src/HOL/Nat.thy
author paulson
Mon Nov 29 14:02:55 2004 +0100 (2004-11-29)
changeset 15341 254f6f00b60e
parent 15281 bd4611956c7b
child 15413 901d1bfedf09
permissions -rw-r--r--
converted to Isar script, simplifying some results
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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 {* Isomorphisms: @{text Abs_Nat} and @{text Rep_Nat} *}
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lemma inj_Rep_Nat: "inj Rep_Nat"
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  apply (rule inj_on_inverseI)
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  apply (rule Rep_Nat_inverse)
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  done
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lemma inj_on_Abs_Nat: "inj_on Abs_Nat Nat"
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  apply (rule inj_on_inverseI)
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  apply (erule Abs_Nat_inverse)
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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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  apply (unfold Zero_nat_def Suc_def)
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  apply (rule inj_on_Abs_Nat [THEN inj_on_contraD])
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  apply (rule Suc_Rep_not_Zero_Rep)
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  apply (rule Rep_Nat Nat.Suc_RepI Nat.Zero_RepI)+
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  done
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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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  apply (unfold Suc_def)
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  apply (rule inj_onI)
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  apply (drule inj_on_Abs_Nat [THEN inj_onD])
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  apply (rule Rep_Nat Nat.Suc_RepI)+
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  apply (drule inj_Suc_Rep [THEN injD])
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  apply (erule inj_Rep_Nat [THEN injD])
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  done
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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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  apply (rule iffI)
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  apply (erule Suc_inject)
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  apply (erule arg_cong)
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  done
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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])
berghofe@13449
   347
paulson@14267
   348
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
berghofe@13449
   349
  by (rule less_Suc_eq_le [THEN iffD2])
berghofe@13449
   350
paulson@14267
   351
lemma le0 [iff]: "(0::nat) \<le> n"
berghofe@13449
   352
  by (unfold le_def, rule not_less0)
berghofe@13449
   353
paulson@14267
   354
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
berghofe@13449
   355
  by (simp add: le_def)
berghofe@13449
   356
paulson@14267
   357
lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)"
berghofe@13449
   358
  by (induct i) (simp_all add: le_def)
berghofe@13449
   359
paulson@14267
   360
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
berghofe@13449
   361
  by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   362
paulson@14267
   363
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
berghofe@13449
   364
  by (drule le_Suc_eq [THEN iffD1], rules+)
berghofe@13449
   365
paulson@14267
   366
lemma leI: "~ n < m ==> m \<le> (n::nat)" by (simp add: le_def)
berghofe@13449
   367
paulson@14267
   368
lemma leD: "m \<le> n ==> ~ n < (m::nat)"
berghofe@13449
   369
  by (simp add: le_def)
berghofe@13449
   370
berghofe@13449
   371
lemmas leE = leD [elim_format]
berghofe@13449
   372
paulson@14267
   373
lemma not_less_iff_le: "(~ n < m) = (m \<le> (n::nat))"
berghofe@13449
   374
  by (blast intro: leI elim: leE)
berghofe@13449
   375
paulson@14267
   376
lemma not_leE: "~ m \<le> n ==> n<(m::nat)"
berghofe@13449
   377
  by (simp add: le_def)
berghofe@13449
   378
paulson@14267
   379
lemma not_le_iff_less: "(~ n \<le> m) = (m < (n::nat))"
berghofe@13449
   380
  by (simp add: le_def)
berghofe@13449
   381
paulson@14267
   382
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
berghofe@13449
   383
  apply (simp add: le_def less_Suc_eq)
berghofe@13449
   384
  apply (blast elim!: less_irrefl less_asym)
berghofe@13449
   385
  done -- {* formerly called lessD *}
berghofe@13449
   386
paulson@14267
   387
lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n"
berghofe@13449
   388
  by (simp add: le_def less_Suc_eq)
berghofe@13449
   389
berghofe@13449
   390
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   391
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
berghofe@13449
   392
  apply (simp add: le_def less_Suc_eq)
berghofe@13449
   393
  using less_linear
berghofe@13449
   394
  apply blast
berghofe@13449
   395
  done
berghofe@13449
   396
paulson@14267
   397
lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)"
berghofe@13449
   398
  by (blast intro: Suc_leI Suc_le_lessD)
berghofe@13449
   399
paulson@14267
   400
lemma le_SucI: "m \<le> n ==> m \<le> Suc n"
berghofe@13449
   401
  by (unfold le_def) (blast dest: Suc_lessD)
berghofe@13449
   402
paulson@14267
   403
lemma less_imp_le: "m < n ==> m \<le> (n::nat)"
berghofe@13449
   404
  by (unfold le_def) (blast elim: less_asym)
berghofe@13449
   405
paulson@14267
   406
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
berghofe@13449
   407
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq
berghofe@13449
   408
berghofe@13449
   409
paulson@14267
   410
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   411
paulson@14267
   412
lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)"
berghofe@13449
   413
  apply (unfold le_def)
berghofe@13449
   414
  using less_linear
berghofe@13449
   415
  apply (blast elim: less_irrefl less_asym)
berghofe@13449
   416
  done
berghofe@13449
   417
paulson@14267
   418
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
berghofe@13449
   419
  apply (unfold le_def)
berghofe@13449
   420
  using less_linear
berghofe@13449
   421
  apply (blast elim!: less_irrefl elim: less_asym)
berghofe@13449
   422
  done
berghofe@13449
   423
paulson@14267
   424
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
berghofe@13449
   425
  by (rules intro: less_or_eq_imp_le le_imp_less_or_eq)
berghofe@13449
   426
berghofe@13449
   427
text {* Useful with @{text Blast}. *}
paulson@14267
   428
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
berghofe@13449
   429
  by (rule less_or_eq_imp_le, rule disjI2)
berghofe@13449
   430
paulson@14267
   431
lemma le_refl: "n \<le> (n::nat)"
berghofe@13449
   432
  by (simp add: le_eq_less_or_eq)
berghofe@13449
   433
paulson@14267
   434
lemma le_less_trans: "[| i \<le> j; j < k |] ==> i < (k::nat)"
berghofe@13449
   435
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
berghofe@13449
   436
paulson@14267
   437
lemma less_le_trans: "[| i < j; j \<le> k |] ==> i < (k::nat)"
berghofe@13449
   438
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
berghofe@13449
   439
paulson@14267
   440
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
berghofe@13449
   441
  by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans)
berghofe@13449
   442
paulson@14267
   443
lemma le_anti_sym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
berghofe@13449
   444
  by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym)
berghofe@13449
   445
paulson@14267
   446
lemma Suc_le_mono [iff]: "(Suc n \<le> Suc m) = (n \<le> m)"
berghofe@13449
   447
  by (simp add: le_simps)
berghofe@13449
   448
berghofe@13449
   449
text {* Axiom @{text order_less_le} of class @{text order}: *}
paulson@14267
   450
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
berghofe@13449
   451
  by (simp add: le_def nat_neq_iff) (blast elim!: less_asym)
berghofe@13449
   452
paulson@14267
   453
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
berghofe@13449
   454
  by (rule iffD2, rule nat_less_le, rule conjI)
berghofe@13449
   455
berghofe@13449
   456
text {* Axiom @{text linorder_linear} of class @{text linorder}: *}
paulson@14267
   457
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
berghofe@13449
   458
  apply (simp add: le_eq_less_or_eq)
berghofe@13449
   459
  using less_linear
berghofe@13449
   460
  apply blast
berghofe@13449
   461
  done
berghofe@13449
   462
paulson@14341
   463
text {* Type {@typ nat} is a wellfounded linear order *}
paulson@14341
   464
wenzelm@14691
   465
instance nat :: "{order, linorder, wellorder}"
wenzelm@14691
   466
  by intro_classes
wenzelm@14691
   467
    (assumption |
wenzelm@14691
   468
      rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+
paulson@14341
   469
berghofe@13449
   470
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
berghofe@13449
   471
  by (blast elim!: less_SucE)
berghofe@13449
   472
berghofe@13449
   473
text {*
berghofe@13449
   474
  Rewrite @{term "n < Suc m"} to @{term "n = m"}
paulson@14267
   475
  if @{term "~ n < m"} or @{term "m \<le> n"} hold.
berghofe@13449
   476
  Not suitable as default simprules because they often lead to looping
berghofe@13449
   477
*}
paulson@14267
   478
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
berghofe@13449
   479
  by (rule not_less_less_Suc_eq, rule leD)
berghofe@13449
   480
berghofe@13449
   481
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   482
berghofe@13449
   483
berghofe@13449
   484
text {*
berghofe@13449
   485
  Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}. 
berghofe@13449
   486
  No longer added as simprules (they loop) 
berghofe@13449
   487
  but via @{text reorient_simproc} in Bin
berghofe@13449
   488
*}
berghofe@13449
   489
berghofe@13449
   490
text {* Polymorphic, not just for @{typ nat} *}
berghofe@13449
   491
lemma zero_reorient: "(0 = x) = (x = 0)"
berghofe@13449
   492
  by auto
berghofe@13449
   493
berghofe@13449
   494
lemma one_reorient: "(1 = x) = (x = 1)"
berghofe@13449
   495
  by auto
berghofe@13449
   496
berghofe@13449
   497
subsection {* Arithmetic operators *}
oheimb@1660
   498
wenzelm@12338
   499
axclass power < type
wenzelm@10435
   500
paulson@3370
   501
consts
berghofe@13449
   502
  power :: "('a::power) => nat => 'a"            (infixr "^" 80)
paulson@3370
   503
wenzelm@9436
   504
berghofe@13449
   505
text {* arithmetic operators @{text "+ -"} and @{text "*"} *}
berghofe@13449
   506
wenzelm@14691
   507
instance nat :: "{plus, minus, times, power}" ..
wenzelm@9436
   508
berghofe@13449
   509
text {* size of a datatype value; overloaded *}
berghofe@13449
   510
consts size :: "'a => nat"
wenzelm@9436
   511
berghofe@13449
   512
primrec
berghofe@13449
   513
  add_0:    "0 + n = n"
berghofe@13449
   514
  add_Suc:  "Suc m + n = Suc (m + n)"
berghofe@13449
   515
berghofe@13449
   516
primrec
berghofe@13449
   517
  diff_0:   "m - 0 = m"
berghofe@13449
   518
  diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
wenzelm@9436
   519
wenzelm@9436
   520
primrec
berghofe@13449
   521
  mult_0:   "0 * n = 0"
berghofe@13449
   522
  mult_Suc: "Suc m * n = n + (m * n)"
berghofe@13449
   523
paulson@14341
   524
text {* These two rules ease the use of primitive recursion. 
paulson@14341
   525
NOTE USE OF @{text "=="} *}
berghofe@13449
   526
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
berghofe@13449
   527
  by simp
berghofe@13449
   528
berghofe@13449
   529
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
berghofe@13449
   530
  by simp
berghofe@13449
   531
paulson@14267
   532
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
berghofe@13449
   533
  by (case_tac n) simp_all
berghofe@13449
   534
paulson@14267
   535
lemma gr_implies_not0: "!!n::nat. m<n ==> n \<noteq> 0"
berghofe@13449
   536
  by (case_tac n) simp_all
berghofe@13449
   537
paulson@14267
   538
lemma neq0_conv [iff]: "!!n::nat. (n \<noteq> 0) = (0 < n)"
berghofe@13449
   539
  by (case_tac n) simp_all
berghofe@13449
   540
berghofe@13449
   541
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   542
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
berghofe@13449
   543
  by (rule iffD1, rule neq0_conv, rules)
berghofe@13449
   544
paulson@14267
   545
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
berghofe@13449
   546
  by (fast intro: not0_implies_Suc)
berghofe@13449
   547
berghofe@13449
   548
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
berghofe@13449
   549
  apply (rule iffI)
paulson@14208
   550
  apply (rule ccontr, simp_all)
berghofe@13449
   551
  done
berghofe@13449
   552
paulson@14267
   553
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
berghofe@13449
   554
  by (induct m') simp_all
berghofe@13449
   555
berghofe@13449
   556
text {* Useful in certain inductive arguments *}
paulson@14267
   557
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
berghofe@13449
   558
  by (case_tac m) simp_all
berghofe@13449
   559
paulson@14341
   560
lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"
berghofe@13449
   561
  apply (rule nat_less_induct)
berghofe@13449
   562
  apply (case_tac n)
berghofe@13449
   563
  apply (case_tac [2] nat)
berghofe@13449
   564
  apply (blast intro: less_trans)+
berghofe@13449
   565
  done
berghofe@13449
   566
paulson@15341
   567
subsection {* @{text LEAST} theorems for type @{typ nat}*}
berghofe@13449
   568
paulson@14267
   569
lemma Least_Suc:
paulson@14267
   570
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
paulson@14208
   571
  apply (case_tac "n", auto)
berghofe@13449
   572
  apply (frule LeastI)
berghofe@13449
   573
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
paulson@14267
   574
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
berghofe@13449
   575
  apply (erule_tac [2] Least_le)
paulson@14208
   576
  apply (case_tac "LEAST x. P x", auto)
berghofe@13449
   577
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
berghofe@13449
   578
  apply (blast intro: order_antisym)
berghofe@13449
   579
  done
berghofe@13449
   580
paulson@14267
   581
lemma Least_Suc2:
paulson@14267
   582
     "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
paulson@14267
   583
  by (erule (1) Least_Suc [THEN ssubst], simp)
berghofe@13449
   584
berghofe@13449
   585
paulson@14265
   586
berghofe@13449
   587
subsection {* @{term min} and @{term max} *}
berghofe@13449
   588
berghofe@13449
   589
lemma min_0L [simp]: "min 0 n = (0::nat)"
berghofe@13449
   590
  by (rule min_leastL) simp
berghofe@13449
   591
berghofe@13449
   592
lemma min_0R [simp]: "min n 0 = (0::nat)"
berghofe@13449
   593
  by (rule min_leastR) simp
berghofe@13449
   594
berghofe@13449
   595
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
berghofe@13449
   596
  by (simp add: min_of_mono)
berghofe@13449
   597
berghofe@13449
   598
lemma max_0L [simp]: "max 0 n = (n::nat)"
berghofe@13449
   599
  by (rule max_leastL) simp
berghofe@13449
   600
berghofe@13449
   601
lemma max_0R [simp]: "max n 0 = (n::nat)"
berghofe@13449
   602
  by (rule max_leastR) simp
berghofe@13449
   603
berghofe@13449
   604
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
berghofe@13449
   605
  by (simp add: max_of_mono)
berghofe@13449
   606
berghofe@13449
   607
berghofe@13449
   608
subsection {* Basic rewrite rules for the arithmetic operators *}
berghofe@13449
   609
berghofe@13449
   610
text {* Difference *}
berghofe@13449
   611
berghofe@14193
   612
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
paulson@15251
   613
  by (induct n) simp_all
berghofe@13449
   614
berghofe@14193
   615
lemma diff_Suc_Suc [simp, code]: "Suc(m) - Suc(n) = m - n"
paulson@15251
   616
  by (induct n) simp_all
berghofe@13449
   617
berghofe@13449
   618
berghofe@13449
   619
text {*
berghofe@13449
   620
  Could be (and is, below) generalized in various ways
berghofe@13449
   621
  However, none of the generalizations are currently in the simpset,
berghofe@13449
   622
  and I dread to think what happens if I put them in
berghofe@13449
   623
*}
berghofe@13449
   624
lemma Suc_pred [simp]: "0 < n ==> Suc (n - Suc 0) = n"
berghofe@13449
   625
  by (simp split add: nat.split)
berghofe@13449
   626
berghofe@14193
   627
declare diff_Suc [simp del, code del]
berghofe@13449
   628
berghofe@13449
   629
berghofe@13449
   630
subsection {* Addition *}
berghofe@13449
   631
berghofe@13449
   632
lemma add_0_right [simp]: "m + 0 = (m::nat)"
berghofe@13449
   633
  by (induct m) simp_all
berghofe@13449
   634
berghofe@13449
   635
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
berghofe@13449
   636
  by (induct m) simp_all
berghofe@13449
   637
berghofe@14193
   638
lemma [code]: "Suc m + n = m + Suc n" by simp
berghofe@14193
   639
berghofe@13449
   640
berghofe@13449
   641
text {* Associative law for addition *}
paulson@14267
   642
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
berghofe@13449
   643
  by (induct m) simp_all
berghofe@13449
   644
berghofe@13449
   645
text {* Commutative law for addition *}
paulson@14267
   646
lemma nat_add_commute: "m + n = n + (m::nat)"
berghofe@13449
   647
  by (induct m) simp_all
berghofe@13449
   648
paulson@14267
   649
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
berghofe@13449
   650
  apply (rule mk_left_commute [of "op +"])
paulson@14267
   651
  apply (rule nat_add_assoc)
paulson@14267
   652
  apply (rule nat_add_commute)
berghofe@13449
   653
  done
berghofe@13449
   654
paulson@14331
   655
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
berghofe@13449
   656
  by (induct k) simp_all
berghofe@13449
   657
paulson@14331
   658
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
berghofe@13449
   659
  by (induct k) simp_all
berghofe@13449
   660
paulson@14331
   661
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
berghofe@13449
   662
  by (induct k) simp_all
berghofe@13449
   663
paulson@14331
   664
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
berghofe@13449
   665
  by (induct k) simp_all
berghofe@13449
   666
berghofe@13449
   667
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
berghofe@13449
   668
berghofe@13449
   669
lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)"
berghofe@13449
   670
  by (case_tac m) simp_all
berghofe@13449
   671
berghofe@13449
   672
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
berghofe@13449
   673
  by (case_tac m) simp_all
berghofe@13449
   674
berghofe@13449
   675
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
berghofe@13449
   676
  by (rule trans, rule eq_commute, rule add_is_1)
berghofe@13449
   677
berghofe@13449
   678
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m | 0 < n)"
berghofe@13449
   679
  by (simp del: neq0_conv add: neq0_conv [symmetric])
berghofe@13449
   680
berghofe@13449
   681
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0"
berghofe@13449
   682
  apply (drule add_0_right [THEN ssubst])
paulson@14267
   683
  apply (simp add: nat_add_assoc del: add_0_right)
berghofe@13449
   684
  done
berghofe@13449
   685
paulson@14267
   686
paulson@14267
   687
subsection {* Multiplication *}
paulson@14267
   688
paulson@14267
   689
text {* right annihilation in product *}
paulson@14267
   690
lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
paulson@14267
   691
  by (induct m) simp_all
paulson@14267
   692
paulson@14267
   693
text {* right successor law for multiplication *}
paulson@14267
   694
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
paulson@14267
   695
  by (induct m) (simp_all add: nat_add_left_commute)
paulson@14267
   696
paulson@14267
   697
text {* Commutative law for multiplication *}
paulson@14267
   698
lemma nat_mult_commute: "m * n = n * (m::nat)"
paulson@14267
   699
  by (induct m) simp_all
paulson@14267
   700
paulson@14267
   701
text {* addition distributes over multiplication *}
paulson@14267
   702
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
paulson@14267
   703
  by (induct m) (simp_all add: nat_add_assoc nat_add_left_commute)
paulson@14267
   704
paulson@14267
   705
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
paulson@14267
   706
  by (induct m) (simp_all add: nat_add_assoc)
paulson@14267
   707
paulson@14267
   708
text {* Associative law for multiplication *}
paulson@14267
   709
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
paulson@14267
   710
  by (induct m) (simp_all add: add_mult_distrib)
paulson@14267
   711
paulson@14267
   712
nipkow@14740
   713
text{*The naturals form a @{text comm_semiring_1_cancel}*}
obua@14738
   714
instance nat :: comm_semiring_1_cancel
paulson@14267
   715
proof
paulson@14267
   716
  fix i j k :: nat
paulson@14267
   717
  show "(i + j) + k = i + (j + k)" by (rule nat_add_assoc)
paulson@14267
   718
  show "i + j = j + i" by (rule nat_add_commute)
paulson@14267
   719
  show "0 + i = i" by simp
paulson@14267
   720
  show "(i * j) * k = i * (j * k)" by (rule nat_mult_assoc)
paulson@14267
   721
  show "i * j = j * i" by (rule nat_mult_commute)
paulson@14267
   722
  show "1 * i = i" by simp
paulson@14267
   723
  show "(i + j) * k = i * k + j * k" by (simp add: add_mult_distrib)
paulson@14267
   724
  show "0 \<noteq> (1::nat)" by simp
paulson@14341
   725
  assume "k+i = k+j" thus "i=j" by simp
paulson@14341
   726
qed
paulson@14341
   727
paulson@14341
   728
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
paulson@15251
   729
  apply (induct m)
paulson@14341
   730
  apply (induct_tac [2] n, simp_all)
paulson@14341
   731
  done
paulson@14341
   732
paulson@14341
   733
subsection {* Monotonicity of Addition *}
paulson@14341
   734
paulson@14341
   735
text {* strict, in 1st argument *}
paulson@14341
   736
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
paulson@14341
   737
  by (induct k) simp_all
paulson@14341
   738
paulson@14341
   739
text {* strict, in both arguments *}
paulson@14341
   740
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   741
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   742
  apply (induct j, simp_all)
paulson@14341
   743
  done
paulson@14341
   744
paulson@14341
   745
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   746
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   747
  apply (induct n)
paulson@14341
   748
  apply (simp_all add: order_le_less)
paulson@14341
   749
  apply (blast elim!: less_SucE 
paulson@14341
   750
               intro!: add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   751
  done
paulson@14341
   752
paulson@14341
   753
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
paulson@14341
   754
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j"
paulson@14341
   755
  apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp)
paulson@14341
   756
  apply (induct_tac x) 
paulson@14341
   757
  apply (simp_all add: add_less_mono)
paulson@14341
   758
  done
paulson@14341
   759
paulson@14341
   760
nipkow@14740
   761
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
obua@14738
   762
instance nat :: ordered_semidom
paulson@14341
   763
proof
paulson@14341
   764
  fix i j k :: nat
paulson@14348
   765
  show "0 < (1::nat)" by simp
paulson@14267
   766
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   767
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   768
qed
paulson@14267
   769
paulson@14267
   770
lemma nat_mult_1: "(1::nat) * n = n"
paulson@14267
   771
  by simp
paulson@14267
   772
paulson@14267
   773
lemma nat_mult_1_right: "n * (1::nat) = n"
paulson@14267
   774
  by simp
paulson@14267
   775
paulson@14267
   776
paulson@14267
   777
subsection {* Additional theorems about "less than" *}
paulson@14267
   778
paulson@14267
   779
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   780
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   781
lemma less_mono_imp_le_mono:
paulson@14267
   782
  assumes lt_mono: "!!i j::nat. i < j ==> f i < f j"
paulson@14267
   783
  and le: "i \<le> j" shows "f i \<le> ((f j)::nat)" using le
paulson@14267
   784
  apply (simp add: order_le_less)
paulson@14267
   785
  apply (blast intro!: lt_mono)
paulson@14267
   786
  done
paulson@14267
   787
paulson@14267
   788
text {* non-strict, in 1st argument *}
paulson@14267
   789
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
paulson@14267
   790
  by (rule add_right_mono)
paulson@14267
   791
paulson@14267
   792
text {* non-strict, in both arguments *}
paulson@14267
   793
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
paulson@14267
   794
  by (rule add_mono)
paulson@14267
   795
paulson@14267
   796
lemma le_add2: "n \<le> ((m + n)::nat)"
paulson@14341
   797
  by (insert add_right_mono [of 0 m n], simp) 
berghofe@13449
   798
paulson@14267
   799
lemma le_add1: "n \<le> ((n + m)::nat)"
paulson@14341
   800
  by (simp add: add_commute, rule le_add2)
berghofe@13449
   801
berghofe@13449
   802
lemma less_add_Suc1: "i < Suc (i + m)"
berghofe@13449
   803
  by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   804
berghofe@13449
   805
lemma less_add_Suc2: "i < Suc (m + i)"
berghofe@13449
   806
  by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   807
paulson@14267
   808
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
berghofe@13449
   809
  by (rules intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   810
paulson@14267
   811
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
berghofe@13449
   812
  by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   813
paulson@14267
   814
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
berghofe@13449
   815
  by (rule le_trans, assumption, rule le_add2)
berghofe@13449
   816
berghofe@13449
   817
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
berghofe@13449
   818
  by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
   819
berghofe@13449
   820
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
berghofe@13449
   821
  by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
   822
berghofe@13449
   823
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
paulson@14341
   824
  apply (rule le_less_trans [of _ "i+j"]) 
paulson@14341
   825
  apply (simp_all add: le_add1)
berghofe@13449
   826
  done
berghofe@13449
   827
berghofe@13449
   828
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
berghofe@13449
   829
  apply (rule notI)
berghofe@13449
   830
  apply (erule add_lessD1 [THEN less_irrefl])
berghofe@13449
   831
  done
berghofe@13449
   832
berghofe@13449
   833
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
berghofe@13449
   834
  by (simp add: add_commute not_add_less1)
berghofe@13449
   835
paulson@14267
   836
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
paulson@14341
   837
  apply (rule order_trans [of _ "m+k"]) 
paulson@14341
   838
  apply (simp_all add: le_add1)
paulson@14341
   839
  done
berghofe@13449
   840
paulson@14267
   841
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
berghofe@13449
   842
  apply (simp add: add_commute)
berghofe@13449
   843
  apply (erule add_leD1)
berghofe@13449
   844
  done
berghofe@13449
   845
paulson@14267
   846
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
berghofe@13449
   847
  by (blast dest: add_leD1 add_leD2)
berghofe@13449
   848
berghofe@13449
   849
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
   850
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
berghofe@13449
   851
  by (force simp del: add_Suc_right
berghofe@13449
   852
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
   853
berghofe@13449
   854
berghofe@13449
   855
berghofe@13449
   856
subsection {* Difference *}
berghofe@13449
   857
berghofe@13449
   858
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
berghofe@13449
   859
  by (induct m) simp_all
berghofe@13449
   860
berghofe@13449
   861
text {* Addition is the inverse of subtraction:
paulson@14267
   862
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
   863
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
berghofe@13449
   864
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   865
paulson@14267
   866
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
berghofe@13449
   867
  by (simp add: add_diff_inverse not_less_iff_le)
berghofe@13449
   868
paulson@14267
   869
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
berghofe@13449
   870
  by (simp add: le_add_diff_inverse add_commute)
berghofe@13449
   871
berghofe@13449
   872
berghofe@13449
   873
subsection {* More results about difference *}
berghofe@13449
   874
paulson@14267
   875
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
berghofe@13449
   876
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   877
berghofe@13449
   878
lemma diff_less_Suc: "m - n < Suc m"
berghofe@13449
   879
  apply (induct m n rule: diff_induct)
berghofe@13449
   880
  apply (erule_tac [3] less_SucE)
berghofe@13449
   881
  apply (simp_all add: less_Suc_eq)
berghofe@13449
   882
  done
berghofe@13449
   883
paulson@14267
   884
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
berghofe@13449
   885
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
   886
berghofe@13449
   887
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
berghofe@13449
   888
  by (rule le_less_trans, rule diff_le_self)
berghofe@13449
   889
berghofe@13449
   890
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
berghofe@13449
   891
  by (induct i j rule: diff_induct) simp_all
berghofe@13449
   892
berghofe@13449
   893
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
berghofe@13449
   894
  by (simp add: diff_diff_left)
berghofe@13449
   895
berghofe@13449
   896
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
paulson@14208
   897
  apply (case_tac "n", safe)
berghofe@13449
   898
  apply (simp add: le_simps)
berghofe@13449
   899
  done
berghofe@13449
   900
berghofe@13449
   901
text {* This and the next few suggested by Florian Kammueller *}
berghofe@13449
   902
lemma diff_commute: "(i::nat) - j - k = i - k - j"
berghofe@13449
   903
  by (simp add: diff_diff_left add_commute)
berghofe@13449
   904
paulson@14267
   905
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
berghofe@13449
   906
  by (induct j k rule: diff_induct) simp_all
berghofe@13449
   907
paulson@14267
   908
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
berghofe@13449
   909
  by (simp add: add_commute diff_add_assoc)
berghofe@13449
   910
berghofe@13449
   911
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
berghofe@13449
   912
  by (induct n) simp_all
berghofe@13449
   913
berghofe@13449
   914
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
berghofe@13449
   915
  by (simp add: diff_add_assoc)
berghofe@13449
   916
paulson@14267
   917
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
berghofe@13449
   918
  apply safe
berghofe@13449
   919
  apply (simp_all add: diff_add_inverse2)
berghofe@13449
   920
  done
berghofe@13449
   921
paulson@14267
   922
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
berghofe@13449
   923
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   924
paulson@14267
   925
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
berghofe@13449
   926
  by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
   927
berghofe@13449
   928
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
berghofe@13449
   929
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   930
paulson@14267
   931
lemma less_imp_add_positive: "i < j  ==> \<exists>k::nat. 0 < k & i + k = j"
berghofe@13449
   932
  apply (rule_tac x = "j - i" in exI)
berghofe@13449
   933
  apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym)
berghofe@13449
   934
  done
wenzelm@9436
   935
berghofe@13449
   936
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
berghofe@13449
   937
  apply (induct k i rule: diff_induct)
berghofe@13449
   938
  apply (simp_all (no_asm))
berghofe@13449
   939
  apply rules
berghofe@13449
   940
  done
berghofe@13449
   941
berghofe@13449
   942
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
berghofe@13449
   943
  apply (rule diff_self_eq_0 [THEN subst])
paulson@14208
   944
  apply (rule zero_induct_lemma, rules+)
berghofe@13449
   945
  done
berghofe@13449
   946
berghofe@13449
   947
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
berghofe@13449
   948
  by (induct k) simp_all
berghofe@13449
   949
berghofe@13449
   950
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
berghofe@13449
   951
  by (simp add: diff_cancel add_commute)
berghofe@13449
   952
berghofe@13449
   953
lemma diff_add_0: "n - (n + m) = (0::nat)"
berghofe@13449
   954
  by (induct n) simp_all
berghofe@13449
   955
berghofe@13449
   956
berghofe@13449
   957
text {* Difference distributes over multiplication *}
berghofe@13449
   958
berghofe@13449
   959
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
berghofe@13449
   960
  by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
berghofe@13449
   961
berghofe@13449
   962
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
berghofe@13449
   963
  by (simp add: diff_mult_distrib mult_commute [of k])
berghofe@13449
   964
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
berghofe@13449
   965
berghofe@13449
   966
lemmas nat_distrib =
berghofe@13449
   967
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
berghofe@13449
   968
berghofe@13449
   969
berghofe@13449
   970
subsection {* Monotonicity of Multiplication *}
berghofe@13449
   971
paulson@14267
   972
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
paulson@14341
   973
  by (simp add: mult_right_mono) 
berghofe@13449
   974
paulson@14267
   975
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
paulson@14341
   976
  by (simp add: mult_left_mono) 
berghofe@13449
   977
paulson@14267
   978
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
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   979
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
paulson@14341
   980
  by (simp add: mult_mono) 
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   981
berghofe@13449
   982
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
paulson@14341
   983
  by (simp add: mult_strict_right_mono) 
berghofe@13449
   984
paulson@14266
   985
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
   986
      there are no negative numbers.*}
paulson@14266
   987
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
   988
  apply (induct m)
paulson@14208
   989
  apply (case_tac [2] n, simp_all)
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   990
  done
berghofe@13449
   991
paulson@14267
   992
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"
berghofe@13449
   993
  apply (induct m)
paulson@14208
   994
  apply (case_tac [2] n, simp_all)
berghofe@13449
   995
  done
berghofe@13449
   996
berghofe@13449
   997
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
paulson@15251
   998
  apply (induct m, simp)
paulson@15251
   999
  apply (induct n, simp, fastsimp)
berghofe@13449
  1000
  done
berghofe@13449
  1001
berghofe@13449
  1002
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
berghofe@13449
  1003
  apply (rule trans)
paulson@14208
  1004
  apply (rule_tac [2] mult_eq_1_iff, fastsimp)
berghofe@13449
  1005
  done
berghofe@13449
  1006
paulson@14341
  1007
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1008
  apply (safe intro!: mult_less_mono1)
paulson@14208
  1009
  apply (case_tac k, auto)
berghofe@13449
  1010
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1011
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1012
  done
berghofe@13449
  1013
berghofe@13449
  1014
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
paulson@14341
  1015
  by (simp add: mult_commute [of k])
berghofe@13449
  1016
paulson@14267
  1017
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
paulson@14208
  1018
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1019
paulson@14267
  1020
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
paulson@14208
  1021
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1022
paulson@14341
  1023
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
paulson@14208
  1024
  apply (cut_tac less_linear, safe, auto)
berghofe@13449
  1025
  apply (drule mult_less_mono1, assumption, simp)+
berghofe@13449
  1026
  done
berghofe@13449
  1027
berghofe@13449
  1028
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
paulson@14341
  1029
  by (simp add: mult_commute [of k])
berghofe@13449
  1030
berghofe@13449
  1031
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
berghofe@13449
  1032
  by (subst mult_less_cancel1) simp
berghofe@13449
  1033
paulson@14267
  1034
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
berghofe@13449
  1035
  by (subst mult_le_cancel1) simp
berghofe@13449
  1036
berghofe@13449
  1037
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
berghofe@13449
  1038
  by (subst mult_cancel1) simp
berghofe@13449
  1039
berghofe@13449
  1040
text {* Lemma for @{text gcd} *}
berghofe@13449
  1041
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1042
  apply (drule sym)
berghofe@13449
  1043
  apply (rule disjCI)
berghofe@13449
  1044
  apply (rule nat_less_cases, erule_tac [2] _)
berghofe@13449
  1045
  apply (fastsimp elim!: less_SucE)
berghofe@13449
  1046
  apply (fastsimp dest: mult_less_mono2)
berghofe@13449
  1047
  done
wenzelm@9436
  1048
clasohm@923
  1049
end