src/HOL/Nat.thy
author paulson
Tue Feb 10 09:58:58 2009 +0000 (2009-02-10)
changeset 29854 708c1c7c87ec
parent 29849 a2baf1b221be
parent 29852 3d4c46f62937
child 29879 4425849f5db7
permissions -rw-r--r--
merged
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(*  Title:      HOL/Nat.thy
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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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  ("Tools/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 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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declare add_0 [code]
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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]
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declare diff_0 [code]
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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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   342
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   343
subsubsection {* Operation definition *}
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   344
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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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   350
  | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
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   351
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declare less_eq_nat.simps [simp del]
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lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps)
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   354
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
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   355
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   356
definition less_nat where
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   357
  less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
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   358
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   359
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
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   360
  by (simp add: less_eq_nat.simps(2))
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   361
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   362
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
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   363
  unfolding less_eq_Suc_le ..
haftmann@26072
   364
haftmann@26072
   365
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
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   366
  by (induct n) (simp_all add: less_eq_nat.simps(2))
haftmann@26072
   367
haftmann@26072
   368
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
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   369
  by (simp add: less_eq_Suc_le)
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   370
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   371
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
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   372
  by simp
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   373
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   374
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
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   375
  by (simp add: less_eq_Suc_le)
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   376
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   377
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
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   378
  by (simp add: less_eq_Suc_le)
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   379
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   380
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
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   381
  by (induct m arbitrary: n)
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   382
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
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   383
haftmann@26072
   384
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
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   385
  by (cases n) (auto intro: le_SucI)
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   386
haftmann@26072
   387
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
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   388
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
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   389
haftmann@26072
   390
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
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   391
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
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   392
wenzelm@26315
   393
instance
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   394
proof
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   395
  fix n m :: nat
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   396
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" 
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   397
  proof (induct n arbitrary: m)
haftmann@27679
   398
    case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   399
  next
haftmann@27679
   400
    case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   401
  qed
haftmann@26072
   402
next
haftmann@26072
   403
  fix n :: nat show "n \<le> n" by (induct n) simp_all
haftmann@26072
   404
next
haftmann@26072
   405
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
haftmann@26072
   406
  then show "n = m"
haftmann@26072
   407
    by (induct n arbitrary: m)
haftmann@26072
   408
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   409
next
haftmann@26072
   410
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
haftmann@26072
   411
  then show "n \<le> q"
haftmann@26072
   412
  proof (induct n arbitrary: m q)
haftmann@26072
   413
    case 0 show ?case by simp
haftmann@26072
   414
  next
haftmann@26072
   415
    case (Suc n) then show ?case
haftmann@26072
   416
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   417
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   418
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   419
  qed
haftmann@26072
   420
next
haftmann@26072
   421
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
haftmann@26072
   422
    by (induct n arbitrary: m)
haftmann@26072
   423
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   424
qed
haftmann@25510
   425
haftmann@25510
   426
end
berghofe@13449
   427
haftmann@29652
   428
instantiation nat :: bot
haftmann@29652
   429
begin
haftmann@29652
   430
haftmann@29652
   431
definition bot_nat :: nat where
haftmann@29652
   432
  "bot_nat = 0"
haftmann@29652
   433
haftmann@29652
   434
instance proof
haftmann@29652
   435
qed (simp add: bot_nat_def)
haftmann@29652
   436
haftmann@29652
   437
end
haftmann@29652
   438
haftmann@26072
   439
subsubsection {* Introduction properties *}
berghofe@13449
   440
haftmann@26072
   441
lemma lessI [iff]: "n < Suc n"
haftmann@26072
   442
  by (simp add: less_Suc_eq_le)
berghofe@13449
   443
haftmann@26072
   444
lemma zero_less_Suc [iff]: "0 < Suc n"
haftmann@26072
   445
  by (simp add: less_Suc_eq_le)
berghofe@13449
   446
berghofe@13449
   447
berghofe@13449
   448
subsubsection {* Elimination properties *}
berghofe@13449
   449
berghofe@13449
   450
lemma less_not_refl: "~ n < (n::nat)"
haftmann@26072
   451
  by (rule order_less_irrefl)
berghofe@13449
   452
wenzelm@26335
   453
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
wenzelm@26335
   454
  by (rule not_sym) (rule less_imp_neq) 
berghofe@13449
   455
paulson@14267
   456
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
haftmann@26072
   457
  by (rule less_imp_neq)
berghofe@13449
   458
wenzelm@26335
   459
lemma less_irrefl_nat: "(n::nat) < n ==> R"
wenzelm@26335
   460
  by (rule notE, rule less_not_refl)
berghofe@13449
   461
berghofe@13449
   462
lemma less_zeroE: "(n::nat) < 0 ==> R"
haftmann@26072
   463
  by (rule notE) (rule not_less0)
berghofe@13449
   464
berghofe@13449
   465
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
haftmann@26072
   466
  unfolding less_Suc_eq_le le_less ..
berghofe@13449
   467
haftmann@26072
   468
lemma less_one [iff, noatp]: "(n < (1::nat)) = (n = 0)"
haftmann@26072
   469
  by (simp add: less_Suc_eq)
berghofe@13449
   470
berghofe@13449
   471
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
haftmann@26072
   472
  by (simp add: less_Suc_eq)
berghofe@13449
   473
berghofe@13449
   474
lemma Suc_mono: "m < n ==> Suc m < Suc n"
haftmann@26072
   475
  by simp
berghofe@13449
   476
nipkow@14302
   477
text {* "Less than" is antisymmetric, sort of *}
nipkow@14302
   478
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
haftmann@26072
   479
  unfolding not_less less_Suc_eq_le by (rule antisym)
nipkow@14302
   480
paulson@14267
   481
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
haftmann@26072
   482
  by (rule linorder_neq_iff)
berghofe@13449
   483
berghofe@13449
   484
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   485
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   486
  shows "P n m"
berghofe@13449
   487
  apply (rule less_linear [THEN disjE])
berghofe@13449
   488
  apply (erule_tac [2] disjE)
berghofe@13449
   489
  apply (erule lessCase)
berghofe@13449
   490
  apply (erule sym [THEN eqCase])
berghofe@13449
   491
  apply (erule major)
berghofe@13449
   492
  done
berghofe@13449
   493
berghofe@13449
   494
berghofe@13449
   495
subsubsection {* Inductive (?) properties *}
berghofe@13449
   496
paulson@14267
   497
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
haftmann@26072
   498
  unfolding less_eq_Suc_le [of m] le_less by simp 
berghofe@13449
   499
haftmann@26072
   500
lemma lessE:
haftmann@26072
   501
  assumes major: "i < k"
haftmann@26072
   502
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
haftmann@26072
   503
  shows P
haftmann@26072
   504
proof -
haftmann@26072
   505
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
haftmann@26072
   506
    unfolding less_eq_Suc_le by (induct k) simp_all
haftmann@26072
   507
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
haftmann@26072
   508
    by (clarsimp simp add: less_le)
haftmann@26072
   509
  with p1 p2 show P by auto
haftmann@26072
   510
qed
haftmann@26072
   511
haftmann@26072
   512
lemma less_SucE: assumes major: "m < Suc n"
haftmann@26072
   513
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
haftmann@26072
   514
  apply (rule major [THEN lessE])
haftmann@26072
   515
  apply (rule eq, blast)
haftmann@26072
   516
  apply (rule less, blast)
berghofe@13449
   517
  done
berghofe@13449
   518
berghofe@13449
   519
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   520
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   521
  apply (rule major [THEN lessE])
berghofe@13449
   522
  apply (erule lessI [THEN minor])
paulson@14208
   523
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   524
  done
berghofe@13449
   525
berghofe@13449
   526
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
haftmann@26072
   527
  by simp
berghofe@13449
   528
berghofe@13449
   529
lemma less_trans_Suc:
berghofe@13449
   530
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   531
  apply (induct k, simp_all)
berghofe@13449
   532
  apply (insert le)
berghofe@13449
   533
  apply (simp add: less_Suc_eq)
berghofe@13449
   534
  apply (blast dest: Suc_lessD)
berghofe@13449
   535
  done
berghofe@13449
   536
berghofe@13449
   537
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
haftmann@26072
   538
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
haftmann@26072
   539
  unfolding not_less less_Suc_eq_le ..
berghofe@13449
   540
haftmann@26072
   541
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   542
  unfolding not_le Suc_le_eq ..
wenzelm@21243
   543
haftmann@24995
   544
text {* Properties of "less than or equal" *}
berghofe@13449
   545
paulson@14267
   546
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
haftmann@26072
   547
  unfolding less_Suc_eq_le .
berghofe@13449
   548
paulson@14267
   549
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
haftmann@26072
   550
  unfolding not_le less_Suc_eq_le ..
berghofe@13449
   551
paulson@14267
   552
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
haftmann@26072
   553
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   554
paulson@14267
   555
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
haftmann@26072
   556
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   557
paulson@14267
   558
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
haftmann@26072
   559
  unfolding Suc_le_eq .
berghofe@13449
   560
berghofe@13449
   561
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   562
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
haftmann@26072
   563
  unfolding Suc_le_eq .
berghofe@13449
   564
wenzelm@26315
   565
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
haftmann@26072
   566
  unfolding less_eq_Suc_le by (rule Suc_leD)
berghofe@13449
   567
paulson@14267
   568
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
wenzelm@26315
   569
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
berghofe@13449
   570
berghofe@13449
   571
paulson@14267
   572
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   573
paulson@14267
   574
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
haftmann@26072
   575
  unfolding le_less .
berghofe@13449
   576
paulson@14267
   577
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
haftmann@26072
   578
  by (rule le_less)
berghofe@13449
   579
wenzelm@22718
   580
text {* Useful with @{text blast}. *}
paulson@14267
   581
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
haftmann@26072
   582
  by auto
berghofe@13449
   583
paulson@14267
   584
lemma le_refl: "n \<le> (n::nat)"
haftmann@26072
   585
  by simp
berghofe@13449
   586
paulson@14267
   587
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
haftmann@26072
   588
  by (rule order_trans)
berghofe@13449
   589
paulson@14267
   590
lemma le_anti_sym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
haftmann@26072
   591
  by (rule antisym)
berghofe@13449
   592
paulson@14267
   593
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
haftmann@26072
   594
  by (rule less_le)
berghofe@13449
   595
paulson@14267
   596
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
haftmann@26072
   597
  unfolding less_le ..
berghofe@13449
   598
haftmann@26072
   599
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
haftmann@26072
   600
  by (rule linear)
paulson@14341
   601
wenzelm@22718
   602
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   603
haftmann@26072
   604
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
haftmann@26072
   605
  unfolding less_Suc_eq_le by auto
berghofe@13449
   606
haftmann@26072
   607
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
haftmann@26072
   608
  unfolding not_less by (rule le_less_Suc_eq)
berghofe@13449
   609
berghofe@13449
   610
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   611
wenzelm@22718
   612
text {* These two rules ease the use of primitive recursion.
paulson@14341
   613
NOTE USE OF @{text "=="} *}
berghofe@13449
   614
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
nipkow@25162
   615
by simp
berghofe@13449
   616
berghofe@13449
   617
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
nipkow@25162
   618
by simp
berghofe@13449
   619
paulson@14267
   620
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   621
by (cases n) simp_all
nipkow@25162
   622
nipkow@25162
   623
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   624
by (cases n) simp_all
berghofe@13449
   625
wenzelm@22718
   626
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
nipkow@25162
   627
by (cases n) simp_all
berghofe@13449
   628
nipkow@25162
   629
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
nipkow@25162
   630
by (cases n) simp_all
nipkow@25140
   631
berghofe@13449
   632
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   633
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@25162
   634
by (rule neq0_conv[THEN iffD1], iprover)
berghofe@13449
   635
paulson@14267
   636
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
nipkow@25162
   637
by (fast intro: not0_implies_Suc)
berghofe@13449
   638
paulson@24286
   639
lemma not_gr0 [iff,noatp]: "!!n::nat. (~ (0 < n)) = (n = 0)"
nipkow@25134
   640
using neq0_conv by blast
berghofe@13449
   641
paulson@14267
   642
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
nipkow@25162
   643
by (induct m') simp_all
berghofe@13449
   644
berghofe@13449
   645
text {* Useful in certain inductive arguments *}
paulson@14267
   646
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
nipkow@25162
   647
by (cases m) simp_all
berghofe@13449
   648
berghofe@13449
   649
haftmann@26072
   650
subsubsection {* @{term min} and @{term max} *}
berghofe@13449
   651
haftmann@25076
   652
lemma mono_Suc: "mono Suc"
nipkow@25162
   653
by (rule monoI) simp
haftmann@25076
   654
berghofe@13449
   655
lemma min_0L [simp]: "min 0 n = (0::nat)"
nipkow@25162
   656
by (rule min_leastL) simp
berghofe@13449
   657
berghofe@13449
   658
lemma min_0R [simp]: "min n 0 = (0::nat)"
nipkow@25162
   659
by (rule min_leastR) simp
berghofe@13449
   660
berghofe@13449
   661
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
nipkow@25162
   662
by (simp add: mono_Suc min_of_mono)
berghofe@13449
   663
paulson@22191
   664
lemma min_Suc1:
paulson@22191
   665
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
nipkow@25162
   666
by (simp split: nat.split)
paulson@22191
   667
paulson@22191
   668
lemma min_Suc2:
paulson@22191
   669
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
nipkow@25162
   670
by (simp split: nat.split)
paulson@22191
   671
berghofe@13449
   672
lemma max_0L [simp]: "max 0 n = (n::nat)"
nipkow@25162
   673
by (rule max_leastL) simp
berghofe@13449
   674
berghofe@13449
   675
lemma max_0R [simp]: "max n 0 = (n::nat)"
nipkow@25162
   676
by (rule max_leastR) simp
berghofe@13449
   677
berghofe@13449
   678
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
nipkow@25162
   679
by (simp add: mono_Suc max_of_mono)
berghofe@13449
   680
paulson@22191
   681
lemma max_Suc1:
paulson@22191
   682
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
nipkow@25162
   683
by (simp split: nat.split)
paulson@22191
   684
paulson@22191
   685
lemma max_Suc2:
paulson@22191
   686
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
nipkow@25162
   687
by (simp split: nat.split)
paulson@22191
   688
berghofe@13449
   689
haftmann@26072
   690
subsubsection {* Monotonicity of Addition *}
berghofe@13449
   691
haftmann@26072
   692
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
haftmann@26072
   693
by (simp add: diff_Suc split: nat.split)
berghofe@13449
   694
paulson@14331
   695
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
nipkow@25162
   696
by (induct k) simp_all
berghofe@13449
   697
paulson@14331
   698
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
nipkow@25162
   699
by (induct k) simp_all
berghofe@13449
   700
nipkow@25162
   701
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
nipkow@25162
   702
by(auto dest:gr0_implies_Suc)
berghofe@13449
   703
paulson@14341
   704
text {* strict, in 1st argument *}
paulson@14341
   705
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
nipkow@25162
   706
by (induct k) simp_all
paulson@14341
   707
paulson@14341
   708
text {* strict, in both arguments *}
paulson@14341
   709
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   710
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   711
  apply (induct j, simp_all)
paulson@14341
   712
  done
paulson@14341
   713
paulson@14341
   714
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   715
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   716
  apply (induct n)
paulson@14341
   717
  apply (simp_all add: order_le_less)
wenzelm@22718
   718
  apply (blast elim!: less_SucE
paulson@14341
   719
               intro!: add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   720
  done
paulson@14341
   721
paulson@14341
   722
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
nipkow@25134
   723
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
nipkow@25134
   724
apply(auto simp: gr0_conv_Suc)
nipkow@25134
   725
apply (induct_tac m)
nipkow@25134
   726
apply (simp_all add: add_less_mono)
nipkow@25134
   727
done
paulson@14341
   728
nipkow@14740
   729
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
obua@14738
   730
instance nat :: ordered_semidom
paulson@14341
   731
proof
paulson@14341
   732
  fix i j k :: nat
paulson@14348
   733
  show "0 < (1::nat)" by simp
paulson@14267
   734
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   735
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   736
qed
paulson@14267
   737
paulson@14267
   738
lemma nat_mult_1: "(1::nat) * n = n"
nipkow@25162
   739
by simp
paulson@14267
   740
paulson@14267
   741
lemma nat_mult_1_right: "n * (1::nat) = n"
nipkow@25162
   742
by simp
paulson@14267
   743
paulson@14267
   744
krauss@26748
   745
subsubsection {* Additional theorems about @{term "op \<le>"} *}
krauss@26748
   746
krauss@26748
   747
text {* Complete induction, aka course-of-values induction *}
krauss@26748
   748
haftmann@27823
   749
instance nat :: wellorder proof
haftmann@27823
   750
  fix P and n :: nat
haftmann@27823
   751
  assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
haftmann@27823
   752
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
haftmann@27823
   753
  proof (induct n)
haftmann@27823
   754
    case (0 n)
krauss@26748
   755
    have "P 0" by (rule step) auto
krauss@26748
   756
    thus ?case using 0 by auto
krauss@26748
   757
  next
haftmann@27823
   758
    case (Suc m n)
haftmann@27823
   759
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
krauss@26748
   760
    thus ?case
krauss@26748
   761
    proof
haftmann@27823
   762
      assume "n \<le> m" thus "P n" by (rule Suc(1))
krauss@26748
   763
    next
haftmann@27823
   764
      assume n: "n = Suc m"
haftmann@27823
   765
      show "P n"
haftmann@27823
   766
        by (rule step) (rule Suc(1), simp add: n le_simps)
krauss@26748
   767
    qed
krauss@26748
   768
  qed
haftmann@27823
   769
  then show "P n" by auto
krauss@26748
   770
qed
krauss@26748
   771
haftmann@27823
   772
lemma Least_Suc:
haftmann@27823
   773
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
haftmann@27823
   774
  apply (case_tac "n", auto)
haftmann@27823
   775
  apply (frule LeastI)
haftmann@27823
   776
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
haftmann@27823
   777
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
haftmann@27823
   778
  apply (erule_tac [2] Least_le)
haftmann@27823
   779
  apply (case_tac "LEAST x. P x", auto)
haftmann@27823
   780
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
haftmann@27823
   781
  apply (blast intro: order_antisym)
haftmann@27823
   782
  done
haftmann@27823
   783
haftmann@27823
   784
lemma Least_Suc2:
haftmann@27823
   785
   "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
haftmann@27823
   786
  apply (erule (1) Least_Suc [THEN ssubst])
haftmann@27823
   787
  apply simp
haftmann@27823
   788
  done
haftmann@27823
   789
haftmann@27823
   790
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
   791
  apply (cases n)
haftmann@27823
   792
   apply blast
haftmann@27823
   793
  apply (rule_tac x="LEAST k. P(k)" in exI)
haftmann@27823
   794
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
haftmann@27823
   795
  done
haftmann@27823
   796
haftmann@27823
   797
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
   798
  apply (cases n)
haftmann@27823
   799
   apply blast
haftmann@27823
   800
  apply (frule (1) ex_least_nat_le)
haftmann@27823
   801
  apply (erule exE)
haftmann@27823
   802
  apply (case_tac k)
haftmann@27823
   803
   apply simp
haftmann@27823
   804
  apply (rename_tac k1)
haftmann@27823
   805
  apply (rule_tac x=k1 in exI)
haftmann@27823
   806
  apply (auto simp add: less_eq_Suc_le)
haftmann@27823
   807
  done
haftmann@27823
   808
krauss@26748
   809
lemma nat_less_induct:
krauss@26748
   810
  assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
krauss@26748
   811
  using assms less_induct by blast
krauss@26748
   812
krauss@26748
   813
lemma measure_induct_rule [case_names less]:
krauss@26748
   814
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   815
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
krauss@26748
   816
  shows "P a"
krauss@26748
   817
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
krauss@26748
   818
krauss@26748
   819
text {* old style induction rules: *}
krauss@26748
   820
lemma measure_induct:
krauss@26748
   821
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   822
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
krauss@26748
   823
  by (rule measure_induct_rule [of f P a]) iprover
krauss@26748
   824
krauss@26748
   825
lemma full_nat_induct:
krauss@26748
   826
  assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
krauss@26748
   827
  shows "P n"
krauss@26748
   828
  by (rule less_induct) (auto intro: step simp:le_simps)
paulson@14267
   829
paulson@19870
   830
text{*An induction rule for estabilishing binary relations*}
wenzelm@22718
   831
lemma less_Suc_induct:
paulson@19870
   832
  assumes less:  "i < j"
paulson@19870
   833
     and  step:  "!!i. P i (Suc i)"
paulson@19870
   834
     and  trans: "!!i j k. P i j ==> P j k ==> P i k"
paulson@19870
   835
  shows "P i j"
paulson@19870
   836
proof -
wenzelm@22718
   837
  from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   838
  have "P i (Suc (i + k))"
paulson@19870
   839
  proof (induct k)
wenzelm@22718
   840
    case 0
wenzelm@22718
   841
    show ?case by (simp add: step)
paulson@19870
   842
  next
paulson@19870
   843
    case (Suc k)
wenzelm@22718
   844
    thus ?case by (auto intro: assms)
paulson@19870
   845
  qed
wenzelm@22718
   846
  thus "P i j" by (simp add: j)
paulson@19870
   847
qed
paulson@19870
   848
krauss@26748
   849
text {* The method of infinite descent, frequently used in number theory.
krauss@26748
   850
Provided by Roelof Oosterhuis.
krauss@26748
   851
$P(n)$ is true for all $n\in\mathbb{N}$ if
krauss@26748
   852
\begin{itemize}
krauss@26748
   853
  \item case ``0'': given $n=0$ prove $P(n)$,
krauss@26748
   854
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
krauss@26748
   855
        a smaller integer $m$ such that $\neg P(m)$.
krauss@26748
   856
\end{itemize} *}
krauss@26748
   857
krauss@26748
   858
text{* A compact version without explicit base case: *}
krauss@26748
   859
lemma infinite_descent:
krauss@26748
   860
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
krauss@26748
   861
by (induct n rule: less_induct, auto)
krauss@26748
   862
krauss@26748
   863
lemma infinite_descent0[case_names 0 smaller]: 
krauss@26748
   864
  "\<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
   865
by (rule infinite_descent) (case_tac "n>0", auto)
krauss@26748
   866
krauss@26748
   867
text {*
krauss@26748
   868
Infinite descent using a mapping to $\mathbb{N}$:
krauss@26748
   869
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
krauss@26748
   870
\begin{itemize}
krauss@26748
   871
\item case ``0'': given $V(x)=0$ prove $P(x)$,
krauss@26748
   872
\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
   873
\end{itemize}
krauss@26748
   874
NB: the proof also shows how to use the previous lemma. *}
krauss@26748
   875
krauss@26748
   876
corollary infinite_descent0_measure [case_names 0 smaller]:
krauss@26748
   877
  assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
krauss@26748
   878
    and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
krauss@26748
   879
  shows "P x"
krauss@26748
   880
proof -
krauss@26748
   881
  obtain n where "n = V x" by auto
krauss@26748
   882
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
krauss@26748
   883
  proof (induct n rule: infinite_descent0)
krauss@26748
   884
    case 0 -- "i.e. $V(x) = 0$"
krauss@26748
   885
    with A0 show "P x" by auto
krauss@26748
   886
  next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
krauss@26748
   887
    case (smaller n)
krauss@26748
   888
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
krauss@26748
   889
    with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
krauss@26748
   890
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
krauss@26748
   891
    then show ?case by auto
krauss@26748
   892
  qed
krauss@26748
   893
  ultimately show "P x" by auto
krauss@26748
   894
qed
krauss@26748
   895
krauss@26748
   896
text{* Again, without explicit base case: *}
krauss@26748
   897
lemma infinite_descent_measure:
krauss@26748
   898
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
krauss@26748
   899
proof -
krauss@26748
   900
  from assms obtain n where "n = V x" by auto
krauss@26748
   901
  moreover have "!!x. V x = n \<Longrightarrow> P x"
krauss@26748
   902
  proof (induct n rule: infinite_descent, auto)
krauss@26748
   903
    fix x assume "\<not> P x"
krauss@26748
   904
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
krauss@26748
   905
  qed
krauss@26748
   906
  ultimately show "P x" by auto
krauss@26748
   907
qed
krauss@26748
   908
paulson@14267
   909
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   910
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   911
lemma less_mono_imp_le_mono:
nipkow@24438
   912
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
   913
by (simp add: order_le_less) (blast)
nipkow@24438
   914
paulson@14267
   915
paulson@14267
   916
text {* non-strict, in 1st argument *}
paulson@14267
   917
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
   918
by (rule add_right_mono)
paulson@14267
   919
paulson@14267
   920
text {* non-strict, in both arguments *}
paulson@14267
   921
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
   922
by (rule add_mono)
paulson@14267
   923
paulson@14267
   924
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
   925
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
   926
paulson@14267
   927
lemma le_add1: "n \<le> ((n + m)::nat)"
nipkow@24438
   928
by (simp add: add_commute, rule le_add2)
berghofe@13449
   929
berghofe@13449
   930
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
   931
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   932
berghofe@13449
   933
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
   934
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   935
paulson@14267
   936
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
   937
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   938
paulson@14267
   939
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
   940
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   941
paulson@14267
   942
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
   943
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
   944
berghofe@13449
   945
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
   946
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
   947
berghofe@13449
   948
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
   949
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
   950
berghofe@13449
   951
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
   952
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
   953
apply (simp_all add: le_add1)
nipkow@24438
   954
done
berghofe@13449
   955
berghofe@13449
   956
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
   957
apply (rule notI)
wenzelm@26335
   958
apply (drule add_lessD1)
wenzelm@26335
   959
apply (erule less_irrefl [THEN notE])
nipkow@24438
   960
done
berghofe@13449
   961
berghofe@13449
   962
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
krauss@26748
   963
by (simp add: add_commute)
berghofe@13449
   964
paulson@14267
   965
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
   966
apply (rule order_trans [of _ "m+k"])
nipkow@24438
   967
apply (simp_all add: le_add1)
nipkow@24438
   968
done
berghofe@13449
   969
paulson@14267
   970
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
nipkow@24438
   971
apply (simp add: add_commute)
nipkow@24438
   972
apply (erule add_leD1)
nipkow@24438
   973
done
berghofe@13449
   974
paulson@14267
   975
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
   976
by (blast dest: add_leD1 add_leD2)
berghofe@13449
   977
berghofe@13449
   978
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
   979
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
   980
by (force simp del: add_Suc_right
berghofe@13449
   981
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
   982
berghofe@13449
   983
haftmann@26072
   984
subsubsection {* More results about difference *}
berghofe@13449
   985
berghofe@13449
   986
text {* Addition is the inverse of subtraction:
paulson@14267
   987
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
   988
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
nipkow@24438
   989
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   990
paulson@14267
   991
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
nipkow@24438
   992
by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
   993
paulson@14267
   994
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
krauss@26748
   995
by (simp add: add_commute)
berghofe@13449
   996
paulson@14267
   997
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
   998
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   999
berghofe@13449
  1000
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
  1001
apply (induct m n rule: diff_induct)
nipkow@24438
  1002
apply (erule_tac [3] less_SucE)
nipkow@24438
  1003
apply (simp_all add: less_Suc_eq)
nipkow@24438
  1004
done
berghofe@13449
  1005
paulson@14267
  1006
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
  1007
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
  1008
haftmann@26072
  1009
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
haftmann@26072
  1010
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
haftmann@26072
  1011
berghofe@13449
  1012
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
  1013
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
  1014
berghofe@13449
  1015
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
  1016
by (cases n) (auto simp add: le_simps)
berghofe@13449
  1017
paulson@14267
  1018
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
  1019
by (induct j k rule: diff_induct) simp_all
berghofe@13449
  1020
paulson@14267
  1021
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
nipkow@24438
  1022
by (simp add: add_commute diff_add_assoc)
berghofe@13449
  1023
paulson@14267
  1024
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
nipkow@24438
  1025
by (auto simp add: diff_add_inverse2)
berghofe@13449
  1026
paulson@14267
  1027
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
  1028
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1029
paulson@14267
  1030
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
  1031
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
  1032
berghofe@13449
  1033
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
  1034
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1035
wenzelm@22718
  1036
lemma less_imp_add_positive:
wenzelm@22718
  1037
  assumes "i < j"
wenzelm@22718
  1038
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
  1039
proof
wenzelm@22718
  1040
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
  1041
    by (simp add: order_less_imp_le)
wenzelm@22718
  1042
qed
wenzelm@9436
  1043
haftmann@26072
  1044
text {* a nice rewrite for bounded subtraction *}
haftmann@26072
  1045
lemma nat_minus_add_max:
haftmann@26072
  1046
  fixes n m :: nat
haftmann@26072
  1047
  shows "n - m + m = max n m"
haftmann@26072
  1048
    by (simp add: max_def not_le order_less_imp_le)
berghofe@13449
  1049
haftmann@26072
  1050
lemma nat_diff_split:
haftmann@26072
  1051
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
haftmann@26072
  1052
    -- {* elimination of @{text -} on @{text nat} *}
haftmann@26072
  1053
by (cases "a < b")
haftmann@26072
  1054
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
haftmann@26072
  1055
    not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)
berghofe@13449
  1056
haftmann@26072
  1057
lemma nat_diff_split_asm:
haftmann@26072
  1058
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
haftmann@26072
  1059
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
haftmann@26072
  1060
by (auto split: nat_diff_split)
berghofe@13449
  1061
berghofe@13449
  1062
haftmann@26072
  1063
subsubsection {* Monotonicity of Multiplication *}
berghofe@13449
  1064
paulson@14267
  1065
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
  1066
by (simp add: mult_right_mono)
berghofe@13449
  1067
paulson@14267
  1068
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
  1069
by (simp add: mult_left_mono)
berghofe@13449
  1070
paulson@14267
  1071
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
  1072
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
  1073
by (simp add: mult_mono)
berghofe@13449
  1074
berghofe@13449
  1075
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
  1076
by (simp add: mult_strict_right_mono)
berghofe@13449
  1077
paulson@14266
  1078
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
  1079
      there are no negative numbers.*}
paulson@14266
  1080
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
  1081
  apply (induct m)
wenzelm@22718
  1082
   apply simp
wenzelm@22718
  1083
  apply (case_tac n)
wenzelm@22718
  1084
   apply simp_all
berghofe@13449
  1085
  done
berghofe@13449
  1086
paulson@14267
  1087
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"
berghofe@13449
  1088
  apply (induct m)
wenzelm@22718
  1089
   apply simp
wenzelm@22718
  1090
  apply (case_tac n)
wenzelm@22718
  1091
   apply simp_all
berghofe@13449
  1092
  done
berghofe@13449
  1093
paulson@14341
  1094
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1095
  apply (safe intro!: mult_less_mono1)
paulson@14208
  1096
  apply (case_tac k, auto)
berghofe@13449
  1097
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1098
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1099
  done
berghofe@13449
  1100
berghofe@13449
  1101
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
nipkow@24438
  1102
by (simp add: mult_commute [of k])
berghofe@13449
  1103
paulson@14267
  1104
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
  1105
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1106
paulson@14267
  1107
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
  1108
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1109
berghofe@13449
  1110
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
  1111
by (subst mult_less_cancel1) simp
berghofe@13449
  1112
paulson@14267
  1113
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
  1114
by (subst mult_le_cancel1) simp
berghofe@13449
  1115
haftmann@26072
  1116
lemma le_square: "m \<le> m * (m::nat)"
haftmann@26072
  1117
  by (cases m) (auto intro: le_add1)
haftmann@26072
  1118
haftmann@26072
  1119
lemma le_cube: "(m::nat) \<le> m * (m * m)"
haftmann@26072
  1120
  by (cases m) (auto intro: le_add1)
berghofe@13449
  1121
berghofe@13449
  1122
text {* Lemma for @{text gcd} *}
berghofe@13449
  1123
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1124
  apply (drule sym)
berghofe@13449
  1125
  apply (rule disjCI)
berghofe@13449
  1126
  apply (rule nat_less_cases, erule_tac [2] _)
paulson@25157
  1127
   apply (drule_tac [2] mult_less_mono2)
nipkow@25162
  1128
    apply (auto)
berghofe@13449
  1129
  done
wenzelm@9436
  1130
haftmann@26072
  1131
text {* the lattice order on @{typ nat} *}
haftmann@24995
  1132
haftmann@26072
  1133
instantiation nat :: distrib_lattice
haftmann@26072
  1134
begin
haftmann@24995
  1135
haftmann@26072
  1136
definition
haftmann@26072
  1137
  "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
haftmann@24995
  1138
haftmann@26072
  1139
definition
haftmann@26072
  1140
  "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
haftmann@24995
  1141
haftmann@26072
  1142
instance by intro_classes
haftmann@26072
  1143
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
haftmann@26072
  1144
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
haftmann@24995
  1145
haftmann@26072
  1146
end
haftmann@24995
  1147
haftmann@24995
  1148
haftmann@25193
  1149
subsection {* Embedding of the Naturals into any
haftmann@25193
  1150
  @{text semiring_1}: @{term of_nat} *}
haftmann@24196
  1151
haftmann@24196
  1152
context semiring_1
haftmann@24196
  1153
begin
haftmann@24196
  1154
haftmann@25559
  1155
primrec
haftmann@25559
  1156
  of_nat :: "nat \<Rightarrow> 'a"
haftmann@25559
  1157
where
haftmann@25559
  1158
  of_nat_0:     "of_nat 0 = 0"
haftmann@25559
  1159
  | of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
haftmann@25193
  1160
haftmann@25193
  1161
lemma of_nat_1 [simp]: "of_nat 1 = 1"
haftmann@25193
  1162
  by simp
haftmann@25193
  1163
haftmann@25193
  1164
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
haftmann@25193
  1165
  by (induct m) (simp_all add: add_ac)
haftmann@25193
  1166
haftmann@25193
  1167
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
haftmann@25193
  1168
  by (induct m) (simp_all add: add_ac left_distrib)
haftmann@25193
  1169
haftmann@28514
  1170
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@28514
  1171
  "of_nat_aux inc 0 i = i"
haftmann@28514
  1172
  | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
haftmann@25928
  1173
haftmann@28514
  1174
lemma of_nat_code [code, code unfold, code inline del]:
haftmann@28514
  1175
  "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
haftmann@28514
  1176
proof (induct n)
haftmann@28514
  1177
  case 0 then show ?case by simp
haftmann@28514
  1178
next
haftmann@28514
  1179
  case (Suc n)
haftmann@28514
  1180
  have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"
haftmann@28514
  1181
    by (induct n) simp_all
haftmann@28514
  1182
  from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"
haftmann@28514
  1183
    by simp
haftmann@28514
  1184
  with Suc show ?case by (simp add: add_commute)
haftmann@28514
  1185
qed
haftmann@28514
  1186
    
haftmann@24196
  1187
end
haftmann@24196
  1188
haftmann@26072
  1189
text{*Class for unital semirings with characteristic zero.
haftmann@26072
  1190
 Includes non-ordered rings like the complex numbers.*}
haftmann@26072
  1191
haftmann@26072
  1192
class semiring_char_0 = semiring_1 +
haftmann@26072
  1193
  assumes of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
haftmann@26072
  1194
begin
haftmann@26072
  1195
haftmann@26072
  1196
text{*Special cases where either operand is zero*}
haftmann@26072
  1197
haftmann@26072
  1198
lemma of_nat_0_eq_iff [simp, noatp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
haftmann@26072
  1199
  by (rule of_nat_eq_iff [of 0, simplified])
haftmann@26072
  1200
haftmann@26072
  1201
lemma of_nat_eq_0_iff [simp, noatp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
haftmann@26072
  1202
  by (rule of_nat_eq_iff [of _ 0, simplified])
haftmann@26072
  1203
haftmann@26072
  1204
lemma inj_of_nat: "inj of_nat"
haftmann@26072
  1205
  by (simp add: inj_on_def)
haftmann@26072
  1206
haftmann@26072
  1207
end
haftmann@26072
  1208
haftmann@25193
  1209
context ordered_semidom
haftmann@25193
  1210
begin
haftmann@25193
  1211
haftmann@25193
  1212
lemma zero_le_imp_of_nat: "0 \<le> of_nat m"
haftmann@25193
  1213
  apply (induct m, simp_all)
haftmann@25193
  1214
  apply (erule order_trans)
haftmann@25193
  1215
  apply (rule ord_le_eq_trans [OF _ add_commute])
haftmann@25193
  1216
  apply (rule less_add_one [THEN less_imp_le])
haftmann@25193
  1217
  done
haftmann@25193
  1218
haftmann@25193
  1219
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
haftmann@25193
  1220
  apply (induct m n rule: diff_induct, simp_all)
haftmann@25193
  1221
  apply (insert add_less_le_mono [OF zero_less_one zero_le_imp_of_nat], force)
haftmann@25193
  1222
  done
haftmann@25193
  1223
haftmann@25193
  1224
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
haftmann@25193
  1225
  apply (induct m n rule: diff_induct, simp_all)
haftmann@25193
  1226
  apply (insert zero_le_imp_of_nat)
haftmann@25193
  1227
  apply (force simp add: not_less [symmetric])
haftmann@25193
  1228
  done
haftmann@25193
  1229
haftmann@25193
  1230
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
haftmann@25193
  1231
  by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
haftmann@25193
  1232
haftmann@26072
  1233
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
haftmann@26072
  1234
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
haftmann@25193
  1235
haftmann@26072
  1236
text{*Every @{text ordered_semidom} has characteristic zero.*}
haftmann@25193
  1237
haftmann@26072
  1238
subclass semiring_char_0
haftmann@28823
  1239
  proof qed (simp add: eq_iff order_eq_iff)
haftmann@25193
  1240
haftmann@25193
  1241
text{*Special cases where either operand is zero*}
haftmann@25193
  1242
haftmann@25193
  1243
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
haftmann@25193
  1244
  by (rule of_nat_le_iff [of 0, simplified])
haftmann@25193
  1245
haftmann@25193
  1246
lemma of_nat_le_0_iff [simp, noatp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
haftmann@25193
  1247
  by (rule of_nat_le_iff [of _ 0, simplified])
haftmann@25193
  1248
haftmann@26072
  1249
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
haftmann@26072
  1250
  by (rule of_nat_less_iff [of 0, simplified])
haftmann@26072
  1251
haftmann@26072
  1252
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
haftmann@26072
  1253
  by (rule of_nat_less_iff [of _ 0, simplified])
haftmann@26072
  1254
haftmann@26072
  1255
end
haftmann@26072
  1256
haftmann@26072
  1257
context ring_1
haftmann@26072
  1258
begin
haftmann@26072
  1259
haftmann@26072
  1260
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
nipkow@29667
  1261
by (simp add: algebra_simps of_nat_add [symmetric])
haftmann@26072
  1262
haftmann@26072
  1263
end
haftmann@26072
  1264
haftmann@26072
  1265
context ordered_idom
haftmann@26072
  1266
begin
haftmann@26072
  1267
haftmann@26072
  1268
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
haftmann@26072
  1269
  unfolding abs_if by auto
haftmann@26072
  1270
haftmann@25193
  1271
end
haftmann@25193
  1272
haftmann@25193
  1273
lemma of_nat_id [simp]: "of_nat n = n"
haftmann@25193
  1274
  by (induct n) auto
haftmann@25193
  1275
haftmann@25193
  1276
lemma of_nat_eq_id [simp]: "of_nat = id"
haftmann@25193
  1277
  by (auto simp add: expand_fun_eq)
haftmann@25193
  1278
haftmann@25193
  1279
haftmann@26149
  1280
subsection {* The Set of Natural Numbers *}
haftmann@25193
  1281
haftmann@26072
  1282
context semiring_1
haftmann@25193
  1283
begin
haftmann@25193
  1284
haftmann@26072
  1285
definition
haftmann@26072
  1286
  Nats  :: "'a set" where
haftmann@28562
  1287
  [code del]: "Nats = range of_nat"
haftmann@26072
  1288
haftmann@26072
  1289
notation (xsymbols)
haftmann@26072
  1290
  Nats  ("\<nat>")
haftmann@25193
  1291
haftmann@26072
  1292
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
haftmann@26072
  1293
  by (simp add: Nats_def)
haftmann@26072
  1294
haftmann@26072
  1295
lemma Nats_0 [simp]: "0 \<in> \<nat>"
haftmann@26072
  1296
apply (simp add: Nats_def)
haftmann@26072
  1297
apply (rule range_eqI)
haftmann@26072
  1298
apply (rule of_nat_0 [symmetric])
haftmann@26072
  1299
done
haftmann@25193
  1300
haftmann@26072
  1301
lemma Nats_1 [simp]: "1 \<in> \<nat>"
haftmann@26072
  1302
apply (simp add: Nats_def)
haftmann@26072
  1303
apply (rule range_eqI)
haftmann@26072
  1304
apply (rule of_nat_1 [symmetric])
haftmann@26072
  1305
done
haftmann@25193
  1306
haftmann@26072
  1307
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
haftmann@26072
  1308
apply (auto simp add: Nats_def)
haftmann@26072
  1309
apply (rule range_eqI)
haftmann@26072
  1310
apply (rule of_nat_add [symmetric])
haftmann@26072
  1311
done
haftmann@26072
  1312
haftmann@26072
  1313
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
haftmann@26072
  1314
apply (auto simp add: Nats_def)
haftmann@26072
  1315
apply (rule range_eqI)
haftmann@26072
  1316
apply (rule of_nat_mult [symmetric])
haftmann@26072
  1317
done
haftmann@25193
  1318
haftmann@25193
  1319
end
haftmann@25193
  1320
haftmann@25193
  1321
wenzelm@21243
  1322
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
wenzelm@21243
  1323
haftmann@22845
  1324
lemma subst_equals:
haftmann@22845
  1325
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1326
  shows "u = s"
haftmann@22845
  1327
  using 2 1 by (rule trans)
haftmann@22845
  1328
haftmann@28952
  1329
use "Tools/arith_data.ML"
haftmann@26101
  1330
declaration {* K ArithData.setup *}
wenzelm@24091
  1331
nipkow@29849
  1332
ML{*
nipkow@29849
  1333
structure ArithFacts =
nipkow@29849
  1334
  NamedThmsFun(val name = "arith"
nipkow@29849
  1335
               val description = "arith facts - only ground formulas");
nipkow@29849
  1336
*}
nipkow@29849
  1337
nipkow@29849
  1338
setup ArithFacts.setup
nipkow@29849
  1339
wenzelm@24091
  1340
use "Tools/lin_arith.ML"
wenzelm@24091
  1341
declaration {* K LinArith.setup *}
wenzelm@24091
  1342
wenzelm@21243
  1343
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1344
nipkow@27625
  1345
context order
nipkow@27625
  1346
begin
nipkow@27625
  1347
nipkow@27625
  1348
lemma lift_Suc_mono_le:
krauss@27627
  1349
  assumes mono: "!!n. f n \<le> f(Suc n)" and "n\<le>n'"
krauss@27627
  1350
  shows "f n \<le> f n'"
krauss@27627
  1351
proof (cases "n < n'")
krauss@27627
  1352
  case True
krauss@27627
  1353
  thus ?thesis
krauss@27627
  1354
    by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)
krauss@27627
  1355
qed (insert `n \<le> n'`, auto) -- {*trivial for @{prop "n = n'"} *}
nipkow@27625
  1356
nipkow@27625
  1357
lemma lift_Suc_mono_less:
krauss@27627
  1358
  assumes mono: "!!n. f n < f(Suc n)" and "n < n'"
krauss@27627
  1359
  shows "f n < f n'"
krauss@27627
  1360
using `n < n'`
krauss@27627
  1361
by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)
nipkow@27625
  1362
nipkow@27789
  1363
lemma lift_Suc_mono_less_iff:
nipkow@27789
  1364
  "(!!n. f n < f(Suc n)) \<Longrightarrow> f(n) < f(m) \<longleftrightarrow> n<m"
nipkow@27789
  1365
by(blast intro: less_asym' lift_Suc_mono_less[of f]
nipkow@27789
  1366
         dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq[THEN iffD1])
nipkow@27789
  1367
nipkow@27625
  1368
end
nipkow@27625
  1369
nipkow@27625
  1370
nipkow@27789
  1371
lemma mono_nat_linear_lb:
nipkow@27789
  1372
  "(!!m n::nat. m<n \<Longrightarrow> f m < f n) \<Longrightarrow> f(m)+k \<le> f(m+k)"
nipkow@27789
  1373
apply(induct_tac k)
nipkow@27789
  1374
 apply simp
nipkow@27789
  1375
apply(erule_tac x="m+n" in meta_allE)
nipkow@27789
  1376
apply(erule_tac x="m+n+1" in meta_allE)
nipkow@27789
  1377
apply simp
nipkow@27789
  1378
done
nipkow@27789
  1379
nipkow@27789
  1380
wenzelm@21243
  1381
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1382
wenzelm@21243
  1383
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1384
by arith
wenzelm@21243
  1385
wenzelm@21243
  1386
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1387
by arith
wenzelm@21243
  1388
wenzelm@21243
  1389
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1390
by arith
wenzelm@21243
  1391
wenzelm@21243
  1392
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
nipkow@24438
  1393
by arith
wenzelm@21243
  1394
wenzelm@21243
  1395
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1396
by arith
wenzelm@21243
  1397
wenzelm@21243
  1398
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
nipkow@24438
  1399
by arith
wenzelm@21243
  1400
wenzelm@21243
  1401
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1402
  second premise n\<le>m*)
wenzelm@21243
  1403
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1404
by arith
wenzelm@21243
  1405
haftmann@26072
  1406
text {* Simplification of relational expressions involving subtraction *}
wenzelm@21243
  1407
wenzelm@21243
  1408
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1409
by (simp split add: nat_diff_split)
wenzelm@21243
  1410
wenzelm@21243
  1411
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1412
by (auto split add: nat_diff_split)
wenzelm@21243
  1413
wenzelm@21243
  1414
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1415
by (auto split add: nat_diff_split)
wenzelm@21243
  1416
wenzelm@21243
  1417
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1418
by (auto split add: nat_diff_split)
wenzelm@21243
  1419
wenzelm@21243
  1420
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1421
wenzelm@21243
  1422
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1423
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1424
by (simp split add: nat_diff_split)
wenzelm@21243
  1425
wenzelm@21243
  1426
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1427
by (simp split add: nat_diff_split)
wenzelm@21243
  1428
wenzelm@21243
  1429
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1430
by (simp split add: nat_diff_split)
wenzelm@21243
  1431
wenzelm@21243
  1432
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1433
by (simp split add: nat_diff_split)
wenzelm@21243
  1434
bulwahn@26143
  1435
lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
bulwahn@26143
  1436
unfolding min_def by auto
bulwahn@26143
  1437
bulwahn@26143
  1438
lemma inj_on_diff_nat: 
bulwahn@26143
  1439
  assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
bulwahn@26143
  1440
  shows "inj_on (\<lambda>n. n - k) N"
bulwahn@26143
  1441
proof (rule inj_onI)
bulwahn@26143
  1442
  fix x y
bulwahn@26143
  1443
  assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
bulwahn@26143
  1444
  with k_le_n have "x - k + k = y - k + k" by auto
bulwahn@26143
  1445
  with a k_le_n show "x = y" by auto
bulwahn@26143
  1446
qed
bulwahn@26143
  1447
haftmann@26072
  1448
text{*Rewriting to pull differences out*}
haftmann@26072
  1449
haftmann@26072
  1450
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
haftmann@26072
  1451
by arith
haftmann@26072
  1452
haftmann@26072
  1453
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
haftmann@26072
  1454
by arith
haftmann@26072
  1455
haftmann@26072
  1456
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
haftmann@26072
  1457
by arith
haftmann@26072
  1458
wenzelm@21243
  1459
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1460
wenzelm@21243
  1461
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1462
by (cases m) auto
wenzelm@21243
  1463
wenzelm@21243
  1464
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1465
by (cases m) auto
wenzelm@21243
  1466
wenzelm@21243
  1467
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1468
by (cases m) auto
wenzelm@21243
  1469
krauss@23001
  1470
text {* Specialized induction principles that work "backwards": *}
krauss@23001
  1471
krauss@23001
  1472
lemma inc_induct[consumes 1, case_names base step]:
krauss@23001
  1473
  assumes less: "i <= j"
krauss@23001
  1474
  assumes base: "P j"
krauss@23001
  1475
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1476
  shows "P i"
krauss@23001
  1477
  using less
krauss@23001
  1478
proof (induct d=="j - i" arbitrary: i)
krauss@23001
  1479
  case (0 i)
krauss@23001
  1480
  hence "i = j" by simp
krauss@23001
  1481
  with base show ?case by simp
krauss@23001
  1482
next
krauss@23001
  1483
  case (Suc d i)
krauss@23001
  1484
  hence "i < j" "P (Suc i)"
krauss@23001
  1485
    by simp_all
krauss@23001
  1486
  thus "P i" by (rule step)
krauss@23001
  1487
qed
krauss@23001
  1488
krauss@23001
  1489
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1490
  assumes less: "i < j"
krauss@23001
  1491
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1492
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1493
  shows "P i"
krauss@23001
  1494
  using less
krauss@23001
  1495
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1496
  case (0 i)
krauss@23001
  1497
  with `i < j` have "j = Suc i" by simp
krauss@23001
  1498
  with base show ?case by simp
krauss@23001
  1499
next
krauss@23001
  1500
  case (Suc d i)
krauss@23001
  1501
  hence "i < j" "P (Suc i)"
krauss@23001
  1502
    by simp_all
krauss@23001
  1503
  thus "P i" by (rule step)
krauss@23001
  1504
qed
krauss@23001
  1505
krauss@23001
  1506
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1507
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1508
krauss@23001
  1509
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1510
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1511
haftmann@26072
  1512
lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False"
haftmann@26072
  1513
  by auto
wenzelm@21243
  1514
wenzelm@21243
  1515
(*The others are
wenzelm@21243
  1516
      i - j - k = i - (j + k),
wenzelm@21243
  1517
      k \<le> j ==> j - k + i = j + i - k,
wenzelm@21243
  1518
      k \<le> j ==> i + (j - k) = i + j - k *)
wenzelm@21243
  1519
lemmas add_diff_assoc = diff_add_assoc [symmetric]
wenzelm@21243
  1520
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
haftmann@26072
  1521
declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
wenzelm@21243
  1522
wenzelm@21243
  1523
text{*At present we prove no analogue of @{text not_less_Least} or @{text
wenzelm@21243
  1524
Least_Suc}, since there appears to be no need.*}
wenzelm@21243
  1525
nipkow@27625
  1526
haftmann@26072
  1527
subsection {* size of a datatype value *}
haftmann@25193
  1528
haftmann@29608
  1529
class size =
krauss@26748
  1530
  fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}
haftmann@23852
  1531
haftmann@25193
  1532
end