src/HOL/Nat.thy
author haftmann
Wed Feb 20 14:52:38 2008 +0100 (2008-02-20)
changeset 26101 a657683e902a
parent 26072 f65a7fa2da6c
child 26143 314c0bcb7df7
permissions -rw-r--r--
tuned structures in arith_data.ML
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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 = "Collect Nat"
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  by (rule exI, rule CollectI, 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 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 CollectD, THEN Nat.induct])
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  apply (iprover elim: Abs_Nat_inverse [OF CollectI, THEN subst])
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  done
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lemma Suc_not_Zero [iff]: "Suc m \<noteq> 0"
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  by (simp add: Zero_nat_def Suc_def
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    Abs_Nat_inject Rep_Nat [THEN CollectD] Suc_RepI Zero_RepI
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      Suc_Rep_not_Zero_Rep)
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lemma Zero_not_Suc [iff]: "0 \<noteq> Suc m"
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  by (rule not_sym, rule Suc_not_Zero not_sym)
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lemma inj_Suc[simp]: "inj_on Suc N"
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  by (simp add: Suc_def inj_on_def Abs_Nat_inject Rep_Nat [THEN CollectD] Suc_RepI
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                inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject)
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lemma Suc_Suc_eq [iff]: "Suc m = Suc n \<longleftrightarrow> m = n"
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  by (rule inj_Suc [THEN inj_eq])
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rep_datatype nat
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  distinct  Suc_not_Zero Zero_not_Suc
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  inject    Suc_Suc_eq
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  induction nat_induct
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declare nat.induct [case_names 0 Suc, induct type: nat]
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declare nat.exhaust [case_names 0 Suc, cases type: nat]
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lemmas nat_rec_0 = nat.recs(1)
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  and nat_rec_Suc = nat.recs(2)
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lemmas nat_case_0 = nat.cases(1)
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  and nat_case_Suc = nat.cases(2)
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text {* Injectiveness and distinctness lemmas *}
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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
haftmann@25510
   343
haftmann@26072
   344
primrec less_eq_nat where
haftmann@26072
   345
  "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
haftmann@26072
   346
  | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
haftmann@26072
   347
haftmann@26072
   348
declare less_eq_nat.simps [simp del, code del]
haftmann@26072
   349
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps)
haftmann@26072
   350
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
haftmann@26072
   351
haftmann@26072
   352
definition less_nat where
haftmann@26072
   353
  less_eq_Suc_le [code func del]: "n < m \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   354
haftmann@26072
   355
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
haftmann@26072
   356
  by (simp add: less_eq_nat.simps(2))
haftmann@26072
   357
haftmann@26072
   358
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
haftmann@26072
   359
  unfolding less_eq_Suc_le ..
haftmann@26072
   360
haftmann@26072
   361
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
haftmann@26072
   362
  by (induct n) (simp_all add: less_eq_nat.simps(2))
haftmann@26072
   363
haftmann@26072
   364
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
haftmann@26072
   365
  by (simp add: less_eq_Suc_le)
haftmann@26072
   366
haftmann@26072
   367
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
haftmann@26072
   368
  by simp
haftmann@26072
   369
haftmann@26072
   370
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
haftmann@26072
   371
  by (simp add: less_eq_Suc_le)
haftmann@26072
   372
haftmann@26072
   373
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
haftmann@26072
   374
  by (simp add: less_eq_Suc_le)
haftmann@26072
   375
haftmann@26072
   376
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
haftmann@26072
   377
  by (induct m arbitrary: n)
haftmann@26072
   378
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   379
haftmann@26072
   380
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
haftmann@26072
   381
  by (cases n) (auto intro: le_SucI)
haftmann@26072
   382
haftmann@26072
   383
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
haftmann@26072
   384
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@24995
   385
haftmann@26072
   386
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
haftmann@26072
   387
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@25510
   388
haftmann@26072
   389
instance proof
haftmann@26072
   390
  fix n m :: nat
haftmann@26072
   391
  have less_imp_le: "n < m \<Longrightarrow> n \<le> m"
haftmann@26072
   392
    unfolding less_eq_Suc_le by (erule Suc_leD)
haftmann@26072
   393
  have irrefl: "\<not> m < m" by (induct m) auto
haftmann@26072
   394
  have strict: "n \<le> m \<Longrightarrow> n \<noteq> m \<Longrightarrow> n < m"
haftmann@26072
   395
  proof (induct n arbitrary: m)
haftmann@26072
   396
    case 0 then show ?case
haftmann@26072
   397
      by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   398
  next
haftmann@26072
   399
    case (Suc n) then show ?case
haftmann@26072
   400
      by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   401
  qed
haftmann@26072
   402
  show "n < m \<longleftrightarrow> n \<le> m \<and> n \<noteq> m"
haftmann@26072
   403
    by (auto simp add: irrefl intro: less_imp_le strict)
haftmann@26072
   404
next
haftmann@26072
   405
  fix n :: nat show "n \<le> n" by (induct n) simp_all
haftmann@26072
   406
next
haftmann@26072
   407
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
haftmann@26072
   408
  then show "n = m"
haftmann@26072
   409
    by (induct n arbitrary: m)
haftmann@26072
   410
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   411
next
haftmann@26072
   412
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
haftmann@26072
   413
  then show "n \<le> q"
haftmann@26072
   414
  proof (induct n arbitrary: m q)
haftmann@26072
   415
    case 0 show ?case by simp
haftmann@26072
   416
  next
haftmann@26072
   417
    case (Suc n) then show ?case
haftmann@26072
   418
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   419
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   420
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   421
  qed
haftmann@26072
   422
next
haftmann@26072
   423
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
haftmann@26072
   424
    by (induct n arbitrary: m)
haftmann@26072
   425
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   426
qed
haftmann@25510
   427
haftmann@25510
   428
end
berghofe@13449
   429
haftmann@26072
   430
subsubsection {* Introduction properties *}
berghofe@13449
   431
haftmann@26072
   432
lemma lessI [iff]: "n < Suc n"
haftmann@26072
   433
  by (simp add: less_Suc_eq_le)
berghofe@13449
   434
haftmann@26072
   435
lemma zero_less_Suc [iff]: "0 < Suc n"
haftmann@26072
   436
  by (simp add: less_Suc_eq_le)
berghofe@13449
   437
berghofe@13449
   438
lemma less_trans: "i < j ==> j < k ==> i < (k::nat)"
haftmann@26072
   439
  by (rule order_less_trans)
berghofe@13449
   440
berghofe@13449
   441
subsubsection {* Elimination properties *}
berghofe@13449
   442
berghofe@13449
   443
lemma less_not_sym: "n < m ==> ~ m < (n::nat)"
haftmann@26072
   444
  by (rule order_less_not_sym)
berghofe@13449
   445
berghofe@13449
   446
lemma less_asym:
berghofe@13449
   447
  assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P
berghofe@13449
   448
  apply (rule contrapos_np)
berghofe@13449
   449
  apply (rule less_not_sym)
berghofe@13449
   450
  apply (rule h1)
berghofe@13449
   451
  apply (erule h2)
berghofe@13449
   452
  done
berghofe@13449
   453
berghofe@13449
   454
lemma less_not_refl: "~ n < (n::nat)"
haftmann@26072
   455
  by (rule order_less_irrefl)
berghofe@13449
   456
berghofe@13449
   457
lemma less_irrefl [elim!]: "(n::nat) < n ==> R"
haftmann@26072
   458
  by (rule notE, rule less_not_refl)
berghofe@13449
   459
paulson@14267
   460
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
haftmann@26072
   461
  by (rule less_imp_neq)
berghofe@13449
   462
haftmann@26072
   463
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
haftmann@26072
   464
  by (rule not_sym) (rule less_imp_neq) 
berghofe@13449
   465
berghofe@13449
   466
lemma less_zeroE: "(n::nat) < 0 ==> R"
haftmann@26072
   467
  by (rule notE) (rule not_less0)
berghofe@13449
   468
berghofe@13449
   469
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
haftmann@26072
   470
  unfolding less_Suc_eq_le le_less ..
berghofe@13449
   471
haftmann@26072
   472
lemma less_one [iff, noatp]: "(n < (1::nat)) = (n = 0)"
haftmann@26072
   473
  by (simp add: less_Suc_eq)
berghofe@13449
   474
berghofe@13449
   475
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
haftmann@26072
   476
  by (simp add: less_Suc_eq)
berghofe@13449
   477
berghofe@13449
   478
lemma Suc_mono: "m < n ==> Suc m < Suc n"
haftmann@26072
   479
  by simp
berghofe@13449
   480
berghofe@13449
   481
lemma less_linear: "m < n | m = n | n < (m::nat)"
haftmann@26072
   482
  by (rule less_linear)
berghofe@13449
   483
nipkow@14302
   484
text {* "Less than" is antisymmetric, sort of *}
nipkow@14302
   485
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
haftmann@26072
   486
  unfolding not_less less_Suc_eq_le by (rule antisym)
nipkow@14302
   487
paulson@14267
   488
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
haftmann@26072
   489
  by (rule linorder_neq_iff)
berghofe@13449
   490
berghofe@13449
   491
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   492
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   493
  shows "P n m"
berghofe@13449
   494
  apply (rule less_linear [THEN disjE])
berghofe@13449
   495
  apply (erule_tac [2] disjE)
berghofe@13449
   496
  apply (erule lessCase)
berghofe@13449
   497
  apply (erule sym [THEN eqCase])
berghofe@13449
   498
  apply (erule major)
berghofe@13449
   499
  done
berghofe@13449
   500
berghofe@13449
   501
berghofe@13449
   502
subsubsection {* Inductive (?) properties *}
berghofe@13449
   503
paulson@14267
   504
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
haftmann@26072
   505
  unfolding less_eq_Suc_le [of m] le_less by simp 
berghofe@13449
   506
haftmann@26072
   507
lemma lessE:
haftmann@26072
   508
  assumes major: "i < k"
haftmann@26072
   509
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
haftmann@26072
   510
  shows P
haftmann@26072
   511
proof -
haftmann@26072
   512
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
haftmann@26072
   513
    unfolding less_eq_Suc_le by (induct k) simp_all
haftmann@26072
   514
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
haftmann@26072
   515
    by (clarsimp simp add: less_le)
haftmann@26072
   516
  with p1 p2 show P by auto
haftmann@26072
   517
qed
haftmann@26072
   518
haftmann@26072
   519
lemma less_SucE: assumes major: "m < Suc n"
haftmann@26072
   520
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
haftmann@26072
   521
  apply (rule major [THEN lessE])
haftmann@26072
   522
  apply (rule eq, blast)
haftmann@26072
   523
  apply (rule less, blast)
berghofe@13449
   524
  done
berghofe@13449
   525
berghofe@13449
   526
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   527
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   528
  apply (rule major [THEN lessE])
berghofe@13449
   529
  apply (erule lessI [THEN minor])
paulson@14208
   530
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   531
  done
berghofe@13449
   532
berghofe@13449
   533
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
haftmann@26072
   534
  by simp
berghofe@13449
   535
berghofe@13449
   536
lemma less_trans_Suc:
berghofe@13449
   537
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   538
  apply (induct k, simp_all)
berghofe@13449
   539
  apply (insert le)
berghofe@13449
   540
  apply (simp add: less_Suc_eq)
berghofe@13449
   541
  apply (blast dest: Suc_lessD)
berghofe@13449
   542
  done
berghofe@13449
   543
berghofe@13449
   544
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
haftmann@26072
   545
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
haftmann@26072
   546
  unfolding not_less less_Suc_eq_le ..
berghofe@13449
   547
haftmann@26072
   548
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   549
  unfolding not_le Suc_le_eq ..
wenzelm@21243
   550
haftmann@24995
   551
text {* Properties of "less than or equal" *}
berghofe@13449
   552
paulson@14267
   553
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
haftmann@26072
   554
  unfolding less_Suc_eq_le .
berghofe@13449
   555
paulson@14267
   556
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
haftmann@26072
   557
  unfolding not_le less_Suc_eq_le ..
berghofe@13449
   558
paulson@14267
   559
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
haftmann@26072
   560
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   561
paulson@14267
   562
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
haftmann@26072
   563
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   564
paulson@14267
   565
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
haftmann@26072
   566
  unfolding Suc_le_eq .
berghofe@13449
   567
berghofe@13449
   568
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   569
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
haftmann@26072
   570
  unfolding Suc_le_eq .
berghofe@13449
   571
paulson@14267
   572
lemma less_imp_le: "m < n ==> m \<le> (n::nat)"
haftmann@26072
   573
  unfolding less_eq_Suc_le by (rule Suc_leD)
berghofe@13449
   574
paulson@14267
   575
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
berghofe@13449
   576
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq
berghofe@13449
   577
berghofe@13449
   578
paulson@14267
   579
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   580
paulson@14267
   581
lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)"
haftmann@26072
   582
  unfolding le_less .
berghofe@13449
   583
paulson@14267
   584
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
haftmann@26072
   585
  unfolding le_less .
berghofe@13449
   586
paulson@14267
   587
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
haftmann@26072
   588
  by (rule le_less)
berghofe@13449
   589
wenzelm@22718
   590
text {* Useful with @{text blast}. *}
paulson@14267
   591
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
haftmann@26072
   592
  by auto
berghofe@13449
   593
paulson@14267
   594
lemma le_refl: "n \<le> (n::nat)"
haftmann@26072
   595
  by simp
berghofe@13449
   596
paulson@14267
   597
lemma le_less_trans: "[| i \<le> j; j < k |] ==> i < (k::nat)"
haftmann@26072
   598
  by (rule order_le_less_trans)
berghofe@13449
   599
paulson@14267
   600
lemma less_le_trans: "[| i < j; j \<le> k |] ==> i < (k::nat)"
haftmann@26072
   601
  by (rule order_less_le_trans)
berghofe@13449
   602
paulson@14267
   603
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
haftmann@26072
   604
  by (rule order_trans)
berghofe@13449
   605
paulson@14267
   606
lemma le_anti_sym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
haftmann@26072
   607
  by (rule antisym)
berghofe@13449
   608
paulson@14267
   609
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
haftmann@26072
   610
  by (rule less_le)
berghofe@13449
   611
paulson@14267
   612
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
haftmann@26072
   613
  unfolding less_le ..
berghofe@13449
   614
haftmann@26072
   615
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
haftmann@26072
   616
  by (rule linear)
paulson@14341
   617
wenzelm@22718
   618
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   619
haftmann@26072
   620
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
haftmann@26072
   621
  unfolding less_Suc_eq_le by auto
berghofe@13449
   622
haftmann@26072
   623
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
haftmann@26072
   624
  unfolding not_less by (rule le_less_Suc_eq)
berghofe@13449
   625
berghofe@13449
   626
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   627
wenzelm@22718
   628
text {* These two rules ease the use of primitive recursion.
paulson@14341
   629
NOTE USE OF @{text "=="} *}
berghofe@13449
   630
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
nipkow@25162
   631
by simp
berghofe@13449
   632
berghofe@13449
   633
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
nipkow@25162
   634
by simp
berghofe@13449
   635
paulson@14267
   636
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   637
by (cases n) simp_all
nipkow@25162
   638
nipkow@25162
   639
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   640
by (cases n) simp_all
berghofe@13449
   641
wenzelm@22718
   642
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
nipkow@25162
   643
by (cases n) simp_all
berghofe@13449
   644
nipkow@25162
   645
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
nipkow@25162
   646
by (cases n) simp_all
nipkow@25140
   647
berghofe@13449
   648
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   649
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@25162
   650
by (rule neq0_conv[THEN iffD1], iprover)
berghofe@13449
   651
paulson@14267
   652
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
nipkow@25162
   653
by (fast intro: not0_implies_Suc)
berghofe@13449
   654
paulson@24286
   655
lemma not_gr0 [iff,noatp]: "!!n::nat. (~ (0 < n)) = (n = 0)"
nipkow@25134
   656
using neq0_conv by blast
berghofe@13449
   657
paulson@14267
   658
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
nipkow@25162
   659
by (induct m') simp_all
berghofe@13449
   660
berghofe@13449
   661
text {* Useful in certain inductive arguments *}
paulson@14267
   662
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
nipkow@25162
   663
by (cases m) simp_all
berghofe@13449
   664
berghofe@13449
   665
haftmann@26072
   666
subsubsection {* @{term min} and @{term max} *}
berghofe@13449
   667
haftmann@25076
   668
lemma mono_Suc: "mono Suc"
nipkow@25162
   669
by (rule monoI) simp
haftmann@25076
   670
berghofe@13449
   671
lemma min_0L [simp]: "min 0 n = (0::nat)"
nipkow@25162
   672
by (rule min_leastL) simp
berghofe@13449
   673
berghofe@13449
   674
lemma min_0R [simp]: "min n 0 = (0::nat)"
nipkow@25162
   675
by (rule min_leastR) simp
berghofe@13449
   676
berghofe@13449
   677
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
nipkow@25162
   678
by (simp add: mono_Suc min_of_mono)
berghofe@13449
   679
paulson@22191
   680
lemma min_Suc1:
paulson@22191
   681
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
nipkow@25162
   682
by (simp split: nat.split)
paulson@22191
   683
paulson@22191
   684
lemma min_Suc2:
paulson@22191
   685
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
nipkow@25162
   686
by (simp split: nat.split)
paulson@22191
   687
berghofe@13449
   688
lemma max_0L [simp]: "max 0 n = (n::nat)"
nipkow@25162
   689
by (rule max_leastL) simp
berghofe@13449
   690
berghofe@13449
   691
lemma max_0R [simp]: "max n 0 = (n::nat)"
nipkow@25162
   692
by (rule max_leastR) simp
berghofe@13449
   693
berghofe@13449
   694
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
nipkow@25162
   695
by (simp add: mono_Suc max_of_mono)
berghofe@13449
   696
paulson@22191
   697
lemma max_Suc1:
paulson@22191
   698
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
nipkow@25162
   699
by (simp split: nat.split)
paulson@22191
   700
paulson@22191
   701
lemma max_Suc2:
paulson@22191
   702
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
nipkow@25162
   703
by (simp split: nat.split)
paulson@22191
   704
berghofe@13449
   705
haftmann@26072
   706
subsubsection {* Monotonicity of Addition *}
berghofe@13449
   707
haftmann@26072
   708
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
haftmann@26072
   709
by (simp add: diff_Suc split: nat.split)
berghofe@13449
   710
paulson@14331
   711
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
nipkow@25162
   712
by (induct k) simp_all
berghofe@13449
   713
paulson@14331
   714
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
nipkow@25162
   715
by (induct k) simp_all
berghofe@13449
   716
nipkow@25162
   717
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
nipkow@25162
   718
by(auto dest:gr0_implies_Suc)
berghofe@13449
   719
paulson@14341
   720
text {* strict, in 1st argument *}
paulson@14341
   721
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
nipkow@25162
   722
by (induct k) simp_all
paulson@14341
   723
paulson@14341
   724
text {* strict, in both arguments *}
paulson@14341
   725
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   726
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   727
  apply (induct j, simp_all)
paulson@14341
   728
  done
paulson@14341
   729
paulson@14341
   730
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   731
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   732
  apply (induct n)
paulson@14341
   733
  apply (simp_all add: order_le_less)
wenzelm@22718
   734
  apply (blast elim!: less_SucE
paulson@14341
   735
               intro!: add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   736
  done
paulson@14341
   737
paulson@14341
   738
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
nipkow@25134
   739
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
nipkow@25134
   740
apply(auto simp: gr0_conv_Suc)
nipkow@25134
   741
apply (induct_tac m)
nipkow@25134
   742
apply (simp_all add: add_less_mono)
nipkow@25134
   743
done
paulson@14341
   744
nipkow@14740
   745
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
obua@14738
   746
instance nat :: ordered_semidom
paulson@14341
   747
proof
paulson@14341
   748
  fix i j k :: nat
paulson@14348
   749
  show "0 < (1::nat)" by simp
paulson@14267
   750
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   751
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   752
qed
paulson@14267
   753
paulson@14267
   754
lemma nat_mult_1: "(1::nat) * n = n"
nipkow@25162
   755
by simp
paulson@14267
   756
paulson@14267
   757
lemma nat_mult_1_right: "n * (1::nat) = n"
nipkow@25162
   758
by simp
paulson@14267
   759
paulson@14267
   760
haftmann@26072
   761
subsubsection {* Additional theorems about "less than" *}
paulson@14267
   762
paulson@19870
   763
text{*An induction rule for estabilishing binary relations*}
wenzelm@22718
   764
lemma less_Suc_induct:
paulson@19870
   765
  assumes less:  "i < j"
paulson@19870
   766
     and  step:  "!!i. P i (Suc i)"
paulson@19870
   767
     and  trans: "!!i j k. P i j ==> P j k ==> P i k"
paulson@19870
   768
  shows "P i j"
paulson@19870
   769
proof -
wenzelm@22718
   770
  from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   771
  have "P i (Suc (i + k))"
paulson@19870
   772
  proof (induct k)
wenzelm@22718
   773
    case 0
wenzelm@22718
   774
    show ?case by (simp add: step)
paulson@19870
   775
  next
paulson@19870
   776
    case (Suc k)
wenzelm@22718
   777
    thus ?case by (auto intro: assms)
paulson@19870
   778
  qed
wenzelm@22718
   779
  thus "P i j" by (simp add: j)
paulson@19870
   780
qed
paulson@19870
   781
paulson@14267
   782
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   783
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   784
lemma less_mono_imp_le_mono:
nipkow@24438
   785
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
   786
by (simp add: order_le_less) (blast)
nipkow@24438
   787
paulson@14267
   788
paulson@14267
   789
text {* non-strict, in 1st argument *}
paulson@14267
   790
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
   791
by (rule add_right_mono)
paulson@14267
   792
paulson@14267
   793
text {* non-strict, in both arguments *}
paulson@14267
   794
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
   795
by (rule add_mono)
paulson@14267
   796
paulson@14267
   797
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
   798
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
   799
paulson@14267
   800
lemma le_add1: "n \<le> ((n + m)::nat)"
nipkow@24438
   801
by (simp add: add_commute, rule le_add2)
berghofe@13449
   802
berghofe@13449
   803
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
   804
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   805
berghofe@13449
   806
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
   807
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   808
paulson@14267
   809
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
   810
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   811
paulson@14267
   812
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
   813
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   814
paulson@14267
   815
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
   816
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
   817
berghofe@13449
   818
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
   819
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
   820
berghofe@13449
   821
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
   822
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
   823
berghofe@13449
   824
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
   825
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
   826
apply (simp_all add: le_add1)
nipkow@24438
   827
done
berghofe@13449
   828
berghofe@13449
   829
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
   830
apply (rule notI)
nipkow@24438
   831
apply (erule add_lessD1 [THEN less_irrefl])
nipkow@24438
   832
done
berghofe@13449
   833
berghofe@13449
   834
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
nipkow@24438
   835
by (simp add: add_commute not_add_less1)
berghofe@13449
   836
paulson@14267
   837
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
   838
apply (rule order_trans [of _ "m+k"])
nipkow@24438
   839
apply (simp_all add: le_add1)
nipkow@24438
   840
done
berghofe@13449
   841
paulson@14267
   842
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
nipkow@24438
   843
apply (simp add: add_commute)
nipkow@24438
   844
apply (erule add_leD1)
nipkow@24438
   845
done
berghofe@13449
   846
paulson@14267
   847
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
   848
by (blast dest: add_leD1 add_leD2)
berghofe@13449
   849
berghofe@13449
   850
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
   851
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
   852
by (force simp del: add_Suc_right
berghofe@13449
   853
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
   854
berghofe@13449
   855
haftmann@26072
   856
subsubsection {* More results about difference *}
berghofe@13449
   857
berghofe@13449
   858
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
nipkow@24438
   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)"
nipkow@24438
   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)"
nipkow@24438
   867
by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
   868
paulson@14267
   869
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
nipkow@24438
   870
by (simp add: le_add_diff_inverse add_commute)
berghofe@13449
   871
paulson@14267
   872
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
   873
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   874
berghofe@13449
   875
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
   876
apply (induct m n rule: diff_induct)
nipkow@24438
   877
apply (erule_tac [3] less_SucE)
nipkow@24438
   878
apply (simp_all add: less_Suc_eq)
nipkow@24438
   879
done
berghofe@13449
   880
paulson@14267
   881
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
   882
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
   883
haftmann@26072
   884
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
haftmann@26072
   885
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
haftmann@26072
   886
berghofe@13449
   887
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
   888
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
   889
berghofe@13449
   890
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
   891
by (cases n) (auto simp add: le_simps)
berghofe@13449
   892
paulson@14267
   893
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
   894
by (induct j k rule: diff_induct) simp_all
berghofe@13449
   895
paulson@14267
   896
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
nipkow@24438
   897
by (simp add: add_commute diff_add_assoc)
berghofe@13449
   898
paulson@14267
   899
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
nipkow@24438
   900
by (auto simp add: diff_add_inverse2)
berghofe@13449
   901
paulson@14267
   902
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
   903
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   904
paulson@14267
   905
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
   906
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
   907
berghofe@13449
   908
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
   909
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   910
wenzelm@22718
   911
lemma less_imp_add_positive:
wenzelm@22718
   912
  assumes "i < j"
wenzelm@22718
   913
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
   914
proof
wenzelm@22718
   915
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
   916
    by (simp add: order_less_imp_le)
wenzelm@22718
   917
qed
wenzelm@9436
   918
haftmann@26072
   919
text {* a nice rewrite for bounded subtraction *}
haftmann@26072
   920
lemma nat_minus_add_max:
haftmann@26072
   921
  fixes n m :: nat
haftmann@26072
   922
  shows "n - m + m = max n m"
haftmann@26072
   923
    by (simp add: max_def not_le order_less_imp_le)
berghofe@13449
   924
haftmann@26072
   925
lemma nat_diff_split:
haftmann@26072
   926
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
haftmann@26072
   927
    -- {* elimination of @{text -} on @{text nat} *}
haftmann@26072
   928
by (cases "a < b")
haftmann@26072
   929
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
haftmann@26072
   930
    not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)
berghofe@13449
   931
haftmann@26072
   932
lemma nat_diff_split_asm:
haftmann@26072
   933
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
haftmann@26072
   934
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
haftmann@26072
   935
by (auto split: nat_diff_split)
berghofe@13449
   936
berghofe@13449
   937
haftmann@26072
   938
subsubsection {* Monotonicity of Multiplication *}
berghofe@13449
   939
paulson@14267
   940
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
   941
by (simp add: mult_right_mono)
berghofe@13449
   942
paulson@14267
   943
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
   944
by (simp add: mult_left_mono)
berghofe@13449
   945
paulson@14267
   946
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
   947
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
   948
by (simp add: mult_mono)
berghofe@13449
   949
berghofe@13449
   950
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
   951
by (simp add: mult_strict_right_mono)
berghofe@13449
   952
paulson@14266
   953
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
   954
      there are no negative numbers.*}
paulson@14266
   955
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
   956
  apply (induct m)
wenzelm@22718
   957
   apply simp
wenzelm@22718
   958
  apply (case_tac n)
wenzelm@22718
   959
   apply simp_all
berghofe@13449
   960
  done
berghofe@13449
   961
paulson@14267
   962
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"
berghofe@13449
   963
  apply (induct m)
wenzelm@22718
   964
   apply simp
wenzelm@22718
   965
  apply (case_tac n)
wenzelm@22718
   966
   apply simp_all
berghofe@13449
   967
  done
berghofe@13449
   968
paulson@14341
   969
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
   970
  apply (safe intro!: mult_less_mono1)
paulson@14208
   971
  apply (case_tac k, auto)
berghofe@13449
   972
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
   973
  apply (blast intro: mult_le_mono1)
berghofe@13449
   974
  done
berghofe@13449
   975
berghofe@13449
   976
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
nipkow@24438
   977
by (simp add: mult_commute [of k])
berghofe@13449
   978
paulson@14267
   979
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
   980
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
   981
paulson@14267
   982
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
   983
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
   984
berghofe@13449
   985
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
   986
by (subst mult_less_cancel1) simp
berghofe@13449
   987
paulson@14267
   988
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
   989
by (subst mult_le_cancel1) simp
berghofe@13449
   990
haftmann@26072
   991
lemma le_square: "m \<le> m * (m::nat)"
haftmann@26072
   992
  by (cases m) (auto intro: le_add1)
haftmann@26072
   993
haftmann@26072
   994
lemma le_cube: "(m::nat) \<le> m * (m * m)"
haftmann@26072
   995
  by (cases m) (auto intro: le_add1)
berghofe@13449
   996
berghofe@13449
   997
text {* Lemma for @{text gcd} *}
berghofe@13449
   998
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
   999
  apply (drule sym)
berghofe@13449
  1000
  apply (rule disjCI)
berghofe@13449
  1001
  apply (rule nat_less_cases, erule_tac [2] _)
paulson@25157
  1002
   apply (drule_tac [2] mult_less_mono2)
nipkow@25162
  1003
    apply (auto)
berghofe@13449
  1004
  done
wenzelm@9436
  1005
haftmann@26072
  1006
text {* the lattice order on @{typ nat} *}
haftmann@24995
  1007
haftmann@26072
  1008
instantiation nat :: distrib_lattice
haftmann@26072
  1009
begin
haftmann@24995
  1010
haftmann@26072
  1011
definition
haftmann@26072
  1012
  "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
haftmann@24995
  1013
haftmann@26072
  1014
definition
haftmann@26072
  1015
  "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
haftmann@24995
  1016
haftmann@26072
  1017
instance by intro_classes
haftmann@26072
  1018
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
haftmann@26072
  1019
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
haftmann@24995
  1020
haftmann@26072
  1021
end
haftmann@24995
  1022
haftmann@24995
  1023
haftmann@25193
  1024
subsection {* Embedding of the Naturals into any
haftmann@25193
  1025
  @{text semiring_1}: @{term of_nat} *}
haftmann@24196
  1026
haftmann@24196
  1027
context semiring_1
haftmann@24196
  1028
begin
haftmann@24196
  1029
haftmann@25559
  1030
primrec
haftmann@25559
  1031
  of_nat :: "nat \<Rightarrow> 'a"
haftmann@25559
  1032
where
haftmann@25559
  1033
  of_nat_0:     "of_nat 0 = 0"
haftmann@25559
  1034
  | of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
haftmann@25193
  1035
haftmann@25193
  1036
lemma of_nat_1 [simp]: "of_nat 1 = 1"
haftmann@25193
  1037
  by simp
haftmann@25193
  1038
haftmann@25193
  1039
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
haftmann@25193
  1040
  by (induct m) (simp_all add: add_ac)
haftmann@25193
  1041
haftmann@25193
  1042
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
haftmann@25193
  1043
  by (induct m) (simp_all add: add_ac left_distrib)
haftmann@25193
  1044
haftmann@25928
  1045
definition
haftmann@25928
  1046
  of_nat_aux :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@25928
  1047
where
haftmann@25928
  1048
  [code func del]: "of_nat_aux n i = of_nat n + i"
haftmann@25928
  1049
haftmann@25928
  1050
lemma of_nat_aux_code [code]:
haftmann@25928
  1051
  "of_nat_aux 0 i = i"
haftmann@25928
  1052
  "of_nat_aux (Suc n) i = of_nat_aux n (i + 1)" -- {* tail recursive *}
haftmann@25928
  1053
  by (simp_all add: of_nat_aux_def add_ac)
haftmann@25928
  1054
haftmann@25928
  1055
lemma of_nat_code [code]:
haftmann@25928
  1056
  "of_nat n = of_nat_aux n 0"
haftmann@25928
  1057
  by (simp add: of_nat_aux_def)
haftmann@25928
  1058
haftmann@24196
  1059
end
haftmann@24196
  1060
haftmann@26072
  1061
text{*Class for unital semirings with characteristic zero.
haftmann@26072
  1062
 Includes non-ordered rings like the complex numbers.*}
haftmann@26072
  1063
haftmann@26072
  1064
class semiring_char_0 = semiring_1 +
haftmann@26072
  1065
  assumes of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
haftmann@26072
  1066
begin
haftmann@26072
  1067
haftmann@26072
  1068
text{*Special cases where either operand is zero*}
haftmann@26072
  1069
haftmann@26072
  1070
lemma of_nat_0_eq_iff [simp, noatp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
haftmann@26072
  1071
  by (rule of_nat_eq_iff [of 0, simplified])
haftmann@26072
  1072
haftmann@26072
  1073
lemma of_nat_eq_0_iff [simp, noatp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
haftmann@26072
  1074
  by (rule of_nat_eq_iff [of _ 0, simplified])
haftmann@26072
  1075
haftmann@26072
  1076
lemma inj_of_nat: "inj of_nat"
haftmann@26072
  1077
  by (simp add: inj_on_def)
haftmann@26072
  1078
haftmann@26072
  1079
end
haftmann@26072
  1080
haftmann@25193
  1081
context ordered_semidom
haftmann@25193
  1082
begin
haftmann@25193
  1083
haftmann@25193
  1084
lemma zero_le_imp_of_nat: "0 \<le> of_nat m"
haftmann@25193
  1085
  apply (induct m, simp_all)
haftmann@25193
  1086
  apply (erule order_trans)
haftmann@25193
  1087
  apply (rule ord_le_eq_trans [OF _ add_commute])
haftmann@25193
  1088
  apply (rule less_add_one [THEN less_imp_le])
haftmann@25193
  1089
  done
haftmann@25193
  1090
haftmann@25193
  1091
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
haftmann@25193
  1092
  apply (induct m n rule: diff_induct, simp_all)
haftmann@25193
  1093
  apply (insert add_less_le_mono [OF zero_less_one zero_le_imp_of_nat], force)
haftmann@25193
  1094
  done
haftmann@25193
  1095
haftmann@25193
  1096
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
haftmann@25193
  1097
  apply (induct m n rule: diff_induct, simp_all)
haftmann@25193
  1098
  apply (insert zero_le_imp_of_nat)
haftmann@25193
  1099
  apply (force simp add: not_less [symmetric])
haftmann@25193
  1100
  done
haftmann@25193
  1101
haftmann@25193
  1102
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
haftmann@25193
  1103
  by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
haftmann@25193
  1104
haftmann@26072
  1105
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
haftmann@26072
  1106
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
haftmann@25193
  1107
haftmann@26072
  1108
text{*Every @{text ordered_semidom} has characteristic zero.*}
haftmann@25193
  1109
haftmann@26072
  1110
subclass semiring_char_0
haftmann@26072
  1111
  by unfold_locales (simp add: eq_iff order_eq_iff)
haftmann@25193
  1112
haftmann@25193
  1113
text{*Special cases where either operand is zero*}
haftmann@25193
  1114
haftmann@25193
  1115
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
haftmann@25193
  1116
  by (rule of_nat_le_iff [of 0, simplified])
haftmann@25193
  1117
haftmann@25193
  1118
lemma of_nat_le_0_iff [simp, noatp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
haftmann@25193
  1119
  by (rule of_nat_le_iff [of _ 0, simplified])
haftmann@25193
  1120
haftmann@26072
  1121
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
haftmann@26072
  1122
  by (rule of_nat_less_iff [of 0, simplified])
haftmann@26072
  1123
haftmann@26072
  1124
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
haftmann@26072
  1125
  by (rule of_nat_less_iff [of _ 0, simplified])
haftmann@26072
  1126
haftmann@26072
  1127
end
haftmann@26072
  1128
haftmann@26072
  1129
context ring_1
haftmann@26072
  1130
begin
haftmann@26072
  1131
haftmann@26072
  1132
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
haftmann@26072
  1133
  by (simp add: compare_rls of_nat_add [symmetric])
haftmann@26072
  1134
haftmann@26072
  1135
end
haftmann@26072
  1136
haftmann@26072
  1137
context ordered_idom
haftmann@26072
  1138
begin
haftmann@26072
  1139
haftmann@26072
  1140
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
haftmann@26072
  1141
  unfolding abs_if by auto
haftmann@26072
  1142
haftmann@25193
  1143
end
haftmann@25193
  1144
haftmann@25193
  1145
lemma of_nat_id [simp]: "of_nat n = n"
haftmann@25193
  1146
  by (induct n) auto
haftmann@25193
  1147
haftmann@25193
  1148
lemma of_nat_eq_id [simp]: "of_nat = id"
haftmann@25193
  1149
  by (auto simp add: expand_fun_eq)
haftmann@25193
  1150
haftmann@25193
  1151
haftmann@26072
  1152
subsection {*The Set of Natural Numbers*}
haftmann@25193
  1153
haftmann@26072
  1154
context semiring_1
haftmann@25193
  1155
begin
haftmann@25193
  1156
haftmann@26072
  1157
definition
haftmann@26072
  1158
  Nats  :: "'a set" where
haftmann@26072
  1159
  "Nats = range of_nat"
haftmann@26072
  1160
haftmann@26072
  1161
notation (xsymbols)
haftmann@26072
  1162
  Nats  ("\<nat>")
haftmann@25193
  1163
haftmann@26072
  1164
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
haftmann@26072
  1165
  by (simp add: Nats_def)
haftmann@26072
  1166
haftmann@26072
  1167
lemma Nats_0 [simp]: "0 \<in> \<nat>"
haftmann@26072
  1168
apply (simp add: Nats_def)
haftmann@26072
  1169
apply (rule range_eqI)
haftmann@26072
  1170
apply (rule of_nat_0 [symmetric])
haftmann@26072
  1171
done
haftmann@25193
  1172
haftmann@26072
  1173
lemma Nats_1 [simp]: "1 \<in> \<nat>"
haftmann@26072
  1174
apply (simp add: Nats_def)
haftmann@26072
  1175
apply (rule range_eqI)
haftmann@26072
  1176
apply (rule of_nat_1 [symmetric])
haftmann@26072
  1177
done
haftmann@25193
  1178
haftmann@26072
  1179
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
haftmann@26072
  1180
apply (auto simp add: Nats_def)
haftmann@26072
  1181
apply (rule range_eqI)
haftmann@26072
  1182
apply (rule of_nat_add [symmetric])
haftmann@26072
  1183
done
haftmann@26072
  1184
haftmann@26072
  1185
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
haftmann@26072
  1186
apply (auto simp add: Nats_def)
haftmann@26072
  1187
apply (rule range_eqI)
haftmann@26072
  1188
apply (rule of_nat_mult [symmetric])
haftmann@26072
  1189
done
haftmann@25193
  1190
haftmann@25193
  1191
end
haftmann@25193
  1192
haftmann@25193
  1193
wenzelm@21243
  1194
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
wenzelm@21243
  1195
haftmann@22845
  1196
lemma subst_equals:
haftmann@22845
  1197
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1198
  shows "u = s"
haftmann@22845
  1199
  using 2 1 by (rule trans)
haftmann@22845
  1200
wenzelm@21243
  1201
use "arith_data.ML"
haftmann@26101
  1202
declaration {* K ArithData.setup *}
wenzelm@24091
  1203
wenzelm@24091
  1204
use "Tools/lin_arith.ML"
wenzelm@24091
  1205
declaration {* K LinArith.setup *}
wenzelm@24091
  1206
wenzelm@21243
  1207
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1208
wenzelm@21243
  1209
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1210
wenzelm@21243
  1211
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1212
by arith
wenzelm@21243
  1213
wenzelm@21243
  1214
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1215
by arith
wenzelm@21243
  1216
wenzelm@21243
  1217
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1218
by arith
wenzelm@21243
  1219
wenzelm@21243
  1220
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
nipkow@24438
  1221
by arith
wenzelm@21243
  1222
wenzelm@21243
  1223
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1224
by arith
wenzelm@21243
  1225
wenzelm@21243
  1226
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
nipkow@24438
  1227
by arith
wenzelm@21243
  1228
wenzelm@21243
  1229
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1230
  second premise n\<le>m*)
wenzelm@21243
  1231
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1232
by arith
wenzelm@21243
  1233
haftmann@26072
  1234
text {* Simplification of relational expressions involving subtraction *}
wenzelm@21243
  1235
wenzelm@21243
  1236
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1237
by (simp split add: nat_diff_split)
wenzelm@21243
  1238
wenzelm@21243
  1239
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1240
by (auto split add: nat_diff_split)
wenzelm@21243
  1241
wenzelm@21243
  1242
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1243
by (auto split add: nat_diff_split)
wenzelm@21243
  1244
wenzelm@21243
  1245
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1246
by (auto split add: nat_diff_split)
wenzelm@21243
  1247
wenzelm@21243
  1248
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1249
wenzelm@21243
  1250
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1251
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1252
by (simp split add: nat_diff_split)
wenzelm@21243
  1253
wenzelm@21243
  1254
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1255
by (simp split add: nat_diff_split)
wenzelm@21243
  1256
wenzelm@21243
  1257
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1258
by (simp split add: nat_diff_split)
wenzelm@21243
  1259
wenzelm@21243
  1260
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1261
by (simp split add: nat_diff_split)
wenzelm@21243
  1262
haftmann@26072
  1263
text{*Rewriting to pull differences out*}
haftmann@26072
  1264
haftmann@26072
  1265
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
haftmann@26072
  1266
by arith
haftmann@26072
  1267
haftmann@26072
  1268
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
haftmann@26072
  1269
by arith
haftmann@26072
  1270
haftmann@26072
  1271
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
haftmann@26072
  1272
by arith
haftmann@26072
  1273
wenzelm@21243
  1274
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1275
wenzelm@21243
  1276
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1277
by (cases m) auto
wenzelm@21243
  1278
wenzelm@21243
  1279
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1280
by (cases m) auto
wenzelm@21243
  1281
wenzelm@21243
  1282
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1283
by (cases m) auto
wenzelm@21243
  1284
krauss@23001
  1285
text {* Specialized induction principles that work "backwards": *}
krauss@23001
  1286
krauss@23001
  1287
lemma inc_induct[consumes 1, case_names base step]:
krauss@23001
  1288
  assumes less: "i <= j"
krauss@23001
  1289
  assumes base: "P j"
krauss@23001
  1290
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1291
  shows "P i"
krauss@23001
  1292
  using less
krauss@23001
  1293
proof (induct d=="j - i" arbitrary: i)
krauss@23001
  1294
  case (0 i)
krauss@23001
  1295
  hence "i = j" by simp
krauss@23001
  1296
  with base show ?case by simp
krauss@23001
  1297
next
krauss@23001
  1298
  case (Suc d i)
krauss@23001
  1299
  hence "i < j" "P (Suc i)"
krauss@23001
  1300
    by simp_all
krauss@23001
  1301
  thus "P i" by (rule step)
krauss@23001
  1302
qed
krauss@23001
  1303
krauss@23001
  1304
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1305
  assumes less: "i < j"
krauss@23001
  1306
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1307
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1308
  shows "P i"
krauss@23001
  1309
  using less
krauss@23001
  1310
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1311
  case (0 i)
krauss@23001
  1312
  with `i < j` have "j = Suc i" by simp
krauss@23001
  1313
  with base show ?case by simp
krauss@23001
  1314
next
krauss@23001
  1315
  case (Suc d i)
krauss@23001
  1316
  hence "i < j" "P (Suc i)"
krauss@23001
  1317
    by simp_all
krauss@23001
  1318
  thus "P i" by (rule step)
krauss@23001
  1319
qed
krauss@23001
  1320
krauss@23001
  1321
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1322
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1323
krauss@23001
  1324
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1325
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1326
haftmann@26072
  1327
lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False"
haftmann@26072
  1328
  by auto
wenzelm@21243
  1329
wenzelm@21243
  1330
(*The others are
wenzelm@21243
  1331
      i - j - k = i - (j + k),
wenzelm@21243
  1332
      k \<le> j ==> j - k + i = j + i - k,
wenzelm@21243
  1333
      k \<le> j ==> i + (j - k) = i + j - k *)
wenzelm@21243
  1334
lemmas add_diff_assoc = diff_add_assoc [symmetric]
wenzelm@21243
  1335
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
haftmann@26072
  1336
declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
wenzelm@21243
  1337
wenzelm@21243
  1338
text{*At present we prove no analogue of @{text not_less_Least} or @{text
wenzelm@21243
  1339
Least_Suc}, since there appears to be no need.*}
wenzelm@21243
  1340
haftmann@26072
  1341
subsection {* size of a datatype value *}
haftmann@25193
  1342
haftmann@26072
  1343
class size = type +
haftmann@26072
  1344
  fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded_Recursion} *}
haftmann@23852
  1345
haftmann@25193
  1346
end