel
paulson@14267
   711
proof
paulson@14267
   712
  fix i j k :: nat
paulson@14267
   713
  show "(i + j) + k = i + (j + k)" by (rule nat_add_assoc)
paulson@14267
   714
  show "i + j = j + i" by (rule nat_add_commute)
paulson@14267
   715
  show "0 + i = i" by simp
paulson@14267
   716
  show "(i * j) * k = i * (j * k)" by (rule nat_mult_assoc)
paulson@14267
   717
  show "i * j = j * i" by (rule nat_mult_commute)
paulson@14267
   718
  show "1 * i = i" by simp
paulson@14267
   719
  show "(i + j) * k = i * k + j * k" by (simp add: add_mult_distrib)
paulson@14267
   720
  show "0 \<noteq> (1::nat)" by simp
paulson@14341
   721
  assume "k+i = k+j" thus "i=j" by simp
paulson@14341
   722
qed
paulson@14341
   723
paulson@14341
   724
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
paulson@15251
   725
  apply (induct m)
wenzelm@22718
   726
   apply (induct_tac [2] n)
wenzelm@22718
   727
    apply simp_all
paulson@14341
   728
  done
paulson@14341
   729
wenzelm@21243
   730
paulson@14341
   731
subsection {* Monotonicity of Addition *}
paulson@14341
   732
paulson@14341
   733
text {* strict, in 1st argument *}
paulson@14341
   734
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
paulson@14341
   735
  by (induct k) simp_all
paulson@14341
   736
paulson@14341
   737
text {* strict, in both arguments *}
paulson@14341
   738
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   739
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   740
  apply (induct j, simp_all)
paulson@14341
   741
  done
paulson@14341
   742
paulson@14341
   743
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   744
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   745
  apply (induct n)
paulson@14341
   746
  apply (simp_all add: order_le_less)
wenzelm@22718
   747
  apply (blast elim!: less_SucE
paulson@14341
   748
               intro!: add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   749
  done
paulson@14341
   750
paulson@14341
   751
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
paulson@14341
   752
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j"
paulson@14341
   753
  apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp)
wenzelm@22718
   754
  apply (induct_tac x)
paulson@14341
   755
  apply (simp_all add: add_less_mono)
paulson@14341
   756
  done
paulson@14341
   757
paulson@14341
   758
nipkow@14740
   759
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
obua@14738
   760
instance nat :: ordered_semidom
paulson@14341
   761
proof
paulson@14341
   762
  fix i j k :: nat
paulson@14348
   763
  show "0 < (1::nat)" by simp
paulson@14267
   764
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   765
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   766
qed
paulson@14267
   767
paulson@14267
   768
lemma nat_mult_1: "(1::nat) * n = n"
paulson@14267
   769
  by simp
paulson@14267
   770
paulson@14267
   771
lemma nat_mult_1_right: "n * (1::nat) = n"
paulson@14267
   772
  by simp
paulson@14267
   773
paulson@14267
   774
paulson@14267
   775
subsection {* Additional theorems about "less than" *}
paulson@14267
   776
paulson@19870
   777
text{*An induction rule for estabilishing binary relations*}
wenzelm@22718
   778
lemma less_Suc_induct:
paulson@19870
   779
  assumes less:  "i < j"
paulson@19870
   780
     and  step:  "!!i. P i (Suc i)"
paulson@19870
   781
     and  trans: "!!i j k. P i j ==> P j k ==> P i k"
paulson@19870
   782
  shows "P i j"
paulson@19870
   783
proof -
wenzelm@22718
   784
  from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   785
  have "P i (Suc (i + k))"
paulson@19870
   786
  proof (induct k)
wenzelm@22718
   787
    case 0
wenzelm@22718
   788
    show ?case by (simp add: step)
paulson@19870
   789
  next
paulson@19870
   790
    case (Suc k)
wenzelm@22718
   791
    thus ?case by (auto intro: assms)
paulson@19870
   792
  qed
wenzelm@22718
   793
  thus "P i j" by (simp add: j)
paulson@19870
   794
qed
paulson@19870
   795
nipkow@24438
   796
text {* The method of infinite descent, frequently used in number theory.
nipkow@24438
   797
Provided by Roelof Oosterhuis.
nipkow@24438
   798
$P(n)$ is true for all $n\in\mathbb{N}$ if
nipkow@24438
   799
\begin{itemize}
nipkow@24438
   800
  \item case ``0'': given $n=0$ prove $P(n)$,
nipkow@24438
   801
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
nipkow@24438
   802
        a smaller integer $m$ such that $\neg P(m)$.
nipkow@24438
   803
\end{itemize} *}
nipkow@24438
   804
nipkow@24438
   805
lemma infinite_descent[case_names 0 smaller]: 
nipkow@24438
   806
  "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
nipkow@24438
   807
by (induct n rule: less_induct, case_tac "n>0", auto)
nipkow@24438
   808
nipkow@24438
   809
nipkow@24438
   810
text {* Infinite descent using a mapping to $\mathbb{N}$:
nipkow@24438
   811
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
nipkow@24438
   812
\begin{itemize}
nipkow@24438
   813
\item case ``0'': given $V(x)=0$ prove $P(x)$,
nipkow@24438
   814
\item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$.
nipkow@24438
   815
\end{itemize}
nipkow@24438
   816
NB: the proof also shows how to use the previous lemma. *}
nipkow@24438
   817
corollary infinite_descent_measure[case_names 0 smaller]:
nipkow@24438
   818
fixes V :: "'a \<Rightarrow> nat" 
nipkow@24438
   819
assumes 0: "!!x. V x = 0 \<Longrightarrow> P x" 
nipkow@24438
   820
and 1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
nipkow@24438
   821
shows "P x"
nipkow@24438
   822
proof -
nipkow@24438
   823
  obtain n where "n = V x" by auto
nipkow@24438
   824
  moreover have "!!x. V x = (n::nat) \<Longrightarrow> P x"
nipkow@24438
   825
  proof (induct n rule: infinite_descent)
nipkow@24438
   826
    case 0 -- "i.e. $V(x) = 0$"
nipkow@24438
   827
    with 0 show "P x" by auto
nipkow@24438
   828
  next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
nipkow@24438
   829
    case (smaller n)
nipkow@24438
   830
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
nipkow@24438
   831
    with 1 obtain y where "V y < V x \<and> \<not> P y" by auto
nipkow@24438
   832
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
nipkow@24438
   833
    thus ?case by auto
nipkow@24438
   834
  qed
nipkow@24438
   835
  ultimately show "P x" by auto
nipkow@24438
   836
qed
paulson@19870
   837
paulson@14267
   838
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   839
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   840
lemma less_mono_imp_le_mono:
nipkow@24438
   841
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
   842
by (simp add: order_le_less) (blast)
nipkow@24438
   843
paulson@14267
   844
paulson@14267
   845
text {* non-strict, in 1st argument *}
paulson@14267
   846
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
   847
by (rule add_right_mono)
paulson@14267
   848
paulson@14267
   849
text {* non-strict, in both arguments *}
paulson@14267
   850
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
   851
by (rule add_mono)
paulson@14267
   852
paulson@14267
   853
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
   854
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
   855
paulson@14267
   856
lemma le_add1: "n \<le> ((n + m)::nat)"
nipkow@24438
   857
by (simp add: add_commute, rule le_add2)
berghofe@13449
   858
berghofe@13449
   859
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
   860
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   861
berghofe@13449
   862
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
   863
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   864
paulson@14267
   865
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
   866
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   867
paulson@14267
   868
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
   869
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   870
paulson@14267
   871
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
   872
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
   873
berghofe@13449
   874
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
   875
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
   876
berghofe@13449
   877
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
   878
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
   879
berghofe@13449
   880
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
   881
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
   882
apply (simp_all add: le_add1)
nipkow@24438
   883
done
berghofe@13449
   884
berghofe@13449
   885
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
   886
apply (rule notI)
nipkow@24438
   887
apply (erule add_lessD1 [THEN less_irrefl])
nipkow@24438
   888
done
berghofe@13449
   889
berghofe@13449
   890
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
nipkow@24438
   891
by (simp add: add_commute not_add_less1)
berghofe@13449
   892
paulson@14267
   893
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
   894
apply (rule order_trans [of _ "m+k"])
nipkow@24438
   895
apply (simp_all add: le_add1)
nipkow@24438
   896
done
berghofe@13449
   897
paulson@14267
   898
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
nipkow@24438
   899
apply (simp add: add_commute)
nipkow@24438
   900
apply (erule add_leD1)
nipkow@24438
   901
done
berghofe@13449
   902
paulson@14267
   903
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
   904
by (blast dest: add_leD1 add_leD2)
berghofe@13449
   905
berghofe@13449
   906
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
   907
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
   908
by (force simp del: add_Suc_right
berghofe@13449
   909
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
   910
berghofe@13449
   911
berghofe@13449
   912
subsection {* Difference *}
berghofe@13449
   913
berghofe@13449
   914
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
nipkow@24438
   915
by (induct m) simp_all
berghofe@13449
   916
berghofe@13449
   917
text {* Addition is the inverse of subtraction:
paulson@14267
   918
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
   919
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
nipkow@24438
   920
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   921
paulson@14267
   922
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
nipkow@24438
   923
by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
   924
paulson@14267
   925
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
nipkow@24438
   926
by (simp add: le_add_diff_inverse add_commute)
berghofe@13449
   927
berghofe@13449
   928
berghofe@13449
   929
subsection {* More results about difference *}
berghofe@13449
   930
paulson@14267
   931
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
   932
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   933
berghofe@13449
   934
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
   935
apply (induct m n rule: diff_induct)
nipkow@24438
   936
apply (erule_tac [3] less_SucE)
nipkow@24438
   937
apply (simp_all add: less_Suc_eq)
nipkow@24438
   938
done
berghofe@13449
   939
paulson@14267
   940
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
   941
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
   942
berghofe@13449
   943
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
   944
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
   945
berghofe@13449
   946
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
nipkow@24438
   947
by (induct i j rule: diff_induct) simp_all
berghofe@13449
   948
berghofe@13449
   949
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
nipkow@24438
   950
by (simp add: diff_diff_left)
berghofe@13449
   951
berghofe@13449
   952
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
   953
by (cases n) (auto simp add: le_simps)
berghofe@13449
   954
berghofe@13449
   955
text {* This and the next few suggested by Florian Kammueller *}
berghofe@13449
   956
lemma diff_commute: "(i::nat) - j - k = i - k - j"
nipkow@24438
   957
by (simp add: diff_diff_left add_commute)
berghofe@13449
   958
paulson@14267
   959
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
   960
by (induct j k rule: diff_induct) simp_all
berghofe@13449
   961
paulson@14267
   962
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
nipkow@24438
   963
by (simp add: add_commute diff_add_assoc)
berghofe@13449
   964
berghofe@13449
   965
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
nipkow@24438
   966
by (induct n) simp_all
berghofe@13449
   967
berghofe@13449
   968
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
nipkow@24438
   969
by (simp add: diff_add_assoc)
berghofe@13449
   970
paulson@14267
   971
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
nipkow@24438
   972
by (auto simp add: diff_add_inverse2)
berghofe@13449
   973
paulson@14267
   974
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
   975
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   976
paulson@14267
   977
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
   978
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
   979
berghofe@13449
   980
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
   981
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   982
wenzelm@22718
   983
lemma less_imp_add_positive:
wenzelm@22718
   984
  assumes "i < j"
wenzelm@22718
   985
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
   986
proof
wenzelm@22718
   987
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
   988
    by (simp add: order_less_imp_le)
wenzelm@22718
   989
qed
wenzelm@9436
   990
berghofe@13449
   991
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
nipkow@24438
   992
by (induct k) simp_all
berghofe@13449
   993
berghofe@13449
   994
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
nipkow@24438
   995
by (simp add: diff_cancel add_commute)
berghofe@13449
   996
berghofe@13449
   997
lemma diff_add_0: "n - (n + m) = (0::nat)"
nipkow@24438
   998
by (induct n) simp_all
berghofe@13449
   999
berghofe@13449
  1000
berghofe@13449
  1001
text {* Difference distributes over multiplication *}
berghofe@13449
  1002
berghofe@13449
  1003
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
nipkow@24438
  1004
by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
berghofe@13449
  1005
berghofe@13449
  1006
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
nipkow@24438
  1007
by (simp add: diff_mult_distrib mult_commute [of k])
berghofe@13449
  1008
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
berghofe@13449
  1009
berghofe@13449
  1010
lemmas nat_distrib =
berghofe@13449
  1011
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
berghofe@13449
  1012
berghofe@13449
  1013
berghofe@13449
  1014
subsection {* Monotonicity of Multiplication *}
berghofe@13449
  1015
paulson@14267
  1016
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
  1017
by (simp add: mult_right_mono)
berghofe@13449
  1018
paulson@14267
  1019
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
  1020
by (simp add: mult_left_mono)
berghofe@13449
  1021
paulson@14267
  1022
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
  1023
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
  1024
by (simp add: mult_mono)
berghofe@13449
  1025
berghofe@13449
  1026
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
  1027
by (simp add: mult_strict_right_mono)
berghofe@13449
  1028
paulson@14266
  1029
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
  1030
      there are no negative numbers.*}
paulson@14266
  1031
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
  1032
  apply (induct m)
wenzelm@22718
  1033
   apply simp
wenzelm@22718
  1034
  apply (case_tac n)
wenzelm@22718
  1035
   apply simp_all
berghofe@13449
  1036
  done
berghofe@13449
  1037
paulson@14267
  1038
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"
berghofe@13449
  1039
  apply (induct m)
wenzelm@22718
  1040
   apply simp
wenzelm@22718
  1041
  apply (case_tac n)
wenzelm@22718
  1042
   apply simp_all
berghofe@13449
  1043
  done
berghofe@13449
  1044
berghofe@13449
  1045
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
wenzelm@22718
  1046
  apply (induct m)
wenzelm@22718
  1047
   apply simp
wenzelm@22718
  1048
  apply (induct n)
wenzelm@22718
  1049
   apply auto
berghofe@13449
  1050
  done
berghofe@13449
  1051
paulson@24286
  1052
lemma one_eq_mult_iff [simp,noatp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
berghofe@13449
  1053
  apply (rule trans)
paulson@14208
  1054
  apply (rule_tac [2] mult_eq_1_iff, fastsimp)
berghofe@13449
  1055
  done
berghofe@13449
  1056
paulson@14341
  1057
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1058
  apply (safe intro!: mult_less_mono1)
paulson@14208
  1059
  apply (case_tac k, auto)
berghofe@13449
  1060
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1061
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1062
  done
berghofe@13449
  1063
berghofe@13449
  1064
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
nipkow@24438
  1065
by (simp add: mult_commute [of k])
berghofe@13449
  1066
paulson@14267
  1067
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
  1068
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1069
paulson@14267
  1070
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
  1071
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1072
paulson@14341
  1073
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
paulson@14208
  1074
  apply (cut_tac less_linear, safe, auto)
berghofe@13449
  1075
  apply (drule mult_less_mono1, assumption, simp)+
berghofe@13449
  1076
  done
berghofe@13449
  1077
berghofe@13449
  1078
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
nipkow@24438
  1079
by (simp add: mult_commute [of k])
berghofe@13449
  1080
berghofe@13449
  1081
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
  1082
by (subst mult_less_cancel1) simp
berghofe@13449
  1083
paulson@14267
  1084
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
  1085
by (subst mult_le_cancel1) simp
berghofe@13449
  1086
berghofe@13449
  1087
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
nipkow@24438
  1088
by (subst mult_cancel1) simp
berghofe@13449
  1089
berghofe@13449
  1090
text {* Lemma for @{text gcd} *}
berghofe@13449
  1091
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1092
  apply (drule sym)
berghofe@13449
  1093
  apply (rule disjCI)
berghofe@13449
  1094
  apply (rule nat_less_cases, erule_tac [2] _)
berghofe@13449
  1095
  apply (fastsimp elim!: less_SucE)
berghofe@13449
  1096
  apply (fastsimp dest: mult_less_mono2)
berghofe@13449
  1097
  done
wenzelm@9436
  1098
haftmann@20588
  1099
haftmann@18702
  1100
subsection {* Code generator setup *}
haftmann@18702
  1101
wenzelm@22718
  1102
lemma one_is_Suc_zero [code inline]: "1 = Suc 0"
nipkow@24438
  1103
by simp
haftmann@20355
  1104
haftmann@20588
  1105
instance nat :: eq ..
haftmann@20588
  1106
haftmann@20588
  1107
lemma [code func]:
wenzelm@22718
  1108
    "(0\<Colon>nat) = 0 \<longleftrightarrow> True"
wenzelm@22718
  1109
    "Suc n = Suc m \<longleftrightarrow> n = m"
wenzelm@22718
  1110
    "Suc n = 0 \<longleftrightarrow> False"
wenzelm@22718
  1111
    "0 = Suc m \<longleftrightarrow> False"
nipkow@24438
  1112
by auto
haftmann@20588
  1113
haftmann@20588
  1114
lemma [code func]:
wenzelm@22718
  1115
    "(0\<Colon>nat) \<le> m \<longleftrightarrow> True"
wenzelm@22718
  1116
    "Suc (n\<Colon>nat) \<le> m \<longleftrightarrow> n < m"
wenzelm@22718
  1117
    "(n\<Colon>nat) < 0 \<longleftrightarrow> False"
wenzelm@22718
  1118
    "(n\<Colon>nat) < Suc m \<longleftrightarrow> n \<le> m"
haftmann@22348
  1119
  using Suc_le_eq less_Suc_eq_le by simp_all
haftmann@20588
  1120
wenzelm@21243
  1121
haftmann@24196
  1122
subsection{*Embedding of the Naturals into any
haftmann@24196
  1123
  @{text semiring_1}: @{term of_nat}*}
haftmann@24196
  1124
haftmann@24196
  1125
context semiring_1
haftmann@24196
  1126
begin
haftmann@24196
  1127
haftmann@24196
  1128
definition
haftmann@24196
  1129
  of_nat_def: "of_nat = nat_rec \<^loc>0 (\<lambda>_. (op \<^loc>+) \<^loc>1)"
haftmann@24196
  1130
haftmann@24196
  1131
end
haftmann@24196
  1132
wenzelm@21243
  1133
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
wenzelm@21243
  1134
haftmann@22845
  1135
lemma subst_equals:
haftmann@22845
  1136
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1137
  shows "u = s"
haftmann@22845
  1138
  using 2 1 by (rule trans)
haftmann@22845
  1139
haftmann@24196
  1140
wenzelm@21243
  1141
use "arith_data.ML"
wenzelm@24091
  1142
declaration {* K arith_data_setup *}
wenzelm@24091
  1143
wenzelm@24091
  1144
use "Tools/lin_arith.ML"
wenzelm@24091
  1145
declaration {* K LinArith.setup *}
wenzelm@24091
  1146
wenzelm@21243
  1147
wenzelm@21243
  1148
text{*The following proofs may rely on the arithmetic proof procedures.*}
wenzelm@21243
  1149
wenzelm@21243
  1150
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
nipkow@24438
  1151
by (auto simp: le_eq_less_or_eq dest: less_imp_Suc_add)
wenzelm@21243
  1152
wenzelm@21243
  1153
lemma pred_nat_trancl_eq_le: "((m, n) : pred_nat^*) = (m \<le> n)"
nipkow@24438
  1154
by (simp add: less_eq reflcl_trancl [symmetric] del: reflcl_trancl, arith)
wenzelm@21243
  1155
wenzelm@21243
  1156
lemma nat_diff_split:
wenzelm@22718
  1157
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
wenzelm@21243
  1158
    -- {* elimination of @{text -} on @{text nat} *}
nipkow@24438
  1159
by (cases "a<b" rule: case_split) (auto simp add: diff_is_0_eq [THEN iffD2])
wenzelm@21243
  1160
wenzelm@21243
  1161
lemma nat_diff_split_asm:
wenzelm@21243
  1162
    "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
wenzelm@21243
  1163
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
nipkow@24438
  1164
by (simp split: nat_diff_split)
wenzelm@21243
  1165
wenzelm@21243
  1166
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1167
wenzelm@21243
  1168
wenzelm@21243
  1169
lemma le_square: "m \<le> m * (m::nat)"
nipkow@24438
  1170
by (induct m) auto
wenzelm@21243
  1171
wenzelm@21243
  1172
lemma le_cube: "(m::nat) \<le> m * (m * m)"
nipkow@24438
  1173
by (induct m) auto
wenzelm@21243
  1174
wenzelm@21243
  1175
wenzelm@21243
  1176
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1177
wenzelm@21243
  1178
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1179
by arith
wenzelm@21243
  1180
wenzelm@21243
  1181
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1182
by arith
wenzelm@21243
  1183
wenzelm@21243
  1184
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1185
by arith
wenzelm@21243
  1186
wenzelm@21243
  1187
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
nipkow@24438
  1188
by arith
wenzelm@21243
  1189
wenzelm@21243
  1190
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1191
by arith
wenzelm@21243
  1192
wenzelm@21243
  1193
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
nipkow@24438
  1194
by arith
wenzelm@21243
  1195
wenzelm@21243
  1196
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1197
  second premise n\<le>m*)
wenzelm@21243
  1198
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1199
by arith
wenzelm@21243
  1200
wenzelm@21243
  1201
wenzelm@21243
  1202
(** Simplification of relational expressions involving subtraction **)
wenzelm@21243
  1203
wenzelm@21243
  1204
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1205
by (simp split add: nat_diff_split)
wenzelm@21243
  1206
wenzelm@21243
  1207
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1208
by (auto split add: nat_diff_split)
wenzelm@21243
  1209
wenzelm@21243
  1210
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1211
by (auto split add: nat_diff_split)
wenzelm@21243
  1212
wenzelm@21243
  1213
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1214
by (auto split add: nat_diff_split)
wenzelm@21243
  1215
wenzelm@21243
  1216
wenzelm@21243
  1217
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1218
wenzelm@21243
  1219
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1220
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1221
by (simp split add: nat_diff_split)
wenzelm@21243
  1222
wenzelm@21243
  1223
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1224
by (simp split add: nat_diff_split)
wenzelm@21243
  1225
wenzelm@21243
  1226
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1227
by (simp split add: nat_diff_split)
wenzelm@21243
  1228
wenzelm@21243
  1229
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1230
by (simp split add: nat_diff_split)
wenzelm@21243
  1231
wenzelm@21243
  1232
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1233
wenzelm@21243
  1234
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1235
by (cases m) auto
wenzelm@21243
  1236
wenzelm@21243
  1237
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1238
by (cases m) auto
wenzelm@21243
  1239
wenzelm@21243
  1240
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1241
by (cases m) auto
wenzelm@21243
  1242
krauss@23001
  1243
text {* Specialized induction principles that work "backwards": *}
krauss@23001
  1244
krauss@23001
  1245
lemma inc_induct[consumes 1, case_names base step]:
krauss@23001
  1246
  assumes less: "i <= j"
krauss@23001
  1247
  assumes base: "P j"
krauss@23001
  1248
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1249
  shows "P i"
krauss@23001
  1250
  using less
krauss@23001
  1251
proof (induct d=="j - i" arbitrary: i)
krauss@23001
  1252
  case (0 i)
krauss@23001
  1253
  hence "i = j" by simp
krauss@23001
  1254
  with base show ?case by simp
krauss@23001
  1255
next
krauss@23001
  1256
  case (Suc d i)
krauss@23001
  1257
  hence "i < j" "P (Suc i)"
krauss@23001
  1258
    by simp_all
krauss@23001
  1259
  thus "P i" by (rule step)
krauss@23001
  1260
qed
krauss@23001
  1261
krauss@23001
  1262
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1263
  assumes less: "i < j"
krauss@23001
  1264
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1265
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1266
  shows "P i"
krauss@23001
  1267
  using less
krauss@23001
  1268
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1269
  case (0 i)
krauss@23001
  1270
  with `i < j` have "j = Suc i" by simp
krauss@23001
  1271
  with base show ?case by simp
krauss@23001
  1272
next
krauss@23001
  1273
  case (Suc d i)
krauss@23001
  1274
  hence "i < j" "P (Suc i)"
krauss@23001
  1275
    by simp_all
krauss@23001
  1276
  thus "P i" by (rule step)
krauss@23001
  1277
qed
krauss@23001
  1278
krauss@23001
  1279
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1280
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1281
krauss@23001
  1282
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1283
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1284
wenzelm@21243
  1285
text{*Rewriting to pull differences out*}
wenzelm@21243
  1286
wenzelm@21243
  1287
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
nipkow@24438
  1288
by arith
wenzelm@21243
  1289
wenzelm@21243
  1290
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
nipkow@24438
  1291
by arith
wenzelm@21243
  1292
wenzelm@21243
  1293
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
nipkow@24438
  1294
by arith
wenzelm@21243
  1295
wenzelm@21243
  1296
(*The others are
wenzelm@21243
  1297
      i - j - k = i - (j + k),
wenzelm@21243
  1298
      k \<le> j ==> j - k + i = j + i - k,
wenzelm@21243
  1299
      k \<le> j ==> i + (j - k) = i + j - k *)
wenzelm@21243
  1300
lemmas add_diff_assoc = diff_add_assoc [symmetric]
wenzelm@21243
  1301
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
wenzelm@21243
  1302
declare diff_diff_left [simp]  add_diff_assoc [simp]  add_diff_assoc2[simp]
wenzelm@21243
  1303
wenzelm@21243
  1304
text{*At present we prove no analogue of @{text not_less_Least} or @{text
wenzelm@21243
  1305
Least_Suc}, since there appears to be no need.*}
wenzelm@21243
  1306
wenzelm@22718
  1307
wenzelm@22718
  1308
subsection{*Embedding of the Naturals into any
huffman@23276
  1309
  @{text semiring_1}: @{term of_nat}*}
wenzelm@21243
  1310
haftmann@24196
  1311
context semiring_1
haftmann@24196
  1312
begin
wenzelm@21243
  1313
haftmann@24196
  1314
lemma of_nat_simps [simp, code]:
haftmann@24196
  1315
  shows of_nat_0:   "of_nat 0 = \<^loc>0"
haftmann@24196
  1316
    and of_nat_Suc: "of_nat (Suc m) = \<^loc>1 \<^loc>+ of_nat m"
haftmann@24196
  1317
  unfolding of_nat_def by simp_all
haftmann@24196
  1318
haftmann@24196
  1319
end
wenzelm@21243
  1320
haftmann@22920
  1321
lemma of_nat_id [simp]: "(of_nat n \<Colon> nat) = n"
nipkow@24438
  1322
by (induct n) auto
haftmann@22920
  1323
wenzelm@21243
  1324
lemma of_nat_1 [simp]: "of_nat 1 = 1"
nipkow@24438
  1325
by simp
wenzelm@21243
  1326
haftmann@24196
  1327
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
nipkow@24438
  1328
by (induct m) (simp_all add: add_ac)
wenzelm@21243
  1329
huffman@23431
  1330
lemma of_nat_mult: "of_nat (m*n) = of_nat m * of_nat n"
nipkow@24438
  1331
by (induct m) (simp_all add: add_ac left_distrib)
wenzelm@21243
  1332
wenzelm@21243
  1333
lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)"
wenzelm@22718
  1334
  apply (induct m, simp_all)
wenzelm@22718
  1335
  apply (erule order_trans)
huffman@23431
  1336
  apply (rule ord_le_eq_trans [OF _ add_commute])
wenzelm@22718
  1337
  apply (rule less_add_one [THEN order_less_imp_le])
wenzelm@22718
  1338
  done
wenzelm@21243
  1339
wenzelm@21243
  1340
lemma less_imp_of_nat_less:
wenzelm@22718
  1341
    "m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)"
wenzelm@22718
  1342
  apply (induct m n rule: diff_induct, simp_all)
huffman@23431
  1343
  apply (insert add_less_le_mono [OF zero_less_one zero_le_imp_of_nat], force)
wenzelm@22718
  1344
  done
wenzelm@21243
  1345
wenzelm@21243
  1346
lemma of_nat_less_imp_less:
wenzelm@22718
  1347
    "of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n"
wenzelm@22718
  1348
  apply (induct m n rule: diff_induct, simp_all)
wenzelm@22718
  1349
  apply (insert zero_le_imp_of_nat)
wenzelm@22718
  1350
  apply (force simp add: linorder_not_less [symmetric])
wenzelm@22718
  1351
  done
wenzelm@21243
  1352
wenzelm@21243
  1353
lemma of_nat_less_iff [simp]:
wenzelm@22718
  1354
    "(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)"
nipkow@24438
  1355
by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
wenzelm@21243
  1356
wenzelm@21243
  1357
text{*Special cases where either operand is zero*}
wenzelm@22718
  1358
wenzelm@22718
  1359
lemma of_nat_0_less_iff [simp]: "((0::'a::ordered_semidom) < of_nat n) = (0 < n)"
nipkow@24438
  1360
by (rule of_nat_less_iff [of 0, simplified])
wenzelm@22718
  1361
wenzelm@22718
  1362
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < (0::'a::ordered_semidom)"
nipkow@24438
  1363
by (rule of_nat_less_iff [of _ 0, simplified])
wenzelm@21243
  1364
wenzelm@21243
  1365
lemma of_nat_le_iff [simp]:
wenzelm@22718
  1366
    "(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)"
nipkow@24438
  1367
by (simp add: linorder_not_less [symmetric])
wenzelm@21243
  1368
wenzelm@21243
  1369
text{*Special cases where either operand is zero*}
wenzelm@22718
  1370
lemma of_nat_0_le_iff [simp]: "(0::'a::ordered_semidom) \<le> of_nat n"
nipkow@24438
  1371
by (rule of_nat_le_iff [of 0, simplified])
paulson@24286
  1372
lemma of_nat_le_0_iff [simp,noatp]: "(of_nat m \<le> (0::'a::ordered_semidom)) = (m = 0)"
nipkow@24438
  1373
by (rule of_nat_le_iff [of _ 0, simplified])
wenzelm@21243
  1374
huffman@23282
  1375
text{*Class for unital semirings with characteristic zero.
huffman@23282
  1376
 Includes non-ordered rings like the complex numbers.*}
haftmann@24196
  1377
haftmann@24196
  1378
class semiring_char_0 = semiring_1 +
haftmann@24196
  1379
  assumes of_nat_eq_iff [simp]:
haftmann@24196
  1380
    "(Nat.semiring_1.of_nat \<^loc>1 \<^loc>0 (op \<^loc>+) m = Nat.semiring_1.of_nat \<^loc>1 \<^loc>0 (op \<^loc>+)  n) = (m = n)"
huffman@23282
  1381
huffman@23282
  1382
text{*Every @{text ordered_semidom} has characteristic zero.*}
huffman@23282
  1383
instance ordered_semidom < semiring_char_0
huffman@23282
  1384
by intro_classes (simp add: order_eq_iff)
wenzelm@21243
  1385
wenzelm@21243
  1386
text{*Special cases where either operand is zero*}
paulson@24286
  1387
lemma of_nat_0_eq_iff [simp,noatp]: "((0::'a::semiring_char_0) = of_nat n) = (0 = n)"
nipkow@24438
  1388
by (rule of_nat_eq_iff [of 0, simplified])
paulson@24286
  1389
lemma of_nat_eq_0_iff [simp,noatp]: "(of_nat m = (0::'a::semiring_char_0)) = (m = 0)"
nipkow@24438
  1390
by (rule of_nat_eq_iff [of _ 0, simplified])
wenzelm@21243
  1391
huffman@23347
  1392
lemma inj_of_nat: "inj (of_nat :: nat \<Rightarrow> 'a::semiring_char_0)"
nipkow@24438
  1393
by (simp add: inj_on_def)
huffman@23347
  1394
huffman@23438
  1395
lemma of_nat_diff:
wenzelm@22718
  1396
    "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::ring_1)"
nipkow@24438
  1397
by (simp del: of_nat_add
wenzelm@22718
  1398
    add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
wenzelm@21243
  1399
haftmann@24196
  1400
lemma abs_of_nat [simp]: "\<bar>of_nat n::'a::ordered_idom\<bar> = of_nat n"
nipkow@24438
  1401
by (rule of_nat_0_le_iff [THEN abs_of_nonneg])
haftmann@24196
  1402
haftmann@23852
  1403
haftmann@23852
  1404
subsection {*The Set of Natural Numbers*}
haftmann@23852
  1405
haftmann@23852
  1406
definition
haftmann@23852
  1407
  Nats  :: "'a::semiring_1 set"
haftmann@23852
  1408
where
haftmann@23852
  1409
  "Nats = range of_nat"
haftmann@23852
  1410
haftmann@23852
  1411
notation (xsymbols)
haftmann@23852
  1412
  Nats  ("\<nat>")
haftmann@23852
  1413
haftmann@23852
  1414
lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"
nipkow@24438
  1415
by (simp add: Nats_def)
haftmann@23852
  1416
haftmann@23852
  1417
lemma Nats_0 [simp]: "0 \<in> Nats"
haftmann@23852
  1418
apply (simp add: Nats_def)
haftmann@23852
  1419
apply (rule range_eqI)
haftmann@23852
  1420
apply (rule of_nat_0 [symmetric])
haftmann@23852
  1421
done
haftmann@23852
  1422
haftmann@23852
  1423
lemma Nats_1 [simp]: "1 \<in> Nats"
haftmann@23852
  1424
apply (simp add: Nats_def)
haftmann@23852
  1425
apply (rule range_eqI)
haftmann@23852
  1426
apply (rule of_nat_1 [symmetric])
haftmann@23852
  1427
done
haftmann@23852
  1428
haftmann@23852
  1429
lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"
haftmann@23852
  1430
apply (auto simp add: Nats_def)
haftmann@23852
  1431
apply (rule range_eqI)
haftmann@23852
  1432
apply (rule of_nat_add [symmetric])
haftmann@23852
  1433
done
haftmann@23852
  1434
haftmann@23852
  1435
lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"
haftmann@23852
  1436
apply (auto simp add: Nats_def)
haftmann@23852
  1437
apply (rule range_eqI)
haftmann@23852
  1438
apply (rule of_nat_mult [symmetric])
haftmann@23852
  1439
done
haftmann@23852
  1440
haftmann@23852
  1441
lemma of_nat_eq_id [simp]: "of_nat = (id :: nat => nat)"
nipkow@24438
  1442
by (auto simp add: expand_fun_eq)
haftmann@23852
  1443
haftmann@23852
  1444
haftmann@22483
  1445
instance nat :: distrib_lattice
haftmann@22483
  1446
  "inf \<equiv> min"
haftmann@22483
  1447
  "sup \<equiv> max"
nipkow@24438
  1448
by intro_classes (auto simp add: inf_nat_def sup_nat_def)
haftmann@22483
  1449
krauss@22157
  1450
krauss@22157
  1451
subsection {* Size function *}
krauss@22157
  1452
haftmann@22920
  1453
lemma nat_size [simp, code func]: "size (n\<Colon>nat) = n"
nipkow@24438
  1454
by (induct n) simp_all
krauss@22157
  1455
haftmann@24196
  1456
subsection {* legacy bindings *}
haftmann@24196
  1457
haftmann@24196
  1458
ML
haftmann@24196
  1459
{*
haftmann@24196
  1460
val pred_nat_trancl_eq_le = thm "pred_nat_trancl_eq_le";
haftmann@24196
  1461
val nat_diff_split = thm "nat_diff_split";
haftmann@24196
  1462
val nat_diff_split_asm = thm "nat_diff_split_asm";
haftmann@24196
  1463
val le_square = thm "le_square";
haftmann@24196
  1464
val le_cube = thm "le_cube";
haftmann@24196
  1465
val diff_less_mono = thm "diff_less_mono";
haftmann@24196
  1466
val less_diff_conv = thm "less_diff_conv";
haftmann@24196
  1467
val le_diff_conv = thm "le_diff_conv";
haftmann@24196
  1468
val le_diff_conv2 = thm "le_diff_conv2";
haftmann@24196
  1469
val diff_diff_cancel = thm "diff_diff_cancel";
haftmann@24196
  1470
val le_add_diff = thm "le_add_diff";
haftmann@24196
  1471
val diff_less = thm "diff_less";
haftmann@24196
  1472
val diff_diff_eq = thm "diff_diff_eq";
haftmann@24196
  1473
val eq_diff_iff = thm "eq_diff_iff";
haftmann@24196
  1474
val less_diff_iff = thm "less_diff_iff";
haftmann@24196
  1475
val le_diff_iff = thm "le_diff_iff";
haftmann@24196
  1476
val diff_le_mono = thm "diff_le_mono";
haftmann@24196
  1477
val diff_le_mono2 = thm "diff_le_mono2";
haftmann@24196
  1478
val diff_less_mono2 = thm "diff_less_mono2";
haftmann@24196
  1479
val diffs0_imp_equal = thm "diffs0_imp_equal";
haftmann@24196
  1480
val one_less_mult = thm "one_less_mult";
haftmann@24196
  1481
val n_less_m_mult_n = thm "n_less_m_mult_n";
haftmann@24196
  1482
val n_less_n_mult_m = thm "n_less_n_mult_m";
haftmann@24196
  1483
val diff_diff_right = thm "diff_diff_right";
haftmann@24196
  1484
val diff_Suc_diff_eq1 = thm "diff_Suc_diff_eq1";
haftmann@24196
  1485
val diff_Suc_diff_eq2 = thm "diff_Suc_diff_eq2";
haftmann@24196
  1486
*}
haftmann@24196
  1487
clasohm@923
  1488
end