src/HOL/Nat.thy
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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 and mod, 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 Product_Type Fields
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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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  ("Tools/nat_arith.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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  Suc_Rep_inject:       "Suc_Rep x = Suc_Rep y ==> x = y" 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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  by (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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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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lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y"
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  by (rule iffI, rule Suc_Rep_inject) simp_all
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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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      Suc_Rep_inject' 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 -- names of variables differ *}
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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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hide (open) fact add_0 add_0_right diff_0
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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)" unfolding One_nat_def by simp
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  show "1 * n = n" unfolding One_nat_def 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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lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
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  unfolding One_nat_def by simp
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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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diff changeset
   300
subsubsection {* Multiplication *}
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   301
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   302
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   303
  by (rule mult_assoc)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   304
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   305
lemma nat_mult_commute: "m * n = n * (m::nat)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   306
  by (rule mult_commute)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   307
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   308
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   309
  by (rule right_distrib)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   310
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   311
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   312
  by (induct m) auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   313
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   314
lemmas nat_distrib =
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   315
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   316
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   317
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   318
  apply (induct m)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   319
   apply simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   320
  apply (induct n)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   321
   apply auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   322
  done
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   323
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   324
lemma one_eq_mult_iff [simp,noatp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   325
  apply (rule trans)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   326
  apply (rule_tac [2] mult_eq_1_iff, fastsimp)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   327
  done
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   328
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   329
lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   330
  unfolding One_nat_def by (rule mult_eq_1_iff)
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   331
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   332
lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   333
  unfolding One_nat_def by (rule one_eq_mult_iff)
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   334
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   335
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   336
proof -
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   337
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   338
  proof (induct n arbitrary: m)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   339
    case 0 then show "m = 0" by simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   340
  next
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   341
    case (Suc n) then show "m = Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   342
      by (cases m) (simp_all add: eq_commute [of "0"])
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   343
  qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   344
  then show ?thesis by auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   345
qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   346
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   347
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   348
  by (simp add: mult_commute)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   349
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   350
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   351
  by (subst mult_cancel1) simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   352
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   353
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   354
subsection {* Orders on @{typ nat} *}
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   355
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   356
subsubsection {* Operation definition *}
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   357
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   358
instantiation nat :: linorder
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   359
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   360
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   361
primrec less_eq_nat where
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   362
  "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   363
  | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   364
28514
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
   365
declare less_eq_nat.simps [simp del]
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   366
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   367
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   368
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   369
definition less_nat where
28514
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
   370
  less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   371
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   372
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   373
  by (simp add: less_eq_nat.simps(2))
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   374
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   375
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   376
  unfolding less_eq_Suc_le ..
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   377
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   378
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   379
  by (induct n) (simp_all add: less_eq_nat.simps(2))
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   380
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   381
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   382
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   383
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   384
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   385
  by simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   386
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   387
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   388
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   389
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   390
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   391
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   392
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   393
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   394
  by (induct m arbitrary: n)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   395
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   396
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   397
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   398
  by (cases n) (auto intro: le_SucI)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   399
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   400
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   401
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   402
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   403
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   404
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   405
26315
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   406
instance
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   407
proof
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   408
  fix n m :: nat
27679
haftmann
parents: 27627
diff changeset
   409
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" 
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   410
  proof (induct n arbitrary: m)
27679
haftmann
parents: 27627
diff changeset
   411
    case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   412
  next
27679
haftmann
parents: 27627
diff changeset
   413
    case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   414
  qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   415
next
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   416
  fix n :: nat show "n \<le> n" by (induct n) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   417
next
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   418
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   419
  then show "n = m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   420
    by (induct n arbitrary: m)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   421
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   422
next
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   423
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   424
  then show "n \<le> q"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   425
  proof (induct n arbitrary: m q)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   426
    case 0 show ?case by simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   427
  next
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   428
    case (Suc n) then show ?case
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   429
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   430
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   431
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   432
  qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   433
next
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   434
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   435
    by (induct n arbitrary: m)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   436
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   437
qed
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   438
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   439
end
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   440
29652
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   441
instantiation nat :: bot
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   442
begin
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   443
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   444
definition bot_nat :: nat where
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   445
  "bot_nat = 0"
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   446
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   447
instance proof
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   448
qed (simp add: bot_nat_def)
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   449
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   450
end
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   451
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   452
subsubsection {* Introduction properties *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   453
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   454
lemma lessI [iff]: "n < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   455
  by (simp add: less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   456
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   457
lemma zero_less_Suc [iff]: "0 < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   458
  by (simp add: less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   459
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   460
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   461
subsubsection {* Elimination properties *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   462
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   463
lemma less_not_refl: "~ n < (n::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   464
  by (rule order_less_irrefl)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   465
26335
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
   466
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
   467
  by (rule not_sym) (rule less_imp_neq) 
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   468
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   469
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   470
  by (rule less_imp_neq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   471
26335
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
   472
lemma less_irrefl_nat: "(n::nat) < n ==> R"
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
   473
  by (rule notE, rule less_not_refl)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   474
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   475
lemma less_zeroE: "(n::nat) < 0 ==> R"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   476
  by (rule notE) (rule not_less0)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   477
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   478
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   479
  unfolding less_Suc_eq_le le_less ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   480
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   481
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   482
  by (simp add: less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   483
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   484
lemma less_one [iff, noatp]: "(n < (1::nat)) = (n = 0)"
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   485
  unfolding One_nat_def by (rule less_Suc0)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   486
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   487
lemma Suc_mono: "m < n ==> Suc m < Suc n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   488
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   489
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   490
text {* "Less than" is antisymmetric, sort of *}
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   491
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   492
  unfolding not_less less_Suc_eq_le by (rule antisym)
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   493
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   494
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   495
  by (rule linorder_neq_iff)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   496
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   497
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   498
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   499
  shows "P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   500
  apply (rule less_linear [THEN disjE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   501
  apply (erule_tac [2] disjE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   502
  apply (erule lessCase)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   503
  apply (erule sym [THEN eqCase])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   504
  apply (erule major)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   505
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   506
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   507
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   508
subsubsection {* Inductive (?) properties *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   509
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   510
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   511
  unfolding less_eq_Suc_le [of m] le_less by simp 
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   512
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   513
lemma lessE:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   514
  assumes major: "i < k"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   515
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   516
  shows P
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   517
proof -
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   518
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   519
    unfolding less_eq_Suc_le by (induct k) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   520
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   521
    by (clarsimp simp add: less_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   522
  with p1 p2 show P by auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   523
qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   524
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   525
lemma less_SucE: assumes major: "m < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   526
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   527
  apply (rule major [THEN lessE])
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   528
  apply (rule eq, blast)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   529
  apply (rule less, blast)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   530
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   531
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   532
lemma Suc_lessE: assumes major: "Suc i < k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   533
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   534
  apply (rule major [THEN lessE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   535
  apply (erule lessI [THEN minor])
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   536
  apply (erule Suc_lessD [THEN minor], assumption)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   537
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   538
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   539
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   540
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   541
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   542
lemma less_trans_Suc:
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   543
  assumes le: "i < j" shows "j < k ==> Suc i < k"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   544
  apply (induct k, simp_all)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   545
  apply (insert le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   546
  apply (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   547
  apply (blast dest: Suc_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   548
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   549
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   550
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   551
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   552
  unfolding not_less less_Suc_eq_le ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   553
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   554
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   555
  unfolding not_le Suc_le_eq ..
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
   556
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   557
text {* Properties of "less than or equal" *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   558
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   559
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   560
  unfolding less_Suc_eq_le .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   561
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   562
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   563
  unfolding not_le less_Suc_eq_le ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   564
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   565
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   566
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   567
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   568
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   569
  by (drule le_Suc_eq [THEN iffD1], iprover+)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   570
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   571
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   572
  unfolding Suc_le_eq .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   573
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   574
text {* Stronger version of @{text Suc_leD} *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   575
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   576
  unfolding Suc_le_eq .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   577
26315
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   578
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   579
  unfolding less_eq_Suc_le by (rule Suc_leD)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   580
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   581
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
26315
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   582
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   583
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   584
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   585
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   586
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   587
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   588
  unfolding le_less .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   589
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   590
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   591
  by (rule le_less)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   592
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   593
text {* Useful with @{text blast}. *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   594
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   595
  by auto
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   596
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   597
lemma le_refl: "n \<le> (n::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   598
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   599
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   600
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   601
  by (rule order_trans)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   602
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33364
diff changeset
   603
lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   604
  by (rule antisym)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   605
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   606
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   607
  by (rule less_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   608
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   609
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   610
  unfolding less_le ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   611
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   612
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   613
  by (rule linear)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   614
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   615
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
15921
b6e345548913 Fixing a problem with lin.arith.
nipkow
parents: 15539
diff changeset
   616
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   617
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   618
  unfolding less_Suc_eq_le by auto
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   619
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   620
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   621
  unfolding not_less by (rule le_less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   622
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   623
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   624
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   625
text {* These two rules ease the use of primitive recursion.
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   626
NOTE USE OF @{text "=="} *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   627
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   628
by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   629
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   630
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   631
by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   632
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   633
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   634
by (cases n) simp_all
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   635
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   636
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   637
by (cases n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   638
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   639
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   640
by (cases n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   641
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   642
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   643
by (cases n) simp_all
25140
273772abbea2 More changes from >0 to ~=0::nat
nipkow
parents: 25134
diff changeset
   644
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   645
text {* This theorem is useful with @{text blast} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   646
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   647
by (rule neq0_conv[THEN iffD1], iprover)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   648
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   649
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   650
by (fast intro: not0_implies_Suc)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   651
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24196
diff changeset
   652
lemma not_gr0 [iff,noatp]: "!!n::nat. (~ (0 < n)) = (n = 0)"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   653
using neq0_conv by blast
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   654
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   655
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   656
by (induct m') simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   657
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   658
text {* Useful in certain inductive arguments *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   659
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   660
by (cases m) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   661
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   662
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   663
subsubsection {* @{term min} and @{term max} *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   664
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   665
lemma mono_Suc: "mono Suc"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   666
by (rule monoI) simp
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   667
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   668
lemma min_0L [simp]: "min 0 n = (0::nat)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   669
by (rule min_leastL) simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   670
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   671
lemma min_0R [simp]: "min n 0 = (0::nat)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   672
by (rule min_leastR) simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   673
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   674
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   675
by (simp add: mono_Suc min_of_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   676
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   677
lemma min_Suc1:
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   678
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   679
by (simp split: nat.split)
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   680
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   681
lemma min_Suc2:
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   682
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   683
by (simp split: nat.split)
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   684
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   685
lemma max_0L [simp]: "max 0 n = (n::nat)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   686
by (rule max_leastL) simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   687
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   688
lemma max_0R [simp]: "max n 0 = (n::nat)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   689
by (rule max_leastR) simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   690
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   691
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   692
by (simp add: mono_Suc max_of_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   693
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   694
lemma max_Suc1:
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   695
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   696
by (simp split: nat.split)
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   697
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   698
lemma max_Suc2:
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   699
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   700
by (simp split: nat.split)
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   701
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   702
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   703
subsubsection {* Monotonicity of Addition *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   704
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   705
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   706
by (simp add: diff_Suc split: nat.split)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   707
30128
365ee7319b86 revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents: 30093
diff changeset
   708
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
365ee7319b86 revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents: 30093
diff changeset
   709
unfolding One_nat_def by (rule Suc_pred)
365ee7319b86 revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents: 30093
diff changeset
   710
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   711
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   712
by (induct k) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   713
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   714
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   715
by (induct k) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   716
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   717
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   718
by(auto dest:gr0_implies_Suc)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   719
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   720
text {* strict, in 1st argument *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   721
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   722
by (induct k) simp_all
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   723
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   724
text {* strict, in both arguments *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   725
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   726
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   727
  apply (induct j, simp_all)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   728
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   729
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   730
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   731
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   732
  apply (induct n)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   733
  apply (simp_all add: order_le_less)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   734
  apply (blast elim!: less_SucE
35047
1b2bae06c796 hide fact Nat.add_0_right; make add_0_right from Groups priority
haftmann
parents: 35028
diff changeset
   735
               intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   736
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   737
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   738
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   739
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   740
apply(auto simp: gr0_conv_Suc)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   741
apply (induct_tac m)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   742
apply (simp_all add: add_less_mono)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   743
done
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   744
14740
c8e1937110c2 fixed latex problems
nipkow
parents: 14738
diff changeset
   745
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34208
diff changeset
   746
instance nat :: linordered_semidom
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   747
proof
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   748
  fix i j k :: nat
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   749
  show "0 < (1::nat)" by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   750
  show "i \<le> j ==> k + i \<le> k + j" by simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   751
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   752
qed
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   753
30056
0a35bee25c20 added lemmas
nipkow
parents: 29879
diff changeset
   754
instance nat :: no_zero_divisors
0a35bee25c20 added lemmas
nipkow
parents: 29879
diff changeset
   755
proof
0a35bee25c20 added lemmas
nipkow
parents: 29879
diff changeset
   756
  fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto
0a35bee25c20 added lemmas
nipkow
parents: 29879
diff changeset
   757
qed
0a35bee25c20 added lemmas
nipkow
parents: 29879
diff changeset
   758
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   759
lemma nat_mult_1: "(1::nat) * n = n"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   760
by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   761
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   762
lemma nat_mult_1_right: "n * (1::nat) = n"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   763
by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   764
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   765
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   766
subsubsection {* Additional theorems about @{term "op \<le>"} *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   767
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   768
text {* Complete induction, aka course-of-values induction *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   769
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   770
instance nat :: wellorder proof
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   771
  fix P and n :: nat
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   772
  assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   773
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   774
  proof (induct n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   775
    case (0 n)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   776
    have "P 0" by (rule step) auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   777
    thus ?case using 0 by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   778
  next
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   779
    case (Suc m n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   780
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   781
    thus ?case
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   782
    proof
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   783
      assume "n \<le> m" thus "P n" by (rule Suc(1))
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   784
    next
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   785
      assume n: "n = Suc m"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   786
      show "P n"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   787
        by (rule step) (rule Suc(1), simp add: n le_simps)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   788
    qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   789
  qed
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   790
  then show "P n" by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   791
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   792
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   793
lemma Least_Suc:
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   794
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   795
  apply (case_tac "n", auto)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   796
  apply (frule LeastI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   797
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   798
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   799
  apply (erule_tac [2] Least_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   800
  apply (case_tac "LEAST x. P x", auto)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   801
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   802
  apply (blast intro: order_antisym)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   803
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   804
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   805
lemma Least_Suc2:
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   806
   "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   807
  apply (erule (1) Least_Suc [THEN ssubst])
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   808
  apply simp
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   809
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   810
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   811
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)"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   812
  apply (cases n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   813
   apply blast
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   814
  apply (rule_tac x="LEAST k. P(k)" in exI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   815
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   816
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   817
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   818
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)"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   819
  unfolding One_nat_def
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   820
  apply (cases n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   821
   apply blast
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   822
  apply (frule (1) ex_least_nat_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   823
  apply (erule exE)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   824
  apply (case_tac k)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   825
   apply simp
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   826
  apply (rename_tac k1)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   827
  apply (rule_tac x=k1 in exI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   828
  apply (auto simp add: less_eq_Suc_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   829
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   830
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   831
lemma nat_less_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   832
  assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   833
  using assms less_induct by blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   834
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   835
lemma measure_induct_rule [case_names less]:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   836
  fixes f :: "'a \<Rightarrow> nat"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   837
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   838
  shows "P a"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   839
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   840
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   841
text {* old style induction rules: *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   842
lemma measure_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   843
  fixes f :: "'a \<Rightarrow> nat"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   844
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   845
  by (rule measure_induct_rule [of f P a]) iprover
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   846
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   847
lemma full_nat_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   848
  assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   849
  shows "P n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   850
  by (rule less_induct) (auto intro: step simp:le_simps)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   851
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   852
text{*An induction rule for estabilishing binary relations*}
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   853
lemma less_Suc_induct:
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   854
  assumes less:  "i < j"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   855
     and  step:  "!!i. P i (Suc i)"
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   856
     and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   857
  shows "P i j"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   858
proof -
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   859
  from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   860
  have "P i (Suc (i + k))"
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   861
  proof (induct k)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   862
    case 0
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   863
    show ?case by (simp add: step)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   864
  next
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   865
    case (Suc k)
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   866
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   867
    hence "i < Suc (i + k)" by (simp add: add_commute)
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   868
    from trans[OF this lessI Suc step]
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   869
    show ?case by simp
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   870
  qed
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   871
  thus "P i j" by (simp add: j)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   872
qed
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   873
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   874
text {* The method of infinite descent, frequently used in number theory.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   875
Provided by Roelof Oosterhuis.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   876
$P(n)$ is true for all $n\in\mathbb{N}$ if
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   877
\begin{itemize}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   878
  \item case ``0'': given $n=0$ prove $P(n)$,
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   879
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   880
        a smaller integer $m$ such that $\neg P(m)$.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   881
\end{itemize} *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   882
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   883
text{* A compact version without explicit base case: *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   884
lemma infinite_descent:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   885
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   886
by (induct n rule: less_induct, auto)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   887
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   888
lemma infinite_descent0[case_names 0 smaller]: 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   889
  "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   890
by (rule infinite_descent) (case_tac "n>0", auto)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   891
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   892
text {*
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   893
Infinite descent using a mapping to $\mathbb{N}$:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   894
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   895
\begin{itemize}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   896
\item case ``0'': given $V(x)=0$ prove $P(x)$,
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   897
\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 $~\neg P(y)$.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   898
\end{itemize}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   899
NB: the proof also shows how to use the previous lemma. *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   900
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   901
corollary infinite_descent0_measure [case_names 0 smaller]:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   902
  assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   903
    and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   904
  shows "P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   905
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   906
  obtain n where "n = V x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   907
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   908
  proof (induct n rule: infinite_descent0)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   909
    case 0 -- "i.e. $V(x) = 0$"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   910
    with A0 show "P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   911
  next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   912
    case (smaller n)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   913
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   914
    with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   915
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   916
    then show ?case by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   917
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   918
  ultimately show "P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   919
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   920
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   921
text{* Again, without explicit base case: *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   922
lemma infinite_descent_measure:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   923
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   924
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   925
  from assms obtain n where "n = V x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   926
  moreover have "!!x. V x = n \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   927
  proof (induct n rule: infinite_descent, auto)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   928
    fix x assume "\<not> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   929
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   930
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   931
  ultimately show "P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   932
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   933
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   934
text {* A [clumsy] way of lifting @{text "<"}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   935
  monotonicity to @{text "\<le>"} monotonicity *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   936
lemma less_mono_imp_le_mono:
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   937
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   938
by (simp add: order_le_less) (blast)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   939
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   940
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   941
text {* non-strict, in 1st argument *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   942
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   943
by (rule add_right_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   944
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   945
text {* non-strict, in both arguments *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   946
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   947
by (rule add_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   948
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   949
lemma le_add2: "n \<le> ((m + n)::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   950
by (insert add_right_mono [of 0 m n], simp)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   951
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   952
lemma le_add1: "n \<le> ((n + m)::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   953
by (simp add: add_commute, rule le_add2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   954
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   955
lemma less_add_Suc1: "i < Suc (i + m)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   956
by (rule le_less_trans, rule le_add1, rule lessI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   957
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   958
lemma less_add_Suc2: "i < Suc (m + i)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   959
by (rule le_less_trans, rule le_add2, rule lessI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   960
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   961
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   962
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   963
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   964
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   965
by (rule le_trans, assumption, rule le_add1)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   966
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   967
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   968
by (rule le_trans, assumption, rule le_add2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   969
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   970
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   971
by (rule less_le_trans, assumption, rule le_add1)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   972
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   973
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   974
by (rule less_le_trans, assumption, rule le_add2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   975
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   976
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   977
apply (rule le_less_trans [of _ "i+j"])
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   978
apply (simp_all add: le_add1)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   979
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   980
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   981
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   982
apply (rule notI)
26335
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
   983
apply (drule add_lessD1)
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
   984
apply (erule less_irrefl [THEN notE])
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   985
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   986
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   987
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   988
by (simp add: add_commute)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   989
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   990
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   991
apply (rule order_trans [of _ "m+k"])
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   992
apply (simp_all add: le_add1)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   993
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   994
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   995
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   996
apply (simp add: add_commute)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   997
apply (erule add_leD1)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   998
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   999
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1000
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1001
by (blast dest: add_leD1 add_leD2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1002
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1003
text {* needs @{text "!!k"} for @{text add_ac} to work *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1004
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1005
by (force simp del: add_Suc_right
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1006
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1007
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1008
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1009
subsubsection {* More results about difference *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1010
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1011
text {* Addition is the inverse of subtraction:
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1012
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1013
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1014
by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1015
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1016
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1017
by (simp add: add_diff_inverse linorder_not_less)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1018
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1019
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1020
by (simp add: add_commute)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1021
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1022
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1023
by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1024
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1025
lemma diff_less_Suc: "m - n < Suc m"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1026
apply (induct m n rule: diff_induct)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1027
apply (erule_tac [3] less_SucE)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1028
apply (simp_all add: less_Suc_eq)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1029
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1030
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1031
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1032
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1033
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1034
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1035
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1036
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1037
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1038
by (rule le_less_trans, rule diff_le_self)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1039
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1040
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1041
by (cases n) (auto simp add: le_simps)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1042
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1043
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1044
by (induct j k rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1045
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1046
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1047
by (simp add: add_commute diff_add_assoc)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1048
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1049
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1050
by (auto simp add: diff_add_inverse2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1051
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1052
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1053
by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1054
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1055
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1056
by (rule iffD2, rule diff_is_0_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1057
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1058
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1059
by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1060
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1061
lemma less_imp_add_positive:
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1062
  assumes "i < j"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1063
  shows "\<exists>k::nat. 0 < k & i + k = j"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1064
proof
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1065
  from assms show "0 < j - i & i + (j - i) = j"
23476
839db6346cc8 fix looping simp rule
huffman
parents: 23438
diff changeset
  1066
    by (simp add: order_less_imp_le)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1067
qed
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1068
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1069
text {* a nice rewrite for bounded subtraction *}
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1070
lemma nat_minus_add_max:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1071
  fixes n m :: nat
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1072
  shows "n - m + m = max n m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1073
    by (simp add: max_def not_le order_less_imp_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1074
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1075
lemma nat_diff_split:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1076
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1077
    -- {* elimination of @{text -} on @{text nat} *}
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1078
by (cases "a < b")
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1079
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1080
    not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1081
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1082
lemma nat_diff_split_asm:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1083
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1084
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1085
by (auto split: nat_diff_split)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1086
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1087
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1088
subsubsection {* Monotonicity of Multiplication *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1089
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1090
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1091
by (simp add: mult_right_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1092
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1093
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1094
by (simp add: mult_left_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1095
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1096
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1097
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1098
by (simp add: mult_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1099
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1100
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1101
by (simp add: mult_strict_right_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1102
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
  1103
text{*Differs from the standard @{text zero_less_mult_iff} in that
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
  1104
      there are no negative numbers.*}
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
  1105
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1106
  apply (induct m)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1107
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1108
  apply (case_tac n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1109
   apply simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1110
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1111
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
  1112
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1113
  apply (induct m)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1114
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1115
  apply (case_tac n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1116
   apply simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1117
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1118
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
  1119
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1120
  apply (safe intro!: mult_less_mono1)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
  1121
  apply (case_tac k, auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1122
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1123
  apply (blast intro: mult_le_mono1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1124
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1125
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1126
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1127
by (simp add: mult_commute [of k])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1128
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1129
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1130
by (simp add: linorder_not_less [symmetric], auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1131
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1132
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1133
by (simp add: linorder_not_less [symmetric], auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1134
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1135
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1136
by (subst mult_less_cancel1) simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1137
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1138
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1139
by (subst mult_le_cancel1) simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1140
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1141
lemma le_square: "m \<le> m * (m::nat)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1142
  by (cases m) (auto intro: le_add1)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1143
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1144
lemma le_cube: "(m::nat) \<le> m * (m * m)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1145
  by (cases m) (auto intro: le_add1)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1146
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1147
text {* Lemma for @{text gcd} *}
30128
365ee7319b86 revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents: 30093
diff changeset
  1148
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1149
  apply (drule sym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1150
  apply (rule disjCI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1151
  apply (rule nat_less_cases, erule_tac [2] _)
25157
8b80535cd017 random tidying of proofs
paulson
parents: 25145
diff changeset
  1152
   apply (drule_tac [2] mult_less_mono2)
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
  1153
    apply (auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1154
  done
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1155
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1156
text {* the lattice order on @{typ nat} *}
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1157
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1158
instantiation nat :: distrib_lattice
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1159
begin
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1160
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1161
definition
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1162
  "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1163
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1164
definition
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1165
  "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1166
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1167
instance by intro_classes
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1168
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1169
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1170
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1171
end
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1172
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1173
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1174
subsection {* Natural operation of natural numbers on functions *}
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1175
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1176
text {*
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1177
  We use the same logical constant for the power operations on
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1178
  functions and relations, in order to share the same syntax.
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1179
*}
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1180
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1181
consts compow :: "nat \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1182
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1183
abbreviation compower :: "('a \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'b" (infixr "^^" 80) where
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1184
  "f ^^ n \<equiv> compow n f"
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1185
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1186
notation (latex output)
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1187
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1188
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1189
notation (HTML output)
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1190
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1191
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1192
text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1193
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1194
overloading
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1195
  funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1196
begin
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1197
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1198
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1199
    "funpow 0 f = id"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1200
  | "funpow (Suc n) f = f o funpow n f"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1201
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1202
end
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1203
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1204
text {* for code generation *}
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1205
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1206
definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
31998
2c7a24f74db9 code attributes use common underscore convention
haftmann
parents: 31714
diff changeset
  1207
  funpow_code_def [code_post]: "funpow = compow"
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1208
31998
2c7a24f74db9 code attributes use common underscore convention
haftmann
parents: 31714
diff changeset
  1209
lemmas [code_unfold] = funpow_code_def [symmetric]
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1210
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1211
lemma [code]:
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1212
  "funpow 0 f = id"
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1213
  "funpow (Suc n) f = f o funpow n f"
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1214
  unfolding funpow_code_def by simp_all
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1215
30975
b2fa60d56735 some jokes are just too bad to appear in a theory file
haftmann
parents: 30971
diff changeset
  1216
hide (open) const funpow
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1217
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1218
lemma funpow_add:
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1219
  "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1220
  by (induct m) simp_all
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1221
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1222
lemma funpow_swap1:
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1223
  "f ((f ^^ n) x) = (f ^^ n) (f x)"
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1224
proof -
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1225
  have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1226
  also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1227
  also have "\<dots> = (f ^^ n) (f x)" by simp
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1228
  finally show ?thesis .
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1229
qed
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1230
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1231
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1232
subsection {* Embedding of the Naturals into any
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1233
  @{text semiring_1}: @{term of_nat} *}
24196
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1234
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1235
context semiring_1
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1236
begin
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1237
25559
f14305fb698c authentic primrec
haftmann
parents: 25534
diff changeset
  1238
primrec
f14305fb698c authentic primrec
haftmann
parents: 25534
diff changeset
  1239
  of_nat :: "nat \<Rightarrow> 'a"
f14305fb698c authentic primrec
haftmann
parents: 25534
diff changeset
  1240
where
f14305fb698c authentic primrec
haftmann
parents: 25534
diff changeset
  1241
  of_nat_0:     "of_nat 0 = 0"
f14305fb698c authentic primrec
haftmann
parents: 25534
diff changeset
  1242
  | of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1243
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1244
lemma of_nat_1 [simp]: "of_nat 1 = 1"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
  1245
  unfolding One_nat_def by simp
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1246
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1247
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1248
  by (induct m) (simp_all add: add_ac)
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1249
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1250
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1251
  by (induct m) (simp_all add: add_ac left_distrib)
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1252
28514
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1253
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1254
  "of_nat_aux inc 0 i = i"
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1255
  | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
25928
042e877d9841 tuned code setup
haftmann
parents: 25690
diff changeset
  1256
30966
55104c664185 avoid local [code]
haftmann
parents: 30954
diff changeset
  1257
lemma of_nat_code:
28514
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1258
  "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1259
proof (induct n)
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1260
  case 0 then show ?case by simp
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1261
next
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1262
  case (Suc n)
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1263
  have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1264
    by (induct n) simp_all
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1265
  from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1266
    by simp
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1267
  with Suc show ?case by (simp add: add_commute)
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
  1268
qed
30966
55104c664185 avoid local [code]
haftmann
parents: 30954
diff changeset
  1269
24196
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1270
end
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1271
31998
2c7a24f74db9 code attributes use common underscore convention
haftmann
parents: 31714
diff changeset
  1272
declare of_nat_code [code, code_unfold, code_inline del]
30966
55104c664185 avoid local [code]
haftmann
parents: 30954
diff changeset
  1273
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1274
text{*Class for unital semirings with characteristic zero.
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1275
 Includes non-ordered rings like the complex numbers.*}
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1276
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1277
class semiring_char_0 = semiring_1 +
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1278
  assumes of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1279
begin
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1280
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1281
text{*Special cases where either operand is zero*}
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1282
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1283
lemma of_nat_0_eq_iff [simp, noatp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
31024
0fdf666e08bf reimplement reorientation simproc using theory data
huffman
parents: 30975
diff changeset
  1284
  by (rule of_nat_eq_iff [of 0 n, unfolded of_nat_0])
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1285
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1286
lemma of_nat_eq_0_iff [simp, noatp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
31024
0fdf666e08bf reimplement reorientation simproc using theory data
huffman
parents: 30975
diff changeset
  1287
  by (rule of_nat_eq_iff [of m 0, unfolded of_nat_0])
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1288
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1289
lemma inj_of_nat: "inj of_nat"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1290
  by (simp add: inj_on_def)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1291
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1292
end
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1293
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34208
diff changeset
  1294
context linordered_semidom
25193
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1295
begin
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1296
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1297
lemma zero_le_imp_of_nat: "0 \<le> of_nat m"
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1298
  apply (induct m, simp_all)
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1299
  apply (erule order_trans)
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1300
  apply (rule ord_le_eq_trans [OF _ add_commute])
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1301
  apply (rule less_add_one [THEN less_imp_le])
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1302
  done
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1303
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1304
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1305
  apply (induct m n rule: diff_induct, simp_all)
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1306
  apply (insert add_less_le_mono [OF zero_less_one zero_le_imp_of_nat], force)
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1307
  done
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1308
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1309
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1310
  apply (induct m n rule: diff_induct, simp_all)
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1311
  apply (insert zero_le_imp_of_nat)
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1312
  apply (force simp add: not_less [symmetric])
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1313
  done
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1314
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1315
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1316
  by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
e2e1a4b00de3 various localizations
haftmann
parents: 25162
diff changeset
  1317
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1318
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"