src/HOL/Nat.thy
author wenzelm
Wed Sep 17 21:27:14 2008 +0200 (2008-09-17)
changeset 28263 69eaa97e7e96
parent 27823 52971512d1a2
child 28514 da83a614c454
permissions -rw-r--r--
moved global ML bindings to global place;
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(*  Title:      HOL/Nat.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
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Type "nat" is a linear order, and a datatype; arithmetic operators + -
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and * (for div, mod and dvd, see theory Divides).
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*)
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header {* Natural numbers *}
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theory Nat
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imports Inductive Ring_and_Field
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uses
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  "~~/src/Tools/rat.ML"
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  "~~/src/Provers/Arith/cancel_sums.ML"
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  ("arith_data.ML")
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  "~~/src/Provers/Arith/fast_lin_arith.ML"
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  ("Tools/lin_arith.ML")
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begin
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subsection {* Type @{text ind} *}
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typedecl ind
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axiomatization
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  Zero_Rep :: ind and
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  Suc_Rep :: "ind => ind"
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where
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  -- {* the axiom of infinity in 2 parts *}
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  inj_Suc_Rep:          "inj Suc_Rep" and
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  Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
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subsection {* Type nat *}
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text {* Type definition *}
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inductive Nat :: "ind \<Rightarrow> bool"
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where
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    Zero_RepI: "Nat Zero_Rep"
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  | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
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global
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typedef (open Nat)
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  nat = Nat
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  by (rule exI, unfold mem_def, rule Nat.Zero_RepI)
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constdefs
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  Suc ::   "nat => nat"
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  Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"
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local
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instantiation nat :: zero
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begin
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definition Zero_nat_def [code func del]:
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  "0 = Abs_Nat Zero_Rep"
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instance ..
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end
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lemma Suc_not_Zero: "Suc m \<noteq> 0"
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  apply (simp add: Zero_nat_def Suc_def Abs_Nat_inject [unfolded mem_def]
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    Rep_Nat [unfolded mem_def] Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep [unfolded mem_def])
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  done
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lemma Zero_not_Suc: "0 \<noteq> Suc m"
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  by (rule not_sym, rule Suc_not_Zero not_sym)
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rep_datatype "0 \<Colon> nat" Suc
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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 [unfolded mem_def, THEN Nat.induct])
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     apply (iprover elim: Abs_Nat_inverse [unfolded mem_def, THEN subst])
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    apply (simp_all add: Abs_Nat_inject [unfolded mem_def] Rep_Nat [unfolded mem_def]
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      Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep [unfolded mem_def]
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      Suc_Rep_not_Zero_Rep [unfolded mem_def, symmetric]
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      inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject)
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  done
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lemma nat_induct [case_names 0 Suc, induct type: nat]:
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  -- {* for backward compatibility -- naming of variables differs *}
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  fixes n
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  assumes "P 0"
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    and "\<And>n. P n \<Longrightarrow> P (Suc n)"
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  shows "P n"
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  using assms by (rule nat.induct) 
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declare nat.exhaust [case_names 0 Suc, cases type: nat]
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lemmas nat_rec_0 = nat.recs(1)
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  and nat_rec_Suc = nat.recs(2)
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lemmas nat_case_0 = nat.cases(1)
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  and nat_case_Suc = nat.cases(2)
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text {* Injectiveness and distinctness lemmas *}
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lemma inj_Suc[simp]: "inj_on Suc N"
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  by (simp add: inj_on_def)
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lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"
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by (rule notE, rule Suc_not_Zero)
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lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"
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by (rule Suc_neq_Zero, erule sym)
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lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"
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by (rule inj_Suc [THEN injD])
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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 n \<noteq> n"
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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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lemma diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
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    (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
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  apply (rule_tac x = m in spec)
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  apply (induct n)
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  prefer 2
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  apply (rule allI)
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  apply (induct_tac x, iprover+)
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  done
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subsection {* Arithmetic operators *}
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instantiation nat :: "{minus, comm_monoid_add}"
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begin
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primrec plus_nat
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where
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  add_0:      "0 + n = (n\<Colon>nat)"
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  | add_Suc:  "Suc m + n = Suc (m + n)"
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lemma add_0_right [simp]: "m + 0 = (m::nat)"
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  by (induct m) simp_all
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lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
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  by (induct m) simp_all
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lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
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  by simp
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primrec minus_nat
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where
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  diff_0:     "m - 0 = (m\<Colon>nat)"
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  | diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
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declare diff_Suc [simp del, code del]
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lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
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  by (induct n) (simp_all add: diff_Suc)
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lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
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  by (induct n) (simp_all add: diff_Suc)
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instance proof
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  fix n m q :: nat
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  show "(n + m) + q = n + (m + q)" by (induct n) simp_all
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  show "n + m = m + n" by (induct n) simp_all
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  show "0 + n = n" by simp
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qed
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end
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instantiation nat :: comm_semiring_1_cancel
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begin
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definition
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  One_nat_def [simp]: "1 = Suc 0"
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primrec times_nat
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where
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  mult_0:     "0 * n = (0\<Colon>nat)"
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  | mult_Suc: "Suc m * n = n + (m * n)"
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lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
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  by (induct m) simp_all
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lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
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  by (induct m) (simp_all add: add_left_commute)
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lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
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  by (induct m) (simp_all add: add_assoc)
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instance proof
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  fix n m q :: nat
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  show "0 \<noteq> (1::nat)" by simp
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  show "1 * n = n" by simp
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  show "n * m = m * n" by (induct n) simp_all
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  show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib)
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  show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib)
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  assume "n + m = n + q" thus "m = q" by (induct n) simp_all
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qed
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end
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subsubsection {* Addition *}
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lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
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  by (rule add_assoc)
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lemma nat_add_commute: "m + n = n + (m::nat)"
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  by (rule add_commute)
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lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
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  by (rule add_left_commute)
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lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
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  by (rule add_left_cancel)
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lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
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  by (rule add_right_cancel)
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text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
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lemma add_is_0 [iff]:
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  fixes m n :: nat
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  shows "(m + n = 0) = (m = 0 & n = 0)"
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  by (cases m) simp_all
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lemma add_is_1:
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  "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
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  by (cases m) simp_all
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lemma one_is_add:
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  "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
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  by (rule trans, rule eq_commute, rule add_is_1)
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lemma add_eq_self_zero:
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  fixes m n :: nat
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  shows "m + n = m \<Longrightarrow> n = 0"
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  by (induct m) simp_all
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lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
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  apply (induct k)
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   apply simp
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  apply(drule comp_inj_on[OF _ inj_Suc])
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  apply (simp add:o_def)
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  done
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subsubsection {* Difference *}
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lemma diff_self_eq_0 [simp]: "(m\<Colon>nat) - m = 0"
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  by (induct m) simp_all
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lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
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  by (induct i j rule: diff_induct) simp_all
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lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
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  by (simp add: diff_diff_left)
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lemma diff_commute: "(i::nat) - j - k = i - k - j"
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  by (simp add: diff_diff_left add_commute)
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lemma diff_add_inverse: "(n + m) - n = (m::nat)"
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  by (induct n) simp_all
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lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
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  by (simp add: diff_add_inverse add_commute [of m n])
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lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
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  by (induct k) simp_all
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lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
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  by (simp add: diff_cancel add_commute)
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lemma diff_add_0: "n - (n + m) = (0::nat)"
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  by (induct n) simp_all
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text {* Difference distributes over multiplication *}
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lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
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by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
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lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
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by (simp add: diff_mult_distrib mult_commute [of k])
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  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
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subsubsection {* Multiplication *}
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lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
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  by (rule mult_assoc)
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lemma nat_mult_commute: "m * n = n * (m::nat)"
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  by (rule mult_commute)
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lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
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  by (rule right_distrib)
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lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
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  by (induct m) auto
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lemmas nat_distrib =
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  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
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lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
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  apply (induct m)
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   apply simp
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  apply (induct n)
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   apply auto
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  done
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lemma one_eq_mult_iff [simp,noatp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
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  apply (rule trans)
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  apply (rule_tac [2] mult_eq_1_iff, fastsimp)
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  done
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lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
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proof -
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  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
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  proof (induct n arbitrary: m)
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    case 0 then show "m = 0" by simp
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  next
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    case (Suc n) then show "m = Suc n"
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      by (cases m) (simp_all add: eq_commute [of "0"])
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  qed
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  then show ?thesis by auto
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qed
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lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
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  by (simp add: mult_commute)
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lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
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  by (subst mult_cancel1) simp
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subsection {* Orders on @{typ nat} *}
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subsubsection {* Operation definition *}
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instantiation nat :: linorder
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begin
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primrec less_eq_nat where
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  "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
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   348
  | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
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   349
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   350
declare less_eq_nat.simps [simp del, code del]
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   351
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps)
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   352
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
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   353
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   354
definition less_nat where
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   355
  less_eq_Suc_le [code func del]: "n < m \<longleftrightarrow> Suc n \<le> m"
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   356
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   357
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
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   358
  by (simp add: less_eq_nat.simps(2))
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   359
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   360
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
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   361
  unfolding less_eq_Suc_le ..
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   362
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   363
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
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   364
  by (induct n) (simp_all add: less_eq_nat.simps(2))
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   365
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   366
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
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   367
  by (simp add: less_eq_Suc_le)
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   368
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   369
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
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   370
  by simp
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   371
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   372
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
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   373
  by (simp add: less_eq_Suc_le)
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   374
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   375
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
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   376
  by (simp add: less_eq_Suc_le)
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   377
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   378
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
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   379
  by (induct m arbitrary: n)
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   380
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
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   381
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   382
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
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   383
  by (cases n) (auto intro: le_SucI)
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   384
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   385
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
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   386
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
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   387
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   388
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
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   389
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
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   390
wenzelm@26315
   391
instance
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   392
proof
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   393
  fix n m :: nat
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   394
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" 
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   395
  proof (induct n arbitrary: m)
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   396
    case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   397
  next
haftmann@27679
   398
    case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   399
  qed
haftmann@26072
   400
next
haftmann@26072
   401
  fix n :: nat show "n \<le> n" by (induct n) simp_all
haftmann@26072
   402
next
haftmann@26072
   403
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
haftmann@26072
   404
  then show "n = m"
haftmann@26072
   405
    by (induct n arbitrary: m)
haftmann@26072
   406
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   407
next
haftmann@26072
   408
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
haftmann@26072
   409
  then show "n \<le> q"
haftmann@26072
   410
  proof (induct n arbitrary: m q)
haftmann@26072
   411
    case 0 show ?case by simp
haftmann@26072
   412
  next
haftmann@26072
   413
    case (Suc n) then show ?case
haftmann@26072
   414
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   415
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   416
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   417
  qed
haftmann@26072
   418
next
haftmann@26072
   419
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
haftmann@26072
   420
    by (induct n arbitrary: m)
haftmann@26072
   421
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   422
qed
haftmann@25510
   423
haftmann@25510
   424
end
berghofe@13449
   425
haftmann@26072
   426
subsubsection {* Introduction properties *}
berghofe@13449
   427
haftmann@26072
   428
lemma lessI [iff]: "n < Suc n"
haftmann@26072
   429
  by (simp add: less_Suc_eq_le)
berghofe@13449
   430
haftmann@26072
   431
lemma zero_less_Suc [iff]: "0 < Suc n"
haftmann@26072
   432
  by (simp add: less_Suc_eq_le)
berghofe@13449
   433
berghofe@13449
   434
berghofe@13449
   435
subsubsection {* Elimination properties *}
berghofe@13449
   436
berghofe@13449
   437
lemma less_not_refl: "~ n < (n::nat)"
haftmann@26072
   438
  by (rule order_less_irrefl)
berghofe@13449
   439
wenzelm@26335
   440
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
wenzelm@26335
   441
  by (rule not_sym) (rule less_imp_neq) 
berghofe@13449
   442
paulson@14267
   443
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
haftmann@26072
   444
  by (rule less_imp_neq)
berghofe@13449
   445
wenzelm@26335
   446
lemma less_irrefl_nat: "(n::nat) < n ==> R"
wenzelm@26335
   447
  by (rule notE, rule less_not_refl)
berghofe@13449
   448
berghofe@13449
   449
lemma less_zeroE: "(n::nat) < 0 ==> R"
haftmann@26072
   450
  by (rule notE) (rule not_less0)
berghofe@13449
   451
berghofe@13449
   452
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
haftmann@26072
   453
  unfolding less_Suc_eq_le le_less ..
berghofe@13449
   454
haftmann@26072
   455
lemma less_one [iff, noatp]: "(n < (1::nat)) = (n = 0)"
haftmann@26072
   456
  by (simp add: less_Suc_eq)
berghofe@13449
   457
berghofe@13449
   458
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
haftmann@26072
   459
  by (simp add: less_Suc_eq)
berghofe@13449
   460
berghofe@13449
   461
lemma Suc_mono: "m < n ==> Suc m < Suc n"
haftmann@26072
   462
  by simp
berghofe@13449
   463
nipkow@14302
   464
text {* "Less than" is antisymmetric, sort of *}
nipkow@14302
   465
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
haftmann@26072
   466
  unfolding not_less less_Suc_eq_le by (rule antisym)
nipkow@14302
   467
paulson@14267
   468
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
haftmann@26072
   469
  by (rule linorder_neq_iff)
berghofe@13449
   470
berghofe@13449
   471
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   472
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   473
  shows "P n m"
berghofe@13449
   474
  apply (rule less_linear [THEN disjE])
berghofe@13449
   475
  apply (erule_tac [2] disjE)
berghofe@13449
   476
  apply (erule lessCase)
berghofe@13449
   477
  apply (erule sym [THEN eqCase])
berghofe@13449
   478
  apply (erule major)
berghofe@13449
   479
  done
berghofe@13449
   480
berghofe@13449
   481
berghofe@13449
   482
subsubsection {* Inductive (?) properties *}
berghofe@13449
   483
paulson@14267
   484
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
haftmann@26072
   485
  unfolding less_eq_Suc_le [of m] le_less by simp 
berghofe@13449
   486
haftmann@26072
   487
lemma lessE:
haftmann@26072
   488
  assumes major: "i < k"
haftmann@26072
   489
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
haftmann@26072
   490
  shows P
haftmann@26072
   491
proof -
haftmann@26072
   492
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
haftmann@26072
   493
    unfolding less_eq_Suc_le by (induct k) simp_all
haftmann@26072
   494
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
haftmann@26072
   495
    by (clarsimp simp add: less_le)
haftmann@26072
   496
  with p1 p2 show P by auto
haftmann@26072
   497
qed
haftmann@26072
   498
haftmann@26072
   499
lemma less_SucE: assumes major: "m < Suc n"
haftmann@26072
   500
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
haftmann@26072
   501
  apply (rule major [THEN lessE])
haftmann@26072
   502
  apply (rule eq, blast)
haftmann@26072
   503
  apply (rule less, blast)
berghofe@13449
   504
  done
berghofe@13449
   505
berghofe@13449
   506
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   507
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   508
  apply (rule major [THEN lessE])
berghofe@13449
   509
  apply (erule lessI [THEN minor])
paulson@14208
   510
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   511
  done
berghofe@13449
   512
berghofe@13449
   513
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
haftmann@26072
   514
  by simp
berghofe@13449
   515
berghofe@13449
   516
lemma less_trans_Suc:
berghofe@13449
   517
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   518
  apply (induct k, simp_all)
berghofe@13449
   519
  apply (insert le)
berghofe@13449
   520
  apply (simp add: less_Suc_eq)
berghofe@13449
   521
  apply (blast dest: Suc_lessD)
berghofe@13449
   522
  done
berghofe@13449
   523
berghofe@13449
   524
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
haftmann@26072
   525
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
haftmann@26072
   526
  unfolding not_less less_Suc_eq_le ..
berghofe@13449
   527
haftmann@26072
   528
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   529
  unfolding not_le Suc_le_eq ..
wenzelm@21243
   530
haftmann@24995
   531
text {* Properties of "less than or equal" *}
berghofe@13449
   532
paulson@14267
   533
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
haftmann@26072
   534
  unfolding less_Suc_eq_le .
berghofe@13449
   535
paulson@14267
   536
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
haftmann@26072
   537
  unfolding not_le less_Suc_eq_le ..
berghofe@13449
   538
paulson@14267
   539
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
haftmann@26072
   540
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   541
paulson@14267
   542
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
haftmann@26072
   543
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   544
paulson@14267
   545
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
haftmann@26072
   546
  unfolding Suc_le_eq .
berghofe@13449
   547
berghofe@13449
   548
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   549
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
haftmann@26072
   550
  unfolding Suc_le_eq .
berghofe@13449
   551
wenzelm@26315
   552
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
haftmann@26072
   553
  unfolding less_eq_Suc_le by (rule Suc_leD)
berghofe@13449
   554
paulson@14267
   555
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
wenzelm@26315
   556
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
berghofe@13449
   557
berghofe@13449
   558
paulson@14267
   559
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   560
paulson@14267
   561
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
haftmann@26072
   562
  unfolding le_less .
berghofe@13449
   563
paulson@14267
   564
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
haftmann@26072
   565
  by (rule le_less)
berghofe@13449
   566
wenzelm@22718
   567
text {* Useful with @{text blast}. *}
paulson@14267
   568
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
haftmann@26072
   569
  by auto
berghofe@13449
   570
paulson@14267
   571
lemma le_refl: "n \<le> (n::nat)"
haftmann@26072
   572
  by simp
berghofe@13449
   573
paulson@14267
   574
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
haftmann@26072
   575
  by (rule order_trans)
berghofe@13449
   576
paulson@14267
   577
lemma le_anti_sym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
haftmann@26072
   578
  by (rule antisym)
berghofe@13449
   579
paulson@14267
   580
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
haftmann@26072
   581
  by (rule less_le)
berghofe@13449
   582
paulson@14267
   583
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
haftmann@26072
   584
  unfolding less_le ..
berghofe@13449
   585
haftmann@26072
   586
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
haftmann@26072
   587
  by (rule linear)
paulson@14341
   588
wenzelm@22718
   589
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   590
haftmann@26072
   591
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
haftmann@26072
   592
  unfolding less_Suc_eq_le by auto
berghofe@13449
   593
haftmann@26072
   594
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
haftmann@26072
   595
  unfolding not_less by (rule le_less_Suc_eq)
berghofe@13449
   596
berghofe@13449
   597
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   598
wenzelm@22718
   599
text {* These two rules ease the use of primitive recursion.
paulson@14341
   600
NOTE USE OF @{text "=="} *}
berghofe@13449
   601
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
nipkow@25162
   602
by simp
berghofe@13449
   603
berghofe@13449
   604
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
nipkow@25162
   605
by simp
berghofe@13449
   606
paulson@14267
   607
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   608
by (cases n) simp_all
nipkow@25162
   609
nipkow@25162
   610
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   611
by (cases n) simp_all
berghofe@13449
   612
wenzelm@22718
   613
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
nipkow@25162
   614
by (cases n) simp_all
berghofe@13449
   615
nipkow@25162
   616
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
nipkow@25162
   617
by (cases n) simp_all
nipkow@25140
   618
berghofe@13449
   619
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   620
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@25162
   621
by (rule neq0_conv[THEN iffD1], iprover)
berghofe@13449
   622
paulson@14267
   623
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
nipkow@25162
   624
by (fast intro: not0_implies_Suc)
berghofe@13449
   625
paulson@24286
   626
lemma not_gr0 [iff,noatp]: "!!n::nat. (~ (0 < n)) = (n = 0)"
nipkow@25134
   627
using neq0_conv by blast
berghofe@13449
   628
paulson@14267
   629
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
nipkow@25162
   630
by (induct m') simp_all
berghofe@13449
   631
berghofe@13449
   632
text {* Useful in certain inductive arguments *}
paulson@14267
   633
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
nipkow@25162
   634
by (cases m) simp_all
berghofe@13449
   635
berghofe@13449
   636
haftmann@26072
   637
subsubsection {* @{term min} and @{term max} *}
berghofe@13449
   638
haftmann@25076
   639
lemma mono_Suc: "mono Suc"
nipkow@25162
   640
by (rule monoI) simp
haftmann@25076
   641
berghofe@13449
   642
lemma min_0L [simp]: "min 0 n = (0::nat)"
nipkow@25162
   643
by (rule min_leastL) simp
berghofe@13449
   644
berghofe@13449
   645
lemma min_0R [simp]: "min n 0 = (0::nat)"
nipkow@25162
   646
by (rule min_leastR) simp
berghofe@13449
   647
berghofe@13449
   648
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
nipkow@25162
   649
by (simp add: mono_Suc min_of_mono)
berghofe@13449
   650
paulson@22191
   651
lemma min_Suc1:
paulson@22191
   652
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
nipkow@25162
   653
by (simp split: nat.split)
paulson@22191
   654
paulson@22191
   655
lemma min_Suc2:
paulson@22191
   656
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
nipkow@25162
   657
by (simp split: nat.split)
paulson@22191
   658
berghofe@13449
   659
lemma max_0L [simp]: "max 0 n = (n::nat)"
nipkow@25162
   660
by (rule max_leastL) simp
berghofe@13449
   661
berghofe@13449
   662
lemma max_0R [simp]: "max n 0 = (n::nat)"
nipkow@25162
   663
by (rule max_leastR) simp
berghofe@13449
   664
berghofe@13449
   665
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
nipkow@25162
   666
by (simp add: mono_Suc max_of_mono)
berghofe@13449
   667
paulson@22191
   668
lemma max_Suc1:
paulson@22191
   669
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
nipkow@25162
   670
by (simp split: nat.split)
paulson@22191
   671
paulson@22191
   672
lemma max_Suc2:
paulson@22191
   673
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
nipkow@25162
   674
by (simp split: nat.split)
paulson@22191
   675
berghofe@13449
   676
haftmann@26072
   677
subsubsection {* Monotonicity of Addition *}
berghofe@13449
   678
haftmann@26072
   679
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
haftmann@26072
   680
by (simp add: diff_Suc split: nat.split)
berghofe@13449
   681
paulson@14331
   682
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
nipkow@25162
   683
by (induct k) simp_all
berghofe@13449
   684
paulson@14331
   685
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
nipkow@25162
   686
by (induct k) simp_all
berghofe@13449
   687
nipkow@25162
   688
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
nipkow@25162
   689
by(auto dest:gr0_implies_Suc)
berghofe@13449
   690
paulson@14341
   691
text {* strict, in 1st argument *}
paulson@14341
   692
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
nipkow@25162
   693
by (induct k) simp_all
paulson@14341
   694
paulson@14341
   695
text {* strict, in both arguments *}
paulson@14341
   696
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   697
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   698
  apply (induct j, simp_all)
paulson@14341
   699
  done
paulson@14341
   700
paulson@14341
   701
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   702
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   703
  apply (induct n)
paulson@14341
   704
  apply (simp_all add: order_le_less)
wenzelm@22718
   705
  apply (blast elim!: less_SucE
paulson@14341
   706
               intro!: add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   707
  done
paulson@14341
   708
paulson@14341
   709
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
nipkow@25134
   710
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
nipkow@25134
   711
apply(auto simp: gr0_conv_Suc)
nipkow@25134
   712
apply (induct_tac m)
nipkow@25134
   713
apply (simp_all add: add_less_mono)
nipkow@25134
   714
done
paulson@14341
   715
nipkow@14740
   716
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
obua@14738
   717
instance nat :: ordered_semidom
paulson@14341
   718
proof
paulson@14341
   719
  fix i j k :: nat
paulson@14348
   720
  show "0 < (1::nat)" by simp
paulson@14267
   721
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   722
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   723
qed
paulson@14267
   724
paulson@14267
   725
lemma nat_mult_1: "(1::nat) * n = n"
nipkow@25162
   726
by simp
paulson@14267
   727
paulson@14267
   728
lemma nat_mult_1_right: "n * (1::nat) = n"
nipkow@25162
   729
by simp
paulson@14267
   730
paulson@14267
   731
krauss@26748
   732
subsubsection {* Additional theorems about @{term "op \<le>"} *}
krauss@26748
   733
krauss@26748
   734
text {* Complete induction, aka course-of-values induction *}
krauss@26748
   735
haftmann@27823
   736
instance nat :: wellorder proof
haftmann@27823
   737
  fix P and n :: nat
haftmann@27823
   738
  assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
haftmann@27823
   739
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
haftmann@27823
   740
  proof (induct n)
haftmann@27823
   741
    case (0 n)
krauss@26748
   742
    have "P 0" by (rule step) auto
krauss@26748
   743
    thus ?case using 0 by auto
krauss@26748
   744
  next
haftmann@27823
   745
    case (Suc m n)
haftmann@27823
   746
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
krauss@26748
   747
    thus ?case
krauss@26748
   748
    proof
haftmann@27823
   749
      assume "n \<le> m" thus "P n" by (rule Suc(1))
krauss@26748
   750
    next
haftmann@27823
   751
      assume n: "n = Suc m"
haftmann@27823
   752
      show "P n"
haftmann@27823
   753
        by (rule step) (rule Suc(1), simp add: n le_simps)
krauss@26748
   754
    qed
krauss@26748
   755
  qed
haftmann@27823
   756
  then show "P n" by auto
krauss@26748
   757
qed
krauss@26748
   758
haftmann@27823
   759
lemma Least_Suc:
haftmann@27823
   760
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
haftmann@27823
   761
  apply (case_tac "n", auto)
haftmann@27823
   762
  apply (frule LeastI)
haftmann@27823
   763
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
haftmann@27823
   764
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
haftmann@27823
   765
  apply (erule_tac [2] Least_le)
haftmann@27823
   766
  apply (case_tac "LEAST x. P x", auto)
haftmann@27823
   767
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
haftmann@27823
   768
  apply (blast intro: order_antisym)
haftmann@27823
   769
  done
haftmann@27823
   770
haftmann@27823
   771
lemma Least_Suc2:
haftmann@27823
   772
   "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
haftmann@27823
   773
  apply (erule (1) Least_Suc [THEN ssubst])
haftmann@27823
   774
  apply simp
haftmann@27823
   775
  done
haftmann@27823
   776
haftmann@27823
   777
lemma ex_least_nat_le: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not>P i) & P(k)"
haftmann@27823
   778
  apply (cases n)
haftmann@27823
   779
   apply blast
haftmann@27823
   780
  apply (rule_tac x="LEAST k. P(k)" in exI)
haftmann@27823
   781
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
haftmann@27823
   782
  done
haftmann@27823
   783
haftmann@27823
   784
lemma ex_least_nat_less: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not>P i) & P(k+1)"
haftmann@27823
   785
  apply (cases n)
haftmann@27823
   786
   apply blast
haftmann@27823
   787
  apply (frule (1) ex_least_nat_le)
haftmann@27823
   788
  apply (erule exE)
haftmann@27823
   789
  apply (case_tac k)
haftmann@27823
   790
   apply simp
haftmann@27823
   791
  apply (rename_tac k1)
haftmann@27823
   792
  apply (rule_tac x=k1 in exI)
haftmann@27823
   793
  apply (auto simp add: less_eq_Suc_le)
haftmann@27823
   794
  done
haftmann@27823
   795
krauss@26748
   796
lemma nat_less_induct:
krauss@26748
   797
  assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
krauss@26748
   798
  using assms less_induct by blast
krauss@26748
   799
krauss@26748
   800
lemma measure_induct_rule [case_names less]:
krauss@26748
   801
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   802
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
krauss@26748
   803
  shows "P a"
krauss@26748
   804
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
krauss@26748
   805
krauss@26748
   806
text {* old style induction rules: *}
krauss@26748
   807
lemma measure_induct:
krauss@26748
   808
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   809
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
krauss@26748
   810
  by (rule measure_induct_rule [of f P a]) iprover
krauss@26748
   811
krauss@26748
   812
lemma full_nat_induct:
krauss@26748
   813
  assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
krauss@26748
   814
  shows "P n"
krauss@26748
   815
  by (rule less_induct) (auto intro: step simp:le_simps)
paulson@14267
   816
paulson@19870
   817
text{*An induction rule for estabilishing binary relations*}
wenzelm@22718
   818
lemma less_Suc_induct:
paulson@19870
   819
  assumes less:  "i < j"
paulson@19870
   820
     and  step:  "!!i. P i (Suc i)"
paulson@19870
   821
     and  trans: "!!i j k. P i j ==> P j k ==> P i k"
paulson@19870
   822
  shows "P i j"
paulson@19870
   823
proof -
wenzelm@22718
   824
  from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   825
  have "P i (Suc (i + k))"
paulson@19870
   826
  proof (induct k)
wenzelm@22718
   827
    case 0
wenzelm@22718
   828
    show ?case by (simp add: step)
paulson@19870
   829
  next
paulson@19870
   830
    case (Suc k)
wenzelm@22718
   831
    thus ?case by (auto intro: assms)
paulson@19870
   832
  qed
wenzelm@22718
   833
  thus "P i j" by (simp add: j)
paulson@19870
   834
qed
paulson@19870
   835
krauss@26748
   836
lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"
krauss@26748
   837
  apply (rule nat_less_induct)
krauss@26748
   838
  apply (case_tac n)
krauss@26748
   839
  apply (case_tac [2] nat)
krauss@26748
   840
  apply (blast intro: less_trans)+
krauss@26748
   841
  done
krauss@26748
   842
krauss@26748
   843
text {* The method of infinite descent, frequently used in number theory.
krauss@26748
   844
Provided by Roelof Oosterhuis.
krauss@26748
   845
$P(n)$ is true for all $n\in\mathbb{N}$ if
krauss@26748
   846
\begin{itemize}
krauss@26748
   847
  \item case ``0'': given $n=0$ prove $P(n)$,
krauss@26748
   848
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
krauss@26748
   849
        a smaller integer $m$ such that $\neg P(m)$.
krauss@26748
   850
\end{itemize} *}
krauss@26748
   851
krauss@26748
   852
text{* A compact version without explicit base case: *}
krauss@26748
   853
lemma infinite_descent:
krauss@26748
   854
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
krauss@26748
   855
by (induct n rule: less_induct, auto)
krauss@26748
   856
krauss@26748
   857
lemma infinite_descent0[case_names 0 smaller]: 
krauss@26748
   858
  "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
krauss@26748
   859
by (rule infinite_descent) (case_tac "n>0", auto)
krauss@26748
   860
krauss@26748
   861
text {*
krauss@26748
   862
Infinite descent using a mapping to $\mathbb{N}$:
krauss@26748
   863
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
krauss@26748
   864
\begin{itemize}
krauss@26748
   865
\item case ``0'': given $V(x)=0$ prove $P(x)$,
krauss@26748
   866
\item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$.
krauss@26748
   867
\end{itemize}
krauss@26748
   868
NB: the proof also shows how to use the previous lemma. *}
krauss@26748
   869
krauss@26748
   870
corollary infinite_descent0_measure [case_names 0 smaller]:
krauss@26748
   871
  assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
krauss@26748
   872
    and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
krauss@26748
   873
  shows "P x"
krauss@26748
   874
proof -
krauss@26748
   875
  obtain n where "n = V x" by auto
krauss@26748
   876
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
krauss@26748
   877
  proof (induct n rule: infinite_descent0)
krauss@26748
   878
    case 0 -- "i.e. $V(x) = 0$"
krauss@26748
   879
    with A0 show "P x" by auto
krauss@26748
   880
  next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
krauss@26748
   881
    case (smaller n)
krauss@26748
   882
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
krauss@26748
   883
    with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
krauss@26748
   884
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
krauss@26748
   885
    then show ?case by auto
krauss@26748
   886
  qed
krauss@26748
   887
  ultimately show "P x" by auto
krauss@26748
   888
qed
krauss@26748
   889
krauss@26748
   890
text{* Again, without explicit base case: *}
krauss@26748
   891
lemma infinite_descent_measure:
krauss@26748
   892
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
krauss@26748
   893
proof -
krauss@26748
   894
  from assms obtain n where "n = V x" by auto
krauss@26748
   895
  moreover have "!!x. V x = n \<Longrightarrow> P x"
krauss@26748
   896
  proof (induct n rule: infinite_descent, auto)
krauss@26748
   897
    fix x assume "\<not> P x"
krauss@26748
   898
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
krauss@26748
   899
  qed
krauss@26748
   900
  ultimately show "P x" by auto
krauss@26748
   901
qed
krauss@26748
   902
paulson@14267
   903
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   904
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   905
lemma less_mono_imp_le_mono:
nipkow@24438
   906
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
   907
by (simp add: order_le_less) (blast)
nipkow@24438
   908
paulson@14267
   909
paulson@14267
   910
text {* non-strict, in 1st argument *}
paulson@14267
   911
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
   912
by (rule add_right_mono)
paulson@14267
   913
paulson@14267
   914
text {* non-strict, in both arguments *}
paulson@14267
   915
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
   916
by (rule add_mono)
paulson@14267
   917
paulson@14267
   918
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
   919
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
   920
paulson@14267
   921
lemma le_add1: "n \<le> ((n + m)::nat)"
nipkow@24438
   922
by (simp add: add_commute, rule le_add2)
berghofe@13449
   923
berghofe@13449
   924
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
   925
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   926
berghofe@13449
   927
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
   928
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   929
paulson@14267
   930
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
   931
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   932
paulson@14267
   933
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
   934
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   935
paulson@14267
   936
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
   937
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
   938
berghofe@13449
   939
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
   940
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
   941
berghofe@13449
   942
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
   943
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
   944
berghofe@13449
   945
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
   946
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
   947
apply (simp_all add: le_add1)
nipkow@24438
   948
done
berghofe@13449
   949
berghofe@13449
   950
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
   951
apply (rule notI)
wenzelm@26335
   952
apply (drule add_lessD1)
wenzelm@26335
   953
apply (erule less_irrefl [THEN notE])
nipkow@24438
   954
done
berghofe@13449
   955
berghofe@13449
   956
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
krauss@26748
   957
by (simp add: add_commute)
berghofe@13449
   958
paulson@14267
   959
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
   960
apply (rule order_trans [of _ "m+k"])
nipkow@24438
   961
apply (simp_all add: le_add1)
nipkow@24438
   962
done
berghofe@13449
   963
paulson@14267
   964
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
nipkow@24438
   965
apply (simp add: add_commute)
nipkow@24438
   966
apply (erule add_leD1)
nipkow@24438
   967
done
berghofe@13449
   968
paulson@14267
   969
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
   970
by (blast dest: add_leD1 add_leD2)
berghofe@13449
   971
berghofe@13449
   972
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
   973
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
   974
by (force simp del: add_Suc_right
berghofe@13449
   975
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
   976
berghofe@13449
   977
haftmann@26072
   978
subsubsection {* More results about difference *}
berghofe@13449
   979
berghofe@13449
   980
text {* Addition is the inverse of subtraction:
paulson@14267
   981
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
   982
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
nipkow@24438
   983
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   984
paulson@14267
   985
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
nipkow@24438
   986
by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
   987
paulson@14267
   988
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
krauss@26748
   989
by (simp add: add_commute)
berghofe@13449
   990
paulson@14267
   991
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
   992
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   993
berghofe@13449
   994
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
   995
apply (induct m n rule: diff_induct)
nipkow@24438
   996
apply (erule_tac [3] less_SucE)
nipkow@24438
   997
apply (simp_all add: less_Suc_eq)
nipkow@24438
   998
done
berghofe@13449
   999
paulson@14267
  1000
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
  1001
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
  1002
haftmann@26072
  1003
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
haftmann@26072
  1004
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
haftmann@26072
  1005
berghofe@13449
  1006
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
  1007
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
  1008
berghofe@13449
  1009
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
  1010
by (cases n) (auto simp add: le_simps)
berghofe@13449
  1011
paulson@14267
  1012
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
  1013
by (induct j k rule: diff_induct) simp_all
berghofe@13449
  1014
paulson@14267
  1015
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
nipkow@24438
  1016
by (simp add: add_commute diff_add_assoc)
berghofe@13449
  1017
paulson@14267
  1018
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
nipkow@24438
  1019
by (auto simp add: diff_add_inverse2)
berghofe@13449
  1020
paulson@14267
  1021
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
  1022
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1023
paulson@14267
  1024
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
  1025
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
  1026
berghofe@13449
  1027
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
  1028
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1029
wenzelm@22718
  1030
lemma less_imp_add_positive:
wenzelm@22718
  1031
  assumes "i < j"
wenzelm@22718
  1032
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
  1033
proof
wenzelm@22718
  1034
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
  1035
    by (simp add: order_less_imp_le)
wenzelm@22718
  1036
qed
wenzelm@9436
  1037
haftmann@26072
  1038
text {* a nice rewrite for bounded subtraction *}
haftmann@26072
  1039
lemma nat_minus_add_max:
haftmann@26072
  1040
  fixes n m :: nat
haftmann@26072
  1041
  shows "n - m + m = max n m"
haftmann@26072
  1042
    by (simp add: max_def not_le order_less_imp_le)
berghofe@13449
  1043
haftmann@26072
  1044
lemma nat_diff_split:
haftmann@26072
  1045
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
haftmann@26072
  1046
    -- {* elimination of @{text -} on @{text nat} *}
haftmann@26072
  1047
by (cases "a < b")
haftmann@26072
  1048
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
haftmann@26072
  1049
    not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)
berghofe@13449
  1050
haftmann@26072
  1051
lemma nat_diff_split_asm:
haftmann@26072
  1052
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
haftmann@26072
  1053
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
haftmann@26072
  1054
by (auto split: nat_diff_split)
berghofe@13449
  1055
berghofe@13449
  1056
haftmann@26072
  1057
subsubsection {* Monotonicity of Multiplication *}
berghofe@13449
  1058
paulson@14267
  1059
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
  1060
by (simp add: mult_right_mono)
berghofe@13449
  1061
paulson@14267
  1062
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
  1063
by (simp add: mult_left_mono)
berghofe@13449
  1064
paulson@14267
  1065
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
  1066
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
  1067
by (simp add: mult_mono)
berghofe@13449
  1068
berghofe@13449
  1069
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
  1070
by (simp add: mult_strict_right_mono)
berghofe@13449
  1071
paulson@14266
  1072
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
  1073
      there are no negative numbers.*}
paulson@14266
  1074
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
  1075
  apply (induct m)
wenzelm@22718
  1076
   apply simp
wenzelm@22718
  1077
  apply (case_tac n)
wenzelm@22718
  1078
   apply simp_all
berghofe@13449
  1079
  done
berghofe@13449
  1080
paulson@14267
  1081
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"
berghofe@13449
  1082
  apply (induct m)
wenzelm@22718
  1083
   apply simp
wenzelm@22718
  1084
  apply (case_tac n)
wenzelm@22718
  1085
   apply simp_all
berghofe@13449
  1086
  done
berghofe@13449
  1087
paulson@14341
  1088
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1089
  apply (safe intro!: mult_less_mono1)
paulson@14208
  1090
  apply (case_tac k, auto)
berghofe@13449
  1091
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1092
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1093
  done
berghofe@13449
  1094
berghofe@13449
  1095
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
nipkow@24438
  1096
by (simp add: mult_commute [of k])
berghofe@13449
  1097
paulson@14267
  1098
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
  1099
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1100
paulson@14267
  1101
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
  1102
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1103
berghofe@13449
  1104
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
  1105
by (subst mult_less_cancel1) simp
berghofe@13449
  1106
paulson@14267
  1107
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
  1108
by (subst mult_le_cancel1) simp
berghofe@13449
  1109
haftmann@26072
  1110
lemma le_square: "m \<le> m * (m::nat)"
haftmann@26072
  1111
  by (cases m) (auto intro: le_add1)
haftmann@26072
  1112
haftmann@26072
  1113
lemma le_cube: "(m::nat) \<le> m * (m * m)"
haftmann@26072
  1114
  by (cases m) (auto intro: le_add1)
berghofe@13449
  1115
berghofe@13449
  1116
text {* Lemma for @{text gcd} *}
berghofe@13449
  1117
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1118
  apply (drule sym)
berghofe@13449
  1119
  apply (rule disjCI)
berghofe@13449
  1120
  apply (rule nat_less_cases, erule_tac [2] _)
paulson@25157
  1121
   apply (drule_tac [2] mult_less_mono2)
nipkow@25162
  1122
    apply (auto)
berghofe@13449
  1123
  done
wenzelm@9436
  1124
haftmann@26072
  1125
text {* the lattice order on @{typ nat} *}
haftmann@24995
  1126
haftmann@26072
  1127
instantiation nat :: distrib_lattice
haftmann@26072
  1128
begin
haftmann@24995
  1129
haftmann@26072
  1130
definition
haftmann@26072
  1131
  "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
haftmann@24995
  1132
haftmann@26072
  1133
definition
haftmann@26072
  1134
  "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
haftmann@24995
  1135
haftmann@26072
  1136
instance by intro_classes
haftmann@26072
  1137
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
haftmann@26072
  1138
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
haftmann@24995
  1139
haftmann@26072
  1140
end
haftmann@24995
  1141
haftmann@24995
  1142
haftmann@25193
  1143
subsection {* Embedding of the Naturals into any
haftmann@25193
  1144
  @{text semiring_1}: @{term of_nat} *}
haftmann@24196
  1145
haftmann@24196
  1146
context semiring_1
haftmann@24196
  1147
begin
haftmann@24196
  1148
haftmann@25559
  1149
primrec
haftmann@25559
  1150
  of_nat :: "nat \<Rightarrow> 'a"
haftmann@25559
  1151
where
haftmann@25559
  1152
  of_nat_0:     "of_nat 0 = 0"
haftmann@25559
  1153
  | of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
haftmann@25193
  1154
haftmann@25193
  1155
lemma of_nat_1 [simp]: "of_nat 1 = 1"
haftmann@25193
  1156
  by simp
haftmann@25193
  1157
haftmann@25193
  1158
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
haftmann@25193
  1159
  by (induct m) (simp_all add: add_ac)
haftmann@25193
  1160
haftmann@25193
  1161
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
haftmann@25193
  1162
  by (induct m) (simp_all add: add_ac left_distrib)
haftmann@25193
  1163
haftmann@25928
  1164
definition
haftmann@25928
  1165
  of_nat_aux :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@25928
  1166
where
haftmann@25928
  1167
  [code func del]: "of_nat_aux n i = of_nat n + i"
haftmann@25928
  1168
haftmann@25928
  1169
lemma of_nat_aux_code [code]:
haftmann@25928
  1170
  "of_nat_aux 0 i = i"
haftmann@25928
  1171
  "of_nat_aux (Suc n) i = of_nat_aux n (i + 1)" -- {* tail recursive *}
haftmann@25928
  1172
  by (simp_all add: of_nat_aux_def add_ac)
haftmann@25928
  1173
haftmann@25928
  1174
lemma of_nat_code [code]:
haftmann@25928
  1175
  "of_nat n = of_nat_aux n 0"
haftmann@25928
  1176
  by (simp add: of_nat_aux_def)
haftmann@25928
  1177
haftmann@24196
  1178
end
haftmann@24196
  1179
haftmann@26072
  1180
text{*Class for unital semirings with characteristic zero.
haftmann@26072
  1181
 Includes non-ordered rings like the complex numbers.*}
haftmann@26072
  1182
haftmann@26072
  1183
class semiring_char_0 = semiring_1 +
haftmann@26072
  1184
  assumes of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
haftmann@26072
  1185
begin
haftmann@26072
  1186
haftmann@26072
  1187
text{*Special cases where either operand is zero*}
haftmann@26072
  1188
haftmann@26072
  1189
lemma of_nat_0_eq_iff [simp, noatp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
haftmann@26072
  1190
  by (rule of_nat_eq_iff [of 0, simplified])
haftmann@26072
  1191
haftmann@26072
  1192
lemma of_nat_eq_0_iff [simp, noatp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
haftmann@26072
  1193
  by (rule of_nat_eq_iff [of _ 0, simplified])
haftmann@26072
  1194
haftmann@26072
  1195
lemma inj_of_nat: "inj of_nat"
haftmann@26072
  1196
  by (simp add: inj_on_def)
haftmann@26072
  1197
haftmann@26072
  1198
end
haftmann@26072
  1199
haftmann@25193
  1200
context ordered_semidom
haftmann@25193
  1201
begin
haftmann@25193
  1202
haftmann@25193
  1203
lemma zero_le_imp_of_nat: "0 \<le> of_nat m"
haftmann@25193
  1204
  apply (induct m, simp_all)
haftmann@25193
  1205
  apply (erule order_trans)
haftmann@25193
  1206
  apply (rule ord_le_eq_trans [OF _ add_commute])
haftmann@25193
  1207
  apply (rule less_add_one [THEN less_imp_le])
haftmann@25193
  1208
  done
haftmann@25193
  1209
haftmann@25193
  1210
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
haftmann@25193
  1211
  apply (induct m n rule: diff_induct, simp_all)
haftmann@25193
  1212
  apply (insert add_less_le_mono [OF zero_less_one zero_le_imp_of_nat], force)
haftmann@25193
  1213
  done
haftmann@25193
  1214
haftmann@25193
  1215
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
haftmann@25193
  1216
  apply (induct m n rule: diff_induct, simp_all)
haftmann@25193
  1217
  apply (insert zero_le_imp_of_nat)
haftmann@25193
  1218
  apply (force simp add: not_less [symmetric])
haftmann@25193
  1219
  done
haftmann@25193
  1220
haftmann@25193
  1221
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
haftmann@25193
  1222
  by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
haftmann@25193
  1223
haftmann@26072
  1224
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
haftmann@26072
  1225
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
haftmann@25193
  1226
haftmann@26072
  1227
text{*Every @{text ordered_semidom} has characteristic zero.*}
haftmann@25193
  1228
haftmann@26072
  1229
subclass semiring_char_0
haftmann@26072
  1230
  by unfold_locales (simp add: eq_iff order_eq_iff)
haftmann@25193
  1231
haftmann@25193
  1232
text{*Special cases where either operand is zero*}
haftmann@25193
  1233
haftmann@25193
  1234
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
haftmann@25193
  1235
  by (rule of_nat_le_iff [of 0, simplified])
haftmann@25193
  1236
haftmann@25193
  1237
lemma of_nat_le_0_iff [simp, noatp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
haftmann@25193
  1238
  by (rule of_nat_le_iff [of _ 0, simplified])
haftmann@25193
  1239
haftmann@26072
  1240
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
haftmann@26072
  1241
  by (rule of_nat_less_iff [of 0, simplified])
haftmann@26072
  1242
haftmann@26072
  1243
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
haftmann@26072
  1244
  by (rule of_nat_less_iff [of _ 0, simplified])
haftmann@26072
  1245
haftmann@26072
  1246
end
haftmann@26072
  1247
haftmann@26072
  1248
context ring_1
haftmann@26072
  1249
begin
haftmann@26072
  1250
haftmann@26072
  1251
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
haftmann@26072
  1252
  by (simp add: compare_rls of_nat_add [symmetric])
haftmann@26072
  1253
haftmann@26072
  1254
end
haftmann@26072
  1255
haftmann@26072
  1256
context ordered_idom
haftmann@26072
  1257
begin
haftmann@26072
  1258
haftmann@26072
  1259
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
haftmann@26072
  1260
  unfolding abs_if by auto
haftmann@26072
  1261
haftmann@25193
  1262
end
haftmann@25193
  1263
haftmann@25193
  1264
lemma of_nat_id [simp]: "of_nat n = n"
haftmann@25193
  1265
  by (induct n) auto
haftmann@25193
  1266
haftmann@25193
  1267
lemma of_nat_eq_id [simp]: "of_nat = id"
haftmann@25193
  1268
  by (auto simp add: expand_fun_eq)
haftmann@25193
  1269
haftmann@25193
  1270
haftmann@26149
  1271
subsection {* The Set of Natural Numbers *}
haftmann@25193
  1272
haftmann@26072
  1273
context semiring_1
haftmann@25193
  1274
begin
haftmann@25193
  1275
haftmann@26072
  1276
definition
haftmann@26072
  1277
  Nats  :: "'a set" where
haftmann@27104
  1278
  [code func del]: "Nats = range of_nat"
haftmann@26072
  1279
haftmann@26072
  1280
notation (xsymbols)
haftmann@26072
  1281
  Nats  ("\<nat>")
haftmann@25193
  1282
haftmann@26072
  1283
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
haftmann@26072
  1284
  by (simp add: Nats_def)
haftmann@26072
  1285
haftmann@26072
  1286
lemma Nats_0 [simp]: "0 \<in> \<nat>"
haftmann@26072
  1287
apply (simp add: Nats_def)
haftmann@26072
  1288
apply (rule range_eqI)
haftmann@26072
  1289
apply (rule of_nat_0 [symmetric])
haftmann@26072
  1290
done
haftmann@25193
  1291
haftmann@26072
  1292
lemma Nats_1 [simp]: "1 \<in> \<nat>"
haftmann@26072
  1293
apply (simp add: Nats_def)
haftmann@26072
  1294
apply (rule range_eqI)
haftmann@26072
  1295
apply (rule of_nat_1 [symmetric])
haftmann@26072
  1296
done
haftmann@25193
  1297
haftmann@26072
  1298
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
haftmann@26072
  1299
apply (auto simp add: Nats_def)
haftmann@26072
  1300
apply (rule range_eqI)
haftmann@26072
  1301
apply (rule of_nat_add [symmetric])
haftmann@26072
  1302
done
haftmann@26072
  1303
haftmann@26072
  1304
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
haftmann@26072
  1305
apply (auto simp add: Nats_def)
haftmann@26072
  1306
apply (rule range_eqI)
haftmann@26072
  1307
apply (rule of_nat_mult [symmetric])
haftmann@26072
  1308
done
haftmann@25193
  1309
haftmann@25193
  1310
end
haftmann@25193
  1311
haftmann@25193
  1312
wenzelm@21243
  1313
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
wenzelm@21243
  1314
haftmann@22845
  1315
lemma subst_equals:
haftmann@22845
  1316
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1317
  shows "u = s"
haftmann@22845
  1318
  using 2 1 by (rule trans)
haftmann@22845
  1319
wenzelm@21243
  1320
use "arith_data.ML"
haftmann@26101
  1321
declaration {* K ArithData.setup *}
wenzelm@24091
  1322
wenzelm@24091
  1323
use "Tools/lin_arith.ML"
wenzelm@24091
  1324
declaration {* K LinArith.setup *}
wenzelm@24091
  1325
wenzelm@21243
  1326
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1327
nipkow@27625
  1328
nipkow@27625
  1329
context order
nipkow@27625
  1330
begin
nipkow@27625
  1331
nipkow@27625
  1332
lemma lift_Suc_mono_le:
krauss@27627
  1333
  assumes mono: "!!n. f n \<le> f(Suc n)" and "n\<le>n'"
krauss@27627
  1334
  shows "f n \<le> f n'"
krauss@27627
  1335
proof (cases "n < n'")
krauss@27627
  1336
  case True
krauss@27627
  1337
  thus ?thesis
krauss@27627
  1338
    by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)
krauss@27627
  1339
qed (insert `n \<le> n'`, auto) -- {*trivial for @{prop "n = n'"} *}
nipkow@27625
  1340
nipkow@27625
  1341
lemma lift_Suc_mono_less:
krauss@27627
  1342
  assumes mono: "!!n. f n < f(Suc n)" and "n < n'"
krauss@27627
  1343
  shows "f n < f n'"
krauss@27627
  1344
using `n < n'`
krauss@27627
  1345
by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)
nipkow@27625
  1346
nipkow@27789
  1347
lemma lift_Suc_mono_less_iff:
nipkow@27789
  1348
  "(!!n. f n < f(Suc n)) \<Longrightarrow> f(n) < f(m) \<longleftrightarrow> n<m"
nipkow@27789
  1349
by(blast intro: less_asym' lift_Suc_mono_less[of f]
nipkow@27789
  1350
         dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq[THEN iffD1])
nipkow@27789
  1351
nipkow@27625
  1352
end
nipkow@27625
  1353
nipkow@27625
  1354
nipkow@27789
  1355
lemma mono_nat_linear_lb:
nipkow@27789
  1356
  "(!!m n::nat. m<n \<Longrightarrow> f m < f n) \<Longrightarrow> f(m)+k \<le> f(m+k)"
nipkow@27789
  1357
apply(induct_tac k)
nipkow@27789
  1358
 apply simp
nipkow@27789
  1359
apply(erule_tac x="m+n" in meta_allE)
nipkow@27789
  1360
apply(erule_tac x="m+n+1" in meta_allE)
nipkow@27789
  1361
apply simp
nipkow@27789
  1362
done
nipkow@27789
  1363
nipkow@27789
  1364
wenzelm@21243
  1365
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1366
wenzelm@21243
  1367
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1368
by arith
wenzelm@21243
  1369
wenzelm@21243
  1370
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1371
by arith
wenzelm@21243
  1372
wenzelm@21243
  1373
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1374
by arith
wenzelm@21243
  1375
wenzelm@21243
  1376
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
nipkow@24438
  1377
by arith
wenzelm@21243
  1378
wenzelm@21243
  1379
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1380
by arith
wenzelm@21243
  1381
wenzelm@21243
  1382
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
nipkow@24438
  1383
by arith
wenzelm@21243
  1384
wenzelm@21243
  1385
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1386
  second premise n\<le>m*)
wenzelm@21243
  1387
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1388
by arith
wenzelm@21243
  1389
haftmann@26072
  1390
text {* Simplification of relational expressions involving subtraction *}
wenzelm@21243
  1391
wenzelm@21243
  1392
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1393
by (simp split add: nat_diff_split)
wenzelm@21243
  1394
wenzelm@21243
  1395
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1396
by (auto split add: nat_diff_split)
wenzelm@21243
  1397
wenzelm@21243
  1398
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1399
by (auto split add: nat_diff_split)
wenzelm@21243
  1400
wenzelm@21243
  1401
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1402
by (auto split add: nat_diff_split)
wenzelm@21243
  1403
wenzelm@21243
  1404
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1405
wenzelm@21243
  1406
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1407
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1408
by (simp split add: nat_diff_split)
wenzelm@21243
  1409
wenzelm@21243
  1410
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1411
by (simp split add: nat_diff_split)
wenzelm@21243
  1412
wenzelm@21243
  1413
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1414
by (simp split add: nat_diff_split)
wenzelm@21243
  1415
wenzelm@21243
  1416
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1417
by (simp split add: nat_diff_split)
wenzelm@21243
  1418
bulwahn@26143
  1419
lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
bulwahn@26143
  1420
unfolding min_def by auto
bulwahn@26143
  1421
bulwahn@26143
  1422
lemma inj_on_diff_nat: 
bulwahn@26143
  1423
  assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
bulwahn@26143
  1424
  shows "inj_on (\<lambda>n. n - k) N"
bulwahn@26143
  1425
proof (rule inj_onI)
bulwahn@26143
  1426
  fix x y
bulwahn@26143
  1427
  assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
bulwahn@26143
  1428
  with k_le_n have "x - k + k = y - k + k" by auto
bulwahn@26143
  1429
  with a k_le_n show "x = y" by auto
bulwahn@26143
  1430
qed
bulwahn@26143
  1431
haftmann@26072
  1432
text{*Rewriting to pull differences out*}
haftmann@26072
  1433
haftmann@26072
  1434
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
haftmann@26072
  1435
by arith
haftmann@26072
  1436
haftmann@26072
  1437
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
haftmann@26072
  1438
by arith
haftmann@26072
  1439
haftmann@26072
  1440
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
haftmann@26072
  1441
by arith
haftmann@26072
  1442
wenzelm@21243
  1443
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1444
wenzelm@21243
  1445
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1446
by (cases m) auto
wenzelm@21243
  1447
wenzelm@21243
  1448
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1449
by (cases m) auto
wenzelm@21243
  1450
wenzelm@21243
  1451
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1452
by (cases m) auto
wenzelm@21243
  1453
krauss@23001
  1454
text {* Specialized induction principles that work "backwards": *}
krauss@23001
  1455
krauss@23001
  1456
lemma inc_induct[consumes 1, case_names base step]:
krauss@23001
  1457
  assumes less: "i <= j"
krauss@23001
  1458
  assumes base: "P j"
krauss@23001
  1459
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1460
  shows "P i"
krauss@23001
  1461
  using less
krauss@23001
  1462
proof (induct d=="j - i" arbitrary: i)
krauss@23001
  1463
  case (0 i)
krauss@23001
  1464
  hence "i = j" by simp
krauss@23001
  1465
  with base show ?case by simp
krauss@23001
  1466
next
krauss@23001
  1467
  case (Suc d i)
krauss@23001
  1468
  hence "i < j" "P (Suc i)"
krauss@23001
  1469
    by simp_all
krauss@23001
  1470
  thus "P i" by (rule step)
krauss@23001
  1471
qed
krauss@23001
  1472
krauss@23001
  1473
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1474
  assumes less: "i < j"
krauss@23001
  1475
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1476
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1477
  shows "P i"
krauss@23001
  1478
  using less
krauss@23001
  1479
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1480
  case (0 i)
krauss@23001
  1481
  with `i < j` have "j = Suc i" by simp
krauss@23001
  1482
  with base show ?case by simp
krauss@23001
  1483
next
krauss@23001
  1484
  case (Suc d i)
krauss@23001
  1485
  hence "i < j" "P (Suc i)"
krauss@23001
  1486
    by simp_all
krauss@23001
  1487
  thus "P i" by (rule step)
krauss@23001
  1488
qed
krauss@23001
  1489
krauss@23001
  1490
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1491
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1492
krauss@23001
  1493
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1494
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1495
haftmann@26072
  1496
lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False"
haftmann@26072
  1497
  by auto
wenzelm@21243
  1498
wenzelm@21243
  1499
(*The others are
wenzelm@21243
  1500
      i - j - k = i - (j + k),
wenzelm@21243
  1501
      k \<le> j ==> j - k + i = j + i - k,
wenzelm@21243
  1502
      k \<le> j ==> i + (j - k) = i + j - k *)
wenzelm@21243
  1503
lemmas add_diff_assoc = diff_add_assoc [symmetric]
wenzelm@21243
  1504
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
haftmann@26072
  1505
declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
wenzelm@21243
  1506
wenzelm@21243
  1507
text{*At present we prove no analogue of @{text not_less_Least} or @{text
wenzelm@21243
  1508
Least_Suc}, since there appears to be no need.*}
wenzelm@21243
  1509
nipkow@27625
  1510
haftmann@26072
  1511
subsection {* size of a datatype value *}
haftmann@25193
  1512
haftmann@26072
  1513
class size = type +
krauss@26748
  1514
  fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}
haftmann@23852
  1515
haftmann@25193
  1516
end