src/HOL/Nat.thy
author paulson
Tue Jun 28 15:27:45 2005 +0200 (2005-06-28)
changeset 16587 b34c8aa657a5
parent 15921 b6e345548913
child 16635 bf7de5723c60
permissions -rw-r--r--
Constant "If" is now local
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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
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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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begin
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subsection {* Type @{text ind} *}
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typedecl ind
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consts
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  Zero_Rep      :: ind
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  Suc_Rep       :: "ind => ind"
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axioms
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  -- {* the axiom of infinity in 2 parts *}
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  inj_Suc_Rep:          "inj Suc_Rep"
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  Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
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subsection {* Type nat *}
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text {* Type definition *}
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consts
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  Nat :: "ind set"
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inductive Nat
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intros
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  Zero_RepI: "Zero_Rep : Nat"
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  Suc_RepI: "i : Nat ==> Suc_Rep i : Nat"
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global
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typedef (open Nat)
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  nat = Nat by (rule exI, rule Nat.Zero_RepI)
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instance nat :: "{ord, zero, one}" ..
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text {* Abstract constants and syntax *}
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consts
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  Suc :: "nat => nat"
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  pred_nat :: "(nat * nat) set"
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local
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defs
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  Zero_nat_def: "0 == Abs_Nat Zero_Rep"
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  Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"
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  One_nat_def [simp]: "1 == Suc 0"
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  -- {* nat operations *}
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  pred_nat_def: "pred_nat == {(m, n). n = Suc m}"
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  less_def: "m < n == (m, n) : trancl pred_nat"
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  le_def: "m \<le> (n::nat) == ~ (n < m)"
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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 (rules 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: "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 {* @{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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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, rules+)
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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]: "(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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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 (rule_tac m = m and n = n in 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 (rule_tac a=n in wf_induct)
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  apply (rule 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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  by (unfold le_def, rule not_less_eq [symmetric])
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lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
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  by (rule less_Suc_eq_le [THEN iffD2])
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lemma le0 [iff]: "(0::nat) \<le> n"
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  by (unfold le_def, rule not_less0)
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lemma Suc_n_not_le_n: "~ Suc n \<le> n"
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  by (simp add: le_def)
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lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)"
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  by (induct i) (simp_all add: le_def)
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lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
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  by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq)
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lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
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  by (drule le_Suc_eq [THEN iffD1], rules+)
berghofe@13449
   341
paulson@14267
   342
lemma leI: "~ n < m ==> m \<le> (n::nat)" by (simp add: le_def)
berghofe@13449
   343
paulson@14267
   344
lemma leD: "m \<le> n ==> ~ n < (m::nat)"
berghofe@13449
   345
  by (simp add: le_def)
berghofe@13449
   346
berghofe@13449
   347
lemmas leE = leD [elim_format]
berghofe@13449
   348
paulson@14267
   349
lemma not_less_iff_le: "(~ n < m) = (m \<le> (n::nat))"
berghofe@13449
   350
  by (blast intro: leI elim: leE)
berghofe@13449
   351
paulson@14267
   352
lemma not_leE: "~ m \<le> n ==> n<(m::nat)"
berghofe@13449
   353
  by (simp add: le_def)
berghofe@13449
   354
paulson@14267
   355
lemma not_le_iff_less: "(~ n \<le> m) = (m < (n::nat))"
berghofe@13449
   356
  by (simp add: le_def)
berghofe@13449
   357
paulson@14267
   358
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
berghofe@13449
   359
  apply (simp add: le_def less_Suc_eq)
berghofe@13449
   360
  apply (blast elim!: less_irrefl less_asym)
berghofe@13449
   361
  done -- {* formerly called lessD *}
berghofe@13449
   362
paulson@14267
   363
lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n"
berghofe@13449
   364
  by (simp add: le_def less_Suc_eq)
berghofe@13449
   365
berghofe@13449
   366
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   367
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
berghofe@13449
   368
  apply (simp add: le_def less_Suc_eq)
berghofe@13449
   369
  using less_linear
berghofe@13449
   370
  apply blast
berghofe@13449
   371
  done
berghofe@13449
   372
paulson@14267
   373
lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)"
berghofe@13449
   374
  by (blast intro: Suc_leI Suc_le_lessD)
berghofe@13449
   375
paulson@14267
   376
lemma le_SucI: "m \<le> n ==> m \<le> Suc n"
berghofe@13449
   377
  by (unfold le_def) (blast dest: Suc_lessD)
berghofe@13449
   378
paulson@14267
   379
lemma less_imp_le: "m < n ==> m \<le> (n::nat)"
berghofe@13449
   380
  by (unfold le_def) (blast elim: less_asym)
berghofe@13449
   381
paulson@14267
   382
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
berghofe@13449
   383
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq
berghofe@13449
   384
berghofe@13449
   385
paulson@14267
   386
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   387
paulson@14267
   388
lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)"
berghofe@13449
   389
  apply (unfold le_def)
berghofe@13449
   390
  using less_linear
berghofe@13449
   391
  apply (blast elim: less_irrefl less_asym)
berghofe@13449
   392
  done
berghofe@13449
   393
paulson@14267
   394
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
berghofe@13449
   395
  apply (unfold le_def)
berghofe@13449
   396
  using less_linear
berghofe@13449
   397
  apply (blast elim!: less_irrefl elim: less_asym)
berghofe@13449
   398
  done
berghofe@13449
   399
paulson@14267
   400
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
berghofe@13449
   401
  by (rules intro: less_or_eq_imp_le le_imp_less_or_eq)
berghofe@13449
   402
berghofe@13449
   403
text {* Useful with @{text Blast}. *}
paulson@14267
   404
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
berghofe@13449
   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)
berghofe@13449
   435
  using less_linear
berghofe@13449
   436
  apply blast
berghofe@13449
   437
  done
berghofe@13449
   438
paulson@14341
   439
text {* Type {@typ nat} is a wellfounded linear order *}
paulson@14341
   440
wenzelm@14691
   441
instance nat :: "{order, linorder, 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
nipkow@15921
   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 {*
berghofe@13449
   463
  Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}. 
berghofe@13449
   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
berghofe@13449
   475
subsection {* Arithmetic operators *}
oheimb@1660
   476
wenzelm@12338
   477
axclass power < type
wenzelm@10435
   478
paulson@3370
   479
consts
berghofe@13449
   480
  power :: "('a::power) => nat => 'a"            (infixr "^" 80)
paulson@3370
   481
wenzelm@9436
   482
berghofe@13449
   483
text {* arithmetic operators @{text "+ -"} and @{text "*"} *}
berghofe@13449
   484
wenzelm@14691
   485
instance nat :: "{plus, minus, times, power}" ..
wenzelm@9436
   486
berghofe@13449
   487
text {* size of a datatype value; overloaded *}
berghofe@13449
   488
consts size :: "'a => nat"
wenzelm@9436
   489
berghofe@13449
   490
primrec
berghofe@13449
   491
  add_0:    "0 + n = n"
berghofe@13449
   492
  add_Suc:  "Suc m + n = Suc (m + n)"
berghofe@13449
   493
berghofe@13449
   494
primrec
berghofe@13449
   495
  diff_0:   "m - 0 = m"
berghofe@13449
   496
  diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
wenzelm@9436
   497
wenzelm@9436
   498
primrec
berghofe@13449
   499
  mult_0:   "0 * n = 0"
berghofe@13449
   500
  mult_Suc: "Suc m * n = n + (m * n)"
berghofe@13449
   501
paulson@14341
   502
text {* These two rules ease the use of primitive recursion. 
paulson@14341
   503
NOTE USE OF @{text "=="} *}
berghofe@13449
   504
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
berghofe@13449
   505
  by simp
berghofe@13449
   506
berghofe@13449
   507
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
berghofe@13449
   508
  by simp
berghofe@13449
   509
paulson@14267
   510
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
berghofe@13449
   511
  by (case_tac n) simp_all
berghofe@13449
   512
paulson@14267
   513
lemma gr_implies_not0: "!!n::nat. m<n ==> n \<noteq> 0"
berghofe@13449
   514
  by (case_tac n) simp_all
berghofe@13449
   515
paulson@14267
   516
lemma neq0_conv [iff]: "!!n::nat. (n \<noteq> 0) = (0 < n)"
berghofe@13449
   517
  by (case_tac n) simp_all
berghofe@13449
   518
berghofe@13449
   519
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   520
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
berghofe@13449
   521
  by (rule iffD1, rule neq0_conv, rules)
berghofe@13449
   522
paulson@14267
   523
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
berghofe@13449
   524
  by (fast intro: not0_implies_Suc)
berghofe@13449
   525
berghofe@13449
   526
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
berghofe@13449
   527
  apply (rule iffI)
paulson@14208
   528
  apply (rule ccontr, simp_all)
berghofe@13449
   529
  done
berghofe@13449
   530
paulson@14267
   531
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
berghofe@13449
   532
  by (induct m') simp_all
berghofe@13449
   533
berghofe@13449
   534
text {* Useful in certain inductive arguments *}
paulson@14267
   535
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
berghofe@13449
   536
  by (case_tac m) simp_all
berghofe@13449
   537
paulson@14341
   538
lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"
berghofe@13449
   539
  apply (rule nat_less_induct)
berghofe@13449
   540
  apply (case_tac n)
berghofe@13449
   541
  apply (case_tac [2] nat)
berghofe@13449
   542
  apply (blast intro: less_trans)+
berghofe@13449
   543
  done
berghofe@13449
   544
paulson@15341
   545
subsection {* @{text LEAST} theorems for type @{typ nat}*}
berghofe@13449
   546
paulson@14267
   547
lemma Least_Suc:
paulson@14267
   548
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
paulson@14208
   549
  apply (case_tac "n", auto)
berghofe@13449
   550
  apply (frule LeastI)
berghofe@13449
   551
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
paulson@14267
   552
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
berghofe@13449
   553
  apply (erule_tac [2] Least_le)
paulson@14208
   554
  apply (case_tac "LEAST x. P x", auto)
berghofe@13449
   555
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
berghofe@13449
   556
  apply (blast intro: order_antisym)
berghofe@13449
   557
  done
berghofe@13449
   558
paulson@14267
   559
lemma Least_Suc2:
paulson@14267
   560
     "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
paulson@14267
   561
  by (erule (1) Least_Suc [THEN ssubst], simp)
berghofe@13449
   562
berghofe@13449
   563
paulson@14265
   564
berghofe@13449
   565
subsection {* @{term min} and @{term max} *}
berghofe@13449
   566
berghofe@13449
   567
lemma min_0L [simp]: "min 0 n = (0::nat)"
berghofe@13449
   568
  by (rule min_leastL) simp
berghofe@13449
   569
berghofe@13449
   570
lemma min_0R [simp]: "min n 0 = (0::nat)"
berghofe@13449
   571
  by (rule min_leastR) simp
berghofe@13449
   572
berghofe@13449
   573
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
berghofe@13449
   574
  by (simp add: min_of_mono)
berghofe@13449
   575
berghofe@13449
   576
lemma max_0L [simp]: "max 0 n = (n::nat)"
berghofe@13449
   577
  by (rule max_leastL) simp
berghofe@13449
   578
berghofe@13449
   579
lemma max_0R [simp]: "max n 0 = (n::nat)"
berghofe@13449
   580
  by (rule max_leastR) simp
berghofe@13449
   581
berghofe@13449
   582
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
berghofe@13449
   583
  by (simp add: max_of_mono)
berghofe@13449
   584
berghofe@13449
   585
berghofe@13449
   586
subsection {* Basic rewrite rules for the arithmetic operators *}
berghofe@13449
   587
berghofe@13449
   588
text {* Difference *}
berghofe@13449
   589
berghofe@14193
   590
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
paulson@15251
   591
  by (induct n) simp_all
berghofe@13449
   592
berghofe@14193
   593
lemma diff_Suc_Suc [simp, code]: "Suc(m) - Suc(n) = m - n"
paulson@15251
   594
  by (induct n) simp_all
berghofe@13449
   595
berghofe@13449
   596
berghofe@13449
   597
text {*
berghofe@13449
   598
  Could be (and is, below) generalized in various ways
berghofe@13449
   599
  However, none of the generalizations are currently in the simpset,
berghofe@13449
   600
  and I dread to think what happens if I put them in
berghofe@13449
   601
*}
berghofe@13449
   602
lemma Suc_pred [simp]: "0 < n ==> Suc (n - Suc 0) = n"
berghofe@13449
   603
  by (simp split add: nat.split)
berghofe@13449
   604
berghofe@14193
   605
declare diff_Suc [simp del, code del]
berghofe@13449
   606
berghofe@13449
   607
berghofe@13449
   608
subsection {* Addition *}
berghofe@13449
   609
berghofe@13449
   610
lemma add_0_right [simp]: "m + 0 = (m::nat)"
berghofe@13449
   611
  by (induct m) simp_all
berghofe@13449
   612
berghofe@13449
   613
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
berghofe@13449
   614
  by (induct m) simp_all
berghofe@13449
   615
berghofe@14193
   616
lemma [code]: "Suc m + n = m + Suc n" by simp
berghofe@14193
   617
berghofe@13449
   618
berghofe@13449
   619
text {* Associative law for addition *}
paulson@14267
   620
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
berghofe@13449
   621
  by (induct m) simp_all
berghofe@13449
   622
berghofe@13449
   623
text {* Commutative law for addition *}
paulson@14267
   624
lemma nat_add_commute: "m + n = n + (m::nat)"
berghofe@13449
   625
  by (induct m) simp_all
berghofe@13449
   626
paulson@14267
   627
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
berghofe@13449
   628
  apply (rule mk_left_commute [of "op +"])
paulson@14267
   629
  apply (rule nat_add_assoc)
paulson@14267
   630
  apply (rule nat_add_commute)
berghofe@13449
   631
  done
berghofe@13449
   632
paulson@14331
   633
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
berghofe@13449
   634
  by (induct k) simp_all
berghofe@13449
   635
paulson@14331
   636
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
berghofe@13449
   637
  by (induct k) simp_all
berghofe@13449
   638
paulson@14331
   639
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
berghofe@13449
   640
  by (induct k) simp_all
berghofe@13449
   641
paulson@14331
   642
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
berghofe@13449
   643
  by (induct k) simp_all
berghofe@13449
   644
berghofe@13449
   645
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
berghofe@13449
   646
berghofe@13449
   647
lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)"
berghofe@13449
   648
  by (case_tac m) simp_all
berghofe@13449
   649
berghofe@13449
   650
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
berghofe@13449
   651
  by (case_tac m) simp_all
berghofe@13449
   652
berghofe@13449
   653
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
berghofe@13449
   654
  by (rule trans, rule eq_commute, rule add_is_1)
berghofe@13449
   655
berghofe@13449
   656
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m | 0 < n)"
berghofe@13449
   657
  by (simp del: neq0_conv add: neq0_conv [symmetric])
berghofe@13449
   658
berghofe@13449
   659
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0"
berghofe@13449
   660
  apply (drule add_0_right [THEN ssubst])
paulson@14267
   661
  apply (simp add: nat_add_assoc del: add_0_right)
berghofe@13449
   662
  done
berghofe@13449
   663
paulson@14267
   664
paulson@14267
   665
subsection {* Multiplication *}
paulson@14267
   666
paulson@14267
   667
text {* right annihilation in product *}
paulson@14267
   668
lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
paulson@14267
   669
  by (induct m) simp_all
paulson@14267
   670
paulson@14267
   671
text {* right successor law for multiplication *}
paulson@14267
   672
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
paulson@14267
   673
  by (induct m) (simp_all add: nat_add_left_commute)
paulson@14267
   674
paulson@14267
   675
text {* Commutative law for multiplication *}
paulson@14267
   676
lemma nat_mult_commute: "m * n = n * (m::nat)"
paulson@14267
   677
  by (induct m) simp_all
paulson@14267
   678
paulson@14267
   679
text {* addition distributes over multiplication *}
paulson@14267
   680
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
paulson@14267
   681
  by (induct m) (simp_all add: nat_add_assoc nat_add_left_commute)
paulson@14267
   682
paulson@14267
   683
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
paulson@14267
   684
  by (induct m) (simp_all add: nat_add_assoc)
paulson@14267
   685
paulson@14267
   686
text {* Associative law for multiplication *}
paulson@14267
   687
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
paulson@14267
   688
  by (induct m) (simp_all add: add_mult_distrib)
paulson@14267
   689
paulson@14267
   690
nipkow@14740
   691
text{*The naturals form a @{text comm_semiring_1_cancel}*}
obua@14738
   692
instance nat :: comm_semiring_1_cancel
paulson@14267
   693
proof
paulson@14267
   694
  fix i j k :: nat
paulson@14267
   695
  show "(i + j) + k = i + (j + k)" by (rule nat_add_assoc)
paulson@14267
   696
  show "i + j = j + i" by (rule nat_add_commute)
paulson@14267
   697
  show "0 + i = i" by simp
paulson@14267
   698
  show "(i * j) * k = i * (j * k)" by (rule nat_mult_assoc)
paulson@14267
   699
  show "i * j = j * i" by (rule nat_mult_commute)
paulson@14267
   700
  show "1 * i = i" by simp
paulson@14267
   701
  show "(i + j) * k = i * k + j * k" by (simp add: add_mult_distrib)
paulson@14267
   702
  show "0 \<noteq> (1::nat)" by simp
paulson@14341
   703
  assume "k+i = k+j" thus "i=j" by simp
paulson@14341
   704
qed
paulson@14341
   705
paulson@14341
   706
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
paulson@15251
   707
  apply (induct m)
paulson@14341
   708
  apply (induct_tac [2] n, simp_all)
paulson@14341
   709
  done
paulson@14341
   710
paulson@14341
   711
subsection {* Monotonicity of Addition *}
paulson@14341
   712
paulson@14341
   713
text {* strict, in 1st argument *}
paulson@14341
   714
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
paulson@14341
   715
  by (induct k) simp_all
paulson@14341
   716
paulson@14341
   717
text {* strict, in both arguments *}
paulson@14341
   718
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   719
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   720
  apply (induct j, simp_all)
paulson@14341
   721
  done
paulson@14341
   722
paulson@14341
   723
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   724
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   725
  apply (induct n)
paulson@14341
   726
  apply (simp_all add: order_le_less)
paulson@14341
   727
  apply (blast elim!: less_SucE 
paulson@14341
   728
               intro!: add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   729
  done
paulson@14341
   730
paulson@14341
   731
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
paulson@14341
   732
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j"
paulson@14341
   733
  apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp)
paulson@14341
   734
  apply (induct_tac x) 
paulson@14341
   735
  apply (simp_all add: add_less_mono)
paulson@14341
   736
  done
paulson@14341
   737
paulson@14341
   738
nipkow@14740
   739
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
obua@14738
   740
instance nat :: ordered_semidom
paulson@14341
   741
proof
paulson@14341
   742
  fix i j k :: nat
paulson@14348
   743
  show "0 < (1::nat)" by simp
paulson@14267
   744
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   745
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   746
qed
paulson@14267
   747
paulson@14267
   748
lemma nat_mult_1: "(1::nat) * n = n"
paulson@14267
   749
  by simp
paulson@14267
   750
paulson@14267
   751
lemma nat_mult_1_right: "n * (1::nat) = n"
paulson@14267
   752
  by simp
paulson@14267
   753
paulson@14267
   754
paulson@14267
   755
subsection {* Additional theorems about "less than" *}
paulson@14267
   756
paulson@14267
   757
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   758
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   759
lemma less_mono_imp_le_mono:
paulson@14267
   760
  assumes lt_mono: "!!i j::nat. i < j ==> f i < f j"
paulson@14267
   761
  and le: "i \<le> j" shows "f i \<le> ((f j)::nat)" using le
paulson@14267
   762
  apply (simp add: order_le_less)
paulson@14267
   763
  apply (blast intro!: lt_mono)
paulson@14267
   764
  done
paulson@14267
   765
paulson@14267
   766
text {* non-strict, in 1st argument *}
paulson@14267
   767
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
paulson@14267
   768
  by (rule add_right_mono)
paulson@14267
   769
paulson@14267
   770
text {* non-strict, in both arguments *}
paulson@14267
   771
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
paulson@14267
   772
  by (rule add_mono)
paulson@14267
   773
paulson@14267
   774
lemma le_add2: "n \<le> ((m + n)::nat)"
paulson@14341
   775
  by (insert add_right_mono [of 0 m n], simp) 
berghofe@13449
   776
paulson@14267
   777
lemma le_add1: "n \<le> ((n + m)::nat)"
paulson@14341
   778
  by (simp add: add_commute, rule le_add2)
berghofe@13449
   779
berghofe@13449
   780
lemma less_add_Suc1: "i < Suc (i + m)"
berghofe@13449
   781
  by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   782
berghofe@13449
   783
lemma less_add_Suc2: "i < Suc (m + i)"
berghofe@13449
   784
  by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   785
paulson@14267
   786
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
berghofe@13449
   787
  by (rules intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   788
paulson@14267
   789
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
berghofe@13449
   790
  by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   791
paulson@14267
   792
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
berghofe@13449
   793
  by (rule le_trans, assumption, rule le_add2)
berghofe@13449
   794
berghofe@13449
   795
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
berghofe@13449
   796
  by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
   797
berghofe@13449
   798
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
berghofe@13449
   799
  by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
   800
berghofe@13449
   801
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
paulson@14341
   802
  apply (rule le_less_trans [of _ "i+j"]) 
paulson@14341
   803
  apply (simp_all add: le_add1)
berghofe@13449
   804
  done
berghofe@13449
   805
berghofe@13449
   806
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
berghofe@13449
   807
  apply (rule notI)
berghofe@13449
   808
  apply (erule add_lessD1 [THEN less_irrefl])
berghofe@13449
   809
  done
berghofe@13449
   810
berghofe@13449
   811
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
berghofe@13449
   812
  by (simp add: add_commute not_add_less1)
berghofe@13449
   813
paulson@14267
   814
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
paulson@14341
   815
  apply (rule order_trans [of _ "m+k"]) 
paulson@14341
   816
  apply (simp_all add: le_add1)
paulson@14341
   817
  done
berghofe@13449
   818
paulson@14267
   819
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
berghofe@13449
   820
  apply (simp add: add_commute)
berghofe@13449
   821
  apply (erule add_leD1)
berghofe@13449
   822
  done
berghofe@13449
   823
paulson@14267
   824
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
berghofe@13449
   825
  by (blast dest: add_leD1 add_leD2)
berghofe@13449
   826
berghofe@13449
   827
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
   828
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
berghofe@13449
   829
  by (force simp del: add_Suc_right
berghofe@13449
   830
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
   831
berghofe@13449
   832
berghofe@13449
   833
berghofe@13449
   834
subsection {* Difference *}
berghofe@13449
   835
berghofe@13449
   836
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
berghofe@13449
   837
  by (induct m) simp_all
berghofe@13449
   838
berghofe@13449
   839
text {* Addition is the inverse of subtraction:
paulson@14267
   840
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
   841
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
berghofe@13449
   842
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   843
paulson@14267
   844
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
berghofe@13449
   845
  by (simp add: add_diff_inverse not_less_iff_le)
berghofe@13449
   846
paulson@14267
   847
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
berghofe@13449
   848
  by (simp add: le_add_diff_inverse add_commute)
berghofe@13449
   849
berghofe@13449
   850
berghofe@13449
   851
subsection {* More results about difference *}
berghofe@13449
   852
paulson@14267
   853
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
berghofe@13449
   854
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   855
berghofe@13449
   856
lemma diff_less_Suc: "m - n < Suc m"
berghofe@13449
   857
  apply (induct m n rule: diff_induct)
berghofe@13449
   858
  apply (erule_tac [3] less_SucE)
berghofe@13449
   859
  apply (simp_all add: less_Suc_eq)
berghofe@13449
   860
  done
berghofe@13449
   861
paulson@14267
   862
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
berghofe@13449
   863
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
   864
berghofe@13449
   865
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
berghofe@13449
   866
  by (rule le_less_trans, rule diff_le_self)
berghofe@13449
   867
berghofe@13449
   868
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
berghofe@13449
   869
  by (induct i j rule: diff_induct) simp_all
berghofe@13449
   870
berghofe@13449
   871
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
berghofe@13449
   872
  by (simp add: diff_diff_left)
berghofe@13449
   873
berghofe@13449
   874
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
paulson@14208
   875
  apply (case_tac "n", safe)
berghofe@13449
   876
  apply (simp add: le_simps)
berghofe@13449
   877
  done
berghofe@13449
   878
berghofe@13449
   879
text {* This and the next few suggested by Florian Kammueller *}
berghofe@13449
   880
lemma diff_commute: "(i::nat) - j - k = i - k - j"
berghofe@13449
   881
  by (simp add: diff_diff_left add_commute)
berghofe@13449
   882
paulson@14267
   883
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
berghofe@13449
   884
  by (induct j k rule: diff_induct) simp_all
berghofe@13449
   885
paulson@14267
   886
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
berghofe@13449
   887
  by (simp add: add_commute diff_add_assoc)
berghofe@13449
   888
berghofe@13449
   889
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
berghofe@13449
   890
  by (induct n) simp_all
berghofe@13449
   891
berghofe@13449
   892
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
berghofe@13449
   893
  by (simp add: diff_add_assoc)
berghofe@13449
   894
paulson@14267
   895
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
berghofe@13449
   896
  apply safe
berghofe@13449
   897
  apply (simp_all add: diff_add_inverse2)
berghofe@13449
   898
  done
berghofe@13449
   899
paulson@14267
   900
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
berghofe@13449
   901
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   902
paulson@14267
   903
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
berghofe@13449
   904
  by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
   905
berghofe@13449
   906
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
berghofe@13449
   907
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   908
paulson@14267
   909
lemma less_imp_add_positive: "i < j  ==> \<exists>k::nat. 0 < k & i + k = j"
berghofe@13449
   910
  apply (rule_tac x = "j - i" in exI)
berghofe@13449
   911
  apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym)
berghofe@13449
   912
  done
wenzelm@9436
   913
berghofe@13449
   914
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
berghofe@13449
   915
  apply (induct k i rule: diff_induct)
berghofe@13449
   916
  apply (simp_all (no_asm))
berghofe@13449
   917
  apply rules
berghofe@13449
   918
  done
berghofe@13449
   919
berghofe@13449
   920
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
berghofe@13449
   921
  apply (rule diff_self_eq_0 [THEN subst])
paulson@14208
   922
  apply (rule zero_induct_lemma, rules+)
berghofe@13449
   923
  done
berghofe@13449
   924
berghofe@13449
   925
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
berghofe@13449
   926
  by (induct k) simp_all
berghofe@13449
   927
berghofe@13449
   928
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
berghofe@13449
   929
  by (simp add: diff_cancel add_commute)
berghofe@13449
   930
berghofe@13449
   931
lemma diff_add_0: "n - (n + m) = (0::nat)"
berghofe@13449
   932
  by (induct n) simp_all
berghofe@13449
   933
berghofe@13449
   934
berghofe@13449
   935
text {* Difference distributes over multiplication *}
berghofe@13449
   936
berghofe@13449
   937
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
berghofe@13449
   938
  by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
berghofe@13449
   939
berghofe@13449
   940
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
berghofe@13449
   941
  by (simp add: diff_mult_distrib mult_commute [of k])
berghofe@13449
   942
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
berghofe@13449
   943
berghofe@13449
   944
lemmas nat_distrib =
berghofe@13449
   945
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
berghofe@13449
   946
berghofe@13449
   947
berghofe@13449
   948
subsection {* Monotonicity of Multiplication *}
berghofe@13449
   949
paulson@14267
   950
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
paulson@14341
   951
  by (simp add: mult_right_mono) 
berghofe@13449
   952
paulson@14267
   953
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
paulson@14341
   954
  by (simp add: mult_left_mono) 
berghofe@13449
   955
paulson@14267
   956
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
   957
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
paulson@14341
   958
  by (simp add: mult_mono) 
berghofe@13449
   959
berghofe@13449
   960
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
paulson@14341
   961
  by (simp add: mult_strict_right_mono) 
berghofe@13449
   962
paulson@14266
   963
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
   964
      there are no negative numbers.*}
paulson@14266
   965
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
   966
  apply (induct m)
paulson@14208
   967
  apply (case_tac [2] n, simp_all)
berghofe@13449
   968
  done
berghofe@13449
   969
paulson@14267
   970
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"
berghofe@13449
   971
  apply (induct m)
paulson@14208
   972
  apply (case_tac [2] n, simp_all)
berghofe@13449
   973
  done
berghofe@13449
   974
berghofe@13449
   975
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
paulson@15251
   976
  apply (induct m, simp)
paulson@15251
   977
  apply (induct n, simp, fastsimp)
berghofe@13449
   978
  done
berghofe@13449
   979
berghofe@13449
   980
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
berghofe@13449
   981
  apply (rule trans)
paulson@14208
   982
  apply (rule_tac [2] mult_eq_1_iff, fastsimp)
berghofe@13449
   983
  done
berghofe@13449
   984
paulson@14341
   985
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
   986
  apply (safe intro!: mult_less_mono1)
paulson@14208
   987
  apply (case_tac k, auto)
berghofe@13449
   988
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
   989
  apply (blast intro: mult_le_mono1)
berghofe@13449
   990
  done
berghofe@13449
   991
berghofe@13449
   992
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
paulson@14341
   993
  by (simp add: mult_commute [of k])
berghofe@13449
   994
paulson@14267
   995
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
paulson@14208
   996
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
   997
paulson@14267
   998
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
paulson@14208
   999
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1000
paulson@14341
  1001
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
paulson@14208
  1002
  apply (cut_tac less_linear, safe, auto)
berghofe@13449
  1003
  apply (drule mult_less_mono1, assumption, simp)+
berghofe@13449
  1004
  done
berghofe@13449
  1005
berghofe@13449
  1006
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
paulson@14341
  1007
  by (simp add: mult_commute [of k])
berghofe@13449
  1008
berghofe@13449
  1009
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
berghofe@13449
  1010
  by (subst mult_less_cancel1) simp
berghofe@13449
  1011
paulson@14267
  1012
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
berghofe@13449
  1013
  by (subst mult_le_cancel1) simp
berghofe@13449
  1014
berghofe@13449
  1015
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
berghofe@13449
  1016
  by (subst mult_cancel1) simp
berghofe@13449
  1017
berghofe@13449
  1018
text {* Lemma for @{text gcd} *}
berghofe@13449
  1019
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1020
  apply (drule sym)
berghofe@13449
  1021
  apply (rule disjCI)
berghofe@13449
  1022
  apply (rule nat_less_cases, erule_tac [2] _)
berghofe@13449
  1023
  apply (fastsimp elim!: less_SucE)
berghofe@13449
  1024
  apply (fastsimp dest: mult_less_mono2)
berghofe@13449
  1025
  done
wenzelm@9436
  1026
nipkow@15539
  1027
clasohm@923
  1028
end