src/HOL/Int.thy
author haftmann
Tue, 27 Oct 2020 16:59:44 +0000
changeset 72512 83b5911c0164
parent 71837 dca11678c495
child 73109 783406dd051e
permissions -rw-r--r--
more lemmas
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(*  Title:      HOL/Int.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Author:     Tobias Nipkow, Florian Haftmann, TU Muenchen
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*)
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section \<open>The Integers as Equivalence Classes over Pairs of Natural Numbers\<close>
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theory Int
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  imports Equiv_Relations Power Quotient Fun_Def
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begin
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subsection \<open>Definition of integers as a quotient type\<close>
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definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool"
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  where "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)"
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lemma intrel_iff [simp]: "intrel (x, y) (u, v) \<longleftrightarrow> x + v = u + y"
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  by (simp add: intrel_def)
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quotient_type int = "nat \<times> nat" / "intrel"
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  morphisms Rep_Integ Abs_Integ
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proof (rule equivpI)
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  show "reflp intrel" by (auto simp: reflp_def)
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  show "symp intrel" by (auto simp: symp_def)
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  show "transp intrel" by (auto simp: transp_def)
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qed
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lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
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  "(\<And>x y. z = Abs_Integ (x, y) \<Longrightarrow> P) \<Longrightarrow> P"
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  by (induct z) auto
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subsection \<open>Integers form a commutative ring\<close>
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instantiation int :: comm_ring_1
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begin
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lift_definition zero_int :: "int" is "(0, 0)" .
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lift_definition one_int :: "int" is "(1, 0)" .
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lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x + u, y + v)"
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  by clarsimp
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lift_definition uminus_int :: "int \<Rightarrow> int"
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  is "\<lambda>(x, y). (y, x)"
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  by clarsimp
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lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x + v, y + u)"
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  by clarsimp
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lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int"
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  is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)"
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proof (clarsimp)
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  fix s t u v w x y z :: nat
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  assume "s + v = u + t" and "w + z = y + x"
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  then have "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x) =
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    (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
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    by simp
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  then show "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
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    by (simp add: algebra_simps)
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qed
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instance
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  by standard (transfer; clarsimp simp: algebra_simps)+
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end
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abbreviation int :: "nat \<Rightarrow> int"
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  where "int \<equiv> of_nat"
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lemma int_def: "int n = Abs_Integ (n, 0)"
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  by (induct n) (simp add: zero_int.abs_eq, simp add: one_int.abs_eq plus_int.abs_eq)
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lemma int_transfer [transfer_rule]:
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  includes lifting_syntax
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  shows "rel_fun (=) pcr_int (\<lambda>n. (n, 0)) int"
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  by (simp add: rel_fun_def int.pcr_cr_eq cr_int_def int_def)
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lemma int_diff_cases: obtains (diff) m n where "z = int m - int n"
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  by transfer clarsimp
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subsection \<open>Integers are totally ordered\<close>
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instantiation int :: linorder
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begin
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lift_definition less_eq_int :: "int \<Rightarrow> int \<Rightarrow> bool"
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  is "\<lambda>(x, y) (u, v). x + v \<le> u + y"
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  by auto
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lift_definition less_int :: "int \<Rightarrow> int \<Rightarrow> bool"
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  is "\<lambda>(x, y) (u, v). x + v < u + y"
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  by auto
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instance
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  by standard (transfer, force)+
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end
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instantiation int :: distrib_lattice
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begin
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definition "(inf :: int \<Rightarrow> int \<Rightarrow> int) = min"
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definition "(sup :: int \<Rightarrow> int \<Rightarrow> int) = max"
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instance
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  by standard (auto simp add: inf_int_def sup_int_def max_min_distrib2)
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end
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subsection \<open>Ordering properties of arithmetic operations\<close>
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instance int :: ordered_cancel_ab_semigroup_add
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proof
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  fix i j k :: int
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  show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
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    by transfer clarsimp
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qed
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text \<open>Strict Monotonicity of Multiplication.\<close>
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text \<open>Strict, in 1st argument; proof is by induction on \<open>k > 0\<close>.\<close>
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lemma zmult_zless_mono2_lemma: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> int k * i < int k * j"
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  for i j :: int
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proof (induct k)
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  case 0
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  then show ?case by simp
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next
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  case (Suc k)
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  then show ?case
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    by (cases "k = 0") (simp_all add: distrib_right add_strict_mono)
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qed
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lemma zero_le_imp_eq_int:
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  assumes "k \<ge> (0::int)" shows "\<exists>n. k = int n"
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proof -
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  have "b \<le> a \<Longrightarrow> \<exists>n::nat. a = n + b" for a b
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    by (rule_tac x="a - b" in exI) simp
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  with assms show ?thesis
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    by transfer auto
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qed
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lemma zero_less_imp_eq_int:
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  assumes "k > (0::int)" shows "\<exists>n>0. k = int n"
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proof -
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  have "b < a \<Longrightarrow> \<exists>n::nat. n>0 \<and> a = n + b" for a b
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   152
    by (rule_tac x="a - b" in exI) simp
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   153
  with assms show ?thesis
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   154
    by transfer auto
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   155
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   156
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   157
lemma zmult_zless_mono2: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   158
  for i j k :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   159
  by (drule zero_less_imp_eq_int) (auto simp add: zmult_zless_mono2_lemma)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   160
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   161
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   162
text \<open>The integers form an ordered integral domain.\<close>
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   163
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   164
instantiation int :: linordered_idom
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   165
begin
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   166
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   167
definition zabs_def: "\<bar>i::int\<bar> = (if i < 0 then - i else i)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   168
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   169
definition zsgn_def: "sgn (i::int) = (if i = 0 then 0 else if 0 < i then 1 else - 1)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   170
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   171
instance
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   172
proof
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   173
  fix i j k :: int
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   174
  show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   175
    by (rule zmult_zless_mono2)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   176
  show "\<bar>i\<bar> = (if i < 0 then -i else i)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   177
    by (simp only: zabs_def)
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
   178
  show "sgn (i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   179
    by (simp only: zsgn_def)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   180
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   181
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   182
end
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   183
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   184
lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + 1 \<le> z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   185
  for w z :: int
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   186
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   187
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   188
lemma zless_iff_Suc_zadd: "w < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   189
  for w z :: int
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   190
proof -
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   191
  have "\<And>a b c d. a + d < c + b \<Longrightarrow> \<exists>n. c + b = Suc (a + n + d)"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   192
    by (rule_tac x="c+b - Suc(a+d)" in exI) arith
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   193
  then show ?thesis
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   194
    by transfer auto
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   195
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   196
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   197
lemma zabs_less_one_iff [simp]: "\<bar>z\<bar> < 1 \<longleftrightarrow> z = 0" (is "?lhs \<longleftrightarrow> ?rhs")
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   198
  for z :: int
62347
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   199
proof
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   200
  assume ?rhs
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   201
  then show ?lhs by simp
62347
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   202
next
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   203
  assume ?lhs
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   204
  with zless_imp_add1_zle [of "\<bar>z\<bar>" 1] have "\<bar>z\<bar> + 1 \<le> 1" by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   205
  then have "\<bar>z\<bar> \<le> 0" by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   206
  then show ?rhs by simp
62347
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   207
qed
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   208
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   209
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61738
diff changeset
   210
subsection \<open>Embedding of the Integers into any \<open>ring_1\<close>: \<open>of_int\<close>\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   211
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   212
context ring_1
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   213
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   214
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   215
lift_definition of_int :: "int \<Rightarrow> 'a"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   216
  is "\<lambda>(i, j). of_nat i - of_nat j"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   217
  by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   218
      of_nat_add [symmetric] simp del: of_nat_add)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   219
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   220
lemma of_int_0 [simp]: "of_int 0 = 0"
48066
c6783c9b87bf transfer method now handles transfer rules for compound terms, e.g. locale-defined constants with hidden parameters
huffman
parents: 48045
diff changeset
   221
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   222
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   223
lemma of_int_1 [simp]: "of_int 1 = 1"
48066
c6783c9b87bf transfer method now handles transfer rules for compound terms, e.g. locale-defined constants with hidden parameters
huffman
parents: 48045
diff changeset
   224
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   225
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   226
lemma of_int_add [simp]: "of_int (w + z) = of_int w + of_int z"
48066
c6783c9b87bf transfer method now handles transfer rules for compound terms, e.g. locale-defined constants with hidden parameters
huffman
parents: 48045
diff changeset
   227
  by transfer (clarsimp simp add: algebra_simps)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   228
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   229
lemma of_int_minus [simp]: "of_int (- z) = - (of_int z)"
48066
c6783c9b87bf transfer method now handles transfer rules for compound terms, e.g. locale-defined constants with hidden parameters
huffman
parents: 48045
diff changeset
   230
  by (transfer fixing: uminus) clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   231
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   232
lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54223
diff changeset
   233
  using of_int_add [of w "- z"] by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   234
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   235
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   236
  by (transfer fixing: times) (clarsimp simp add: algebra_simps)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   237
61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   238
lemma mult_of_int_commute: "of_int x * y = y * of_int x"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   239
  by (transfer fixing: times) (auto simp: algebra_simps mult_of_nat_commute)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61524
diff changeset
   240
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   241
text \<open>Collapse nested embeddings.\<close>
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   242
lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   243
  by (induct n) auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   244
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   245
lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   246
  by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   247
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   248
lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   249
  by simp
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   250
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   251
lemma of_int_power [simp]: "of_int (z ^ n) = of_int z ^ n"
31015
555f4033cd97 reorganization of power lemmas
haftmann
parents: 31010
diff changeset
   252
  by (induct n) simp_all
555f4033cd97 reorganization of power lemmas
haftmann
parents: 31010
diff changeset
   253
66816
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   254
lemma of_int_of_bool [simp]:
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   255
  "of_int (of_bool P) = of_bool P"
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   256
  by auto
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   257
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   258
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   259
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   260
context ring_char_0
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   261
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   262
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   263
lemma of_int_eq_iff [simp]: "of_int w = of_int z \<longleftrightarrow> w = z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   264
  by transfer (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   265
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   266
text \<open>Special cases where either operand is zero.\<close>
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   267
lemma of_int_eq_0_iff [simp]: "of_int z = 0 \<longleftrightarrow> z = 0"
36424
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   268
  using of_int_eq_iff [of z 0] by simp
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   269
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   270
lemma of_int_0_eq_iff [simp]: "0 = of_int z \<longleftrightarrow> z = 0"
36424
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   271
  using of_int_eq_iff [of 0 z] by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   272
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   273
lemma of_int_eq_1_iff [iff]: "of_int z = 1 \<longleftrightarrow> z = 1"
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   274
  using of_int_eq_iff [of z 1] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   275
66912
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   276
lemma numeral_power_eq_of_int_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   277
  "numeral x ^ n = of_int y \<longleftrightarrow> numeral x ^ n = y"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   278
  using of_int_eq_iff[of "numeral x ^ n" y, unfolded of_int_numeral of_int_power] .
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   279
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   280
lemma of_int_eq_numeral_power_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   281
  "of_int y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   282
  using numeral_power_eq_of_int_cancel_iff [of x n y] by (metis (mono_tags))
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   283
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   284
lemma neg_numeral_power_eq_of_int_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   285
  "(- numeral x) ^ n = of_int y \<longleftrightarrow> (- numeral x) ^ n = y"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   286
  using of_int_eq_iff[of "(- numeral x) ^ n" y]
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   287
  by simp
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   288
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   289
lemma of_int_eq_neg_numeral_power_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   290
  "of_int y = (- numeral x) ^ n \<longleftrightarrow> y = (- numeral x) ^ n"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   291
  using neg_numeral_power_eq_of_int_cancel_iff[of x n y] by (metis (mono_tags))
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   292
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   293
lemma of_int_eq_of_int_power_cancel_iff[simp]: "(of_int b) ^ w = of_int x \<longleftrightarrow> b ^ w = x"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   294
  by (metis of_int_power of_int_eq_iff)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   295
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   296
lemma of_int_power_eq_of_int_cancel_iff[simp]: "of_int x = (of_int b) ^ w \<longleftrightarrow> x = b ^ w"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   297
  by (metis of_int_eq_of_int_power_cancel_iff)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   298
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   299
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   300
36424
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   301
context linordered_idom
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   302
begin
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   303
63652
804b80a80016 misc tuning and modernization;
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parents: 63648
diff changeset
   304
text \<open>Every \<open>linordered_idom\<close> has characteristic zero.\<close>
36424
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   305
subclass ring_char_0 ..
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   306
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   307
lemma of_int_le_iff [simp]: "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   308
  by (transfer fixing: less_eq)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   309
    (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add)
36424
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   310
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   311
lemma of_int_less_iff [simp]: "of_int w < of_int z \<longleftrightarrow> w < z"
36424
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   312
  by (simp add: less_le order_less_le)
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   313
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   314
lemma of_int_0_le_iff [simp]: "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
36424
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   315
  using of_int_le_iff [of 0 z] by simp
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   316
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   317
lemma of_int_le_0_iff [simp]: "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
36424
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   318
  using of_int_le_iff [of z 0] by simp
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   319
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   320
lemma of_int_0_less_iff [simp]: "0 < of_int z \<longleftrightarrow> 0 < z"
36424
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   321
  using of_int_less_iff [of 0 z] by simp
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   322
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   323
lemma of_int_less_0_iff [simp]: "of_int z < 0 \<longleftrightarrow> z < 0"
36424
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   324
  using of_int_less_iff [of z 0] by simp
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   325
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   326
lemma of_int_1_le_iff [simp]: "1 \<le> of_int z \<longleftrightarrow> 1 \<le> z"
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   327
  using of_int_le_iff [of 1 z] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   328
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   329
lemma of_int_le_1_iff [simp]: "of_int z \<le> 1 \<longleftrightarrow> z \<le> 1"
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   330
  using of_int_le_iff [of z 1] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   331
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   332
lemma of_int_1_less_iff [simp]: "1 < of_int z \<longleftrightarrow> 1 < z"
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   333
  using of_int_less_iff [of 1 z] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   334
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   335
lemma of_int_less_1_iff [simp]: "of_int z < 1 \<longleftrightarrow> z < 1"
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   336
  using of_int_less_iff [of z 1] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   337
62128
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61944
diff changeset
   338
lemma of_int_pos: "z > 0 \<Longrightarrow> of_int z > 0"
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61944
diff changeset
   339
  by simp
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61944
diff changeset
   340
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61944
diff changeset
   341
lemma of_int_nonneg: "z \<ge> 0 \<Longrightarrow> of_int z \<ge> 0"
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61944
diff changeset
   342
  by simp
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61944
diff changeset
   343
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   344
lemma of_int_abs [simp]: "of_int \<bar>x\<bar> = \<bar>of_int x\<bar>"
62347
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   345
  by (auto simp add: abs_if)
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   346
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   347
lemma of_int_lessD:
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   348
  assumes "\<bar>of_int n\<bar> < x"
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   349
  shows "n = 0 \<or> x > 1"
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   350
proof (cases "n = 0")
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   351
  case True
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   352
  then show ?thesis by simp
62347
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   353
next
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   354
  case False
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   355
  then have "\<bar>n\<bar> \<noteq> 0" by simp
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   356
  then have "\<bar>n\<bar> > 0" by simp
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   357
  then have "\<bar>n\<bar> \<ge> 1"
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   358
    using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   359
  then have "\<bar>of_int n\<bar> \<ge> 1"
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   360
    unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   361
  then have "1 < x" using assms by (rule le_less_trans)
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   362
  then show ?thesis ..
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   363
qed
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   364
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   365
lemma of_int_leD:
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   366
  assumes "\<bar>of_int n\<bar> \<le> x"
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   367
  shows "n = 0 \<or> 1 \<le> x"
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   368
proof (cases "n = 0")
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   369
  case True
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   370
  then show ?thesis by simp
62347
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   371
next
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   372
  case False
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   373
  then have "\<bar>n\<bar> \<noteq> 0" by simp
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   374
  then have "\<bar>n\<bar> > 0" by simp
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   375
  then have "\<bar>n\<bar> \<ge> 1"
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   376
    using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   377
  then have "\<bar>of_int n\<bar> \<ge> 1"
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   378
    unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   379
  then have "1 \<le> x" using assms by (rule order_trans)
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   380
  then show ?thesis ..
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   381
qed
2230b7047376 generalized some lemmas;
haftmann
parents: 62128
diff changeset
   382
66912
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   383
lemma numeral_power_le_of_int_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   384
  "numeral x ^ n \<le> of_int a \<longleftrightarrow> numeral x ^ n \<le> a"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   385
  by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_le_iff)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   386
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   387
lemma of_int_le_numeral_power_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   388
  "of_int a \<le> numeral x ^ n \<longleftrightarrow> a \<le> numeral x ^ n"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   389
  by (metis (mono_tags) local.numeral_power_eq_of_int_cancel_iff of_int_le_iff)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   390
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   391
lemma numeral_power_less_of_int_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   392
  "numeral x ^ n < of_int a \<longleftrightarrow> numeral x ^ n < a"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   393
  by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_less_iff)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   394
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   395
lemma of_int_less_numeral_power_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   396
  "of_int a < numeral x ^ n \<longleftrightarrow> a < numeral x ^ n"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   397
  by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_less_iff)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   398
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   399
lemma neg_numeral_power_le_of_int_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   400
  "(- numeral x) ^ n \<le> of_int a \<longleftrightarrow> (- numeral x) ^ n \<le> a"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   401
  by (metis (mono_tags) of_int_le_iff of_int_neg_numeral of_int_power)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   402
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   403
lemma of_int_le_neg_numeral_power_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   404
  "of_int a \<le> (- numeral x) ^ n \<longleftrightarrow> a \<le> (- numeral x) ^ n"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   405
  by (metis (mono_tags) of_int_le_iff of_int_neg_numeral of_int_power)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   406
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   407
lemma neg_numeral_power_less_of_int_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   408
  "(- numeral x) ^ n < of_int a \<longleftrightarrow> (- numeral x) ^ n < a"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   409
  using of_int_less_iff[of "(- numeral x) ^ n" a]
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   410
  by simp
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   411
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   412
lemma of_int_less_neg_numeral_power_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   413
  "of_int a < (- numeral x) ^ n \<longleftrightarrow> a < (- numeral x::int) ^ n"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   414
  using of_int_less_iff[of a "(- numeral x) ^ n"]
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   415
  by simp
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   416
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   417
lemma of_int_le_of_int_power_cancel_iff[simp]: "(of_int b) ^ w \<le> of_int x \<longleftrightarrow> b ^ w \<le> x"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   418
  by (metis (mono_tags) of_int_le_iff of_int_power)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   419
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   420
lemma of_int_power_le_of_int_cancel_iff[simp]: "of_int x \<le> (of_int b) ^ w\<longleftrightarrow> x \<le> b ^ w"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   421
  by (metis (mono_tags) of_int_le_iff of_int_power)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   422
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   423
lemma of_int_less_of_int_power_cancel_iff[simp]: "(of_int b) ^ w < of_int x \<longleftrightarrow> b ^ w < x"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   424
  by (metis (mono_tags) of_int_less_iff of_int_power)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   425
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   426
lemma of_int_power_less_of_int_cancel_iff[simp]: "of_int x < (of_int b) ^ w\<longleftrightarrow> x < b ^ w"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   427
  by (metis (mono_tags) of_int_less_iff of_int_power)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
   428
67969
83c8cafdebe8 Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   429
lemma of_int_max: "of_int (max x y) = max (of_int x) (of_int y)"
83c8cafdebe8 Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   430
  by (auto simp: max_def)
83c8cafdebe8 Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   431
83c8cafdebe8 Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   432
lemma of_int_min: "of_int (min x y) = min (of_int x) (of_int y)"
83c8cafdebe8 Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   433
  by (auto simp: min_def)
83c8cafdebe8 Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   434
36424
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   435
end
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   436
69791
195aeee8b30a Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents: 69700
diff changeset
   437
context division_ring
195aeee8b30a Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents: 69700
diff changeset
   438
begin
195aeee8b30a Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents: 69700
diff changeset
   439
195aeee8b30a Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents: 69700
diff changeset
   440
lemmas mult_inverse_of_int_commute =
195aeee8b30a Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents: 69700
diff changeset
   441
  mult_commute_imp_mult_inverse_commute[OF mult_of_int_commute]
195aeee8b30a Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents: 69700
diff changeset
   442
195aeee8b30a Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents: 69700
diff changeset
   443
end
195aeee8b30a Formal Laurent series and overhaul of Formal power series (due to Jeremy Sylvestre)
Manuel Eberl <eberlm@in.tum.de>
parents: 69700
diff changeset
   444
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69182
diff changeset
   445
text \<open>Comparisons involving \<^term>\<open>of_int\<close>.\<close>
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   446
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   447
lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) \<longleftrightarrow> z = numeral n"
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   448
  using of_int_eq_iff by fastforce
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   449
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
   450
lemma of_int_le_numeral_iff [simp]:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   451
  "of_int z \<le> (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z \<le> numeral n"
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   452
  using of_int_le_iff [of z "numeral n"] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   453
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
   454
lemma of_int_numeral_le_iff [simp]:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   455
  "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   456
  using of_int_le_iff [of "numeral n"] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   457
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
   458
lemma of_int_less_numeral_iff [simp]:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   459
  "of_int z < (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z < numeral n"
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   460
  using of_int_less_iff [of z "numeral n"] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   461
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61609
diff changeset
   462
lemma of_int_numeral_less_iff [simp]:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   463
  "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   464
  using of_int_less_iff [of "numeral n" z] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   465
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   466
lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
56889
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56525
diff changeset
   467
  by (metis of_int_of_nat_eq of_int_less_iff)
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56525
diff changeset
   468
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   469
lemma of_int_eq_id [simp]: "of_int = id"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   470
proof
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   471
  show "of_int z = id z" for z
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   472
    by (cases z rule: int_diff_cases) simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   473
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   474
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   475
instance int :: no_top
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   476
proof
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   477
  show "\<And>x::int. \<exists>y. x < y"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   478
    by (rule_tac x="x + 1" in exI) simp
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   479
qed
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   480
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   481
instance int :: no_bot
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   482
proof
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   483
  show "\<And>x::int. \<exists>y. y < x"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   484
    by (rule_tac x="x - 1" in exI) simp
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   485
qed
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   486
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   487
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61738
diff changeset
   488
subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   489
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   490
lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   491
  by auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   492
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   493
lemma nat_int [simp]: "nat (int n) = n"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   494
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   495
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   496
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   497
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   498
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   499
lemma nat_0_le: "0 \<le> z \<Longrightarrow> int (nat z) = z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   500
  by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   501
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   502
lemma nat_le_0 [simp]: "z \<le> 0 \<Longrightarrow> nat z = 0"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   503
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   504
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   505
lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> nat w \<le> nat z \<longleftrightarrow> w \<le> z"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   506
  by transfer (clarsimp, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   507
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69182
diff changeset
   508
text \<open>An alternative condition is \<^term>\<open>0 \<le> w\<close>.\<close>
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   509
lemma nat_mono_iff: "0 < z \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   510
  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   511
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   512
lemma nat_less_eq_zless: "0 \<le> w \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   513
  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   514
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   515
lemma zless_nat_conj [simp]: "nat w < nat z \<longleftrightarrow> 0 < z \<and> w < z"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   516
  by transfer (clarsimp, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   517
64714
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   518
lemma nonneg_int_cases:
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   519
  assumes "0 \<le> k"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   520
  obtains n where "k = int n"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   521
proof -
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   522
  from assms have "k = int (nat k)"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   523
    by simp
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   524
  then show thesis
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   525
    by (rule that)
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   526
qed
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   527
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   528
lemma pos_int_cases:
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   529
  assumes "0 < k"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   530
  obtains n where "k = int n" and "n > 0"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   531
proof -
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   532
  from assms have "0 \<le> k"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   533
    by simp
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   534
  then obtain n where "k = int n"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   535
    by (rule nonneg_int_cases)
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   536
  moreover have "n > 0"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   537
    using \<open>k = int n\<close> assms by simp
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   538
  ultimately show thesis
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   539
    by (rule that)
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   540
qed
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   541
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   542
lemma nonpos_int_cases:
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   543
  assumes "k \<le> 0"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   544
  obtains n where "k = - int n"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   545
proof -
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   546
  from assms have "- k \<ge> 0"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   547
    by simp
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   548
  then obtain n where "- k = int n"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   549
    by (rule nonneg_int_cases)
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   550
  then have "k = - int n"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   551
    by simp
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   552
  then show thesis
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   553
    by (rule that)
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   554
qed
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   555
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   556
lemma neg_int_cases:
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   557
  assumes "k < 0"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   558
  obtains n where "k = - int n" and "n > 0"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   559
proof -
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   560
  from assms have "- k > 0"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   561
    by simp
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   562
  then obtain n where "- k = int n" and "- k > 0"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   563
    by (blast elim: pos_int_cases)
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   564
  then have "k = - int n" and "n > 0"
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   565
    by simp_all
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   566
  then show thesis
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   567
    by (rule that)
53bab28983f1 complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents: 64272
diff changeset
   568
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   569
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   570
lemma nat_eq_iff: "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   571
  by transfer (clarsimp simp add: le_imp_diff_is_add)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   572
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   573
lemma nat_eq_iff2: "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   574
  using nat_eq_iff [of w m] by auto
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   575
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   576
lemma nat_0 [simp]: "nat 0 = 0"
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   577
  by (simp add: nat_eq_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   578
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   579
lemma nat_1 [simp]: "nat 1 = Suc 0"
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   580
  by (simp add: nat_eq_iff)
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   581
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   582
lemma nat_numeral [simp]: "nat (numeral k) = numeral k"
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   583
  by (simp add: nat_eq_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   584
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   585
lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0"
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   586
  by simp
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   587
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   588
lemma nat_2: "nat 2 = Suc (Suc 0)"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   589
  by simp
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   590
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   591
lemma nat_less_iff: "0 \<le> w \<Longrightarrow> nat w < m \<longleftrightarrow> w < of_nat m"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   592
  by transfer (clarsimp, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   593
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   594
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   595
  by transfer (clarsimp simp add: le_diff_conv)
44707
487ae6317f7b move lemmas nat_le_iff and nat_mono into Int.thy
huffman
parents: 44695
diff changeset
   596
487ae6317f7b move lemmas nat_le_iff and nat_mono into Int.thy
huffman
parents: 44695
diff changeset
   597
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   598
  by transfer auto
44707
487ae6317f7b move lemmas nat_le_iff and nat_mono into Int.thy
huffman
parents: 44695
diff changeset
   599
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   600
lemma nat_0_iff[simp]: "nat i = 0 \<longleftrightarrow> i \<le> 0"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   601
  for i :: int
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   602
  by transfer clarsimp
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
   603
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   604
lemma int_eq_iff: "of_nat m = z \<longleftrightarrow> m = nat z \<and> 0 \<le> z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   605
  by (auto simp add: nat_eq_iff2)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   606
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   607
lemma zero_less_nat_eq [simp]: "0 < nat z \<longleftrightarrow> 0 < z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   608
  using zless_nat_conj [of 0] by auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   609
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   610
lemma nat_add_distrib: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   611
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   612
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   613
lemma nat_diff_distrib': "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   614
  by transfer clarsimp
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   615
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   616
lemma nat_diff_distrib: "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   617
  by (rule nat_diff_distrib') auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   618
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   619
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   620
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   621
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   622
lemma le_nat_iff: "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
53065
de1816a7293e added lemma
haftmann
parents: 52435
diff changeset
   623
  by transfer auto
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   624
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   625
lemma zless_nat_eq_int_zless: "m < nat z \<longleftrightarrow> int m < z"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   626
  by transfer (clarsimp simp add: less_diff_conv)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   627
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   628
lemma (in ring_1) of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
48066
c6783c9b87bf transfer method now handles transfer rules for compound terms, e.g. locale-defined constants with hidden parameters
huffman
parents: 48045
diff changeset
   629
  by transfer (clarsimp simp add: of_nat_diff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   630
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   631
lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
54249
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   632
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   633
66886
960509bfd47e added lemmas and tuned proofs
haftmann
parents: 66836
diff changeset
   634
lemma nat_abs_triangle_ineq:
960509bfd47e added lemmas and tuned proofs
haftmann
parents: 66836
diff changeset
   635
  "nat \<bar>k + l\<bar> \<le> nat \<bar>k\<bar> + nat \<bar>l\<bar>"
960509bfd47e added lemmas and tuned proofs
haftmann
parents: 66836
diff changeset
   636
  by (simp add: nat_add_distrib [symmetric] nat_le_eq_zle abs_triangle_ineq)
960509bfd47e added lemmas and tuned proofs
haftmann
parents: 66836
diff changeset
   637
66816
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   638
lemma nat_of_bool [simp]:
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   639
  "nat (of_bool P) = of_bool P"
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   640
  by auto
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   641
66836
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   642
lemma split_nat [arith_split]: "P (nat i) \<longleftrightarrow> ((\<forall>n. i = int n \<longrightarrow> P n) \<and> (i < 0 \<longrightarrow> P 0))"
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   643
  (is "?P = (?L \<and> ?R)")
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   644
  for i :: int
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   645
proof (cases "i < 0")
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   646
  case True
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   647
  then show ?thesis
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   648
    by auto
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   649
next
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   650
  case False
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   651
  have "?P = ?L"
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   652
  proof
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   653
    assume ?P
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   654
    then show ?L using False by auto
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   655
  next
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   656
    assume ?L
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   657
    moreover from False have "int (nat i) = i"
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   658
      by (simp add: not_less)
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   659
    ultimately show ?P
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   660
      by simp
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   661
  qed
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   662
  with False show ?thesis by simp
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   663
qed
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   664
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   665
lemma all_nat: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x\<ge>0. P (nat x))"
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   666
  by (auto split: split_nat)
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   667
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   668
lemma ex_nat: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> P (nat x))"
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   669
proof
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   670
  assume "\<exists>x. P x"
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   671
  then obtain x where "P x" ..
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   672
  then have "int x \<ge> 0 \<and> P (nat (int x))" by simp
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   673
  then show "\<exists>x\<ge>0. P (nat x)" ..
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   674
next
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   675
  assume "\<exists>x\<ge>0. P (nat x)"
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   676
  then show "\<exists>x. P x" by auto
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   677
qed
4eb431c3f974 tuned imports
haftmann
parents: 66816
diff changeset
   678
54249
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   679
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   680
text \<open>For termination proofs:\<close>
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   681
lemma measure_function_int[measure_function]: "is_measure (nat \<circ> abs)" ..
29779
2786b348c376 declare "nat o abs" as default measure for int
krauss
parents: 29700
diff changeset
   682
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   683
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69182
diff changeset
   684
subsection \<open>Lemmas about the Function \<^term>\<open>of_nat\<close> and Orderings\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   685
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
   686
lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   687
  by (simp add: order_less_le del: of_nat_Suc)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   688
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   689
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   690
  by (rule negative_zless_0 [THEN order_less_le_trans], simp)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   691
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   692
lemma negative_zle_0: "- int n \<le> 0"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   693
  by (simp add: minus_le_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   694
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   695
lemma negative_zle [iff]: "- int n \<le> int m"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   696
  by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   697
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   698
lemma not_zle_0_negative [simp]: "\<not> 0 \<le> - int (Suc n)"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   699
  by (subst le_minus_iff) (simp del: of_nat_Suc)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   700
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   701
lemma int_zle_neg: "int n \<le> - int m \<longleftrightarrow> n = 0 \<and> m = 0"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   702
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   703
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   704
lemma not_int_zless_negative [simp]: "\<not> int n < - int m"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   705
  by (simp add: linorder_not_less)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   706
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   707
lemma negative_eq_positive [simp]: "- int n = of_nat m \<longleftrightarrow> n = 0 \<and> m = 0"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   708
  by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   709
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   710
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   711
  (is "?lhs \<longleftrightarrow> ?rhs")
62348
9a5f43dac883 dropped various legacy fact bindings
haftmann
parents: 62347
diff changeset
   712
proof
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   713
  assume ?rhs
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   714
  then show ?lhs by auto
62348
9a5f43dac883 dropped various legacy fact bindings
haftmann
parents: 62347
diff changeset
   715
next
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   716
  assume ?lhs
62348
9a5f43dac883 dropped various legacy fact bindings
haftmann
parents: 62347
diff changeset
   717
  then have "0 \<le> z - w" by simp
9a5f43dac883 dropped various legacy fact bindings
haftmann
parents: 62347
diff changeset
   718
  then obtain n where "z - w = int n"
9a5f43dac883 dropped various legacy fact bindings
haftmann
parents: 62347
diff changeset
   719
    using zero_le_imp_eq_int [of "z - w"] by blast
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   720
  then have "z = w + int n" by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   721
  then show ?rhs ..
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   722
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   723
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   724
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   725
  by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   726
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   727
text \<open>
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   728
  This version is proved for all ordered rings, not just integers!
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   729
  It is proved here because attribute \<open>arith_split\<close> is not available
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   730
  in theory \<open>Rings\<close>.
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   731
  But is it really better than just rewriting with \<open>abs_if\<close>?
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   732
\<close>
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   733
lemma abs_split [arith_split, no_atp]: "P \<bar>a\<bar> \<longleftrightarrow> (0 \<le> a \<longrightarrow> P a) \<and> (a < 0 \<longrightarrow> P (- a))"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   734
  for a :: "'a::linordered_idom"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   735
  by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   736
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   737
lemma negD:
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   738
  assumes "x < 0" shows "\<exists>n. x = - (int (Suc n))"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   739
proof -
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   740
  have "\<And>a b. a < b \<Longrightarrow> \<exists>n. Suc (a + n) = b"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   741
    by (rule_tac x="b - Suc a" in exI) arith
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   742
  with assms show ?thesis
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   743
    by transfer auto
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   744
qed
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   745
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   746
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   747
subsection \<open>Cases and induction\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   748
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   749
text \<open>
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   750
  Now we replace the case analysis rule by a more conventional one:
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   751
  whether an integer is negative or not.
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   752
\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   753
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   754
text \<open>This version is symmetric in the two subgoals.\<close>
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   755
lemma int_cases2 [case_names nonneg nonpos, cases type: int]:
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   756
  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int n) \<Longrightarrow> P) \<Longrightarrow> P"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   757
  by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   758
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   759
text \<open>This is the default, with a negative case.\<close>
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   760
lemma int_cases [case_names nonneg neg, cases type: int]:
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   761
  assumes pos: "\<And>n. z = int n \<Longrightarrow> P" and neg: "\<And>n. z = - (int (Suc n)) \<Longrightarrow> P"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   762
  shows P
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   763
proof (cases "z < 0")
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   764
  case True
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   765
  with neg show ?thesis
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   766
    by (blast dest!: negD)
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   767
next
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   768
  case False
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   769
  with pos show ?thesis
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   770
    by (force simp add: linorder_not_less dest: nat_0_le [THEN sym])
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   771
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   772
60868
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   773
lemma int_cases3 [case_names zero pos neg]:
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   774
  fixes k :: int
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   775
  assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 61169
diff changeset
   776
    and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
60868
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   777
  shows "P"
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   778
proof (cases k "0::int" rule: linorder_cases)
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   779
  case equal
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   780
  with assms(1) show P by simp
60868
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   781
next
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   782
  case greater
63539
70d4d9e5707b tuned proofs -- avoid improper use of "this";
wenzelm
parents: 62390
diff changeset
   783
  then have *: "nat k > 0" by simp
70d4d9e5707b tuned proofs -- avoid improper use of "this";
wenzelm
parents: 62390
diff changeset
   784
  moreover from * have "k = int (nat k)" by auto
60868
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   785
  ultimately show P using assms(2) by blast
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   786
next
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   787
  case less
63539
70d4d9e5707b tuned proofs -- avoid improper use of "this";
wenzelm
parents: 62390
diff changeset
   788
  then have *: "nat (- k) > 0" by simp
70d4d9e5707b tuned proofs -- avoid improper use of "this";
wenzelm
parents: 62390
diff changeset
   789
  moreover from * have "k = - int (nat (- k))" by auto
60868
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   790
  ultimately show P using assms(3) by blast
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   791
qed
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   792
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   793
lemma int_of_nat_induct [case_names nonneg neg, induct type: int]:
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   794
  "(\<And>n. P (int n)) \<Longrightarrow> (\<And>n. P (- (int (Suc n)))) \<Longrightarrow> P z"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   795
  by (cases z) auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   796
66816
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   797
lemma sgn_mult_dvd_iff [simp]:
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   798
  "sgn r * l dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   799
  by (cases r rule: int_cases3) auto
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   800
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   801
lemma mult_sgn_dvd_iff [simp]:
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   802
  "l * sgn r dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   803
  using sgn_mult_dvd_iff [of r l k] by (simp add: ac_simps)
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   804
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   805
lemma dvd_sgn_mult_iff [simp]:
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   806
  "l dvd sgn r * k \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   807
  by (cases r rule: int_cases3) simp_all
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   808
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   809
lemma dvd_mult_sgn_iff [simp]:
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   810
  "l dvd k * sgn r \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   811
  using dvd_sgn_mult_iff [of l r k] by (simp add: ac_simps)
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   812
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   813
lemma int_sgnE:
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   814
  fixes k :: int
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   815
  obtains n and l where "k = sgn l * int n"
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   816
proof -
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   817
  have "k = sgn k * int (nat \<bar>k\<bar>)"
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   818
    by (simp add: sgn_mult_abs)
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   819
  then show ?thesis ..
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   820
qed
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
   821
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   822
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   823
subsubsection \<open>Binary comparisons\<close>
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   824
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   825
text \<open>Preliminaries\<close>
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   826
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   827
lemma le_imp_0_less:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   828
  fixes z :: int
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   829
  assumes le: "0 \<le> z"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   830
  shows "0 < 1 + z"
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   831
proof -
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   832
  have "0 \<le> z" by fact
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   833
  also have "\<dots> < z + 1" by (rule less_add_one)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   834
  also have "\<dots> = 1 + z" by (simp add: ac_simps)
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   835
  finally show "0 < 1 + z" .
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   836
qed
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   837
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   838
lemma odd_less_0_iff: "1 + z + z < 0 \<longleftrightarrow> z < 0"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   839
  for z :: int
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   840
proof (cases z)
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   841
  case (nonneg n)
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   842
  then show ?thesis
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   843
    by (simp add: linorder_not_less add.assoc add_increasing le_imp_0_less [THEN order_less_imp_le])
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   844
next
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   845
  case (neg n)
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   846
  then show ?thesis
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   847
    by (simp del: of_nat_Suc of_nat_add of_nat_1
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   848
        add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   849
qed
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   850
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   851
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   852
subsubsection \<open>Comparisons, for Ordered Rings\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   853
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   854
lemma odd_nonzero: "1 + z + z \<noteq> 0"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   855
  for z :: int
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   856
proof (cases z)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   857
  case (nonneg n)
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   858
  have le: "0 \<le> z + z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   859
    by (simp add: nonneg add_increasing)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   860
  then show ?thesis
67116
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
   861
    using le_imp_0_less [OF le] by (auto simp: ac_simps)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   862
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   863
  case (neg n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   864
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   865
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   866
    assume eq: "1 + z + z = 0"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   867
    have "0 < 1 + (int n + int n)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   868
      by (simp add: le_imp_0_less add_increasing)
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   869
    also have "\<dots> = - (1 + z + z)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   870
      by (simp add: neg add.assoc [symmetric])
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   871
    also have "\<dots> = 0" by (simp add: eq)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   872
    finally have "0<0" ..
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   873
    then show False by blast
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   874
  qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   875
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   876
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
   877
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   878
subsection \<open>The Set of Integers\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   879
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   880
context ring_1
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   881
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   882
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60868
diff changeset
   883
definition Ints :: "'a set"  ("\<int>")
b72a990adfe2 prefer symbols;
wenzelm
parents: 60868
diff changeset
   884
  where "\<int> = range of_int"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   885
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   886
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   887
  by (simp add: Ints_def)
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   888
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   889
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
45533
af3690f6bd79 simplify some proofs
huffman
parents: 45532
diff changeset
   890
  using Ints_of_int [of "of_nat n"] by simp
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   891
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   892
lemma Ints_0 [simp]: "0 \<in> \<int>"
45533
af3690f6bd79 simplify some proofs
huffman
parents: 45532
diff changeset
   893
  using Ints_of_int [of "0"] by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   894
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   895
lemma Ints_1 [simp]: "1 \<in> \<int>"
45533
af3690f6bd79 simplify some proofs
huffman
parents: 45532
diff changeset
   896
  using Ints_of_int [of "1"] by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   897
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   898
lemma Ints_numeral [simp]: "numeral n \<in> \<int>"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   899
  by (subst of_nat_numeral [symmetric], rule Ints_of_nat)
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   900
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   901
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   902
  by (force simp add: Ints_def simp flip: of_int_add intro: range_eqI)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   903
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   904
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   905
  by (force simp add: Ints_def simp flip: of_int_minus intro: range_eqI)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   906
68721
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 67969
diff changeset
   907
lemma minus_in_Ints_iff: "-x \<in> \<int> \<longleftrightarrow> x \<in> \<int>"
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 67969
diff changeset
   908
  using Ints_minus[of x] Ints_minus[of "-x"] by auto
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 67969
diff changeset
   909
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   910
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   911
  by (force simp add: Ints_def simp flip: of_int_diff intro: range_eqI)
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   912
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   913
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
   914
  by (force simp add: Ints_def simp flip: of_int_mult intro: range_eqI)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   915
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   916
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   917
  by (induct n) simp_all
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   918
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   919
lemma Ints_cases [cases set: Ints]:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   920
  assumes "q \<in> \<int>"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   921
  obtains (of_int) z where "q = of_int z"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   922
  unfolding Ints_def
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   923
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   924
  from \<open>q \<in> \<int>\<close> have "q \<in> range of_int" unfolding Ints_def .
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   925
  then obtain z where "q = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   926
  then show thesis ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   927
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   928
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   929
lemma Ints_induct [case_names of_int, induct set: Ints]:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   930
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   931
  by (rule Ints_cases) auto
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   932
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   933
lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   934
  unfolding Nats_def Ints_def
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   935
  by (rule subsetI, elim imageE, hypsubst, subst of_int_of_nat_eq[symmetric], rule imageI) simp_all
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   936
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   937
lemma Nats_altdef1: "\<nat> = {of_int n |n. n \<ge> 0}"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   938
proof (intro subsetI equalityI)
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   939
  fix x :: 'a
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   940
  assume "x \<in> {of_int n |n. n \<ge> 0}"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   941
  then obtain n where "x = of_int n" "n \<ge> 0"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   942
    by (auto elim!: Ints_cases)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   943
  then have "x = of_nat (nat n)"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   944
    by (subst of_nat_nat) simp_all
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   945
  then show "x \<in> \<nat>"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   946
    by simp
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   947
next
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   948
  fix x :: 'a
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   949
  assume "x \<in> \<nat>"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   950
  then obtain n where "x = of_nat n"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   951
    by (auto elim!: Nats_cases)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   952
  then have "x = of_int (int n)" by simp
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   953
  also have "int n \<ge> 0" by simp
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   954
  then have "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   955
  finally show "x \<in> {of_int n |n. n \<ge> 0}" .
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   956
qed
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   957
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   958
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   959
64758
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
   960
lemma (in linordered_idom) Ints_abs [simp]:
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
   961
  shows "a \<in> \<int> \<Longrightarrow> abs a \<in> \<int>"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
   962
  by (auto simp: abs_if)
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
   963
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   964
lemma (in linordered_idom) Nats_altdef2: "\<nat> = {n \<in> \<int>. n \<ge> 0}"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   965
proof (intro subsetI equalityI)
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   966
  fix x :: 'a
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   967
  assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   968
  then obtain n where "x = of_int n" "n \<ge> 0"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   969
    by (auto elim!: Ints_cases)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   970
  then have "x = of_nat (nat n)"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   971
    by (subst of_nat_nat) simp_all
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   972
  then show "x \<in> \<nat>"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   973
    by simp
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   974
qed (auto elim!: Nats_cases)
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   975
64849
766db3539859 moved some lemmas to appropriate places
haftmann
parents: 64758
diff changeset
   976
lemma (in idom_divide) of_int_divide_in_Ints: 
766db3539859 moved some lemmas to appropriate places
haftmann
parents: 64758
diff changeset
   977
  "of_int a div of_int b \<in> \<int>" if "b dvd a"
766db3539859 moved some lemmas to appropriate places
haftmann
parents: 64758
diff changeset
   978
proof -
766db3539859 moved some lemmas to appropriate places
haftmann
parents: 64758
diff changeset
   979
  from that obtain c where "a = b * c" ..
766db3539859 moved some lemmas to appropriate places
haftmann
parents: 64758
diff changeset
   980
  then show ?thesis
766db3539859 moved some lemmas to appropriate places
haftmann
parents: 64758
diff changeset
   981
    by (cases "of_int b = 0") simp_all
766db3539859 moved some lemmas to appropriate places
haftmann
parents: 64758
diff changeset
   982
qed
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   983
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69182
diff changeset
   984
text \<open>The premise involving \<^term>\<open>Ints\<close> prevents \<^term>\<open>a = 1/2\<close>.\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   985
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   986
lemma Ints_double_eq_0_iff:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   987
  fixes a :: "'a::ring_char_0"
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60868
diff changeset
   988
  assumes in_Ints: "a \<in> \<int>"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   989
  shows "a + a = 0 \<longleftrightarrow> a = 0"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   990
    (is "?lhs \<longleftrightarrow> ?rhs")
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   991
proof -
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   992
  from in_Ints have "a \<in> range of_int"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   993
    unfolding Ints_def [symmetric] .
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   994
  then obtain z where a: "a = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   995
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   996
  proof
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   997
    assume ?rhs
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
   998
    then show ?lhs by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   999
  next
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1000
    assume ?lhs
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1001
    with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1002
    then have "z + z = 0" by (simp only: of_int_eq_iff)
67116
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1003
    then have "z = 0" by (simp only: double_zero)
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1004
    with a show ?rhs by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1005
  qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1006
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1007
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1008
lemma Ints_odd_nonzero:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1009
  fixes a :: "'a::ring_char_0"
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60868
diff changeset
  1010
  assumes in_Ints: "a \<in> \<int>"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1011
  shows "1 + a + a \<noteq> 0"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1012
proof -
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1013
  from in_Ints have "a \<in> range of_int"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1014
    unfolding Ints_def [symmetric] .
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1015
  then obtain z where a: "a = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1016
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1017
  proof
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1018
    assume "1 + a + a = 0"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1019
    with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1020
    then have "1 + z + z = 0" by (simp only: of_int_eq_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1021
    with odd_nonzero show False by blast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1022
  qed
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1023
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1024
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60868
diff changeset
  1025
lemma Nats_numeral [simp]: "numeral w \<in> \<nat>"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1026
  using of_nat_in_Nats [of "numeral w"] by simp
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
  1027
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1028
lemma Ints_odd_less_0:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1029
  fixes a :: "'a::linordered_idom"
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60868
diff changeset
  1030
  assumes in_Ints: "a \<in> \<int>"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1031
  shows "1 + a + a < 0 \<longleftrightarrow> a < 0"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1032
proof -
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1033
  from in_Ints have "a \<in> range of_int"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1034
    unfolding Ints_def [symmetric] .
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1035
  then obtain z where a: "a = of_int z" ..
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1036
  with a have "1 + a + a < 0 \<longleftrightarrow> of_int (1 + z + z) < (of_int 0 :: 'a)"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1037
    by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1038
  also have "\<dots> \<longleftrightarrow> z < 0"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1039
    by (simp only: of_int_less_iff odd_less_0_iff)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1040
  also have "\<dots> \<longleftrightarrow> a < 0"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1041
    by (simp add: a)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1042
  finally show ?thesis .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1043
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1044
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1045
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69182
diff changeset
  1046
subsection \<open>\<^term>\<open>sum\<close> and \<^term>\<open>prod\<close>\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1047
69182
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1048
context semiring_1
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1049
begin
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1050
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1051
lemma of_nat_sum [simp]:
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1052
  "of_nat (sum f A) = (\<Sum>x\<in>A. of_nat (f x))"
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1053
  by (induction A rule: infinite_finite_induct) auto
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1054
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1055
end
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1056
69182
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1057
context ring_1
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1058
begin
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1059
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1060
lemma of_int_sum [simp]:
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1061
  "of_int (sum f A) = (\<Sum>x\<in>A. of_int (f x))"
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1062
  by (induction A rule: infinite_finite_induct) auto
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1063
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1064
end
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1065
69182
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1066
context comm_semiring_1
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1067
begin
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1068
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1069
lemma of_nat_prod [simp]:
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1070
  "of_nat (prod f A) = (\<Prod>x\<in>A. of_nat (f x))"
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1071
  by (induction A rule: infinite_finite_induct) auto
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1072
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1073
end
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1074
69182
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1075
context comm_ring_1
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1076
begin
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1077
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1078
lemma of_int_prod [simp]:
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1079
  "of_int (prod f A) = (\<Prod>x\<in>A. of_int (f x))"
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1080
  by (induction A rule: infinite_finite_induct) auto
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1081
2424301cc73d more and generalized lemmas
haftmann
parents: 68721
diff changeset
  1082
end
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1083
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1084
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1085
subsection \<open>Setting up simplification procedures\<close>
30802
f9e9e800d27e simplify theorem references
huffman
parents: 30796
diff changeset
  1086
70356
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1087
ML_file \<open>Tools/int_arith.ML\<close>
54249
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
  1088
70356
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1089
declaration \<open>K (
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1090
  Lin_Arith.add_discrete_type \<^type_name>\<open>Int.int\<close>
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1091
  #> Lin_Arith.add_lessD @{thm zless_imp_add1_zle}
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1092
  #> Lin_Arith.add_inj_thms @{thms of_nat_le_iff [THEN iffD2] of_nat_eq_iff [THEN iffD2]}
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1093
  #> Lin_Arith.add_inj_const (\<^const_name>\<open>of_nat\<close>, \<^typ>\<open>nat \<Rightarrow> int\<close>)
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1094
  #> Lin_Arith.add_simps
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1095
      @{thms of_int_0 of_int_1 of_int_add of_int_mult of_int_numeral of_int_neg_numeral nat_0 nat_1 diff_nat_numeral nat_numeral
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1096
      neg_less_iff_less
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1097
      True_implies_equals
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1098
      distrib_left [where a = "numeral v" for v]
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1099
      distrib_left [where a = "- numeral v" for v]
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1100
      div_by_1 div_0
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1101
      times_divide_eq_right times_divide_eq_left
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1102
      minus_divide_left [THEN sym] minus_divide_right [THEN sym]
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1103
      add_divide_distrib diff_divide_distrib
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1104
      of_int_minus of_int_diff
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1105
      of_int_of_nat_eq}
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1106
  #> Lin_Arith.add_simprocs [Int_Arith.zero_one_idom_simproc]
4a327c061870 streamlined setup for linear algebra, particularly removed redundant rule declarations
haftmann
parents: 70354
diff changeset
  1107
)\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1108
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1109
simproc_setup fast_arith
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1110
  ("(m::'a::linordered_idom) < n" |
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1111
    "(m::'a::linordered_idom) \<le> n" |
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1112
    "(m::'a::linordered_idom) = n") =
61144
5e94dfead1c2 simplified simproc programming interfaces;
wenzelm
parents: 61076
diff changeset
  1113
  \<open>K Lin_Arith.simproc\<close>
43595
7ae4a23b5be6 modernized some simproc setup;
wenzelm
parents: 43531
diff changeset
  1114
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1115
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1116
subsection\<open>More Inequality Reasoning\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1117
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1118
lemma zless_add1_eq: "w < z + 1 \<longleftrightarrow> w < z \<or> w = z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1119
  for w z :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1120
  by arith
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1121
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1122
lemma add1_zle_eq: "w + 1 \<le> z \<longleftrightarrow> w < z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1123
  for w z :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1124
  by arith
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1125
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1126
lemma zle_diff1_eq [simp]: "w \<le> z - 1 \<longleftrightarrow> w < z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1127
  for w z :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1128
  by arith
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1129
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1130
lemma zle_add1_eq_le [simp]: "w < z + 1 \<longleftrightarrow> w \<le> z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1131
  for w z :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1132
  by arith
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1133
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1134
lemma int_one_le_iff_zero_less: "1 \<le> z \<longleftrightarrow> 0 < z"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1135
  for z :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1136
  by arith
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1137
64758
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1138
lemma Ints_nonzero_abs_ge1:
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1139
  fixes x:: "'a :: linordered_idom"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1140
    assumes "x \<in> Ints" "x \<noteq> 0"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1141
    shows "1 \<le> abs x"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1142
proof (rule Ints_cases [OF \<open>x \<in> Ints\<close>])
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1143
  fix z::int
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1144
  assume "x = of_int z"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1145
    with \<open>x \<noteq> 0\<close> 
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1146
  show "1 \<le> \<bar>x\<bar>"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1147
    apply (auto simp add: abs_if)
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1148
    by (metis diff_0 of_int_1 of_int_le_iff of_int_minus zle_diff1_eq)
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1149
qed
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1150
  
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1151
lemma Ints_nonzero_abs_less1:
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1152
  fixes x:: "'a :: linordered_idom"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1153
  shows "\<lbrakk>x \<in> Ints; abs x < 1\<rbrakk> \<Longrightarrow> x = 0"
3b33d2fc5fc0 A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents: 64714
diff changeset
  1154
    using Ints_nonzero_abs_ge1 [of x] by auto
70365
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70356
diff changeset
  1155
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70356
diff changeset
  1156
lemma Ints_eq_abs_less1:
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70356
diff changeset
  1157
  fixes x:: "'a :: linordered_idom"
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70356
diff changeset
  1158
  shows "\<lbrakk>x \<in> Ints; y \<in> Ints\<rbrakk> \<Longrightarrow> x = y \<longleftrightarrow> abs (x-y) < 1"
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70356
diff changeset
  1159
  using eq_iff_diff_eq_0 by (fastforce intro: Ints_nonzero_abs_less1)
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70356
diff changeset
  1160
 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1161
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69182
diff changeset
  1162
subsection \<open>The functions \<^term>\<open>nat\<close> and \<^term>\<open>int\<close>\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1163
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69182
diff changeset
  1164
text \<open>Simplify the term \<^term>\<open>w + - z\<close>.\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1165
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1166
lemma one_less_nat_eq [simp]: "Suc 0 < nat z \<longleftrightarrow> 1 < z"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1167
  using zless_nat_conj [of 1 z] by auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1168
67116
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1169
lemma int_eq_iff_numeral [simp]:
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1170
  "int m = numeral v \<longleftrightarrow> m = numeral v"
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1171
  by (simp add: int_eq_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1172
67116
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1173
lemma nat_abs_int_diff:
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1174
  "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
  1175
  by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
  1176
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
  1177
lemma nat_int_add: "nat (int a + int b) = a + b"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
  1178
  by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
  1179
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1180
context ring_1
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1181
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1182
33056
791a4655cae3 renamed "nitpick_const_xxx" attributes to "nitpick_xxx" and "nitpick_ind_intros" to "nitpick_intros"
blanchet
parents: 32437
diff changeset
  1183
lemma of_int_of_nat [nitpick_simp]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1184
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1185
proof (cases "k < 0")
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1186
  case True
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1187
  then have "0 \<le> - k" by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1188
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1189
  with True show ?thesis by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1190
next
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1191
  case False
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1192
  then show ?thesis by (simp add: not_less)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1193
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1194
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1195
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1196
64014
ca1239a3277b more lemmas
haftmann
parents: 63652
diff changeset
  1197
lemma transfer_rule_of_int:
70927
cc204e10385c tuned syntax
haftmann
parents: 70365
diff changeset
  1198
  includes lifting_syntax
64014
ca1239a3277b more lemmas
haftmann
parents: 63652
diff changeset
  1199
  fixes R :: "'a::ring_1 \<Rightarrow> 'b::ring_1 \<Rightarrow> bool"
ca1239a3277b more lemmas
haftmann
parents: 63652
diff changeset
  1200
  assumes [transfer_rule]: "R 0 0" "R 1 1"
70927
cc204e10385c tuned syntax
haftmann
parents: 70365
diff changeset
  1201
    "(R ===> R ===> R) (+) (+)"
cc204e10385c tuned syntax
haftmann
parents: 70365
diff changeset
  1202
    "(R ===> R) uminus uminus"
cc204e10385c tuned syntax
haftmann
parents: 70365
diff changeset
  1203
  shows "((=) ===> R) of_int of_int"
64014
ca1239a3277b more lemmas
haftmann
parents: 63652
diff changeset
  1204
proof -
70927
cc204e10385c tuned syntax
haftmann
parents: 70365
diff changeset
  1205
  note assms
64014
ca1239a3277b more lemmas
haftmann
parents: 63652
diff changeset
  1206
  note transfer_rule_of_nat [transfer_rule]
70927
cc204e10385c tuned syntax
haftmann
parents: 70365
diff changeset
  1207
  have [transfer_rule]: "((=) ===> R) of_nat of_nat"
64014
ca1239a3277b more lemmas
haftmann
parents: 63652
diff changeset
  1208
    by transfer_prover
ca1239a3277b more lemmas
haftmann
parents: 63652
diff changeset
  1209
  show ?thesis
ca1239a3277b more lemmas
haftmann
parents: 63652
diff changeset
  1210
    by (unfold of_int_of_nat [abs_def]) transfer_prover
ca1239a3277b more lemmas
haftmann
parents: 63652
diff changeset
  1211
qed
ca1239a3277b more lemmas
haftmann
parents: 63652
diff changeset
  1212
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1213
lemma nat_mult_distrib:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1214
  fixes z z' :: int
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1215
  assumes "0 \<le> z"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1216
  shows "nat (z * z') = nat z * nat z'"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1217
proof (cases "0 \<le> z'")
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1218
  case False
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1219
  with assms have "z * z' \<le> 0"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1220
    by (simp add: not_le mult_le_0_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1221
  then have "nat (z * z') = 0" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1222
  moreover from False have "nat z' = 0" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1223
  ultimately show ?thesis by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1224
next
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1225
  case True
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1226
  with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1227
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1228
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1229
      (simp only: of_nat_mult of_nat_nat [OF True]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1230
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1231
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1232
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1233
lemma nat_mult_distrib_neg:
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1234
  assumes "z \<le> (0::int)" shows "nat (z * z') = nat (- z) * nat (- z')" (is "?L = ?R")
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1235
proof -
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1236
  have "?L = nat (- z * - z')"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1237
    using assms by auto
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1238
  also have "... = ?R"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1239
    by (rule nat_mult_distrib) (use assms in auto)
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1240
  finally show ?thesis .
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1241
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1242
61944
5d06ecfdb472 prefer symbols for "abs";
wenzelm
parents: 61799
diff changeset
  1243
lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1244
  by (cases "z = 0 \<or> w = 0")
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1245
    (auto simp add: abs_if nat_mult_distrib [symmetric]
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1246
      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1247
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1248
lemma int_in_range_abs [simp]: "int n \<in> range abs"
60570
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
  1249
proof (rule range_eqI)
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1250
  show "int n = \<bar>int n\<bar>" by simp
60570
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
  1251
qed
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
  1252
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1253
lemma range_abs_Nats [simp]: "range abs = (\<nat> :: int set)"
60570
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
  1254
proof -
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
  1255
  have "\<bar>k\<bar> \<in> \<nat>" for k :: int
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
  1256
    by (cases k) simp_all
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
  1257
  moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
  1258
    using that by induct simp
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
  1259
  ultimately show ?thesis by blast
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 61169
diff changeset
  1260
qed
60570
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
  1261
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1262
lemma Suc_nat_eq_nat_zadd1: "0 \<le> z \<Longrightarrow> Suc (nat z) = nat (1 + z)"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1263
  for z :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1264
  by (rule sym) (simp add: nat_eq_iff)
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
  1265
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
  1266
lemma diff_nat_eq_if:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1267
  "nat z - nat z' =
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1268
    (if z' < 0 then nat z
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1269
     else
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1270
      let d = z - z'
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1271
      in if d < 0 then 0 else nat d)"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1272
  by (simp add: Let_def nat_diff_distrib [symmetric])
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
  1273
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1274
lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)"
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
  1275
  using diff_nat_numeral [of v Num.One] by simp
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
  1276
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1277
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1278
subsection \<open>Induction principles for int\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1279
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1280
text \<open>Well-founded segments of the integers.\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1281
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1282
definition int_ge_less_than :: "int \<Rightarrow> (int \<times> int) set"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1283
  where "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1284
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1285
lemma wf_int_ge_less_than: "wf (int_ge_less_than d)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1286
proof -
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1287
  have "int_ge_less_than d \<subseteq> measure (\<lambda>z. nat (z - d))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1288
    by (auto simp add: int_ge_less_than_def)
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1289
  then show ?thesis
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1290
    by (rule wf_subset [OF wf_measure])
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1291
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1292
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1293
text \<open>
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1294
  This variant looks odd, but is typical of the relations suggested
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1295
  by RankFinder.\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1296
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1297
definition int_ge_less_than2 :: "int \<Rightarrow> (int \<times> int) set"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1298
  where "int_ge_less_than2 d = {(z',z). d \<le> z \<and> z' < z}"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1299
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1300
lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1301
proof -
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1302
  have "int_ge_less_than2 d \<subseteq> measure (\<lambda>z. nat (1 + z - d))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1303
    by (auto simp add: int_ge_less_than2_def)
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1304
  then show ?thesis
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1305
    by (rule wf_subset [OF wf_measure])
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1306
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1307
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1308
(* `set:int': dummy construction *)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1309
theorem int_ge_induct [case_names base step, induct set: int]:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1310
  fixes i :: int
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1311
  assumes ge: "k \<le> i"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1312
    and base: "P k"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1313
    and step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1314
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1315
proof -
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1316
  have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" for n
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1317
  proof (induct n)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1318
    case 0
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1319
    then have "i = k" by arith
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1320
    with base show "P i" by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1321
  next
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1322
    case (Suc n)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1323
    then have "n = nat ((i - 1) - k)" by arith
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1324
    moreover have k: "k \<le> i - 1" using Suc.prems by arith
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1325
    ultimately have "P (i - 1)" by (rule Suc.hyps)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1326
    from step [OF k this] show ?case by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1327
  qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1328
  with ge show ?thesis by fast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1329
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1330
25928
042e877d9841 tuned code setup
haftmann
parents: 25919
diff changeset
  1331
(* `set:int': dummy construction *)
042e877d9841 tuned code setup
haftmann
parents: 25919
diff changeset
  1332
theorem int_gr_induct [case_names base step, induct set: int]:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1333
  fixes i k :: int
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1334
  assumes "k < i" "P (k + 1)" "\<And>i. k < i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1335
  shows "P i"
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1336
proof -
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1337
  have "k+1 \<le> i"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1338
    using assms by auto
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1339
  then show ?thesis
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1340
    by (induction i rule: int_ge_induct) (auto simp: assms)
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1341
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1342
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1343
theorem int_le_induct [consumes 1, case_names base step]:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1344
  fixes i k :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1345
  assumes le: "i \<le> k"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1346
    and base: "P k"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1347
    and step: "\<And>i. i \<le> k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1348
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1349
proof -
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1350
  have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" for n
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1351
  proof (induct n)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1352
    case 0
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1353
    then have "i = k" by arith
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1354
    with base show "P i" by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1355
  next
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1356
    case (Suc n)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1357
    then have "n = nat (k - (i + 1))" by arith
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1358
    moreover have k: "i + 1 \<le> k" using Suc.prems by arith
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1359
    ultimately have "P (i + 1)" by (rule Suc.hyps)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1360
    from step[OF k this] show ?case by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1361
  qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1362
  with le show ?thesis by fast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1363
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1364
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1365
theorem int_less_induct [consumes 1, case_names base step]:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1366
  fixes i k :: int
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1367
  assumes "i < k" "P (k - 1)" "\<And>i. i < k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1368
  shows "P i"
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1369
proof -
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1370
  have "i \<le> k-1"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1371
    using assms by auto
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1372
  then show ?thesis
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1373
    by (induction i rule: int_le_induct) (auto simp: assms)
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1374
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1375
36811
4ab4aa5bee1c renamed former Int.int_induct to Int.int_of_nat_induct, former Presburger.int_induct to Int.int_induct: is more conservative and more natural than the intermediate solution
haftmann
parents: 36801
diff changeset
  1376
theorem int_induct [case_names base step1 step2]:
36801
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1377
  fixes k :: int
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1378
  assumes base: "P k"
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1379
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1380
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1381
  shows "P i"
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1382
proof -
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1383
  have "i \<le> k \<or> i \<ge> k" by arith
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1384
  then show ?thesis
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1385
  proof
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1386
    assume "i \<ge> k"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1387
    then show ?thesis
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1388
      using base by (rule int_ge_induct) (fact step1)
36801
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1389
  next
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1390
    assume "i \<le> k"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1391
    then show ?thesis
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1392
      using base by (rule int_le_induct) (fact step2)
36801
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1393
  qed
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1394
qed
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1395
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1396
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1397
subsection \<open>Intermediate value theorems\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1398
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1399
lemma nat_ivt_aux: 
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1400
  "\<lbrakk>\<forall>i<n. \<bar>f (Suc i) - f i\<bar> \<le> 1; f 0 \<le> k; k \<le> f n\<rbrakk> \<Longrightarrow> \<exists>i \<le> n. f i = k"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1401
  for m n :: nat and k :: int
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1402
proof (induct n)
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1403
  case (Suc n)
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1404
  show ?case
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1405
  proof (cases "k = f (Suc n)")
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1406
    case False
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1407
    with Suc have "k \<le> f n"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1408
      by auto
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1409
    with Suc show ?thesis
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1410
      by (auto simp add: abs_if split: if_split_asm intro: le_SucI)
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1411
  qed (use Suc in auto)
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1412
qed auto
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1413
67116
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1414
lemma nat_intermed_int_val:
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1415
  fixes m n :: nat and k :: int
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1416
  assumes "\<forall>i. m \<le> i \<and> i < n \<longrightarrow> \<bar>f (Suc i) - f i\<bar> \<le> 1" "m \<le> n" "f m \<le> k" "k \<le> f n"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1417
  shows "\<exists>i. m \<le> i \<and> i \<le> n \<and> f i = k"
67116
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1418
proof -
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1419
  obtain i where "i \<le> n - m" "k = f (m + i)"
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1420
    using nat_ivt_aux [of "n - m" "f \<circ> plus m" k] assms by auto
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1421
  with assms show ?thesis
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1422
    by (rule_tac x = "m + i" in exI) auto
67116
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1423
qed
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1424
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1425
lemma nat0_intermed_int_val:
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1426
  "\<exists>i\<le>n. f i = k"
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1427
  if "\<forall>i<n. \<bar>f (i + 1) - f i\<bar> \<le> 1" "f 0 \<le> k" "k \<le> f n"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1428
  for n :: nat and k :: int
67116
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1429
  using nat_intermed_int_val [of 0 n f k] that by auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1430
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1431
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1432
subsection \<open>Products and 1, by T. M. Rasmussen\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1433
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1434
lemma abs_zmult_eq_1:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1435
  fixes m n :: int
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1436
  assumes mn: "\<bar>m * n\<bar> = 1"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1437
  shows "\<bar>m\<bar> = 1"
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1438
proof -
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1439
  from mn have 0: "m \<noteq> 0" "n \<noteq> 0" by auto
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1440
  have "\<not> 2 \<le> \<bar>m\<bar>"
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1441
  proof
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1442
    assume "2 \<le> \<bar>m\<bar>"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1443
    then have "2 * \<bar>n\<bar> \<le> \<bar>m\<bar> * \<bar>n\<bar>" by (simp add: mult_mono 0)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1444
    also have "\<dots> = \<bar>m * n\<bar>" by (simp add: abs_mult)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1445
    also from mn have "\<dots> = 1" by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1446
    finally have "2 * \<bar>n\<bar> \<le> 1" .
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1447
    with 0 show "False" by arith
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1448
  qed
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1449
  with 0 show ?thesis by auto
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1450
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1451
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1452
lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 \<Longrightarrow> m = 1 \<or> m = - 1"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1453
  for m n :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1454
  using abs_zmult_eq_1 [of m n] by arith
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1455
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1456
lemma pos_zmult_eq_1_iff:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1457
  fixes m n :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1458
  assumes "0 < m"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1459
  shows "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1460
proof -
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1461
  from assms have "m * n = 1 \<Longrightarrow> m = 1"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1462
    by (auto dest: pos_zmult_eq_1_iff_lemma)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1463
  then show ?thesis
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1464
    by (auto dest: pos_zmult_eq_1_iff_lemma)
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1465
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1466
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1467
lemma zmult_eq_1_iff: "m * n = 1 \<longleftrightarrow> (m = 1 \<and> n = 1) \<or> (m = - 1 \<and> n = - 1)" (is "?L = ?R")
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1468
  for m n :: int
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1469
proof
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1470
  assume L: ?L show ?R
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1471
    using pos_zmult_eq_1_iff_lemma [OF L] L by force
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1472
qed auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1473
69700
7a92cbec7030 new material about summations and powers, along with some tweaks
paulson <lp15@cam.ac.uk>
parents: 69605
diff changeset
  1474
lemma infinite_UNIV_int [simp]: "\<not> finite (UNIV::int set)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1475
proof
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1476
  assume "finite (UNIV::int set)"
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
  1477
  moreover have "inj (\<lambda>i::int. 2 * i)"
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1478
    by (rule injI) simp
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
  1479
  ultimately have "surj (\<lambda>i::int. 2 * i)"
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1480
    by (rule finite_UNIV_inj_surj)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1481
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1482
  then show False by (simp add: pos_zmult_eq_1_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1483
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1484
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1485
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1486
subsection \<open>The divides relation\<close>
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1487
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1488
lemma zdvd_antisym_nonneg: "0 \<le> m \<Longrightarrow> 0 \<le> n \<Longrightarrow> m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1489
  for m n :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1490
  by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1491
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1492
lemma zdvd_antisym_abs:
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1493
  fixes a b :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1494
  assumes "a dvd b" and "b dvd a"
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1495
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1496
proof (cases "a = 0")
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1497
  case True
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1498
  with assms show ?thesis by simp
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1499
next
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1500
  case False
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1501
  from \<open>a dvd b\<close> obtain k where k: "b = a * k"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1502
    unfolding dvd_def by blast
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1503
  from \<open>b dvd a\<close> obtain k' where k': "a = b * k'"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1504
    unfolding dvd_def by blast
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1505
  from k k' have "a = a * k * k'" by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1506
  with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1507
    using \<open>a \<noteq> 0\<close> by (simp add: mult.assoc)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1508
  then have "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1509
    by (simp add: zmult_eq_1_iff)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1510
  with k k' show ?thesis by auto
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1511
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1512
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1513
lemma zdvd_zdiffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> k dvd m"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1514
  for k m n :: int
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1515
  using dvd_add_right_iff [of k "- n" m] by simp
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1516
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1517
lemma zdvd_reduce: "k dvd n + k * m \<longleftrightarrow> k dvd n"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1518
  for k m n :: int
58649
a62065b5e1e2 generalized and consolidated some theorems concerning divisibility
haftmann
parents: 58512
diff changeset
  1519
  using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1520
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1521
lemma dvd_imp_le_int:
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1522
  fixes d i :: int
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1523
  assumes "i \<noteq> 0" and "d dvd i"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1524
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1525
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1526
  from \<open>d dvd i\<close> obtain k where "i = d * k" ..
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1527
  with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1528
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1529
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1530
  with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1531
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1532
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1533
lemma zdvd_not_zless:
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1534
  fixes m n :: int
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1535
  assumes "0 < m" and "m < n"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1536
  shows "\<not> n dvd m"
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1537
proof
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1538
  from assms have "0 < n" by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1539
  assume "n dvd m" then obtain k where k: "m = n * k" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1540
  with \<open>0 < m\<close> have "0 < n * k" by auto
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1541
  with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff)
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1542
  with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1543
  with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1544
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1545
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1546
lemma zdvd_mult_cancel:
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1547
  fixes k m n :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1548
  assumes d: "k * m dvd k * n"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1549
    and "k \<noteq> 0"
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1550
  shows "m dvd n"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1551
proof -
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1552
  from d obtain h where h: "k * n = k * m * h"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1553
    unfolding dvd_def by blast
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1554
  have "n = m * h"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1555
  proof (rule ccontr)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1556
    assume "\<not> ?thesis"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1557
    with \<open>k \<noteq> 0\<close> have "k * n \<noteq> k * (m * h)" by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1558
    with h show False
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1559
      by (simp add: mult.assoc)
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1560
  qed
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1561
  then show ?thesis by simp
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1562
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1563
67118
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1564
lemma int_dvd_int_iff [simp]:
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1565
  "int m dvd int n \<longleftrightarrow> m dvd n"
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1566
proof -
67118
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1567
  have "m dvd n" if "int n = int m * k" for k
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1568
  proof (cases k)
67118
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1569
    case (nonneg q)
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1570
    with that have "n = m * q"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1571
      by (simp del: of_nat_mult add: of_nat_mult [symmetric])
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1572
    then show ?thesis ..
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1573
  next
67118
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1574
    case (neg q)
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1575
    with that have "int n = int m * (- int (Suc q))"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1576
      by simp
67118
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1577
    also have "\<dots> = - (int m * int (Suc q))"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1578
      by (simp only: mult_minus_right)
67118
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1579
    also have "\<dots> = - int (m * Suc q)"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1580
      by (simp only: of_nat_mult [symmetric])
67118
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1581
    finally have "- int (m * Suc q) = int n" ..
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1582
    then show ?thesis
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1583
      by (simp only: negative_eq_positive) auto
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1584
  qed
67118
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1585
  then show ?thesis by (auto simp add: dvd_def)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1586
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1587
67118
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1588
lemma dvd_nat_abs_iff [simp]:
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1589
  "n dvd nat \<bar>k\<bar> \<longleftrightarrow> int n dvd k"
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1590
proof -
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1591
  have "n dvd nat \<bar>k\<bar> \<longleftrightarrow> int n dvd int (nat \<bar>k\<bar>)"
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1592
    by (simp only: int_dvd_int_iff)
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1593
  then show ?thesis
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1594
    by simp
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1595
qed
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1596
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1597
lemma nat_abs_dvd_iff [simp]:
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1598
  "nat \<bar>k\<bar> dvd n \<longleftrightarrow> k dvd int n"
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1599
proof -
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1600
  have "nat \<bar>k\<bar> dvd n \<longleftrightarrow> int (nat \<bar>k\<bar>) dvd int n"
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1601
    by (simp only: int_dvd_int_iff)
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1602
  then show ?thesis
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1603
    by simp
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1604
qed
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1605
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1606
lemma zdvd1_eq [simp]: "x dvd 1 \<longleftrightarrow> \<bar>x\<bar> = 1" (is "?lhs \<longleftrightarrow> ?rhs")
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1607
  for x :: int
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1608
proof
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1609
  assume ?lhs
67118
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1610
  then have "nat \<bar>x\<bar> dvd nat \<bar>1\<bar>"
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1611
    by (simp only: nat_abs_dvd_iff) simp
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1612
  then have "nat \<bar>x\<bar> = 1"
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1613
    by simp
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1614
  then show ?rhs
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1615
    by (cases "x < 0") simp_all
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1616
next
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1617
  assume ?rhs
67118
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1618
  then have "x = 1 \<or> x = - 1"
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1619
    by auto
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1620
  then show ?lhs
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1621
    by (auto intro: dvdI)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1622
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1623
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1624
lemma zdvd_mult_cancel1:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1625
  fixes m :: int
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1626
  assumes mp: "m \<noteq> 0"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1627
  shows "m * n dvd m \<longleftrightarrow> \<bar>n\<bar> = 1"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1628
    (is "?lhs \<longleftrightarrow> ?rhs")
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1629
proof
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1630
  assume ?rhs
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1631
  then show ?lhs
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1632
    by (cases "n > 0") (auto simp add: minus_equation_iff)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1633
next
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1634
  assume ?lhs
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1635
  then have "m * n dvd m * 1" by simp
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1636
  from zdvd_mult_cancel[OF this mp] show ?rhs
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1637
    by (simp only: zdvd1_eq)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1638
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1639
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1640
lemma nat_dvd_iff: "nat z dvd m \<longleftrightarrow> (if 0 \<le> z then z dvd int m else m = 0)"
67118
ccab07d1196c more simplification rules
haftmann
parents: 67116
diff changeset
  1641
  using nat_abs_dvd_iff [of z m] by (cases "z \<ge> 0") auto
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1642
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1643
lemma eq_nat_nat_iff: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
67116
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  1644
  by (auto elim: nonneg_int_cases)
33341
5a989586d102 moved some dvd [int] facts to Int
haftmann
parents: 33320
diff changeset
  1645
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1646
lemma nat_power_eq: "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
33341
5a989586d102 moved some dvd [int] facts to Int
haftmann
parents: 33320
diff changeset
  1647
  by (induct n) (simp_all add: nat_mult_distrib)
5a989586d102 moved some dvd [int] facts to Int
haftmann
parents: 33320
diff changeset
  1648
66912
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1649
lemma numeral_power_eq_nat_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1650
  "numeral x ^ n = nat y \<longleftrightarrow> numeral x ^ n = y"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1651
  using nat_eq_iff2 by auto
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1652
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1653
lemma nat_eq_numeral_power_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1654
  "nat y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1655
  using numeral_power_eq_nat_cancel_iff[of x n y]
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1656
  by (metis (mono_tags))
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1657
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1658
lemma numeral_power_le_nat_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1659
  "numeral x ^ n \<le> nat a \<longleftrightarrow> numeral x ^ n \<le> a"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1660
  using nat_le_eq_zle[of "numeral x ^ n" a]
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1661
  by (auto simp: nat_power_eq)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1662
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1663
lemma nat_le_numeral_power_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1664
  "nat a \<le> numeral x ^ n \<longleftrightarrow> a \<le> numeral x ^ n"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1665
  by (simp add: nat_le_iff)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1666
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1667
lemma numeral_power_less_nat_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1668
  "numeral x ^ n < nat a \<longleftrightarrow> numeral x ^ n < a"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1669
  using nat_less_eq_zless[of "numeral x ^ n" a]
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1670
  by (auto simp: nat_power_eq)
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1671
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1672
lemma nat_less_numeral_power_cancel_iff [simp]:
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1673
  "nat a < numeral x ^ n \<longleftrightarrow> a < numeral x ^ n"
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1674
  using nat_less_eq_zless[of a "numeral x ^ n"]
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1675
  by (cases "a < 0") (auto simp: nat_power_eq less_le_trans[where y=0])
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66886
diff changeset
  1676
71616
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1677
lemma zdvd_imp_le: "z \<le> n" if "z dvd n" "0 < n" for n z :: int
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1678
proof (cases n)
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1679
  case (nonneg n)
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1680
  show ?thesis
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1681
    by (cases z) (use nonneg dvd_imp_le that in auto)
a9de39608b1a more tidying up of old apply-proofs
paulson <lp15@cam.ac.uk>
parents: 70927
diff changeset
  1682
qed (use that in auto)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1683
36749
a8dc19a352e6 moved lemma zdvd_period to theory Int
haftmann
parents: 36719
diff changeset
  1684
lemma zdvd_period:
a8dc19a352e6 moved lemma zdvd_period to theory Int
haftmann
parents: 36719
diff changeset
  1685
  fixes a d :: int
a8dc19a352e6 moved lemma zdvd_period to theory Int
haftmann
parents: 36719
diff changeset
  1686
  assumes "a dvd d"
a8dc19a352e6 moved lemma zdvd_period to theory Int
haftmann
parents: 36719
diff changeset
  1687
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  1688
    (is "?lhs \<longleftrightarrow> ?rhs")
36749
a8dc19a352e6 moved lemma zdvd_period to theory Int
haftmann
parents: 36719
diff changeset
  1689
proof -
66816
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
  1690
  from assms have "a dvd (x + t) \<longleftrightarrow> a dvd ((x + t) + c * d)"
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
  1691
    by (simp add: dvd_add_left_iff)
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
  1692
  then show ?thesis
212a3334e7da more fundamental definition of div and mod on int
haftmann
parents: 66035
diff changeset
  1693
    by (simp add: ac_simps)
36749
a8dc19a352e6 moved lemma zdvd_period to theory Int
haftmann
parents: 36719
diff changeset
  1694
qed
a8dc19a352e6 moved lemma zdvd_period to theory Int
haftmann
parents: 36719
diff changeset
  1695
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1696
71837
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1697
subsection \<open>Powers with integer exponents\<close>
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1698
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1699
text \<open>
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1700
  The following allows writing powers with an integer exponent. While the type signature
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1701
  is very generic, most theorems will assume that the underlying type is a division ring or
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1702
  a field.
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1703
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1704
  The notation `powi' is inspired by the `powr' notation for real/complex exponentiation.
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1705
\<close>
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1706
definition power_int :: "'a :: {inverse, power} \<Rightarrow> int \<Rightarrow> 'a" (infixr "powi" 80) where
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1707
  "power_int x n = (if n \<ge> 0 then x ^ nat n else inverse x ^ (nat (-n)))"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1708
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1709
lemma power_int_0_right [simp]: "power_int x 0 = 1"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1710
  and power_int_1_right [simp]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1711
        "power_int (y :: 'a :: {power, inverse, monoid_mult}) 1 = y"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1712
  and power_int_minus1_right [simp]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1713
        "power_int (y :: 'a :: {power, inverse, monoid_mult}) (-1) = inverse y"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1714
  by (simp_all add: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1715
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1716
lemma power_int_of_nat [simp]: "power_int x (int n) = x ^ n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1717
  by (simp add: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1718
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1719
lemma power_int_numeral [simp]: "power_int x (numeral n) = x ^ numeral n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1720
  by (simp add: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1721
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1722
lemma int_cases4 [case_names nonneg neg]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1723
  fixes m :: int
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1724
  obtains n where "m = int n" | n where "n > 0" "m = -int n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1725
proof (cases "m \<ge> 0")
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1726
  case True
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1727
  thus ?thesis using that(1)[of "nat m"] by auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1728
next
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1729
  case False
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1730
  thus ?thesis using that(2)[of "nat (-m)"] by auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1731
qed
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1732
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1733
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1734
context
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1735
  assumes "SORT_CONSTRAINT('a::division_ring)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1736
begin
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1737
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1738
lemma power_int_minus: "power_int (x::'a) (-n) = inverse (power_int x n)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1739
  by (auto simp: power_int_def power_inverse)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1740
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1741
lemma power_int_eq_0_iff [simp]: "power_int (x::'a) n = 0 \<longleftrightarrow> x = 0 \<and> n \<noteq> 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1742
  by (auto simp: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1743
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1744
lemma power_int_0_left_If: "power_int (0 :: 'a) m = (if m = 0 then 1 else 0)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1745
  by (auto simp: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1746
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1747
lemma power_int_0_left [simp]: "m \<noteq> 0 \<Longrightarrow> power_int (0 :: 'a) m = 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1748
  by (simp add: power_int_0_left_If)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1749
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1750
lemma power_int_1_left [simp]: "power_int 1 n = (1 :: 'a :: division_ring)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1751
  by (auto simp: power_int_def) 
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1752
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1753
lemma power_diff_conv_inverse: "x \<noteq> 0 \<Longrightarrow> m \<le> n \<Longrightarrow> (x :: 'a) ^ (n - m) = x ^ n * inverse x ^ m"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1754
  by (simp add: field_simps flip: power_add)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1755
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1756
lemma power_mult_inverse_distrib: "x ^ m * inverse (x :: 'a) = inverse x * x ^ m"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1757
proof (cases "x = 0")
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1758
  case [simp]: False
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1759
  show ?thesis
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1760
  proof (cases m)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1761
    case (Suc m')
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1762
    have "x ^ Suc m' * inverse x = x ^ m'"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1763
      by (subst power_Suc2) (auto simp: mult.assoc)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1764
    also have "\<dots> = inverse x * x ^ Suc m'"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1765
      by (subst power_Suc) (auto simp: mult.assoc [symmetric])
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1766
    finally show ?thesis using Suc by simp
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1767
  qed auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1768
qed auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1769
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1770
lemma power_mult_power_inverse_commute:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1771
  "x ^ m * inverse (x :: 'a) ^ n = inverse x ^ n * x ^ m"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1772
proof (induction n)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1773
  case (Suc n)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1774
  have "x ^ m * inverse x ^ Suc n = (x ^ m * inverse x ^ n) * inverse x"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1775
    by (simp only: power_Suc2 mult.assoc)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1776
  also have "x ^ m * inverse x ^ n = inverse x ^ n * x ^ m"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1777
    by (rule Suc)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1778
  also have "\<dots> * inverse x = (inverse x ^ n * inverse x) * x ^ m"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1779
    by (simp add: mult.assoc power_mult_inverse_distrib)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1780
  also have "\<dots> = inverse x ^ (Suc n) * x ^ m"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1781
    by (simp only: power_Suc2)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1782
  finally show ?case .
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1783
qed auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1784
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1785
lemma power_int_add:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1786
  assumes "x \<noteq> 0 \<or> m + n \<noteq> 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1787
  shows   "power_int (x::'a) (m + n) = power_int x m * power_int x n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1788
proof (cases "x = 0")
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1789
  case True
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1790
  thus ?thesis using assms by (auto simp: power_int_0_left_If)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1791
next
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1792
  case [simp]: False
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1793
  show ?thesis
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1794
  proof (cases m n rule: int_cases4[case_product int_cases4])
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1795
    case (nonneg_nonneg a b)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1796
    thus ?thesis
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1797
      by (auto simp: power_int_def nat_add_distrib power_add)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1798
  next
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1799
    case (nonneg_neg a b)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1800
    thus ?thesis
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1801
      by (auto simp: power_int_def nat_diff_distrib not_le power_diff_conv_inverse
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1802
                     power_mult_power_inverse_commute)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1803
  next
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1804
    case (neg_nonneg a b)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1805
    thus ?thesis
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1806
      by (auto simp: power_int_def nat_diff_distrib not_le power_diff_conv_inverse
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1807
                     power_mult_power_inverse_commute)    
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1808
  next
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1809
    case (neg_neg a b)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1810
    thus ?thesis
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1811
      by (auto simp: power_int_def nat_add_distrib add.commute simp flip: power_add)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1812
  qed
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1813
qed
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1814
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1815
lemma power_int_add_1:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1816
  assumes "x \<noteq> 0 \<or> m \<noteq> -1"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1817
  shows   "power_int (x::'a) (m + 1) = power_int x m * x"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1818
  using assms by (subst power_int_add) auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1819
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1820
lemma power_int_add_1':
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1821
  assumes "x \<noteq> 0 \<or> m \<noteq> -1"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1822
  shows   "power_int (x::'a) (m + 1) = x * power_int x m"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1823
  using assms by (subst add.commute, subst power_int_add) auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1824
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1825
lemma power_int_commutes: "power_int (x :: 'a) n * x = x * power_int x n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1826
  by (cases "x = 0") (auto simp flip: power_int_add_1 power_int_add_1')
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1827
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1828
lemma power_int_inverse [field_simps, field_split_simps, divide_simps]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1829
  "power_int (inverse (x :: 'a)) n = inverse (power_int x n)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1830
  by (auto simp: power_int_def power_inverse)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1831
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1832
lemma power_int_mult: "power_int (x :: 'a) (m * n) = power_int (power_int x m) n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1833
  by (auto simp: power_int_def zero_le_mult_iff simp flip: power_mult power_inverse nat_mult_distrib)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1834
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1835
end
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1836
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1837
context
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1838
  assumes "SORT_CONSTRAINT('a::field)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1839
begin
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1840
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1841
lemma power_int_diff:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1842
  assumes "x \<noteq> 0 \<or> m \<noteq> n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1843
  shows   "power_int (x::'a) (m - n) = power_int x m / power_int x n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1844
  using power_int_add[of x m "-n"] assms by (auto simp: field_simps power_int_minus)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1845
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1846
lemma power_int_minus_mult: "x \<noteq> 0 \<or> n \<noteq> 0 \<Longrightarrow> power_int (x :: 'a) (n - 1) * x = power_int x n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1847
  by (auto simp flip: power_int_add_1)  
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1848
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1849
lemma power_int_mult_distrib: "power_int (x * y :: 'a) m = power_int x m * power_int y m"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1850
  by (auto simp: power_int_def power_mult_distrib)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1851
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1852
lemmas power_int_mult_distrib_numeral1 = power_int_mult_distrib [where x = "numeral w" for w, simp]
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1853
lemmas power_int_mult_distrib_numeral2 = power_int_mult_distrib [where y = "numeral w" for w, simp]
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1854
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1855
lemma power_int_divide_distrib: "power_int (x / y :: 'a) m = power_int x m / power_int y m"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1856
  using power_int_mult_distrib[of x "inverse y" m] unfolding power_int_inverse
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1857
  by (simp add: field_simps)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1858
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1859
end
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1860
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1861
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1862
lemma power_int_add_numeral [simp]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1863
  "power_int x (numeral m) * power_int x (numeral n) = power_int x (numeral (m + n))"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1864
  for x :: "'a :: division_ring"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1865
  by (simp add: power_int_add [symmetric])
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1866
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1867
lemma power_int_add_numeral2 [simp]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1868
  "power_int x (numeral m) * (power_int x (numeral n) * b) = power_int x (numeral (m + n)) * b"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1869
  for x :: "'a :: division_ring"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1870
  by (simp add: mult.assoc [symmetric])
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1871
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1872
lemma power_int_mult_numeral [simp]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1873
  "power_int (power_int x (numeral m)) (numeral n) = power_int x (numeral (m * n))"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1874
  for x :: "'a :: division_ring"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1875
  by (simp only: numeral_mult power_int_mult)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1876
  
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1877
lemma power_int_not_zero: "(x :: 'a :: division_ring) \<noteq> 0 \<or> n = 0 \<Longrightarrow> power_int x n \<noteq> 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1878
  by (subst power_int_eq_0_iff) auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1879
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1880
lemma power_int_one_over [field_simps, field_split_simps, divide_simps]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1881
  "power_int (1 / x :: 'a :: division_ring) n = 1 / power_int x n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1882
  using power_int_inverse[of x] by (simp add: divide_inverse)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1883
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1884
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1885
context
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1886
  assumes "SORT_CONSTRAINT('a :: linordered_field)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1887
begin
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1888
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1889
lemma power_int_numeral_neg_numeral [simp]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1890
  "power_int (numeral m) (-numeral n) = (inverse (numeral (Num.pow m n)) :: 'a)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1891
  by (simp add: power_int_minus)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1892
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1893
lemma zero_less_power_int [simp]: "0 < (x :: 'a) \<Longrightarrow> 0 < power_int x n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1894
  by (auto simp: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1895
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1896
lemma zero_le_power_int [simp]: "0 \<le> (x :: 'a) \<Longrightarrow> 0 \<le> power_int x n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1897
  by (auto simp: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1898
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1899
lemma power_int_mono: "(x :: 'a) \<le> y \<Longrightarrow> n \<ge> 0 \<Longrightarrow> 0 \<le> x \<Longrightarrow> power_int x n \<le> power_int y n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1900
  by (cases n rule: int_cases4) (auto intro: power_mono)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1901
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1902
lemma one_le_power_int [simp]: "1 \<le> (x :: 'a) \<Longrightarrow> n \<ge> 0 \<Longrightarrow> 1 \<le> power_int x n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1903
  using power_int_mono [of 1 x n] by simp
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1904
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1905
lemma power_int_le_one: "0 \<le> (x :: 'a) \<Longrightarrow> n \<ge> 0 \<Longrightarrow> x \<le> 1 \<Longrightarrow> power_int x n \<le> 1"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1906
  using power_int_mono [of x 1 n] by simp
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1907
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1908
lemma power_int_le_imp_le_exp:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1909
  assumes gt1: "1 < (x :: 'a :: linordered_field)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1910
  assumes "power_int x m \<le> power_int x n" "n \<ge> 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1911
  shows   "m \<le> n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1912
proof (cases "m < 0")
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1913
  case True
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1914
  with \<open>n \<ge> 0\<close> show ?thesis by simp
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1915
next
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1916
  case False
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1917
  with assms have "x ^ nat m \<le> x ^ nat n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1918
    by (simp add: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1919
  from gt1 and this show ?thesis
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1920
    using False \<open>n \<ge> 0\<close> by auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1921
qed
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1922
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1923
lemma power_int_le_imp_less_exp:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1924
  assumes gt1: "1 < (x :: 'a :: linordered_field)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1925
  assumes "power_int x m < power_int x n" "n \<ge> 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1926
  shows   "m < n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1927
proof (cases "m < 0")
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1928
  case True
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1929
  with \<open>n \<ge> 0\<close> show ?thesis by simp
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1930
next
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1931
  case False
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1932
  with assms have "x ^ nat m < x ^ nat n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1933
    by (simp add: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1934
  from gt1 and this show ?thesis
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1935
    using False \<open>n \<ge> 0\<close> by auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1936
qed
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1937
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1938
lemma power_int_strict_mono:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1939
  "(a :: 'a :: linordered_field) < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> power_int a n < power_int b n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1940
  by (auto simp: power_int_def intro!: power_strict_mono)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1941
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1942
lemma power_int_mono_iff [simp]:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1943
  fixes a b :: "'a :: linordered_field"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1944
  shows "\<lbrakk>a \<ge> 0; b \<ge> 0; n > 0\<rbrakk> \<Longrightarrow> power_int a n \<le> power_int b n \<longleftrightarrow> a \<le> b"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1945
  by (auto simp: power_int_def intro!: power_strict_mono)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1946
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1947
lemma power_int_strict_increasing:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1948
  fixes a :: "'a :: linordered_field"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1949
  assumes "n < N" "1 < a"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1950
  shows   "power_int a N > power_int a n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1951
proof -
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1952
  have *: "a ^ nat (N - n) > a ^ 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1953
    using assms by (intro power_strict_increasing) auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1954
  have "power_int a N = power_int a n * power_int a (N - n)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1955
    using assms by (simp flip: power_int_add)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1956
  also have "\<dots> > power_int a n * 1"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1957
    using assms *
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1958
    by (intro mult_strict_left_mono zero_less_power_int) (auto simp: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1959
  finally show ?thesis by simp
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1960
qed
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1961
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1962
lemma power_int_increasing:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1963
  fixes a :: "'a :: linordered_field"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1964
  assumes "n \<le> N" "a \<ge> 1"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1965
  shows   "power_int a N \<ge> power_int a n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1966
proof -
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1967
  have *: "a ^ nat (N - n) \<ge> a ^ 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1968
    using assms by (intro power_increasing) auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1969
  have "power_int a N = power_int a n * power_int a (N - n)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1970
    using assms by (simp flip: power_int_add)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1971
  also have "\<dots> \<ge> power_int a n * 1"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1972
    using assms * by (intro mult_left_mono) (auto simp: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1973
  finally show ?thesis by simp
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1974
qed
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1975
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1976
lemma power_int_strict_decreasing:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1977
  fixes a :: "'a :: linordered_field"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1978
  assumes "n < N" "0 < a" "a < 1"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1979
  shows   "power_int a N < power_int a n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1980
proof -
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1981
  have *: "a ^ nat (N - n) < a ^ 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1982
    using assms by (intro power_strict_decreasing) auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1983
  have "power_int a N = power_int a n * power_int a (N - n)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1984
    using assms by (simp flip: power_int_add)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1985
  also have "\<dots> < power_int a n * 1"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1986
    using assms *
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1987
    by (intro mult_strict_left_mono zero_less_power_int) (auto simp: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1988
  finally show ?thesis by simp
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1989
qed
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1990
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1991
lemma power_int_decreasing:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1992
  fixes a :: "'a :: linordered_field"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1993
  assumes "n \<le> N" "0 \<le> a" "a \<le> 1" "a \<noteq> 0 \<or> N \<noteq> 0 \<or> n = 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1994
  shows   "power_int a N \<le> power_int a n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1995
proof (cases "a = 0")
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1996
  case False
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1997
  have *: "a ^ nat (N - n) \<le> a ^ 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1998
    using assms by (intro power_decreasing) auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  1999
  have "power_int a N = power_int a n * power_int a (N - n)"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2000
    using assms False by (simp flip: power_int_add)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2001
  also have "\<dots> \<le> power_int a n * 1"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2002
    using assms * by (intro mult_left_mono) (auto simp: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2003
  finally show ?thesis by simp
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2004
qed (use assms in \<open>auto simp: power_int_0_left_If\<close>)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2005
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2006
lemma one_less_power_int: "1 < (a :: 'a) \<Longrightarrow> 0 < n \<Longrightarrow> 1 < power_int a n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2007
  using power_int_strict_increasing[of 0 n a] by simp
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2008
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2009
lemma power_int_abs: "\<bar>power_int a n :: 'a\<bar> = power_int \<bar>a\<bar> n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2010
  by (auto simp: power_int_def power_abs)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2011
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2012
lemma power_int_sgn [simp]: "sgn (power_int a n :: 'a) = power_int (sgn a) n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2013
  by (auto simp: power_int_def)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2014
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2015
lemma abs_power_int_minus [simp]: "\<bar>power_int (- a) n :: 'a\<bar> = \<bar>power_int a n\<bar>"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2016
  by (simp add: power_int_abs)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2017
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2018
lemma power_int_strict_antimono:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2019
  assumes "(a :: 'a :: linordered_field) < b" "0 < a" "n < 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2020
  shows   "power_int a n > power_int b n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2021
proof -
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2022
  have "inverse (power_int a (-n)) > inverse (power_int b (-n))"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2023
    using assms by (intro less_imp_inverse_less power_int_strict_mono zero_less_power_int) auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2024
  thus ?thesis by (simp add: power_int_minus)
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2025
qed
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2026
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2027
lemma power_int_antimono:
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2028
  assumes "(a :: 'a :: linordered_field) \<le> b" "0 < a" "n < 0"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2029
  shows   "power_int a n \<ge> power_int b n"
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2030
  using power_int_strict_antimono[of a b n] assms by (cases "a = b") auto
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2031
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2032
end
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2033
dca11678c495 new constant power_int in HOL
Manuel Eberl <eberlm@in.tum.de>
parents: 71616
diff changeset
  2034
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  2035
subsection \<open>Finiteness of intervals\<close>
46756
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2036
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2037
lemma finite_interval_int1 [iff]: "finite {i :: int. a \<le> i \<and> i \<le> b}"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2038
proof (cases "a \<le> b")
46756
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2039
  case True
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2040
  then show ?thesis
46756
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2041
  proof (induct b rule: int_ge_induct)
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2042
    case base
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2043
    have "{i. a \<le> i \<and> i \<le> a} = {a}" by auto
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2044
    then show ?case by simp
46756
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2045
  next
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2046
    case (step b)
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2047
    then have "{i. a \<le> i \<and> i \<le> b + 1} = {i. a \<le> i \<and> i \<le> b} \<union> {b + 1}" by auto
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2048
    with step show ?case by simp
46756
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2049
  qed
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2050
next
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2051
  case False
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2052
  then show ?thesis
46756
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2053
    by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2054
qed
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2055
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2056
lemma finite_interval_int2 [iff]: "finite {i :: int. a \<le> i \<and> i < b}"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2057
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
46756
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2058
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2059
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i \<and> i \<le> b}"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2060
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
46756
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2061
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2062
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i \<and> i < b}"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2063
  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
46756
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2064
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46027
diff changeset
  2065
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  2066
subsection \<open>Configuration of the code generator\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  2067
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  2068
text \<open>Constructors\<close>
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2069
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2070
definition Pos :: "num \<Rightarrow> int"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2071
  where [simp, code_abbrev]: "Pos = numeral"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2072
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2073
definition Neg :: "num \<Rightarrow> int"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2074
  where [simp, code_abbrev]: "Neg n = - (Pos n)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2075
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2076
code_datatype "0::int" Pos Neg
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2077
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2078
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2079
text \<open>Auxiliary operations.\<close>
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2080
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2081
definition dup :: "int \<Rightarrow> int"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2082
  where [simp]: "dup k = k + k"
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26300
diff changeset
  2083
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2084
lemma dup_code [code]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2085
  "dup 0 = 0"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2086
  "dup (Pos n) = Pos (Num.Bit0 n)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2087
  "dup (Neg n) = Neg (Num.Bit0 n)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2088
  by (simp_all add: numeral_Bit0)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2089
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2090
definition sub :: "num \<Rightarrow> num \<Rightarrow> int"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2091
  where [simp]: "sub m n = numeral m - numeral n"
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26300
diff changeset
  2092
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2093
lemma sub_code [code]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2094
  "sub Num.One Num.One = 0"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2095
  "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2096
  "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2097
  "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2098
  "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2099
  "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2100
  "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2101
  "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2102
  "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
66035
de6cd60b1226 replace non-arithmetic terms by fresh variables before replaying linear-arithmetic proofs: avoid failed proof replays due to an overambitious simpset which may cause proof replay to diverge from the pre-computed proof trace
boehmes
parents: 64996
diff changeset
  2103
  by (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2104
72512
83b5911c0164 more lemmas
haftmann
parents: 71837
diff changeset
  2105
lemma sub_BitM_One_eq:
83b5911c0164 more lemmas
haftmann
parents: 71837
diff changeset
  2106
  \<open>(Num.sub (Num.BitM n) num.One) = 2 * (Num.sub n Num.One :: int)\<close>
83b5911c0164 more lemmas
haftmann
parents: 71837
diff changeset
  2107
  by (cases n) simp_all
83b5911c0164 more lemmas
haftmann
parents: 71837
diff changeset
  2108
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2109
text \<open>Implementations.\<close>
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2110
64996
b316cd527a11 dropped superfluous preprocessing rule
haftmann
parents: 64849
diff changeset
  2111
lemma one_int_code [code]: "1 = Pos Num.One"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2112
  by simp
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2113
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2114
lemma plus_int_code [code]:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2115
  "k + 0 = k"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2116
  "0 + l = l"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2117
  "Pos m + Pos n = Pos (m + n)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2118
  "Pos m + Neg n = sub m n"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2119
  "Neg m + Pos n = sub n m"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2120
  "Neg m + Neg n = Neg (m + n)"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2121
  for k l :: int
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2122
  by simp_all
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26300
diff changeset
  2123
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2124
lemma uminus_int_code [code]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2125
  "uminus 0 = (0::int)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2126
  "uminus (Pos m) = Neg m"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2127
  "uminus (Neg m) = Pos m"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2128
  by simp_all
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2129
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2130
lemma minus_int_code [code]:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2131
  "k - 0 = k"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2132
  "0 - l = uminus l"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2133
  "Pos m - Pos n = sub m n"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2134
  "Pos m - Neg n = Pos (m + n)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2135
  "Neg m - Pos n = Neg (m + n)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2136
  "Neg m - Neg n = sub n m"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2137
  for k l :: int
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2138
  by simp_all
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2139
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2140
lemma times_int_code [code]:
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2141
  "k * 0 = 0"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2142
  "0 * l = 0"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2143
  "Pos m * Pos n = Pos (m * n)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2144
  "Pos m * Neg n = Neg (m * n)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2145
  "Neg m * Pos n = Neg (m * n)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2146
  "Neg m * Neg n = Pos (m * n)"
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2147
  for k l :: int
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2148
  by simp_all
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26300
diff changeset
  2149
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 37887
diff changeset
  2150
instantiation int :: equal
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26300
diff changeset
  2151
begin
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26300
diff changeset
  2152
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2153
definition "HOL.equal k l \<longleftrightarrow> k = (l::int)"
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 37887
diff changeset
  2154
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 61144
diff changeset
  2155
instance
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 61144
diff changeset
  2156
  by standard (rule equal_int_def)
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26300
diff changeset
  2157
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26300
diff changeset
  2158
end
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26300
diff changeset
  2159
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2160
lemma equal_int_code [code]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2161
  "HOL.equal 0 (0::int) \<longleftrightarrow> True"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2162
  "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2163
  "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2164
  "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2165
  "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2166
  "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2167
  "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2168
  "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2169
  "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2170
  by (auto simp add: equal)
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26300
diff changeset
  2171
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2172
lemma equal_int_refl [code nbe]: "HOL.equal k k \<longleftrightarrow> True"
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2173
  for k :: int
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2174
  by (fact equal_refl)
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26300
diff changeset
  2175
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28537
diff changeset
  2176
lemma less_eq_int_code [code]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2177
  "0 \<le> (0::int) \<longleftrightarrow> True"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2178
  "0 \<le> Pos l \<longleftrightarrow> True"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2179
  "0 \<le> Neg l \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2180
  "Pos k \<le> 0 \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2181
  "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2182
  "Pos k \<le> Neg l \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2183
  "Neg k \<le> 0 \<longleftrightarrow> True"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2184
  "Neg k \<le> Pos l \<longleftrightarrow> True"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2185
  "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
  2186
  by simp_all
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26300
diff changeset
  2187
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28537
diff changeset
  2188
lemma less_int_code [code]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2189
  "0 < (0::int) \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2190
  "0 < Pos l \<longleftrightarrow> True"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2191
  "0 < Neg l \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2192
  "Pos k < 0 \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2193
  "Pos k < Pos l \<longleftrightarrow> k < l"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2194
  "Pos k < Neg l \<longleftrightarrow> False"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2195
  "Neg k < 0 \<longleftrightarrow> True"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2196
  "Neg k < Pos l \<longleftrightarrow> True"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2197
  "Neg k < Neg l \<longleftrightarrow> l < k"
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
  2198
  by simp_all
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  2199
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2200
lemma nat_code [code]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2201
  "nat (Int.Neg k) = 0"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2202
  "nat 0 = 0"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2203
  "nat (Int.Pos k) = nat_of_num k"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  2204
  by (simp_all add: nat_of_num_numeral)
25928
042e877d9841 tuned code setup
haftmann
parents: 25919
diff changeset
  2205
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2206
lemma (in ring_1) of_int_code [code]:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  2207
  "of_int (Int.Neg k) = - numeral k"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2208
  "of_int 0 = 0"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2209
  "of_int (Int.Pos k) = numeral k"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2210
  by simp_all
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  2211
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2212
63652
804b80a80016 misc tuning and modernization;
wenzelm
parents: 63648
diff changeset
  2213
text \<open>Serializer setup.\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  2214
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51994
diff changeset
  2215
code_identifier
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 51994
diff changeset
  2216
  code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  2217
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  2218
quickcheck_params [default_type = int]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  2219
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  2220
hide_const (open) Pos Neg sub dup
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  2221
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  2222
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61738
diff changeset
  2223
text \<open>De-register \<open>int\<close> as a quotient type:\<close>
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
  2224
53652
18fbca265e2e use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents: 53065
diff changeset
  2225
lifting_update int.lifting
18fbca265e2e use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents: 53065
diff changeset
  2226
lifting_forget int.lifting
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
  2227
67116
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2228
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2229
subsection \<open>Duplicates\<close>
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2230
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2231
lemmas int_sum = of_nat_sum [where 'a=int]
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2232
lemmas int_prod = of_nat_prod [where 'a=int]
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2233
lemmas zle_int = of_nat_le_iff [where 'a=int]
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2234
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2235
lemmas nonneg_eq_int = nonneg_int_cases
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2236
lemmas double_eq_0_iff = double_zero
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2237
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2238
lemmas int_distrib =
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2239
  distrib_right [of z1 z2 w]
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2240
  distrib_left [of w z1 z2]
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2241
  left_diff_distrib [of z1 z2 w]
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2242
  right_diff_distrib [of w z1 z2]
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2243
  for z1 z2 w :: int
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2244
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  2245
end
67116
7397a6df81d8 cleaned up and tuned
haftmann
parents: 66912
diff changeset
  2246