src/HOL/Nat.thy
author nipkow
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(*  Title:      HOL/Nat.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
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Type "nat" is a linear order, and a datatype; arithmetic operators + -
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and * (for div, mod and dvd, see theory Divides).
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*)
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header {* Natural numbers *}
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theory Nat
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imports Wellfounded_Recursion Ring_and_Field
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uses
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  "~~/src/Tools/rat.ML"
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  "~~/src/Provers/Arith/cancel_sums.ML"
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  ("arith_data.ML")
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  "~~/src/Provers/Arith/fast_lin_arith.ML"
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  ("Tools/lin_arith.ML")
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  ("Tools/function_package/size.ML")
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begin
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subsection {* Type @{text ind} *}
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typedecl ind
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axiomatization
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  Zero_Rep :: ind and
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  Suc_Rep :: "ind => ind"
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where
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  -- {* the axiom of infinity in 2 parts *}
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  inj_Suc_Rep:          "inj Suc_Rep" and
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  Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
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subsection {* Type nat *}
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text {* Type definition *}
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inductive_set Nat :: "ind set"
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where
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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
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proof
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  show "Zero_Rep : Nat" by (rule Nat.Zero_RepI)
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qed
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consts
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  Suc :: "nat => nat"
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local
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instance nat :: zero
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  Zero_nat_def: "0 == Abs_Nat Zero_Rep" ..
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lemmas [code func del] = Zero_nat_def
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defs
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  Suc_def:      "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"
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theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n"
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  apply (unfold Zero_nat_def Suc_def)
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  apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}
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  apply (erule Rep_Nat [THEN Nat.induct])
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  apply (iprover elim: Abs_Nat_inverse [THEN subst])
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  done
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lemma Suc_not_Zero [iff]: "Suc m \<noteq> 0"
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  by (simp add: Zero_nat_def Suc_def Abs_Nat_inject Rep_Nat Suc_RepI Zero_RepI
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                Suc_Rep_not_Zero_Rep)
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lemma Zero_not_Suc [iff]: "0 \<noteq> Suc m"
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  by (rule not_sym, rule Suc_not_Zero not_sym)
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lemma inj_Suc[simp]: "inj_on Suc N"
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  by (simp add: Suc_def inj_on_def Abs_Nat_inject Rep_Nat Suc_RepI
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                inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject)
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lemma Suc_Suc_eq [iff]: "(Suc m = Suc n) = (m = n)"
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  by (rule inj_Suc [THEN inj_eq])
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rep_datatype nat
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  distinct  Suc_not_Zero Zero_not_Suc
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  inject    Suc_Suc_eq
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  induction nat_induct
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declare nat.induct [case_names 0 Suc, induct type: nat]
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declare nat.exhaust [case_names 0 Suc, cases type: nat]
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lemmas nat_rec_0 = nat.recs(1)
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  and nat_rec_Suc = nat.recs(2)
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lemmas nat_case_0 = nat.cases(1)
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  and nat_case_Suc = nat.cases(2)
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text {* Injectiveness and distinctness lemmas *}
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lemma Suc_neq_Zero: "Suc m = 0 ==> 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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lemma Suc_inject: "Suc x = Suc y ==> x = y"
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  by (rule inj_Suc [THEN injD])
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lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False"
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  by auto
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lemma n_not_Suc_n: "n \<noteq> Suc n"
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  by (induct n) simp_all
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lemma Suc_n_not_n: "Suc t \<noteq> t"
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  by (rule not_sym, rule n_not_Suc_n)
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text {* A special form of induction for reasoning
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  about @{term "m < n"} and @{term "m - n"} *}
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theorem diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
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    (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
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  apply (rule_tac x = m in spec)
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  apply (induct n)
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  prefer 2
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  apply (rule allI)
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  apply (induct_tac x, iprover+)
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  done
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subsection {* Arithmetic operators *}
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instance nat :: "{one, plus, minus, times}"
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  One_nat_def [simp]: "1 == Suc 0" ..
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primrec
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  add_0:    "0 + n = n"
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  add_Suc:  "Suc m + n = Suc (m + n)"
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primrec
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  diff_0:   "m - 0 = m"
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  diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
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primrec
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  mult_0:   "0 * n = 0"
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  mult_Suc: "Suc m * n = n + (m * n)"
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subsection {* Orders on @{typ nat} *}
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definition
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  pred_nat :: "(nat * nat) set" where
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  "pred_nat = {(m, n). n = Suc m}"
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instance nat :: ord
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  less_def: "m < n == (m, n) : pred_nat^+"
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  le_def:   "m \<le> (n::nat) == ~ (n < m)" ..
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lemmas [code func del] = less_def le_def
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lemma wf_pred_nat: "wf pred_nat"
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parents: 14193
diff changeset
   165
  apply (unfold wf_def pred_nat_def, clarify)
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   166
  apply (induct_tac x, blast+)
13449
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   167
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   168
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   169
lemma wf_less: "wf {(x, y::nat). x < y}"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   170
  apply (unfold less_def)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   171
  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset], blast)
13449
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   172
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   173
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   174
lemma less_eq: "((m, n) : pred_nat^+) = (m < n)"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   175
  apply (unfold less_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   176
  apply (rule refl)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   177
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   178
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   179
subsubsection {* Introduction properties *}
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parents: 12338
diff changeset
   180
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   181
lemma less_trans: "i < j ==> j < k ==> i < (k::nat)"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   182
  apply (unfold less_def)
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144f45277d5a misc tidying
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parents: 14193
diff changeset
   183
  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
   184
  done
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   185
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   186
lemma lessI [iff]: "n < Suc n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   187
  apply (unfold less_def pred_nat_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   188
  apply (simp add: r_into_trancl)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   189
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   190
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   191
lemma less_SucI: "i < j ==> i < Suc j"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   192
  apply (rule less_trans, assumption)
13449
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berghofe
parents: 12338
diff changeset
   193
  apply (rule lessI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   194
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   195
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   196
lemma zero_less_Suc [iff]: "0 < Suc n"
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   197
  apply (induct n)
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berghofe
parents: 12338
diff changeset
   198
  apply (rule lessI)
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berghofe
parents: 12338
diff changeset
   199
  apply (erule less_trans)
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berghofe
parents: 12338
diff changeset
   200
  apply (rule lessI)
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berghofe
parents: 12338
diff changeset
   201
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   202
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   203
subsubsection {* Elimination properties *}
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parents: 12338
diff changeset
   204
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   205
lemma less_not_sym: "n < m ==> ~ m < (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   206
  apply (unfold less_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   207
  apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   208
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   209
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   210
lemma less_asym:
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parents: 12338
diff changeset
   211
  assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   212
  apply (rule contrapos_np)
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berghofe
parents: 12338
diff changeset
   213
  apply (rule less_not_sym)
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berghofe
parents: 12338
diff changeset
   214
  apply (rule h1)
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berghofe
parents: 12338
diff changeset
   215
  apply (erule h2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   216
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   217
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   218
lemma less_not_refl: "~ n < (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   219
  apply (unfold less_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   220
  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   221
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   222
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   223
lemma less_irrefl [elim!]: "(n::nat) < n ==> R"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   224
  by (rule notE, rule less_not_refl)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   225
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   226
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)" by blast
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   227
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   228
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   229
  by (rule not_sym, rule less_not_refl2)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   230
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   231
lemma lessE:
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   232
  assumes major: "i < k"
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parents: 12338
diff changeset
   233
  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
   234
  shows P
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   235
  apply (rule major [unfolded less_def pred_nat_def, THEN tranclE], simp_all)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   236
  apply (erule p1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   237
  apply (rule p2)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   238
  apply (simp add: less_def pred_nat_def, assumption)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   239
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   240
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   241
lemma not_less0 [iff]: "~ n < (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   242
  by (blast elim: lessE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   243
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   244
lemma less_zeroE: "(n::nat) < 0 ==> R"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   245
  by (rule notE, rule not_less0)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   246
43c9ec498291 - Converted to new theory format
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parents: 12338
diff changeset
   247
lemma less_SucE: assumes major: "m < Suc n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   248
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   249
  apply (rule major [THEN lessE])
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   250
  apply (rule eq, blast)
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   251
  apply (rule less, blast)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   252
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   253
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   254
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   255
  by (blast elim!: less_SucE intro: less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   256
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24196
diff changeset
   257
lemma less_one [iff,noatp]: "(n < (1::nat)) = (n = 0)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   258
  by (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   259
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   260
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   261
  by (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   262
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   263
lemma Suc_mono: "m < n ==> Suc m < Suc n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   264
  by (induct n) (fast elim: less_trans lessE)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   265
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   266
text {* "Less than" is a linear ordering *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   267
lemma less_linear: "m < n | m = n | n < (m::nat)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   268
  apply (induct m)
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   269
  apply (induct n)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   270
  apply (rule refl [THEN disjI1, THEN disjI2])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   271
  apply (rule zero_less_Suc [THEN disjI1])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   272
  apply (blast intro: Suc_mono less_SucI elim: lessE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   273
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   274
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   275
text {* "Less than" is antisymmetric, sort of *}
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   276
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   277
  apply(simp only:less_Suc_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   278
  apply blast
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   279
  done
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   280
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   281
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   282
  using less_linear by blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   283
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   284
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   285
  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
   286
  shows "P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   287
  apply (rule less_linear [THEN disjE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   288
  apply (erule_tac [2] disjE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   289
  apply (erule lessCase)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   290
  apply (erule sym [THEN eqCase])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   291
  apply (erule major)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   292
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   293
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   294
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   295
subsubsection {* Inductive (?) properties *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   296
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   297
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   298
  apply (simp add: nat_neq_iff)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   299
  apply (blast elim!: less_irrefl less_SucE elim: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   300
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   301
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   302
lemma Suc_lessD: "Suc m < n ==> m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   303
  apply (induct n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   304
  apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   305
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   306
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   307
lemma Suc_lessE: assumes major: "Suc i < k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   308
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   309
  apply (rule major [THEN lessE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   310
  apply (erule lessI [THEN minor])
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   311
  apply (erule Suc_lessD [THEN minor], assumption)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   312
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   313
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   314
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   315
  by (blast elim: lessE dest: Suc_lessD)
4104
84433b1ab826 nat datatype_info moved to Nat.thy;
wenzelm
parents: 3370
diff changeset
   316
16635
bf7de5723c60 Moved some code lemmas from Main to Nat.
berghofe
parents: 15921
diff changeset
   317
lemma Suc_less_eq [iff, code]: "(Suc m < Suc n) = (m < n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   318
  apply (rule iffI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   319
  apply (erule Suc_less_SucD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   320
  apply (erule Suc_mono)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   321
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   322
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   323
lemma less_trans_Suc:
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   324
  assumes le: "i < j" shows "j < k ==> Suc i < k"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   325
  apply (induct k, simp_all)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   326
  apply (insert le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   327
  apply (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   328
  apply (blast dest: Suc_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   329
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   330
16635
bf7de5723c60 Moved some code lemmas from Main to Nat.
berghofe
parents: 15921
diff changeset
   331
lemma [code]: "((n::nat) < 0) = False" by simp
bf7de5723c60 Moved some code lemmas from Main to Nat.
berghofe
parents: 15921
diff changeset
   332
lemma [code]: "(0 < Suc n) = True" by simp
bf7de5723c60 Moved some code lemmas from Main to Nat.
berghofe
parents: 15921
diff changeset
   333
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   334
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
   335
lemma not_less_eq: "(~ m < n) = (n < Suc m)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   336
  by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   337
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   338
text {* Complete induction, aka course-of-values induction *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   339
lemma nat_less_induct:
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   340
  assumes prem: "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   341
  apply (induct n rule: wf_induct [OF wf_pred_nat [THEN wf_trancl]])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   342
  apply (rule prem)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   343
  apply (unfold less_def, assumption)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   344
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   345
14131
a4fc8b1af5e7 declarations moved from PreList.thy
paulson
parents: 13596
diff changeset
   346
lemmas less_induct = nat_less_induct [rule_format, case_names less]
a4fc8b1af5e7 declarations moved from PreList.thy
paulson
parents: 13596
diff changeset
   347
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
   348
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   349
text {* Properties of "less than or equal" *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   350
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   351
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
   352
lemma less_Suc_eq_le: "(m < Suc n) = (m \<le> n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   353
  unfolding le_def by (rule not_less_eq [symmetric])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   354
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   355
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   356
  by (rule less_Suc_eq_le [THEN iffD2])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   357
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   358
lemma le0 [iff]: "(0::nat) \<le> n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   359
  unfolding le_def by (rule not_less0)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   360
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   361
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   362
  by (simp add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   363
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   364
lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   365
  by (induct i) (simp_all add: le_def)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   366
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   367
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
   368
  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
   369
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   370
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 16796
diff changeset
   371
  by (drule le_Suc_eq [THEN iffD1], iprover+)
13449
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 Suc_leI: "m < n ==> Suc(m) \<le> n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   374
  apply (simp add: le_def less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   375
  apply (blast elim!: less_irrefl less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   376
  done -- {* formerly called lessD *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   377
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   378
lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   379
  by (simp add: le_def less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   380
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   381
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
   382
lemma Suc_le_lessD: "Suc m \<le> n ==> m < 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
  using less_linear
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   385
  apply blast
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   386
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   387
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   388
lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   389
  by (blast intro: Suc_leI Suc_le_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   390
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   391
lemma le_SucI: "m \<le> n ==> m \<le> Suc n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   392
  by (unfold le_def) (blast dest: Suc_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   393
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   394
lemma less_imp_le: "m < n ==> m \<le> (n::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   395
  by (unfold le_def) (blast elim: less_asym)
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
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
   398
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   399
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   400
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   401
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
   402
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   403
lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   404
  unfolding le_def
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   405
  using less_linear
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   406
  by (blast elim: less_irrefl less_asym)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   407
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   408
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   409
  unfolding le_def
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   410
  using less_linear
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   411
  by (blast elim!: less_irrefl elim: less_asym)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   412
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   413
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 16796
diff changeset
   414
  by (iprover intro: less_or_eq_imp_le le_imp_less_or_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   415
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   416
text {* Useful with @{text blast}. *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   417
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   418
  by (rule less_or_eq_imp_le) (rule disjI2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   419
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   420
lemma le_refl: "n \<le> (n::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   421
  by (simp add: le_eq_less_or_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   422
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   423
lemma le_less_trans: "[| i \<le> j; j < k |] ==> i < (k::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   424
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   425
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   426
lemma less_le_trans: "[| i < j; j \<le> k |] ==> i < (k::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   427
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   428
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   429
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
   430
  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
   431
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   432
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
   433
  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
   434
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   435
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
   436
  by (simp add: le_simps)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   437
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   438
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
   439
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
   440
  by (simp add: le_def nat_neq_iff) (blast elim!: less_asym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   441
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   442
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
   443
  by (rule iffD2, rule nat_less_le, rule conjI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   444
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   445
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
   446
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   447
  apply (simp add: le_eq_less_or_eq)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   448
  using less_linear by blast
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   449
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   450
text {* Type @{typ nat} is a wellfounded linear order *}
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   451
22318
6efe70ab7add simpliefied instance statement
haftmann
parents: 22262
diff changeset
   452
instance nat :: wellorder
14691
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14348
diff changeset
   453
  by intro_classes
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14348
diff changeset
   454
    (assumption |
e1eedc8cad37 tuned instance statements;
wenzelm
parents: 14348
diff changeset
   455
      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
   456
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   457
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
15921
b6e345548913 Fixing a problem with lin.arith.
nipkow
parents: 15539
diff changeset
   458
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   459
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   460
  by (blast elim!: less_SucE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   461
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   462
text {*
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   463
  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
   464
  if @{term "~ n < m"} or @{term "m \<le> n"} hold.
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   465
  Not suitable as default simprules because they often lead to looping
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   466
*}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   467
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
   468
  by (rule not_less_less_Suc_eq, rule leD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   469
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   470
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   471
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 {*
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   474
  Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}.
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   475
  No longer added as simprules (they loop)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   476
  but via @{text reorient_simproc} in Bin
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   477
*}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   478
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   479
text {* Polymorphic, not just for @{typ nat} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   480
lemma zero_reorient: "(0 = x) = (x = 0)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   481
  by auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   482
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   483
lemma one_reorient: "(1 = x) = (x = 1)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   484
  by auto
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   485
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   486
text {* These two rules ease the use of primitive recursion.
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   487
NOTE USE OF @{text "=="} *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   488
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
   489
  by simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   490
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   491
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
   492
  by simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   493
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   494
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   495
  by (cases n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   496
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   497
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   498
  by (cases n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   499
25111
d52a58b51f1f neq0_conv removed from [iff] -- causes problems by simple goals with blast, auto etc...
chaieb
parents: 25076
diff changeset
   500
lemma neq0_conv: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   501
  by (cases n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   502
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   503
text {* This theorem is useful with @{text blast} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   504
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 16796
diff changeset
   505
  by (rule iffD1, rule neq0_conv, iprover)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   506
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   507
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   508
  by (fast intro: not0_implies_Suc)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   509
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24196
diff changeset
   510
lemma not_gr0 [iff,noatp]: "!!n::nat. (~ (0 < n)) = (n = 0)"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   511
using neq0_conv by blast
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   512
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   513
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   514
  by (induct m') simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   515
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   516
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
   517
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   518
  by (cases m) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   519
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   520
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
   521
  apply (rule nat_less_induct)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   522
  apply (case_tac n)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   523
  apply (case_tac [2] nat)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   524
  apply (blast intro: less_trans)+
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   525
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   526
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
   527
15341
254f6f00b60e converted to Isar script, simplifying some results
paulson
parents: 15281
diff changeset
   528
subsection {* @{text LEAST} theorems for type @{typ nat}*}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   529
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   530
lemma Least_Suc:
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   531
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   532
  apply (case_tac "n", auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   533
  apply (frule LeastI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   534
  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
   535
  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
   536
  apply (erule_tac [2] Least_le)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   537
  apply (case_tac "LEAST x. P x", auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   538
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   539
  apply (blast intro: order_antisym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   540
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   541
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   542
lemma Least_Suc2:
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   543
     "[|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
   544
  by (erule (1) Least_Suc [THEN ssubst], simp)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   545
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   546
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   547
subsection {* @{term min} and @{term max} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   548
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   549
lemma mono_Suc: "mono Suc"
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   550
  by (rule monoI) simp
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   551
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   552
lemma min_0L [simp]: "min 0 n = (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   553
  by (rule min_leastL) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   554
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   555
lemma min_0R [simp]: "min n 0 = (0::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   556
  by (rule min_leastR) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   557
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   558
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   559
  by (simp add: mono_Suc min_of_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   560
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   561
lemma min_Suc1:
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   562
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   563
  by (simp split: nat.split)
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   564
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   565
lemma min_Suc2:
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   566
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   567
  by (simp split: nat.split)
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   568
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   569
lemma max_0L [simp]: "max 0 n = (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   570
  by (rule max_leastL) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   571
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   572
lemma max_0R [simp]: "max n 0 = (n::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   573
  by (rule max_leastR) simp
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   574
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   575
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
25076
a50b36401c61 localized mono predicate
haftmann
parents: 25062
diff changeset
   576
  by (simp add: mono_Suc max_of_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   577
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   578
lemma max_Suc1:
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   579
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   580
  by (simp split: nat.split)
22191
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   581
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   582
lemma max_Suc2:
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   583
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   584
  by (simp split: nat.split)
9c07aab3a653 min/max lemmas (actually unused!)
paulson
parents: 22157
diff changeset
   585
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   586
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   587
subsection {* Basic rewrite rules for the arithmetic operators *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   588
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   589
text {* Difference *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   590
14193
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   591
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   592
  by (induct n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   593
14193
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   594
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
   595
  by (induct n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   596
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   597
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   598
text {*
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   599
  Could be (and is, below) generalized in various ways
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   600
  However, none of the generalizations are currently in the simpset,
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   601
  and I dread to think what happens if I put them in
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   602
*}
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   603
lemma Suc_pred [simp]: "0 \<noteq> n ==> Suc (n - Suc 0) = n"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   604
by (simp split add: nat.split)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   605
14193
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   606
declare diff_Suc [simp del, code del]
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   607
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   608
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   609
subsection {* Addition *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   610
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   611
lemma add_0_right [simp]: "m + 0 = (m::nat)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   612
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   613
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   614
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   615
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   616
19890
1aad48bcc674 slight adaption for code generator
haftmann
parents: 19870
diff changeset
   617
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
1aad48bcc674 slight adaption for code generator
haftmann
parents: 19870
diff changeset
   618
  by simp
14193
30e41f63712e Improved efficiency of code generated for + and -
berghofe
parents: 14131
diff changeset
   619
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   620
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   621
text {* Associative law for addition *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   622
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   623
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   624
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   625
text {* Commutative law for addition *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   626
lemma nat_add_commute: "m + n = n + (m::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   627
  by (induct m) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   628
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   629
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   630
  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
   631
  apply (rule nat_add_assoc)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   632
  apply (rule nat_add_commute)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   633
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   634
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   635
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
   636
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   637
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   638
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
   639
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   640
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   641
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
   642
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   643
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   644
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
   645
  by (induct k) simp_all
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   646
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   647
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   648
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   649
lemma add_is_0 [iff]: fixes m :: nat shows "(m + n = 0) = (m = 0 & n = 0)"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   650
  by (cases m) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   651
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   652
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   653
  by (cases m) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   654
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   655
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
   656
  by (rule trans, rule eq_commute, rule add_is_1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   657
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   658
lemma add_gr_0 [iff]: "!!m::nat. (m + n \<noteq> 0) = (m\<noteq>0 | n\<noteq>0)"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   659
by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   660
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   661
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   662
  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
   663
  apply (simp add: nat_add_assoc del: add_0_right)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   664
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   665
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16635
diff changeset
   666
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   667
  apply (induct k)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   668
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   669
  apply(drule comp_inj_on[OF _ inj_Suc])
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   670
  apply (simp add:o_def)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   671
  done
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16635
diff changeset
   672
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16635
diff changeset
   673
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   674
subsection {* Multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   675
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   676
text {* right annihilation in product *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   677
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
   678
  by (induct m) simp_all
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   679
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   680
text {* right successor law for multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   681
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
   682
  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
   683
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   684
text {* Commutative law for multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   685
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
   686
  by (induct m) simp_all
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   687
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   688
text {* addition distributes over multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   689
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
   690
  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
   691
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   692
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
   693
  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
   694
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   695
text {* Associative law for multiplication *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   696
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
   697
  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
   698
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   699
14740
c8e1937110c2 fixed latex problems
nipkow
parents: 14738
diff changeset
   700
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
   701
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
   702
proof
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   703
  fix i j k :: nat
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   704
  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
   705
  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
   706
  show "0 + i = i" by simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   707
  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
   708
  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
   709
  show "1 * i = i" by simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   710
  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
   711
  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
   712
  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
   713
qed
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   714
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   715
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
   716
  apply (induct m)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   717
   apply (induct_tac [2] n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   718
    apply simp_all
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   719
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   720
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
   721
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   722
subsection {* Monotonicity of Addition *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   723
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   724
text {* strict, in 1st argument *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   725
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
   726
  by (induct k) simp_all
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   727
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   728
text {* strict, in both arguments *}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   729
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
   730
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   731
  apply (induct j, simp_all)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   732
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   733
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   734
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
   735
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
   736
  apply (induct n)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   737
  apply (simp_all add: order_le_less)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   738
  apply (blast elim!: less_SucE
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   739
               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
   740
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   741
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   742
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   743
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   744
apply(auto simp: gr0_conv_Suc)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   745
apply (induct_tac m)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   746
apply (simp_all add: add_less_mono)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   747
done
14341
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
14740
c8e1937110c2 fixed latex problems
nipkow
parents: 14738
diff changeset
   750
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
   751
instance nat :: ordered_semidom
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   752
proof
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   753
  fix i j k :: nat
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   754
  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
   755
  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
   756
  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
   757
qed
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   758
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   759
lemma nat_mult_1: "(1::nat) * n = n"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   760
  by simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   761
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   762
lemma nat_mult_1_right: "n * (1::nat) = n"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   763
  by simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   764
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   765
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   766
subsection {* Additional theorems about "less than" *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   767
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   768
text{*An induction rule for estabilishing binary relations*}
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   769
lemma less_Suc_induct:
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   770
  assumes less:  "i < j"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   771
     and  step:  "!!i. P i (Suc i)"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   772
     and  trans: "!!i j k. P i j ==> P j k ==> P i k"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   773
  shows "P i j"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   774
proof -
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   775
  from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   776
  have "P i (Suc (i + k))"
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   777
  proof (induct k)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   778
    case 0
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   779
    show ?case by (simp add: step)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   780
  next
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   781
    case (Suc k)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   782
    thus ?case by (auto intro: assms)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   783
  qed
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   784
  thus "P i j" by (simp add: j)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   785
qed
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   786
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   787
text {* The method of infinite descent, frequently used in number theory.
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   788
Provided by Roelof Oosterhuis.
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   789
$P(n)$ is true for all $n\in\mathbb{N}$ if
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   790
\begin{itemize}
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   791
  \item case ``0'': given $n=0$ prove $P(n)$,
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   792
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   793
        a smaller integer $m$ such that $\neg P(m)$.
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   794
\end{itemize} *}
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   795
24523
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   796
lemma infinite_descent0[case_names 0 smaller]: 
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   797
  "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   798
by (induct n rule: less_induct, case_tac "n>0", auto)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   799
24523
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   800
text{* A compact version without explicit base case: *}
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   801
lemma infinite_descent:
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   802
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   803
by (induct n rule: less_induct, auto)
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   804
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   805
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   806
text {* Infinite descent using a mapping to $\mathbb{N}$:
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   807
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   808
\begin{itemize}
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   809
\item case ``0'': given $V(x)=0$ prove $P(x)$,
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   810
\item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$.
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   811
\end{itemize}
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   812
NB: the proof also shows how to use the previous lemma. *}
24523
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   813
corollary infinite_descent0_measure[case_names 0 smaller]:
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   814
assumes 0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   815
and     1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   816
shows "P x"
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   817
proof -
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   818
  obtain n where "n = V x" by auto
24523
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   819
  moreover have "!!x. V x = n \<Longrightarrow> P x"
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   820
  proof (induct n rule: infinite_descent0)
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   821
    case 0 -- "i.e. $V(x) = 0$"
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   822
    with 0 show "P x" by auto
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   823
  next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   824
    case (smaller n)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   825
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   826
    with 1 obtain y where "V y < V x \<and> \<not> P y" by auto
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   827
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   828
    thus ?case by auto
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   829
  qed
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   830
  ultimately show "P x" by auto
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   831
qed
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   832
24523
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   833
text{* Again, without explicit base case: *}
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   834
lemma infinite_descent_measure:
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   835
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   836
proof -
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   837
  from assms obtain n where "n = V x" by auto
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   838
  moreover have "!!x. V x = n \<Longrightarrow> P x"
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   839
  proof (induct n rule: infinite_descent, auto)
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   840
    fix x assume "\<not> P x"
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   841
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   842
  qed
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   843
  ultimately show "P x" by auto
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   844
qed
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   845
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   846
cd723b2209ea added variations on infinite descent
nipkow
parents: 24438
diff changeset
   847
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   848
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
   849
  monotonicity to @{text "\<le>"} monotonicity *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   850
lemma less_mono_imp_le_mono:
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   851
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   852
by (simp add: order_le_less) (blast)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   853
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   854
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   855
text {* non-strict, in 1st argument *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   856
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   857
by (rule add_right_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   858
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   859
text {* non-strict, in both arguments *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   860
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   861
by (rule add_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   862
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   863
lemma le_add2: "n \<le> ((m + n)::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   864
by (insert add_right_mono [of 0 m n], simp)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   865
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   866
lemma le_add1: "n \<le> ((n + m)::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   867
by (simp add: add_commute, rule le_add2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   868
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   869
lemma less_add_Suc1: "i < Suc (i + m)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   870
by (rule le_less_trans, rule le_add1, rule lessI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   871
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   872
lemma less_add_Suc2: "i < Suc (m + i)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   873
by (rule le_less_trans, rule le_add2, rule lessI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   874
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   875
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   876
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   877
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   878
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   879
by (rule le_trans, assumption, rule le_add1)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   880
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   881
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   882
by (rule le_trans, assumption, rule le_add2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   883
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   884
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   885
by (rule less_le_trans, assumption, rule le_add1)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   886
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   887
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   888
by (rule less_le_trans, assumption, rule le_add2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   889
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   890
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   891
apply (rule le_less_trans [of _ "i+j"])
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   892
apply (simp_all add: le_add1)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   893
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   894
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   895
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   896
apply (rule notI)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   897
apply (erule add_lessD1 [THEN less_irrefl])
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   898
done
13449
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 not_add_less2 [iff]: "~ (j + i < (i::nat))"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   901
by (simp add: add_commute not_add_less1)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   902
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   903
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   904
apply (rule order_trans [of _ "m+k"])
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   905
apply (simp_all add: le_add1)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   906
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   907
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   908
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   909
apply (simp add: add_commute)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   910
apply (erule add_leD1)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   911
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   912
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   913
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   914
by (blast dest: add_leD1 add_leD2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   915
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   916
text {* needs @{text "!!k"} for @{text add_ac} to work *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   917
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   918
by (force simp del: add_Suc_right
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   919
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   920
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   921
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   922
subsection {* Difference *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   923
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   924
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   925
by (induct m) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   926
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   927
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
   928
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   929
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   930
by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   931
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   932
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   933
by (simp add: add_diff_inverse linorder_not_less)
13449
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 le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   936
by (simp add: le_add_diff_inverse add_commute)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   937
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   938
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   939
subsection {* More results about difference *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   940
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   941
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   942
by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   943
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   944
lemma diff_less_Suc: "m - n < Suc m"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   945
apply (induct m n rule: diff_induct)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   946
apply (erule_tac [3] less_SucE)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   947
apply (simp_all add: less_Suc_eq)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   948
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   949
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   950
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   951
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   952
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   953
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   954
by (rule le_less_trans, rule diff_le_self)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   955
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   956
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   957
by (induct i j rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   958
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   959
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   960
by (simp add: diff_diff_left)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   961
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   962
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   963
by (cases n) (auto simp add: le_simps)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   964
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   965
text {* This and the next few suggested by Florian Kammueller *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   966
lemma diff_commute: "(i::nat) - j - k = i - k - j"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   967
by (simp add: diff_diff_left add_commute)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   968
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   969
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   970
by (induct j k rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   971
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   972
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   973
by (simp add: add_commute diff_add_assoc)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   974
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   975
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   976
by (induct n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   977
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   978
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   979
by (simp add: diff_add_assoc)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   980
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   981
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   982
by (auto simp add: diff_add_inverse2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   983
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   984
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   985
by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   986
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   987
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   988
by (rule iffD2, rule diff_is_0_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   989
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   990
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
   991
by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   992
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   993
lemma less_imp_add_positive:
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   994
  assumes "i < j"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   995
  shows "\<exists>k::nat. 0 < k & i + k = j"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   996
proof
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   997
  from assms show "0 < j - i & i + (j - i) = j"
23476
839db6346cc8 fix looping simp rule
huffman
parents: 23438
diff changeset
   998
    by (simp add: order_less_imp_le)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   999
qed
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1000
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1001
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1002
by (induct k) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1003
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1004
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1005
by (simp add: diff_cancel add_commute)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1006
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1007
lemma diff_add_0: "n - (n + m) = (0::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1008
by (induct n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1009
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1010
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1011
text {* Difference distributes over multiplication *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1012
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1013
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1014
by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1015
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1016
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1017
by (simp add: diff_mult_distrib mult_commute [of k])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1018
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1019
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1020
lemmas nat_distrib =
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1021
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1022
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1023
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1024
subsection {* Monotonicity of Multiplication *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1025
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1026
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1027
by (simp add: mult_right_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1028
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1029
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1030
by (simp add: mult_left_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1031
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1032
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1033
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1034
by (simp add: mult_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1035
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1036
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1037
by (simp add: mult_strict_right_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1038
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
  1039
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
  1040
      there are no negative numbers.*}
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
  1041
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
  1042
  apply (induct m)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1043
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1044
  apply (case_tac n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1045
   apply simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1046
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1047
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1048
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
  1049
  apply (induct m)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1050
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1051
  apply (case_tac n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1052
   apply simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1053
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1054
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1055
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1056
  apply (induct m)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1057
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1058
  apply (induct n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1059
   apply auto
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1060
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1061
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24196
diff changeset
  1062
lemma one_eq_mult_iff [simp,noatp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1063
  apply (rule trans)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
  1064
  apply (rule_tac [2] mult_eq_1_iff, fastsimp)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1065
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1066
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
  1067
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
  1068
  apply (safe intro!: mult_less_mono1)
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
  1069
  apply (case_tac k, auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1070
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1071
  apply (blast intro: mult_le_mono1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1072
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1073
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1074
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1075
by (simp add: mult_commute [of k])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1076
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1077
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1078
by (simp add: linorder_not_less [symmetric], auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1079
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1080
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1081
by (simp add: linorder_not_less [symmetric], auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1082
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
  1083
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
  1084
apply (cut_tac less_linear, safe, auto simp add: neq0_conv)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
  1085
apply (drule mult_less_mono1, assumption, simp)+
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
  1086
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1087
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1088
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1089
by (simp add: mult_commute [of k])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1090
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1091
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1092
by (subst mult_less_cancel1) simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1093
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1094
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1095
by (subst mult_le_cancel1) simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1096
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1097
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1098
by (subst mult_cancel1) simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1099
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1100
text {* Lemma for @{text gcd} *}
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1101
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
  1102
  apply (drule sym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1103
  apply (rule disjCI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1104
  apply (rule nat_less_cases, erule_tac [2] _)
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
  1105
  apply (fastsimp elim!: less_SucE)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
  1106
  apply (auto simp add: neq0_conv dest: mult_less_mono2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1107
  done
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1108
20588
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1109
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1110
subsection {* size of a datatype value *}
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1111
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1112
class size = type +
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1113
  fixes size :: "'a \<Rightarrow> nat"
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1114
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1115
use "Tools/function_package/size.ML"
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1116
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1117
setup Size.setup
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1118
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1119
lemma nat_size [simp, code func]: "size (n\<Colon>nat) = n"
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1120
  by (induct n) simp_all
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1121
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1122
lemma size_bool [code func]:
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1123
  "size (b\<Colon>bool) = 0" by (cases b) auto
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1124
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1125
declare "*.size" [noatp]
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1126
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1127
18702
7dc7dcd63224 substantial improvements in code generator
haftmann
parents: 18648
diff changeset
  1128
subsection {* Code generator setup *}
7dc7dcd63224 substantial improvements in code generator
haftmann
parents: 18648
diff changeset
  1129
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1130
lemma one_is_Suc_zero [code inline]: "1 = Suc 0"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1131
by simp
20355
50aaae6ae4db cleanup code generation for Numerals
haftmann
parents: 19890
diff changeset
  1132
20588
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1133
instance nat :: eq ..
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1134
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1135
lemma [code func]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1136
    "(0\<Colon>nat) = 0 \<longleftrightarrow> True"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1137
    "Suc n = Suc m \<longleftrightarrow> n = m"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1138
    "Suc n = 0 \<longleftrightarrow> False"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1139
    "0 = Suc m \<longleftrightarrow> False"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1140
by auto
20588
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1141
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1142
lemma [code func]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1143
    "(0\<Colon>nat) \<le> m \<longleftrightarrow> True"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1144
    "Suc (n\<Colon>nat) \<le> m \<longleftrightarrow> n < m"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1145
    "(n\<Colon>nat) < 0 \<longleftrightarrow> False"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1146
    "(n\<Colon>nat) < Suc m \<longleftrightarrow> n \<le> m"
22348
ab505d281015 adjusted code lemmas
haftmann
parents: 22318
diff changeset
  1147
  using Suc_le_eq less_Suc_eq_le by simp_all
20588
c847c56edf0c added operational equality
haftmann
parents: 20380
diff changeset
  1148
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1149
24196
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1150
subsection{*Embedding of the Naturals into any
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1151
  @{text semiring_1}: @{term of_nat}*}
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1152
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1153
context semiring_1
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1154
begin
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1155
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1156
definition
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24995
diff changeset
  1157
  of_nat_def: "of_nat = nat_rec 0 (\<lambda>_. (op +) 1)"
24196
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1158
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1159
end
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1160
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1161
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1162
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
  1163
lemma subst_equals:
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
  1164
  assumes 1: "t = s" and 2: "u = t"
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
  1165
  shows "u = s"
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
  1166
  using 2 1 by (rule trans)
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
  1167
24196
f1dbfd7e3223 localized of_nat
haftmann
parents: 24162
diff changeset
  1168
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1169
use "arith_data.ML"
24091
109f19a13872 added Tools/lin_arith.ML;
wenzelm
parents: 24075
diff changeset
  1170
declaration {* K arith_data_setup *}
109f19a13872 added Tools/lin_arith.ML;
wenzelm
parents: 24075
diff changeset
  1171
109f19a13872 added Tools/lin_arith.ML;
wenzelm
parents: 24075
diff changeset
  1172
use "Tools/lin_arith.ML"
109f19a13872 added Tools/lin_arith.ML;
wenzelm
parents: 24075
diff changeset
  1173
declaration {* K LinArith.setup *}
109f19a13872 added Tools/lin_arith.ML;
wenzelm
parents: 24075
diff changeset
  1174
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1175
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1176
text{*The following proofs may rely on the arithmetic proof procedures.*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1177
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1178
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1179
by (auto simp: le_eq_less_or_eq dest: less_imp_Suc_add)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1180
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1181
lemma pred_nat_trancl_eq_le: "((m, n) : pred_nat^*) = (m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1182
by (simp add: less_eq reflcl_trancl [symmetric] del: reflcl_trancl, arith)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1183
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1184
lemma nat_diff_split:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1185
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1186
    -- {* elimination of @{text -} on @{text nat} *}
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1187
by (cases "a<b" rule: case_split) (auto simp add: diff_is_0_eq [THEN iffD2])
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1188
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1189
lemma nat_diff_split_asm:
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1190
    "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1191
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1192
by (simp split: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1193
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1194
lemmas [arith_split] = nat_diff_split split_min split_max
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1195
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1196
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1197
lemma le_square: "m \<le> m * (m::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1198
by (induct m) auto
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1199
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1200
lemma le_cube: "(m::nat) \<le> m * (m * m)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1201
by (induct m) auto
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1202
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1203
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1204
text{*Subtraction laws, mostly by Clemens Ballarin*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1205
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1206
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1207
by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1208
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1209
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1210
by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1211
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1212
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1213
by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1214
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1215
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1216
by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1217
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1218
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1219
by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1220
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1221
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1222
by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1223
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1224
(*Replaces the previous diff_less and le_diff_less, which had the stronger
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1225
  second premise n\<le>m*)
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1226
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1227
by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1228
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1229
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1230
(** Simplification of relational expressions involving subtraction **)
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1231
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1232
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1233
by (simp split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1234
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1235
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1236
by (auto split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1237
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1238
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1239
by (auto split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1240
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1241
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1242
by (auto split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1244
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1245
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1246
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1247
(* Monotonicity of subtraction in first argument *)
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1248
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1249
by (simp split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1250
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1251
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1252
by (simp split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1253
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1254
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1255
by (simp split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1256
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1257
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1258
by (simp split add: nat_diff_split)
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1259
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1260
text{*Lemmas for ex/Factorization*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1261
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1262
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1263
by (cases m) auto
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1264
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1265
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1266
by (cases m) auto
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1267
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1268
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1269
by (cases m) auto
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1270
23001
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1271
text {* Specialized induction principles that work "backwards": *}
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1272
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1273
lemma inc_induct[consumes 1, case_names base step]:
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1274
  assumes less: "i <= j"
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1275
  assumes base: "P j"
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1276
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1277
  shows "P i"
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1278
  using less
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1279
proof (induct d=="j - i" arbitrary: i)
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1280
  case (0 i)
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1281
  hence "i = j" by simp
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1282
  with base show ?case by simp
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1283
next
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1284
  case (Suc d i)
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1285
  hence "i < j" "P (Suc i)"
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1286
    by simp_all
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1287
  thus "P i" by (rule step)
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1288
qed
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1289
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1290
lemma strict_inc_induct[consumes 1, case_names base step]:
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1291
  assumes less: "i < j"
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1292
  assumes base: "!!i. j = Suc i ==> P i"
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1293
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1294
  shows "P i"
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1295
  using less
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1296
proof (induct d=="j - i - 1" arbitrary: i)
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1297
  case (0 i)
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1298
  with `i < j` have "j = Suc i" by simp
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1299
  with base show ?case by simp
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1300
next
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1301
  case (Suc d i)
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1302
  hence "i < j" "P (Suc i)"
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1303
    by simp_all
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1304
  thus "P i" by (rule step)
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1305
qed
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1306
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1307
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1308
  using inc_induct[of "k - i" k P, simplified] by blast
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1309
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1310
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
3608f0362a91 added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents: 22920
diff changeset
  1311
  using inc_induct[of 0 k P] by blast
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1312
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1313
text{*Rewriting to pull differences out*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1314
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1315
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1316
by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1317
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1318
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1319
by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1320
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1321
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1322
by arith
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1323
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1324
(*The others are
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1325
      i - j - k = i - (j + k),
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1326
      k \<le> j ==> j - k + i = j + i - k,
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1327
      k \<le> j ==> i + (j - k) = i + j - k *)
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1328
lemmas add_diff_assoc = diff_add_assoc [symmetric]
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1329
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1330
declare diff_diff_left [simp]  add_diff_assoc [simp]  add_diff_assoc2[simp]
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1331
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1332
text{*At present we prove no analogue of @{text not_less_Least} or @{text
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1333
Least_Suc}, since there appears to be no need.*}
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
  1334
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1335
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1336
subsection{*Embedding of the Naturals into any
23276
a70934b63910 generalize of_nat and related constants to class semiring_1
huffman
parents: 23263
diff changeset
  1337
  @{text semiring_1}: @{term of_nat}*}
21243