src/HOL/Nat.thy
author paulson
Tue, 19 Oct 2004 18:18:45 +0200
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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
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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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begin
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subsection {* Type @{text ind} *}
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typedecl ind
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consts
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  Zero_Rep      :: ind
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  Suc_Rep       :: "ind => ind"
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axioms
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  -- {* the axiom of infinity in 2 parts *}
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  inj_Suc_Rep:          "inj Suc_Rep"
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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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consts
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  Nat :: "ind set"
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inductive Nat
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intros
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  Zero_RepI: "Zero_Rep : Nat"
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  Suc_RepI: "i : Nat ==> Suc_Rep i : Nat"
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global
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typedef (open Nat)
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  nat = Nat by (rule exI, rule Nat.Zero_RepI)
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instance nat :: "{ord, zero, one}" ..
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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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  pred_nat :: "(nat * nat) set"
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local
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defs
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  Zero_nat_def: "0 == Abs_Nat Zero_Rep"
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  Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"
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  One_nat_def [simp]: "1 == Suc 0"
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  -- {* nat operations *}
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  pred_nat_def: "pred_nat == {(m, n). n = Suc m}"
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  less_def: "m < n == (m, n) : trancl pred_nat"
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  le_def: "m \<le> (n::nat) == ~ (n < m)"
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text {* Induction *}
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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 (rules elim: Abs_Nat_inverse [THEN subst])
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  done
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text {* Isomorphisms: @{text Abs_Nat} and @{text Rep_Nat} *}
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lemma inj_Rep_Nat: "inj Rep_Nat"
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  apply (rule inj_on_inverseI)
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  apply (rule Rep_Nat_inverse)
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  done
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lemma inj_on_Abs_Nat: "inj_on Abs_Nat Nat"
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  apply (rule inj_on_inverseI)
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  apply (erule Abs_Nat_inverse)
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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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  apply (unfold Zero_nat_def Suc_def)
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  apply (rule inj_on_Abs_Nat [THEN inj_on_contraD])
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  apply (rule Suc_Rep_not_Zero_Rep)
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  apply (rule Rep_Nat Nat.Suc_RepI Nat.Zero_RepI)+
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  done
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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: "inj Suc"
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  apply (unfold Suc_def)
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  apply (rule inj_onI)
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  apply (drule inj_on_Abs_Nat [THEN inj_onD])
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  apply (rule Rep_Nat Nat.Suc_RepI)+
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  apply (drule inj_Suc_Rep [THEN injD])
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  apply (erule inj_Rep_Nat [THEN injD])
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  done
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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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  apply (rule iffI)
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  apply (erule Suc_inject)
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  apply (erule arg_cong)
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  done
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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 {* @{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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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, rules+)
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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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  apply (unfold less_def)
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parents: 12338
diff changeset
   172
  apply (rule refl)
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parents: 12338
diff changeset
   173
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   174
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   175
subsubsection {* Introduction properties *}
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   176
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   177
lemma less_trans: "i < j ==> j < k ==> i < (k::nat)"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   178
  apply (unfold less_def)
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144f45277d5a misc tidying
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parents: 14193
diff changeset
   179
  apply (rule trans_trancl [THEN transD], assumption+)
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43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   180
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   181
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   182
lemma lessI [iff]: "n < Suc n"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   183
  apply (unfold less_def pred_nat_def)
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parents: 12338
diff changeset
   184
  apply (simp add: r_into_trancl)
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   185
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   186
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   187
lemma less_SucI: "i < j ==> i < Suc j"
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parents: 14193
diff changeset
   188
  apply (rule less_trans, assumption)
13449
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parents: 12338
diff changeset
   189
  apply (rule lessI)
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parents: 12338
diff changeset
   190
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   191
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   192
lemma zero_less_Suc [iff]: "0 < Suc n"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   193
  apply (induct n)
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parents: 12338
diff changeset
   194
  apply (rule lessI)
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parents: 12338
diff changeset
   195
  apply (erule less_trans)
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parents: 12338
diff changeset
   196
  apply (rule lessI)
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parents: 12338
diff changeset
   197
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   198
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diff changeset
   199
subsubsection {* Elimination properties *}
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parents: 12338
diff changeset
   200
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   201
lemma less_not_sym: "n < m ==> ~ m < (n::nat)"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   202
  apply (unfold less_def)
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   203
  apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym])
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parents: 12338
diff changeset
   204
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   205
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   206
lemma less_asym:
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parents: 12338
diff changeset
   207
  assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   208
  apply (rule contrapos_np)
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parents: 12338
diff changeset
   209
  apply (rule less_not_sym)
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parents: 12338
diff changeset
   210
  apply (rule h1)
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parents: 12338
diff changeset
   211
  apply (erule h2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   212
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   213
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   214
lemma less_not_refl: "~ n < (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   215
  apply (unfold less_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   216
  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   217
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   218
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   219
lemma less_irrefl [elim!]: "(n::nat) < n ==> R"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   220
  by (rule notE, rule less_not_refl)
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   221
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   222
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)" by blast
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   223
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   224
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   225
  by (rule not_sym, rule less_not_refl2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   226
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   227
lemma lessE:
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parents: 12338
diff changeset
   228
  assumes major: "i < k"
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parents: 12338
diff changeset
   229
  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
   230
  shows P
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   231
  apply (rule major [unfolded less_def pred_nat_def, THEN tranclE], simp_all)
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43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   232
  apply (erule p1)
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parents: 12338
diff changeset
   233
  apply (rule p2)
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144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   234
  apply (simp add: less_def pred_nat_def, assumption)
13449
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   235
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   236
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   237
lemma not_less0 [iff]: "~ n < (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   238
  by (blast elim: lessE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   239
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   240
lemma less_zeroE: "(n::nat) < 0 ==> R"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   241
  by (rule notE, rule not_less0)
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   242
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   243
lemma less_SucE: assumes major: "m < Suc n"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   244
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   245
  apply (rule major [THEN lessE])
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   246
  apply (rule eq, blast)
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   247
  apply (rule less, blast)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   248
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   249
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   250
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   251
  by (blast elim!: less_SucE intro: less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   252
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   253
lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   254
  by (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   255
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   256
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   257
  by (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   258
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   259
lemma Suc_mono: "m < n ==> Suc m < Suc n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   260
  by (induct n) (fast elim: less_trans lessE)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   261
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   262
text {* "Less than" is a linear ordering *}
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   263
lemma less_linear: "m < n | m = n | n < (m::nat)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   264
  apply (induct m)
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   265
  apply (induct n)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   266
  apply (rule refl [THEN disjI1, THEN disjI2])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   267
  apply (rule zero_less_Suc [THEN disjI1])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   268
  apply (blast intro: Suc_mono less_SucI elim: lessE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   269
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   270
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   271
text {* "Less than" is antisymmetric, sort of *}
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   272
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   273
apply(simp only:less_Suc_eq)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   274
apply blast
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   275
done
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   276
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   277
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   278
  using less_linear by blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   279
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   280
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   281
  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
   282
  shows "P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   283
  apply (rule less_linear [THEN disjE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   284
  apply (erule_tac [2] disjE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   285
  apply (erule lessCase)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   286
  apply (erule sym [THEN eqCase])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   287
  apply (erule major)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   288
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   289
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   290
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   291
subsubsection {* Inductive (?) properties *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   292
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   293
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   294
  apply (simp add: nat_neq_iff)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   295
  apply (blast elim!: less_irrefl less_SucE elim: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   296
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   297
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   298
lemma Suc_lessD: "Suc m < n ==> m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   299
  apply (induct n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   300
  apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   301
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   302
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   303
lemma Suc_lessE: assumes major: "Suc i < k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   304
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   305
  apply (rule major [THEN lessE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   306
  apply (erule lessI [THEN minor])
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   307
  apply (erule Suc_lessD [THEN minor], assumption)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   308
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   309
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   310
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   311
  by (blast elim: lessE dest: Suc_lessD)
4104
84433b1ab826 nat datatype_info moved to Nat.thy;
wenzelm
parents: 3370
diff changeset
   312
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   313
lemma Suc_less_eq [iff]: "(Suc m < Suc n) = (m < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   314
  apply (rule iffI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   315
  apply (erule Suc_less_SucD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   316
  apply (erule Suc_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   317
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   318
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   319
lemma less_trans_Suc:
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   320
  assumes le: "i < j" shows "j < k ==> Suc i < k"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   321
  apply (induct k, simp_all)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   322
  apply (insert le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   323
  apply (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   324
  apply (blast dest: Suc_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   325
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   326
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   327
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
   328
lemma not_less_eq: "(~ m < n) = (n < Suc m)"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   329
by (rule_tac m = m and n = n in diff_induct, simp_all)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   330
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   331
text {* Complete induction, aka course-of-values induction *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   332
lemma nat_less_induct:
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   333
  assumes prem: "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   334
  apply (rule_tac a=n in wf_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   335
  apply (rule wf_pred_nat [THEN wf_trancl])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   336
  apply (rule prem)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   337
  apply (unfold less_def, assumption)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   338
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   339
14131
a4fc8b1af5e7 declarations moved from PreList.thy
paulson
parents: 13596
diff changeset
   340
lemmas less_induct = nat_less_induct [rule_format, case_names less]
a4fc8b1af5e7 declarations moved from PreList.thy
paulson
parents: 13596
diff changeset
   341
a4fc8b1af5e7 declarations moved from PreList.thy
paulson
parents: 13596
diff changeset
   342
subsection {* Properties of "less than or equal" *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   343
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   344
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
   345
lemma less_Suc_eq_le: "(m < Suc n) = (m \<le> n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   346
  by (unfold le_def, rule not_less_eq [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   347
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   348
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   349
  by (rule less_Suc_eq_le [THEN iffD2])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   350
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   351
lemma le0 [iff]: "(0::nat) \<le> n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   352
  by (unfold le_def, rule not_less0)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   353
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   354
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   355
  by (simp add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   356
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   357
lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   358
  by (induct i) (simp_all add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   359
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   360
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
   361
  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
   362
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   363
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   364
  by (drule le_Suc_eq [THEN iffD1], rules+)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   365
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   366
lemma leI: "~ n < m ==> m \<le> (n::nat)" by (simp add: le_def)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   367
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   368
lemma leD: "m \<le> n ==> ~ n < (m::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   369
  by (simp add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   370
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   371
lemmas leE = leD [elim_format]
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   372
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   373
lemma not_less_iff_le: "(~ n < m) = (m \<le> (n::nat))"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   374
  by (blast intro: leI elim: leE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   375
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   376
lemma not_leE: "~ m \<le> n ==> n<(m::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   377
  by (simp add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   378
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   379
lemma not_le_iff_less: "(~ n \<le> m) = (m < (n::nat))"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   380
  by (simp add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   381
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   382
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   383
  apply (simp add: le_def less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   384
  apply (blast elim!: less_irrefl less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   385
  done -- {* formerly called lessD *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   386
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   387
lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   388
  by (simp add: le_def less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   389
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   390
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
   391
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   392
  apply (simp add: le_def less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   393
  using less_linear
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   394
  apply blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   395
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   396
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   397
lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   398
  by (blast intro: Suc_leI Suc_le_lessD)
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 le_SucI: "m \<le> n ==> m \<le> Suc n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   401
  by (unfold le_def) (blast dest: Suc_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   402
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   403
lemma less_imp_le: "m < n ==> m \<le> (n::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   404
  by (unfold le_def) (blast elim: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   405
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   406
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
   407
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   408
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   409
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   410
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
   411
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   412
lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   413
  apply (unfold le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   414
  using less_linear
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   415
  apply (blast elim: less_irrefl less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   416
  done
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_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   419
  apply (unfold le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   420
  using less_linear
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   421
  apply (blast elim!: less_irrefl elim: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   422
  done
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_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   425
  by (rules intro: less_or_eq_imp_le le_imp_less_or_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   426
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   427
text {* Useful with @{text Blast}. *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   428
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   429
  by (rule less_or_eq_imp_le, rule disjI2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   430
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   431
lemma le_refl: "n \<le> (n::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   432
  by (simp add: le_eq_less_or_eq)
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_less_trans: "[| i \<le> j; j < k |] ==> i < (k::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   435
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   436
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   437
lemma less_le_trans: "[| i < j; j \<le> k |] ==> i < (k::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   438
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   439
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   440
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
   441
  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
   442
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   443
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
   444
  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
   445
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   446
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
   447
  by (simp add: le_simps)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   448
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   449
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
   450
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
   451
  by (simp add: le_def nat_neq_iff) (blast elim!: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   452
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   453
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
   454
  by (rule iffD2, rule nat_less_le, rule conjI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   455
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   456
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
   457
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   458
  apply (simp add: le_eq_less_or_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   459
  using less_linear
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   460
  apply blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   461
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   462
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   463
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
   464
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14348
diff changeset
   465
instance nat :: "{order, linorder, wellorder}"
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14348
diff changeset
   466
  by intro_classes
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14348
diff changeset
   467
    (assumption |
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14348
diff changeset
   468
      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
   469
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   470
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   471
  by (blast elim!: less_SucE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   472
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   473
text {*
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   474
  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
   475
  if @{term "~ n < m"} or @{term "m \<le> n"} hold.
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   476
  Not suitable as default simprules because they often lead to looping
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   477
*}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   478
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
   479
  by (rule not_less_less_Suc_eq, rule leD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   480
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   481
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   482
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   483
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   484
text {*
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   485
  Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}. 
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   486
  No longer added as simprules (they loop) 
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   487
  but via @{text reorient_simproc} in Bin
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   488
*}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   489
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   490
text {* Polymorphic, not just for @{typ nat} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   491
lemma zero_reorient: "(0 = x) = (x = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   492
  by auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   493
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   494
lemma one_reorient: "(1 = x) = (x = 1)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   495
  by auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   496
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   497
subsection {* Arithmetic operators *}
1660
8cb42cd97579 *** empty log message ***
oheimb
parents: 1625
diff changeset
   498
12338
de0f4a63baa5 renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents: 11451
diff changeset
   499
axclass power < type
10435
b100e8d2c355 added axclass power (from HOL.thy);
wenzelm
parents: 9436
diff changeset
   500
3370
5c5fdce3a4e4 Overloading of "^" requires new type class "power", with types "nat" and
paulson
parents: 2608
diff changeset
   501
consts
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   502
  power :: "('a::power) => nat => 'a"            (infixr "^" 80)
3370
5c5fdce3a4e4 Overloading of "^" requires new type class "power", with types "nat" and
paulson
parents: 2608
diff changeset
   503
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   504
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   505
text {* arithmetic operators @{text "+ -"} and @{text "*"} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   506
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14348
diff changeset
   507
instance nat :: "{plus, minus, times, power}" ..
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   508
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   509
text {* size of a datatype value; overloaded *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   510
consts size :: "'a => nat"
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   511
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   512
primrec
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   513
  add_0:    "0 + n = n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   514
  add_Suc:  "Suc m + n = Suc (m + n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   515
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   516
primrec
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   517
  diff_0:   "m - 0 = m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   518
  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
   519
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   520
primrec
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   521
  mult_0:   "0 * n = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   522
  mult_Suc: "Suc m * n = n + (m * n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   523
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   524
text {* These two rules ease the use of primitive recursion. 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   525
NOTE USE OF @{text "=="} *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   526
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
   527
  by simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   528
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   529
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
   530
  by simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   531
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   532
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   533
  by (case_tac n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   534
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   535
lemma gr_implies_not0: "!!n::nat. m<n ==> n \<noteq> 0"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   536
  by (case_tac n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   537
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   538
lemma neq0_conv [iff]: "!!n::nat. (n \<noteq> 0) = (0 < n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   539
  by (case_tac n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   540
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   541
text {* This theorem is useful with @{text blast} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   542
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   543
  by (rule iffD1, rule neq0_conv, rules)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   544
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   545
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   546
  by (fast intro: not0_implies_Suc)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   547
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   548
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   549
  apply (rule iffI)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   550
  apply (rule ccontr, simp_all)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   551
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   552
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   553
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   554
  by (induct m') simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   555
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   556
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
   557
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   558
  by (case_tac m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   559
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   560
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
   561
  apply (rule nat_less_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   562
  apply (case_tac n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   563
  apply (case_tac [2] nat)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   564
  apply (blast intro: less_trans)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   565
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   566
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   567
subsection {* @{text LEAST} theorems for type @{typ nat} by specialization *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   568
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   569
lemmas LeastI = wellorder_LeastI
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   570
lemmas Least_le = wellorder_Least_le
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   571
lemmas not_less_Least = wellorder_not_less_Least
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   572
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   573
lemma Least_Suc:
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   574
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   575
  apply (case_tac "n", auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   576
  apply (frule LeastI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   577
  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
   578
  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
   579
  apply (erule_tac [2] Least_le)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   580
  apply (case_tac "LEAST x. P x", auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   581
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   582
  apply (blast intro: order_antisym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   583
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   584
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   585
lemma Least_Suc2:
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   586
     "[|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
   587
  by (erule (1) Least_Suc [THEN ssubst], simp)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   588
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   589
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents: 14208
diff changeset
   590
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   591
subsection {* @{term min} and @{term max} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   592
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   593
lemma min_0L [simp]: "min 0 n = (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   594
  by (rule min_leastL) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   595
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   596
lemma min_0R [simp]: "min n 0 = (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   597
  by (rule min_leastR) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   598
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   599
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
   600
  by (simp add: min_of_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   601
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   602
lemma max_0L [simp]: "max 0 n = (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   603
  by (rule max_leastL) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   604
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   605
lemma max_0R [simp]: "max n 0 = (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   606
  by (rule max_leastR) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   607
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   608
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
   609
  by (simp add: max_of_mono)
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
subsection {* Basic rewrite rules for the arithmetic operators *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   613
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   614
text {* Difference *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   615
14193
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   616
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   617
  by (induct n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   618
14193
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   619
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
   620
  by (induct n) simp_all
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
text {*
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   624
  Could be (and is, below) generalized in various ways
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   625
  However, none of the generalizations are currently in the simpset,
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   626
  and I dread to think what happens if I put them in
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 Suc_pred [simp]: "0 < n ==> Suc (n - Suc 0) = n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   629
  by (simp split add: nat.split)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   630
14193
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   631
declare diff_Suc [simp del, code del]
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   632
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   633
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   634
subsection {* Addition *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   635
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   636
lemma add_0_right [simp]: "m + 0 = (m::nat)"
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
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   640
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   641
14193
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   642
lemma [code]: "Suc m + n = m + Suc n" by simp
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   643
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   644
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   645
text {* Associative law for addition *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   646
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   647
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   648
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   649
text {* Commutative law for addition *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   650
lemma nat_add_commute: "m + n = n + (m::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   651
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   652
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   653
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   654
  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
   655
  apply (rule nat_add_assoc)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   656
  apply (rule nat_add_commute)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   657
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   658
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   659
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
   660
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   661
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   662
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
   663
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   664
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   665
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
   666
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   667
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   668
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
   669
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   670
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   671
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   672
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   673
lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   674
  by (case_tac m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   675
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   676
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   677
  by (case_tac m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   678
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   679
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
   680
  by (rule trans, rule eq_commute, rule add_is_1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   681
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   682
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
   683
  by (simp del: neq0_conv add: neq0_conv [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   684
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   685
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   686
  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
   687
  apply (simp add: nat_add_assoc del: add_0_right)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   688
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   689
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   690
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   691
subsection {* Multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   692
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   693
text {* right annihilation in product *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   694
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
   695
  by (induct m) simp_all
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   696
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   697
text {* right successor law for multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   698
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
   699
  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
   700
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   701
text {* Commutative law for multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   702
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
   703
  by (induct m) simp_all
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   704
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   705
text {* addition distributes over multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   706
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
   707
  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
   708
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   709
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
   710
  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
   711
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   712
text {* Associative law for multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   713
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
   714
  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
   715
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   716
14740
c8e1937110c2 fixed latex problems
nipkow
parents: 14738
diff changeset
   717
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
   718
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
   719
proof
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   720
  fix i j k :: nat
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_add_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_add_commute)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   723
  show "0 + 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 * (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
   725
  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
   726
  show "1 * i = i" by simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   727
  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
   728
  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
   729
  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
   730
qed
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   731
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   732
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
   733
  apply (induct m)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   734
  apply (induct_tac [2] n, simp_all)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   735
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   736
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   737
subsection {* Monotonicity of Addition *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   738
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   739
text {* strict, in 1st argument *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   740
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
   741
  by (induct k) simp_all
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   742
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   743
text {* strict, in both arguments *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   744
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
   745
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   746
  apply (induct j, simp_all)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   747
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   748
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   749
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
   750
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
   751
  apply (induct n)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   752
  apply (simp_all add: order_le_less)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   753
  apply (blast elim!: less_SucE 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   754
               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
   755
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   756
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   757
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
   758
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
   759
  apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   760
  apply (induct_tac x) 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   761
  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
   762
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   763
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   764
14740
c8e1937110c2 fixed latex problems
nipkow
parents: 14738
diff changeset
   765
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
   766
instance nat :: ordered_semidom
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   767
proof
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   768
  fix i j k :: nat
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   769
  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
   770
  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
   771
  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
   772
qed
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   773
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   774
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
   775
  by simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   776
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   777
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
   778
  by simp
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
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   781
subsection {* Additional theorems about "less than" *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   782
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   783
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
   784
  monotonicity to @{text "\<le>"} monotonicity *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   785
lemma less_mono_imp_le_mono:
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   786
  assumes lt_mono: "!!i j::nat. i < j ==> f i < f j"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   787
  and le: "i \<le> j" shows "f i \<le> ((f j)::nat)" using le
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   788
  apply (simp add: order_le_less)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   789
  apply (blast intro!: lt_mono)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   790
  done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   791
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   792
text {* non-strict, in 1st argument *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   793
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
   794
  by (rule add_right_mono)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   795
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   796
text {* non-strict, in both arguments *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   797
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
   798
  by (rule add_mono)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   799
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   800
lemma le_add2: "n \<le> ((m + n)::nat)"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   801
  by (insert add_right_mono [of 0 m n], simp) 
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   802
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   803
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
   804
  by (simp add: add_commute, rule le_add2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   805
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   806
lemma less_add_Suc1: "i < Suc (i + m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   807
  by (rule le_less_trans, rule le_add1, rule lessI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   808
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   809
lemma less_add_Suc2: "i < Suc (m + i)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   810
  by (rule le_less_trans, rule le_add2, rule lessI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   811
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   812
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   813
  by (rules intro!: less_add_Suc1 less_imp_Suc_add)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   814
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   815
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   816
  by (rule le_trans, assumption, rule le_add1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   817
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   818
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   819
  by (rule le_trans, assumption, rule le_add2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   820
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   821
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   822
  by (rule less_le_trans, assumption, rule le_add1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   823
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   824
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   825
  by (rule less_le_trans, assumption, rule le_add2)
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 add_lessD1: "i + j < (k::nat) ==> i < k"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   828
  apply (rule le_less_trans [of _ "i+j"]) 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   829
  apply (simp_all add: le_add1)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   830
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   831
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   832
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   833
  apply (rule notI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   834
  apply (erule add_lessD1 [THEN less_irrefl])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   835
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   836
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   837
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   838
  by (simp add: add_commute not_add_less1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   839
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   840
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   841
  apply (rule order_trans [of _ "m+k"]) 
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   842
  apply (simp_all add: le_add1)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   843
  done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   844
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   845
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   846
  apply (simp add: add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   847
  apply (erule add_leD1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   848
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   849
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   850
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
   851
  by (blast dest: add_leD1 add_leD2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   852
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   853
text {* needs @{text "!!k"} for @{text add_ac} to work *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   854
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
   855
  by (force simp del: add_Suc_right
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   856
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   857
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   858
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   859
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   860
subsection {* Difference *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   861
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   862
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   863
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   864
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   865
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
   866
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   867
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   868
  by (induct m n rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   869
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   870
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   871
  by (simp add: add_diff_inverse not_less_iff_le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   872
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   873
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
   874
  by (simp add: le_add_diff_inverse add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   875
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   876
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   877
subsection {* More results about difference *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   878
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   879
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
   880
  by (induct m n rule: diff_induct) simp_all
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_less_Suc: "m - n < Suc m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   883
  apply (induct m n rule: diff_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   884
  apply (erule_tac [3] less_SucE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   885
  apply (simp_all add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   886
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   887
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   888
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   889
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   890
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   891
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   892
  by (rule le_less_trans, rule diff_le_self)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   893
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   894
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   895
  by (induct i j rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   896
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   897
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   898
  by (simp add: diff_diff_left)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   899
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   900
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   901
  apply (case_tac "n", safe)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   902
  apply (simp add: le_simps)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   903
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   904
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   905
text {* This and the next few suggested by Florian Kammueller *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   906
lemma diff_commute: "(i::nat) - j - k = i - k - j"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   907
  by (simp add: diff_diff_left add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   908
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   909
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
   910
  by (induct j k rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   911
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   912
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
   913
  by (simp add: add_commute diff_add_assoc)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   914
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   915
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   916
  by (induct n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   917
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   918
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   919
  by (simp add: diff_add_assoc)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   920
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   921
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   922
  apply safe
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   923
  apply (simp_all add: diff_add_inverse2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   924
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   925
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   926
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
   927
  by (induct m n rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   928
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   929
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
   930
  by (rule iffD2, rule diff_is_0_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   931
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   932
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   933
  by (induct m n rule: diff_induct) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   934
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   935
lemma less_imp_add_positive: "i < j  ==> \<exists>k::nat. 0 < k & i + k = j"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   936
  apply (rule_tac x = "j - i" in exI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   937
  apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   938
  done
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
   939
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   940
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
   941
  apply (induct k i rule: diff_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   942
  apply (simp_all (no_asm))
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   943
  apply rules
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   944
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   945
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   946
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   947
  apply (rule diff_self_eq_0 [THEN subst])
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   948
  apply (rule zero_induct_lemma, rules+)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   949
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   950
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   951
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   952
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   953
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   954
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   955
  by (simp add: diff_cancel add_commute)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   956
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   957
lemma diff_add_0: "n - (n + m) = (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   958
  by (induct n) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   959
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   960
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   961
text {* Difference distributes over multiplication *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   962
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   963
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   964
  by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   965
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   966
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   967
  by (simp add: diff_mult_distrib mult_commute [of k])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   968
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   969
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   970
lemmas nat_distrib =
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   971
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   972
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   973
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   974
subsection {* Monotonicity of Multiplication *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   975
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   976
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   977
  by (simp add: mult_right_mono) 
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   978
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   979
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   980
  by (simp add: mult_left_mono) 
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   981
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   982
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   983
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   984
  by (simp add: mult_mono) 
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   985
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   986
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   987
  by (simp add: mult_strict_right_mono) 
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   988
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
   989
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
   990
      there are no negative numbers.*}
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
   991
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
   992
  apply (induct m)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   993
  apply (case_tac [2] n, simp_all)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   994
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   995
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   996
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
   997
  apply (induct m)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   998
  apply (case_tac [2] n, simp_all)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   999
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1000
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1001
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1002
  apply (induct m, simp)
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1003
  apply (induct n, simp, fastsimp)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1004
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1005
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1006
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
  1007
  apply (rule trans)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
  1008
  apply (rule_tac [2] mult_eq_1_iff, fastsimp)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1009
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1010
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
  1011
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
  1012
  apply (safe intro!: mult_less_mono1)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
  1013
  apply (case_tac k, auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1014
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1015
  apply (blast intro: mult_le_mono1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1016
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1017
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1018
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
  1019
  by (simp add: mult_commute [of k])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1020
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1021
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
  1022
by (simp add: linorder_not_less [symmetric], auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1023
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1024
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
  1025
by (simp add: linorder_not_less [symmetric], auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1026
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
  1027
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
  1028
  apply (cut_tac less_linear, safe, auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1029
  apply (drule mult_less_mono1, assumption, simp)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1030
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1031
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1032
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
  1033
  by (simp add: mult_commute [of k])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1034
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1035
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1036
  by (subst mult_less_cancel1) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1037
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1038
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
  1039
  by (subst mult_le_cancel1) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1040
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1041
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1042
  by (subst mult_cancel1) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1043
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1044
text {* Lemma for @{text gcd} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1045
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
  1046
  apply (drule sym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1047
  apply (rule disjCI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1048
  apply (rule nat_less_cases, erule_tac [2] _)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1049
  apply (fastsimp elim!: less_SucE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1050
  apply (fastsimp dest: mult_less_mono2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1051
  done
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1052
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
  1053
end