src/HOL/Nat.thy
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Thu, 10 May 2007 10:21:50 +0200
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size [nat] is identity
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(*  Title:      HOL/Nat.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
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Type "nat" is a linear order, and a datatype; arithmetic operators + -
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and * (for div, mod and dvd, see theory Divides).
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*)
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header {* Natural numbers *}
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theory Nat
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imports Wellfounded_Recursion Ring_and_Field
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uses ("arith_data.ML")
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begin
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subsection {* Type @{text ind} *}
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typedecl ind
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axiomatization
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  Zero_Rep :: ind and
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  Suc_Rep :: "ind => ind"
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where
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  -- {* the axiom of infinity in 2 parts *}
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  inj_Suc_Rep:          "inj Suc_Rep" and
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  Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
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subsection {* Type nat *}
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text {* Type definition *}
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inductive2 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 ==> Nat (Suc_Rep i)"
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global
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typedef (open Nat)
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  nat = "Collect Nat"
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proof
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  from Nat.Zero_RepI
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  show "Zero_Rep : Collect Nat" ..
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qed
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text {* Abstract constants and syntax *}
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consts
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  Suc :: "nat => nat"
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local
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defs
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  Suc_def:      "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"
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definition
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  pred_nat :: "(nat * nat) set" where
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  "pred_nat = {(m, n). n = Suc m}"
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instance nat :: "{ord, zero, one}"
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  Zero_nat_def: "0 == Abs_Nat Zero_Rep"
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  One_nat_def [simp]: "1 == Suc 0"
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  less_def: "m < n == (m, n) : pred_nat^+"
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  le_def:   "m \<le> (n::nat) == ~ (n < m)" ..
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lemmas [code func del] = less_def le_def
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text {* Induction *}
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lemma Rep_Nat': "Nat (Rep_Nat x)"
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  by (rule Rep_Nat [simplified mem_Collect_eq])
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lemma Abs_Nat_inverse': "Nat y \<Longrightarrow> Rep_Nat (Abs_Nat y) = y"
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  by (rule Abs_Nat_inverse [simplified mem_Collect_eq])
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theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n"
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  apply (unfold Zero_nat_def Suc_def)
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  apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}
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  apply (erule Rep_Nat' [THEN Nat.induct])
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  apply (iprover elim: Abs_Nat_inverse' [THEN subst])
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  done
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text {* Distinctness of constructors *}
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lemma Suc_not_Zero [iff]: "Suc m \<noteq> 0"
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  by (simp add: Zero_nat_def Suc_def Abs_Nat_inject Rep_Nat' Suc_RepI Zero_RepI
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                Suc_Rep_not_Zero_Rep)
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lemma Zero_not_Suc [iff]: "0 \<noteq> Suc m"
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  by (rule not_sym, rule Suc_not_Zero not_sym)
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lemma Suc_neq_Zero: "Suc m = 0 ==> R"
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  by (rule notE, rule Suc_not_Zero)
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lemma Zero_neq_Suc: "0 = Suc m ==> R"
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  by (rule Suc_neq_Zero, erule sym)
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text {* Injectiveness of @{term Suc} *}
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lemma inj_Suc[simp]: "inj_on Suc N"
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  by (simp add: Suc_def inj_on_def Abs_Nat_inject Rep_Nat' Suc_RepI
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                inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject)
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lemma Suc_inject: "Suc x = Suc y ==> x = y"
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  by (rule inj_Suc [THEN injD])
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lemma Suc_Suc_eq [iff]: "(Suc m = Suc n) = (m = n)"
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  by (rule inj_Suc [THEN inj_eq])
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lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False"
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  by auto
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text {* size of a datatype value *}
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class size = type +
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  fixes size :: "'a \<Rightarrow> nat"
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text {* @{typ nat} is a datatype *}
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rep_datatype nat
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  distinct  Suc_not_Zero Zero_not_Suc
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  inject    Suc_Suc_eq
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  induction nat_induct
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declare nat.induct [case_names 0 Suc, induct type: nat]
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declare nat.exhaust [case_names 0 Suc, cases type: nat]
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lemmas nat_rec_0 = nat.recs(1)
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  and nat_rec_Suc = nat.recs(2)
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lemmas nat_case_0 = nat.cases(1)
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  and nat_case_Suc = nat.cases(2)
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lemma n_not_Suc_n: "n \<noteq> Suc n"
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  by (induct n) simp_all
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lemma Suc_n_not_n: "Suc t \<noteq> t"
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  by (rule not_sym, rule n_not_Suc_n)
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text {* A special form of induction for reasoning
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  about @{term "m < n"} and @{term "m - n"} *}
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theorem diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
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    (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
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  apply (rule_tac x = m in spec)
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  apply (induct n)
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  prefer 2
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  apply (rule allI)
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  apply (induct_tac x, iprover+)
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  done
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subsection {* Basic properties of "less than" *}
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lemma wf_pred_nat: "wf pred_nat"
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  apply (unfold wf_def pred_nat_def, clarify)
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  apply (induct_tac x, blast+)
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  done
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lemma wf_less: "wf {(x, y::nat). x < y}"
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  apply (unfold less_def)
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  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset], blast)
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  done
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lemma less_eq: "((m, n) : pred_nat^+) = (m < n)"
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parents: 12338
diff changeset
   167
  apply (unfold less_def)
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parents: 12338
diff changeset
   168
  apply (rule refl)
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parents: 12338
diff changeset
   169
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   170
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parents: 12338
diff changeset
   171
subsubsection {* Introduction properties *}
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parents: 12338
diff changeset
   172
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parents: 12338
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   173
lemma less_trans: "i < j ==> j < k ==> i < (k::nat)"
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parents: 12338
diff changeset
   174
  apply (unfold less_def)
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144f45277d5a misc tidying
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parents: 14193
diff changeset
   175
  apply (rule trans_trancl [THEN transD], assumption+)
13449
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   176
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   177
43c9ec498291 - Converted to new theory format
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parents: 12338
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   178
lemma lessI [iff]: "n < Suc n"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   179
  apply (unfold less_def pred_nat_def)
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parents: 12338
diff changeset
   180
  apply (simp add: r_into_trancl)
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parents: 12338
diff changeset
   181
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   182
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   183
lemma less_SucI: "i < j ==> i < Suc j"
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parents: 14193
diff changeset
   184
  apply (rule less_trans, assumption)
13449
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parents: 12338
diff changeset
   185
  apply (rule lessI)
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   186
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   187
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   188
lemma zero_less_Suc [iff]: "0 < Suc n"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   189
  apply (induct n)
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parents: 12338
diff changeset
   190
  apply (rule lessI)
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parents: 12338
diff changeset
   191
  apply (erule less_trans)
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parents: 12338
diff changeset
   192
  apply (rule lessI)
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parents: 12338
diff changeset
   193
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   194
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diff changeset
   195
subsubsection {* Elimination properties *}
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parents: 12338
diff changeset
   196
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parents: 12338
diff changeset
   197
lemma less_not_sym: "n < m ==> ~ m < (n::nat)"
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parents: 12338
diff changeset
   198
  apply (unfold less_def)
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parents: 12338
diff changeset
   199
  apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym])
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parents: 12338
diff changeset
   200
  done
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parents: 12338
diff changeset
   201
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   202
lemma less_asym:
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parents: 12338
diff changeset
   203
  assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P
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parents: 12338
diff changeset
   204
  apply (rule contrapos_np)
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parents: 12338
diff changeset
   205
  apply (rule less_not_sym)
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parents: 12338
diff changeset
   206
  apply (rule h1)
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parents: 12338
diff changeset
   207
  apply (erule h2)
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parents: 12338
diff changeset
   208
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   209
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   210
lemma less_not_refl: "~ n < (n::nat)"
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parents: 12338
diff changeset
   211
  apply (unfold less_def)
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parents: 12338
diff changeset
   212
  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl])
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berghofe
parents: 12338
diff changeset
   213
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   214
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   215
lemma less_irrefl [elim!]: "(n::nat) < n ==> R"
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parents: 12338
diff changeset
   216
  by (rule notE, rule less_not_refl)
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parents: 12338
diff changeset
   217
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   218
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)" by blast
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   219
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   220
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   221
  by (rule not_sym, rule less_not_refl2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   222
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   223
lemma lessE:
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parents: 12338
diff changeset
   224
  assumes major: "i < k"
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parents: 12338
diff changeset
   225
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   226
  shows P
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   227
  apply (rule major [unfolded less_def pred_nat_def, THEN tranclE], simp_all)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   228
  apply (erule p1)
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parents: 12338
diff changeset
   229
  apply (rule p2)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   230
  apply (simp add: less_def pred_nat_def, assumption)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   231
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   232
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   233
lemma not_less0 [iff]: "~ n < (0::nat)"
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parents: 12338
diff changeset
   234
  by (blast elim: lessE)
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parents: 12338
diff changeset
   235
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   236
lemma less_zeroE: "(n::nat) < 0 ==> R"
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parents: 12338
diff changeset
   237
  by (rule notE, rule not_less0)
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   238
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   239
lemma less_SucE: assumes major: "m < Suc n"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   240
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   241
  apply (rule major [THEN lessE])
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   242
  apply (rule eq, blast)
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   243
  apply (rule less, blast)
13449
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   244
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   245
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   246
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
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parents: 12338
diff changeset
   247
  by (blast elim!: less_SucE intro: less_trans)
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parents: 12338
diff changeset
   248
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   249
lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   250
  by (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   251
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   252
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   253
  by (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   254
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   255
lemma Suc_mono: "m < n ==> Suc m < Suc n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   256
  by (induct n) (fast elim: less_trans lessE)+
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   257
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   258
text {* "Less than" is a linear ordering *}
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   259
lemma less_linear: "m < n | m = n | n < (m::nat)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   260
  apply (induct m)
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   261
  apply (induct n)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   262
  apply (rule refl [THEN disjI1, THEN disjI2])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   263
  apply (rule zero_less_Suc [THEN disjI1])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   264
  apply (blast intro: Suc_mono less_SucI elim: lessE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   265
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   266
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   267
text {* "Less than" is antisymmetric, sort of *}
6c24235e8d5d *** empty log message ***
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parents: 14267
diff changeset
   268
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   269
  apply(simp only:less_Suc_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   270
  apply blast
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   271
  done
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   272
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   273
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   274
  using less_linear by blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   275
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   276
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   277
  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
   278
  shows "P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   279
  apply (rule less_linear [THEN disjE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   280
  apply (erule_tac [2] disjE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   281
  apply (erule lessCase)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   282
  apply (erule sym [THEN eqCase])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   283
  apply (erule major)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   284
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   285
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   286
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   287
subsubsection {* Inductive (?) properties *}
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   288
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   289
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   290
  apply (simp add: nat_neq_iff)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   291
  apply (blast elim!: less_irrefl less_SucE elim: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   292
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   293
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   294
lemma Suc_lessD: "Suc m < n ==> m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   295
  apply (induct n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   296
  apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   297
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   298
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   299
lemma Suc_lessE: assumes major: "Suc i < k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   300
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   301
  apply (rule major [THEN lessE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   302
  apply (erule lessI [THEN minor])
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   303
  apply (erule Suc_lessD [THEN minor], assumption)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   304
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   305
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   306
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   307
  by (blast elim: lessE dest: Suc_lessD)
4104
84433b1ab826 nat datatype_info moved to Nat.thy;
wenzelm
parents: 3370
diff changeset
   308
16635
bf7de5723c60 Moved some code lemmas from Main to Nat.
berghofe
parents: 15921
diff changeset
   309
lemma Suc_less_eq [iff, code]: "(Suc m < Suc n) = (m < n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   310
  apply (rule iffI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   311
  apply (erule Suc_less_SucD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   312
  apply (erule Suc_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   313
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   314
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   315
lemma less_trans_Suc:
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   316
  assumes le: "i < j" shows "j < k ==> Suc i < k"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   317
  apply (induct k, simp_all)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   318
  apply (insert le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   319
  apply (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   320
  apply (blast dest: Suc_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   321
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   322
16635
bf7de5723c60 Moved some code lemmas from Main to Nat.
berghofe
parents: 15921
diff changeset
   323
lemma [code]: "((n::nat) < 0) = False" by simp
bf7de5723c60 Moved some code lemmas from Main to Nat.
berghofe
parents: 15921
diff changeset
   324
lemma [code]: "(0 < Suc n) = True" by simp
bf7de5723c60 Moved some code lemmas from Main to Nat.
berghofe
parents: 15921
diff changeset
   325
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   326
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   327
lemma not_less_eq: "(~ m < n) = (n < Suc m)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   328
  by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   329
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   330
text {* Complete induction, aka course-of-values induction *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   331
lemma nat_less_induct:
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   332
  assumes prem: "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   333
  apply (induct n rule: wf_induct [OF wf_pred_nat [THEN wf_trancl]])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   334
  apply (rule prem)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   335
  apply (unfold less_def, assumption)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   336
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   337
14131
a4fc8b1af5e7 declarations moved from PreList.thy
paulson
parents: 13596
diff changeset
   338
lemmas less_induct = nat_less_induct [rule_format, case_names less]
a4fc8b1af5e7 declarations moved from PreList.thy
paulson
parents: 13596
diff changeset
   339
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
   340
14131
a4fc8b1af5e7 declarations moved from PreList.thy
paulson
parents: 13596
diff changeset
   341
subsection {* Properties of "less than or equal" *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   342
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   343
text {* Was @{text le_eq_less_Suc}, but this orientation is more useful *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   344
lemma less_Suc_eq_le: "(m < Suc n) = (m \<le> n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   345
  unfolding le_def by (rule not_less_eq [symmetric])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   346
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   347
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   348
  by (rule less_Suc_eq_le [THEN iffD2])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   349
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   350
lemma le0 [iff]: "(0::nat) \<le> n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   351
  unfolding le_def by (rule not_less0)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   352
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   353
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   354
  by (simp add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   355
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   356
lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   357
  by (induct i) (simp_all add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   358
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   359
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   360
  by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   361
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   362
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 16796
diff changeset
   363
  by (drule le_Suc_eq [THEN iffD1], iprover+)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   364
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   365
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   366
  apply (simp add: le_def less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   367
  apply (blast elim!: less_irrefl less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   368
  done -- {* formerly called lessD *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   369
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   370
lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   371
  by (simp add: le_def less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   372
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   373
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
   374
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   375
  apply (simp add: le_def less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   376
  using less_linear
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   377
  apply blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   378
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   379
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   380
lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   381
  by (blast intro: Suc_leI Suc_le_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   382
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   383
lemma le_SucI: "m \<le> n ==> m \<le> Suc n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   384
  by (unfold le_def) (blast dest: Suc_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   385
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   386
lemma less_imp_le: "m < n ==> m \<le> (n::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   387
  by (unfold le_def) (blast elim: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   388
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   389
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   390
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   391
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   392
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   393
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
   394
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   395
lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   396
  unfolding le_def
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   397
  using less_linear
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   398
  by (blast elim: less_irrefl less_asym)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   399
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   400
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   401
  unfolding le_def
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   402
  using less_linear
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   403
  by (blast elim!: less_irrefl elim: less_asym)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   404
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   405
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 16796
diff changeset
   406
  by (iprover intro: less_or_eq_imp_le le_imp_less_or_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   407
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   408
text {* Useful with @{text blast}. *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   409
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   410
  by (rule less_or_eq_imp_le) (rule disjI2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   411
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   412
lemma le_refl: "n \<le> (n::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   413
  by (simp add: le_eq_less_or_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   414
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   415
lemma le_less_trans: "[| i \<le> j; j < k |] ==> i < (k::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   416
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   417
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   418
lemma less_le_trans: "[| i < j; j \<le> k |] ==> i < (k::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   419
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   420
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   421
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   422
  by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   423
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   424
lemma le_anti_sym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   425
  by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   426
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   427
lemma Suc_le_mono [iff]: "(Suc n \<le> Suc m) = (n \<le> m)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   428
  by (simp add: le_simps)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   429
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   430
text {* Axiom @{text order_less_le} of class @{text order}: *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   431
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   432
  by (simp add: le_def nat_neq_iff) (blast elim!: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   433
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   434
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   435
  by (rule iffD2, rule nat_less_le, rule conjI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   436
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   437
text {* Axiom @{text linorder_linear} of class @{text linorder}: *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   438
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   439
  apply (simp add: le_eq_less_or_eq)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   440
  using less_linear by blast
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   441
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   442
text {* Type {@typ nat} is a wellfounded linear order *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   443
22318
6efe70ab7add simpliefied instance statement
haftmann
parents: 22262
diff changeset
   444
instance nat :: wellorder
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14348
diff changeset
   445
  by intro_classes
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14348
diff changeset
   446
    (assumption |
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14348
diff changeset
   447
      rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   448
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   449
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
15921
b6e345548913 Fixing a problem with lin.arith.
nipkow
parents: 15539
diff changeset
   450
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   451
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   452
  by (blast elim!: less_SucE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   453
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   454
text {*
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   455
  Rewrite @{term "n < Suc m"} to @{term "n = m"}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   456
  if @{term "~ n < m"} or @{term "m \<le> n"} hold.
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   457
  Not suitable as default simprules because they often lead to looping
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   458
*}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   459
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   460
  by (rule not_less_less_Suc_eq, rule leD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   461
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   462
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   463
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   464
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   465
text {*
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   466
  Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}.
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   467
  No longer added as simprules (they loop)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   468
  but via @{text reorient_simproc} in Bin
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   469
*}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   470
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   471
text {* Polymorphic, not just for @{typ nat} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   472
lemma zero_reorient: "(0 = x) = (x = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   473
  by auto
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 one_reorient: "(1 = x) = (x = 1)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   476
  by auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   477
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
   478
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   479
subsection {* Arithmetic operators *}
1660
8cb42cd97579 *** empty log message ***
oheimb
parents: 1625
diff changeset
   480
22473
753123c89d72 explizit "type" superclass
haftmann
parents: 22348
diff changeset
   481
class power = type +
21411
a9671d4f7c03 power is now a class
haftmann
parents: 21252
diff changeset
   482
  fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"            (infixr "\<^loc>^" 80)
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   483
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   484
text {* arithmetic operators @{text "+ -"} and @{text "*"} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   485
21456
1c2b9df41e98 cleanup
haftmann
parents: 21411
diff changeset
   486
instance nat :: "{plus, minus, times}" ..
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   487
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   488
primrec
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   489
  add_0:    "0 + n = n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   490
  add_Suc:  "Suc m + n = Suc (m + n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   491
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   492
primrec
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   493
  diff_0:   "m - 0 = m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   494
  diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   495
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   496
primrec
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   497
  mult_0:   "0 * n = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   498
  mult_Suc: "Suc m * n = n + (m * n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   499
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   500
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
   501
NOTE USE OF @{text "=="} *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   502
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   503
  by simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   504
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   505
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   506
  by simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   507
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   508
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   509
  by (cases n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   510
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   511
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   512
  by (cases n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   513
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   514
lemma neq0_conv [iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   515
  by (cases n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   516
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   517
text {* This theorem is useful with @{text blast} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   518
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 16796
diff changeset
   519
  by (rule iffD1, rule neq0_conv, iprover)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   520
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   521
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   522
  by (fast intro: not0_implies_Suc)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   523
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   524
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   525
  apply (rule iffI)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   526
  apply (rule ccontr)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   527
  apply simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   528
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   529
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   530
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   531
  by (induct m') simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   532
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   533
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
   534
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   535
  by (cases m) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   536
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   537
lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   538
  apply (rule nat_less_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   539
  apply (case_tac n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   540
  apply (case_tac [2] nat)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   541
  apply (blast intro: less_trans)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   542
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   543
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
   544
15341
254f6f00b60e converted to Isar script, simplifying some results
paulson
parents: 15281
diff changeset
   545
subsection {* @{text LEAST} theorems for type @{typ nat}*}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   546
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   547
lemma Least_Suc:
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   548
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   549
  apply (case_tac "n", auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   550
  apply (frule LeastI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   551
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   552
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   553
  apply (erule_tac [2] Least_le)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   554
  apply (case_tac "LEAST x. P x", auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   555
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   556
  apply (blast intro: order_antisym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   557
  done
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 Least_Suc2:
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   560
     "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   561
  by (erule (1) Least_Suc [THEN ssubst], simp)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   562
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   563
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   564
subsection {* @{term min} and @{term max} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   565
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   566
lemma min_0L [simp]: "min 0 n = (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   567
  by (rule min_leastL) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   568
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   569
lemma min_0R [simp]: "min n 0 = (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   570
  by (rule min_leastR) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   571
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   572
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   573
  by (simp add: min_of_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   574
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   575
lemma min_Suc1:
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   576
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   577
  by (simp split: nat.split)
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   578
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   579
lemma min_Suc2:
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   580
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   581
  by (simp split: nat.split)
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   582
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   583
lemma max_0L [simp]: "max 0 n = (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   584
  by (rule max_leastL) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   585
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   586
lemma max_0R [simp]: "max n 0 = (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   587
  by (rule max_leastR) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   588
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   589
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   590
  by (simp add: max_of_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   591
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   592
lemma max_Suc1:
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   593
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   594
  by (simp split: nat.split)
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   595
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   596
lemma max_Suc2:
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   597
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   598
  by (simp split: nat.split)
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   599
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   600
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   601
subsection {* Basic rewrite rules for the arithmetic operators *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   602
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   603
text {* Difference *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   604
14193
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   605
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   606
  by (induct n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   607
14193
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   608
lemma diff_Suc_Suc [simp, code]: "Suc(m) - Suc(n) = m - n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   609
  by (induct n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   610
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   611
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   612
text {*
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   613
  Could be (and is, below) generalized in various ways
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   614
  However, none of the generalizations are currently in the simpset,
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   615
  and I dread to think what happens if I put them in
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   616
*}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   617
lemma Suc_pred [simp]: "0 < n ==> Suc (n - Suc 0) = n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   618
  by (simp split add: nat.split)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   619
14193
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   620
declare diff_Suc [simp del, code del]
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   621
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   622
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   623
subsection {* Addition *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   624
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   625
lemma add_0_right [simp]: "m + 0 = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   626
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   627
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   628
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   629
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   630
19890
1aad48bcc674 slight adaption for code generator
haftmann
parents: 19870
diff changeset
   631
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
1aad48bcc674 slight adaption for code generator
haftmann
parents: 19870
diff changeset
   632
  by simp
14193
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   633
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   634
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   635
text {* Associative law for addition *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   636
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   637
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   638
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   639
text {* Commutative law for addition *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   640
lemma nat_add_commute: "m + n = n + (m::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   641
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   642
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   643
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   644
  apply (rule mk_left_commute [of "op +"])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   645
  apply (rule nat_add_assoc)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   646
  apply (rule nat_add_commute)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   647
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   648
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   649
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   650
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   651
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   652
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   653
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   654
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   655
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   656
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   657
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   658
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   659
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   660
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   661
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   662
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   663
lemma add_is_0 [iff]: fixes m :: nat shows "(m + n = 0) = (m = 0 & n = 0)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   664
  by (cases m) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   665
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   666
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   667
  by (cases m) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   668
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   669
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   670
  by (rule trans, rule eq_commute, rule add_is_1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   671
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   672
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m | 0 < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   673
  by (simp del: neq0_conv add: neq0_conv [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   674
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   675
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   676
  apply (drule add_0_right [THEN ssubst])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   677
  apply (simp add: nat_add_assoc del: add_0_right)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   678
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   679
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16635
diff changeset
   680
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   681
  apply (induct k)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   682
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   683
  apply(drule comp_inj_on[OF _ inj_Suc])
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   684
  apply (simp add:o_def)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   685
  done
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16635
diff changeset
   686
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16635
diff changeset
   687
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   688
subsection {* Multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   689
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   690
text {* right annihilation in product *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   691
lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   692
  by (induct m) simp_all
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   693
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   694
text {* right successor law for multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   695
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   696
  by (induct m) (simp_all add: nat_add_left_commute)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   697
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   698
text {* Commutative law for multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   699
lemma nat_mult_commute: "m * n = n * (m::nat)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   700
  by (induct m) simp_all
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   701
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   702
text {* addition distributes over multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   703
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   704
  by (induct m) (simp_all add: nat_add_assoc nat_add_left_commute)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   705
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   706
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   707
  by (induct m) (simp_all add: nat_add_assoc)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   708
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   709
text {* Associative law for multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   710
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   711
  by (induct m) (simp_all add: add_mult_distrib)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   712
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   713
14740
c8e1937110c2 fixed latex problems
nipkow
parents: 14738
diff changeset
   714
text{*The naturals form a @{text comm_semiring_1_cancel}*}
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   715
instance nat :: comm_semiring_1_cancel
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   716
proof
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   717
  fix i j k :: nat
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   718
  show "(i + j) + k = i + (j + k)" by (rule nat_add_assoc)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   719
  show "i + j = j + i" by (rule nat_add_commute)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   720
  show "0 + i = i" by simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   721
  show "(i * j) * k = i * (j * k)" by (rule nat_mult_assoc)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   722
  show "i * j = j * i" by (rule nat_mult_commute)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   723
  show "1 * i = i" by simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   724
  show "(i + j) * k = i * k + j * k" by (simp add: add_mult_distrib)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   725
  show "0 \<noteq> (1::nat)" by simp
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   726
  assume "k+i = k+j" thus "i=j" by simp
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   727
qed
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   728
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   729
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   730
  apply (induct m)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   731
   apply (induct_tac [2] n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   732
    apply simp_all
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   733
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   734
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
   735
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   736
subsection {* Monotonicity of Addition *}
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 *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   739
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   740
  by (induct k) simp_all
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   741
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   742
text {* strict, in both arguments *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   743
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
   744
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   745
  apply (induct j, simp_all)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   746
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   747
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   748
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
   749
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
   750
  apply (induct n)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   751
  apply (simp_all add: order_le_less)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   752
  apply (blast elim!: less_SucE
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   753
               intro!: add_0_right [symmetric] add_Suc_right [symmetric])
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   754
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   755
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   756
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   757
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   758
  apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   759
  apply (induct_tac x)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   760
  apply (simp_all add: add_less_mono)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   761
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   762
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   763
14740
c8e1937110c2 fixed latex problems
nipkow
parents: 14738
diff changeset
   764
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   765
instance nat :: ordered_semidom
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   766
proof
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   767
  fix i j k :: nat
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   768
  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
   769
  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
   770
  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
   771
qed
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   772
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   773
lemma nat_mult_1: "(1::nat) * n = n"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   774
  by simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   775
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   776
lemma nat_mult_1_right: "n * (1::nat) = n"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   777
  by simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   778
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   779
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   780
subsection {* Additional theorems about "less than" *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   781
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   782
text{*An induction rule for estabilishing binary relations*}
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   783
lemma less_Suc_induct:
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   784
  assumes less:  "i < j"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   785
     and  step:  "!!i. P i (Suc i)"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   786
     and  trans: "!!i j k. P i j ==> P j k ==> P i k"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   787
  shows "P i j"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   788
proof -
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   789
  from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   790
  have "P i (Suc (i + k))"
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   791
  proof (induct k)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   792
    case 0
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   793
    show ?case by (simp add: step)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   794
  next
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   795
    case (Suc k)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   796
    thus ?case by (auto intro: assms)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   797
  qed
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   798
  thus "P i j" by (simp add: j)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   799
qed
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   800
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   801
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   802
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
   803
  monotonicity to @{text "\<le>"} monotonicity *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   804
lemma less_mono_imp_le_mono:
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   805
  assumes lt_mono: "!!i j::nat. i < j ==> f i < f j"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   806
    and le: "i \<le> j"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   807
  shows "f i \<le> ((f j)::nat)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   808
  using le
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   809
  apply (simp add: order_le_less)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   810
  apply (blast intro!: lt_mono)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   811
  done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   812
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   813
text {* non-strict, in 1st argument *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   814
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   815
  by (rule add_right_mono)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   816
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   817
text {* non-strict, in both arguments *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   818
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   819
  by (rule add_mono)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   820
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   821
lemma le_add2: "n \<le> ((m + n)::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   822
  by (insert add_right_mono [of 0 m n], simp)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   823
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   824
lemma le_add1: "n \<le> ((n + m)::nat)"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   825
  by (simp add: add_commute, rule le_add2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   826
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   827
lemma less_add_Suc1: "i < Suc (i + m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   828
  by (rule le_less_trans, rule le_add1, rule lessI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   829
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   830
lemma less_add_Suc2: "i < Suc (m + i)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   831
  by (rule le_less_trans, rule le_add2, rule lessI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   832
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   833
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 16796
diff changeset
   834
  by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   835
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   836
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   837
  by (rule le_trans, assumption, rule le_add1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   838
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   839
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   840
  by (rule le_trans, assumption, rule le_add2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   841
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   842
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   843
  by (rule less_le_trans, assumption, rule le_add1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   844
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   845
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   846
  by (rule less_le_trans, assumption, rule le_add2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   847
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   848
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   849
  apply (rule le_less_trans [of _ "i+j"])
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   850
  apply (simp_all add: le_add1)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   851
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   852
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   853
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   854
  apply (rule notI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   855
  apply (erule add_lessD1 [THEN less_irrefl])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   856
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   857
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   858
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   859
  by (simp add: add_commute not_add_less1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   860
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   861
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   862
  apply (rule order_trans [of _ "m+k"])
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   863
  apply (simp_all add: le_add1)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   864
  done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   865
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   866
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   867
  apply (simp add: add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   868
  apply (erule add_leD1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   869
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   870
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   871
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   872
  by (blast dest: add_leD1 add_leD2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   873
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   874
text {* needs @{text "!!k"} for @{text add_ac} to work *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   875
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   876
  by (force simp del: add_Suc_right
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   877
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   878
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   879
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   880
subsection {* Difference *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   881
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   882
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   883
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   884
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   885
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
   886
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   887
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   888
  by (induct m n rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   889
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   890
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
   891
  by (simp add: add_diff_inverse linorder_not_less)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   892
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   893
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   894
  by (simp add: le_add_diff_inverse add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   895
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   896
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   897
subsection {* More results about difference *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   898
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   899
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   900
  by (induct m n rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   901
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   902
lemma diff_less_Suc: "m - n < Suc m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   903
  apply (induct m n rule: diff_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   904
  apply (erule_tac [3] less_SucE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   905
  apply (simp_all add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   906
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   907
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   908
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   909
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   910
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   911
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   912
  by (rule le_less_trans, rule diff_le_self)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   913
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   914
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   915
  by (induct i j rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   916
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   917
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   918
  by (simp add: diff_diff_left)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   919
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   920
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   921
  by (cases n) (auto simp add: le_simps)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   922
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   923
text {* This and the next few suggested by Florian Kammueller *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   924
lemma diff_commute: "(i::nat) - j - k = i - k - j"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   925
  by (simp add: diff_diff_left add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   926
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   927
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   928
  by (induct j k rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   929
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   930
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   931
  by (simp add: add_commute diff_add_assoc)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   932
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   933
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   934
  by (induct n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   935
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   936
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   937
  by (simp add: diff_add_assoc)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   938
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   939
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   940
  by (auto simp add: diff_add_inverse2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   941
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   942
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   943
  by (induct m n rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   944
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   945
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   946
  by (rule iffD2, rule diff_is_0_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   947
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   948
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   949
  by (induct m n rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   950
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   951
lemma less_imp_add_positive:
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   952
  assumes "i < j"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   953
  shows "\<exists>k::nat. 0 < k & i + k = j"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   954
proof
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   955
  from assms show "0 < j - i & i + (j - i) = j"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   956
    by (simp add: add_diff_inverse less_not_sym)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   957
qed
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   958
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   959
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   960
  apply (induct k i rule: diff_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   961
  apply (simp_all (no_asm))
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 16796
diff changeset
   962
  apply iprover
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   963
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   964
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   965
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   966
  apply (rule diff_self_eq_0 [THEN subst])
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 16796
diff changeset
   967
  apply (rule zero_induct_lemma, iprover+)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   968
  done
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 diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   971
  by (induct k) simp_all
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 diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   974
  by (simp add: diff_cancel add_commute)
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 diff_add_0: "n - (n + m) = (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   977
  by (induct n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   978
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   979
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   980
text {* Difference distributes over multiplication *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   981
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   982
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   983
  by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   984
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   985
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   986
  by (simp add: diff_mult_distrib mult_commute [of k])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   987
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   988
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   989
lemmas nat_distrib =
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   990
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   991
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   992
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   993
subsection {* Monotonicity of Multiplication *}
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 mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   996
  by (simp add: mult_right_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   997
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   998
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   999
  by (simp add: mult_left_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1000
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1001
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1002
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1003
  by (simp add: mult_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1004
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1005
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1006
  by (simp add: mult_strict_right_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1007
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
  1008
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
  1009
      there are no negative numbers.*}
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
  1010
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
  1011
  apply (induct m)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1012
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1013
  apply (case_tac n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1014
   apply simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1015
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1016
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1017
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1018
  apply (induct m)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1019
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1020
  apply (case_tac n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1021
   apply simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1022
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1023
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1024
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1025
  apply (induct m)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1026
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1027
  apply (induct n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1028
   apply auto
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1029
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1030
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1031
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1032
  apply (rule trans)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
  1033
  apply (rule_tac [2] mult_eq_1_iff, fastsimp)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1034
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1035
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
  1036
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
  1037
  apply (safe intro!: mult_less_mono1)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
  1038
  apply (case_tac k, auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1039
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1040
  apply (blast intro: mult_le_mono1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1041
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1042
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1043
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
  1044
  by (simp add: mult_commute [of k])
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 mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1047
  by (simp add: linorder_not_less [symmetric], auto)
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 mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1050
  by (simp add: linorder_not_less [symmetric], auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1051
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
  1052
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
  1053
  apply (cut_tac less_linear, safe, auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1054
  apply (drule mult_less_mono1, assumption, simp)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1055
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1056
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1057
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
  1058
  by (simp add: mult_commute [of k])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1059
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1060
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1061
  by (subst mult_less_cancel1) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1062
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1063
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1064
  by (subst mult_le_cancel1) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1065
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1066
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1067
  by (subst mult_cancel1) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1068
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1069
text {* Lemma for @{text gcd} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1070
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1071
  apply (drule sym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1072
  apply (rule disjCI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1073
  apply (rule nat_less_cases, erule_tac [2] _)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1074
  apply (fastsimp elim!: less_SucE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1075
  apply (fastsimp dest: mult_less_mono2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1076
  done
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1077
20588
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1078
18702
7dc7dcd63224 substantial improvements in code generator
haftmann
parents: 18648
diff changeset
  1079
subsection {* Code generator setup *}
7dc7dcd63224 substantial improvements in code generator
haftmann
parents: 18648
diff changeset
  1080
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1081
lemma one_is_Suc_zero [code inline]: "1 = Suc 0"
20355
50aaae6ae4db cleanup code generation for Numerals
haftmann
parents: 19890
diff changeset
  1082
  by simp
50aaae6ae4db cleanup code generation for Numerals
haftmann
parents: 19890
diff changeset
  1083
20588
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1084
instance nat :: eq ..
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1085
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1086
lemma [code func]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1087
    "(0\<Colon>nat) = 0 \<longleftrightarrow> True"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1088
    "Suc n = Suc m \<longleftrightarrow> n = m"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1089
    "Suc n = 0 \<longleftrightarrow> False"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1090
    "0 = Suc m \<longleftrightarrow> False"
22348
ab505d281015 adjusted code lemmas
haftmann
parents: 22318
diff changeset
  1091
  by auto
20588
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1092
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1093
lemma [code func]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1094
    "(0\<Colon>nat) \<le> m \<longleftrightarrow> True"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1095
    "Suc (n\<Colon>nat) \<le> m \<longleftrightarrow> n < m"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1096
    "(n\<Colon>nat) < 0 \<longleftrightarrow> False"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1097
    "(n\<Colon>nat) < Suc m \<longleftrightarrow> n \<le> m"
22348
ab505d281015 adjusted code lemmas
haftmann
parents: 22318
diff changeset
  1098
  using Suc_le_eq less_Suc_eq_le by simp_all
20588
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1099
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1100
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1101
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1102
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
  1103
lemma subst_equals:
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
  1104
  assumes 1: "t = s" and 2: "u = t"
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
  1105
  shows "u = s"
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
  1106
  using 2 1 by (rule trans)
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
  1107
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1108
use "arith_data.ML"
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1109
setup arith_setup
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1110
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1111
text{*The following proofs may rely on the arithmetic proof procedures.*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1112
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1113
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1114
  by (auto simp: le_eq_less_or_eq dest: less_imp_Suc_add)
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1115
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1116
lemma pred_nat_trancl_eq_le: "((m, n) : pred_nat^*) = (m \<le> n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1117
  by (simp add: less_eq reflcl_trancl [symmetric] del: reflcl_trancl, arith)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1118
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1119
lemma nat_diff_split:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1120
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1121
    -- {* elimination of @{text -} on @{text nat} *}
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1122
  by (cases "a<b" rule: case_split) (auto simp add: diff_is_0_eq [THEN iffD2])
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1123
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1124
lemma nat_diff_split_asm:
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1125
    "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1126
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1127
  by (simp split: nat_diff_split)
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1128
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1129
lemmas [arith_split] = nat_diff_split split_min split_max
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1130
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1131
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1132
lemma le_square: "m \<le> m * (m::nat)"
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1133
  by (induct m) auto
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1134
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1135
lemma le_cube: "(m::nat) \<le> m * (m * m)"
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1136
  by (induct m) auto
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1137
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1138
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1139
text{*Subtraction laws, mostly by Clemens Ballarin*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1140
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1141
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1142
  by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1143
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1144
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1145
  by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1146
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1147
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1148
  by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1149
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1150
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1151
  by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1152
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1153
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1154
  by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1155
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1156
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1157
  by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1158
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1159
(*Replaces the previous diff_less and le_diff_less, which had the stronger
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1160
  second premise n\<le>m*)
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1161
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1162
  by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1163
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1164
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1165
(** Simplification of relational expressions involving subtraction **)
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1166
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1167
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1168
  by (simp split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1169
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1170
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1171
  by (auto split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1172
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1173
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1174
  by (auto split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1175
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1176
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1177
  by (auto split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1178
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1179
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1180
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1181
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1182
(* Monotonicity of subtraction in first argument *)
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1183
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1184
  by (simp split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1185
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1186
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1187
  by (simp split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1188
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1189
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1190
  by (simp split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1191
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1192
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1193
  by (simp split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1194
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1195
text{*Lemmas for ex/Factorization*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1196
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1197
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1198
  by (cases m) auto
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1199
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1200
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1201
  by (cases m) auto
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1202
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1203
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1204
  by (cases m) auto
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1205
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1206
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1207
text{*Rewriting to pull differences out*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1208
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1209
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1210
  by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1211
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1212
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1213
  by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1214
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1215
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1216
  by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1217
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1218
(*The others are
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1219
      i - j - k = i - (j + k),
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1220
      k \<le> j ==> j - k + i = j + i - k,
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1221
      k \<le> j ==> i + (j - k) = i + j - k *)
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1222
lemmas add_diff_assoc = diff_add_assoc [symmetric]
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1223
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1224
declare diff_diff_left [simp]  add_diff_assoc [simp]  add_diff_assoc2[simp]
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1225
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1226
text{*At present we prove no analogue of @{text not_less_Least} or @{text
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1227
Least_Suc}, since there appears to be no need.*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1228
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1229
ML
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1230
{*
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1231
val pred_nat_trancl_eq_le = thm "pred_nat_trancl_eq_le";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1232
val nat_diff_split = thm "nat_diff_split";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1233
val nat_diff_split_asm = thm "nat_diff_split_asm";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1234
val le_square = thm "le_square";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1235
val le_cube = thm "le_cube";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1236
val diff_less_mono = thm "diff_less_mono";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1237
val less_diff_conv = thm "less_diff_conv";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1238
val le_diff_conv = thm "le_diff_conv";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1239
val le_diff_conv2 = thm "le_diff_conv2";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1240
val diff_diff_cancel = thm "diff_diff_cancel";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1241
val le_add_diff = thm "le_add_diff";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1242
val diff_less = thm "diff_less";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1243
val diff_diff_eq = thm "diff_diff_eq";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1244
val eq_diff_iff = thm "eq_diff_iff";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1245
val less_diff_iff = thm "less_diff_iff";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1246
val le_diff_iff = thm "le_diff_iff";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1247
val diff_le_mono = thm "diff_le_mono";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1248
val diff_le_mono2 = thm "diff_le_mono2";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1249
val diff_less_mono2 = thm "diff_less_mono2";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1250
val diffs0_imp_equal = thm "diffs0_imp_equal";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1251
val one_less_mult = thm "one_less_mult";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1252
val n_less_m_mult_n = thm "n_less_m_mult_n";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1253
val n_less_n_mult_m = thm "n_less_n_mult_m";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1254
val diff_diff_right = thm "diff_diff_right";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1255
val diff_Suc_diff_eq1 = thm "diff_Suc_diff_eq1";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1256
val diff_Suc_diff_eq2 = thm "diff_Suc_diff_eq2";
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1257
*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1258
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1259
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1260
subsection{*Embedding of the Naturals into any
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1261
  @{text semiring_1_cancel}: @{term of_nat}*}
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1262
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1263
consts of_nat :: "nat => 'a::semiring_1_cancel"
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1264
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1265
primrec
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1266
  of_nat_0:   "of_nat 0 = 0"
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1267
  of_nat_Suc: "of_nat (Suc m) = of_nat m + 1"
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1268
22920
0dbcb73bf9bf size [nat] is identity
haftmann
parents: 22845
diff changeset
  1269
lemma of_nat_id [simp]: "(of_nat n \<Colon> nat) = n"
0dbcb73bf9bf size [nat] is identity
haftmann
parents: 22845
diff changeset
  1270
  by (induct n) auto
0dbcb73bf9bf size [nat] is identity
haftmann
parents: 22845
diff changeset
  1271
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1272
lemma of_nat_1 [simp]: "of_nat 1 = 1"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1273
  by simp
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1274
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1275
lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1276
  by (induct m) (simp_all add: add_ac)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1277
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1278
lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1279
  by (induct m) (simp_all add: add_ac left_distrib)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1280
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1281
lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1282
  apply (induct m, simp_all)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1283
  apply (erule order_trans)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1284
  apply (rule less_add_one [THEN order_less_imp_le])
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1285
  done
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1286
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1287
lemma less_imp_of_nat_less:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1288
    "m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1289
  apply (induct m n rule: diff_induct, simp_all)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1290
  apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1291
  done
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1292
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1293
lemma of_nat_less_imp_less:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1294
    "of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1295
  apply (induct m n rule: diff_induct, simp_all)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1296
  apply (insert zero_le_imp_of_nat)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1297
  apply (force simp add: linorder_not_less [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1298
  done
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1299
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1300
lemma of_nat_less_iff [simp]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1301
    "(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1302
  by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1303
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1304
text{*Special cases where either operand is zero*}
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1305
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1306
lemma of_nat_0_less_iff [simp]: "((0::'a::ordered_semidom) < of_nat n) = (0 < n)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1307
  by (rule of_nat_less_iff [of 0, simplified])
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1308
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1309
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < (0::'a::ordered_semidom)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1310
  by (rule of_nat_less_iff [of _ 0, simplified])
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1311
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1312
lemma of_nat_le_iff [simp]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1313
    "(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1314
  by (simp add: linorder_not_less [symmetric])
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1315
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1316
text{*Special cases where either operand is zero*}
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1317
lemma of_nat_0_le_iff [simp]: "(0::'a::ordered_semidom) \<le> of_nat n"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1318
  by (rule of_nat_le_iff [of 0, simplified])
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1319
lemma of_nat_le_0_iff [simp]: "(of_nat m \<le> (0::'a::ordered_semidom)) = (m = 0)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1320
  by (rule of_nat_le_iff [of _ 0, simplified])
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1321
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1322
text{*The ordering on the @{text semiring_1_cancel} is necessary
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1323
to exclude the possibility of a finite field, which indeed wraps back to
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1324
zero.*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1325
lemma of_nat_eq_iff [simp]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1326
    "(of_nat m = (of_nat n::'a::ordered_semidom)) = (m = n)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1327
  by (simp add: order_eq_iff)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1328
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1329
text{*Special cases where either operand is zero*}
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1330
lemma of_nat_0_eq_iff [simp]: "((0::'a::ordered_semidom) = of_nat n) = (0 = n)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1331
  by (rule of_nat_eq_iff [of 0, simplified])
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1332
lemma of_nat_eq_0_iff [simp]: "(of_nat m = (0::'a::ordered_semidom)) = (m = 0)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1333
  by (rule of_nat_eq_iff [of _ 0, simplified])
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1334
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1335
lemma of_nat_diff [simp]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1336
    "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::ring_1)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1337
  by (simp del: of_nat_add
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1338
    add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1339
22483
86064f2f2188 added instance for lattice
haftmann
parents: 22473
diff changeset
  1340
instance nat :: distrib_lattice
86064f2f2188 added instance for lattice
haftmann
parents: 22473
diff changeset
  1341
  "inf \<equiv> min"
86064f2f2188 added instance for lattice
haftmann
parents: 22473
diff changeset
  1342
  "sup \<equiv> max"
86064f2f2188 added instance for lattice
haftmann
parents: 22473
diff changeset
  1343
  by intro_classes (auto simp add: inf_nat_def sup_nat_def)
86064f2f2188 added instance for lattice
haftmann
parents: 22473
diff changeset
  1344
22157
e1d68715ed09 Added lemma nat_size[simp]: "size (n::nat) = n"
krauss
parents: 21672