src/HOL/Int.thy
author paulson <lp15@cam.ac.uk>
Tue, 17 Nov 2015 12:32:08 +0000
changeset 61694 6571c78c9667
parent 61649 268d88ec9087
child 61738 c4f6031f1310
permissions -rw-r--r--
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
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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" where
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  "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"
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    unfolding reflp_def by auto
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  show "symp intrel"
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    unfolding symp_def by auto
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  show "transp intrel"
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    unfolding transp_def by auto
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qed
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lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
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     "(!!x y. z = Abs_Integ (x, y) ==> P) ==> 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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  hence "(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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  thus "(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" where
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  "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,
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    simp add: one_int.abs_eq plus_int.abs_eq)
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lemma int_transfer [transfer_rule]:
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  "(rel_fun (op =) pcr_int) (\<lambda>n. (n, 0)) int"
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  unfolding rel_fun_def int.pcr_cr_eq cr_int_def int_def by simp
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lemma int_diff_cases:
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  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
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  "(inf :: int \<Rightarrow> int \<Rightarrow> int) = min"
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definition
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  "(sup :: int \<Rightarrow> int \<Rightarrow> int) = max"
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instance
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  by intro_classes
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    (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 k>0\<close>
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lemma zmult_zless_mono2_lemma:
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     "(i::int)<j ==> 0<k ==> int k * i < int k * j"
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apply (induct k)
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apply simp
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apply (simp add: distrib_right)
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apply (case_tac "k=0")
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apply (simp_all add: add_strict_mono)
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done
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lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = int n"
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apply transfer
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apply clarsimp
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apply (rule_tac x="a - b" in exI, simp)
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done
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lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = int n"
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apply transfer
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apply clarsimp
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apply (rule_tac x="a - b" in exI, simp)
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done
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lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
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apply (drule zero_less_imp_eq_int)
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apply (auto simp add: zmult_zless_mono2_lemma)
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done
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text\<open>The integers form an ordered integral domain\<close>
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instantiation int :: linordered_idom
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begin
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definition
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  zabs_def: "\<bar>i::int\<bar> = (if i < 0 then - i else i)"
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definition
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  zsgn_def: "sgn (i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
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instance proof
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  fix i j k :: int
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  show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
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    by (rule zmult_zless_mono2)
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  show "\<bar>i\<bar> = (if i < 0 then -i else i)"
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    by (simp only: zabs_def)
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  show "sgn (i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
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    by (simp only: zsgn_def)
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qed
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end
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lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1::int) \<le> z"
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  by transfer clarsimp
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lemma zless_iff_Suc_zadd:
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  "(w :: int) < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
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apply transfer
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apply auto
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apply (rename_tac a b c d)
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apply (rule_tac x="c+b - Suc(a+d)" in exI)
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apply arith
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done
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lemmas int_distrib =
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  distrib_right [of z1 z2 w]
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  distrib_left [of w z1 z2]
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  left_diff_distrib [of z1 z2 w]
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  right_diff_distrib [of w z1 z2]
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  for z1 z2 w :: int
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subsection \<open>Embedding of the Integers into any @{text ring_1}: @{text of_int}\<close>
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context ring_1
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begin
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lift_definition of_int :: "int \<Rightarrow> 'a" is "\<lambda>(i, j). of_nat i - of_nat j"
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  by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq
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    of_nat_add [symmetric] simp del: of_nat_add)
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lemma of_int_0 [simp]: "of_int 0 = 0"
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  by transfer simp
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lemma of_int_1 [simp]: "of_int 1 = 1"
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  by transfer simp
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lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
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  by transfer (clarsimp simp add: algebra_simps)
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lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
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  by (transfer fixing: uminus) clarsimp
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lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
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  using of_int_add [of w "- z"] by simp
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lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
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  by (transfer fixing: times) (clarsimp simp add: algebra_simps of_nat_mult)
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61531
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
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lemma mult_of_int_commute: "of_int x * y = y * of_int x"
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  by (transfer fixing: times) (auto simp: algebra_simps mult_of_nat_commute)
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text\<open>Collapse nested embeddings\<close>
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lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
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by (induct n) auto
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lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
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  by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])
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lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
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  by simp
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61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
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lemma of_int_power [simp]:
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  "of_int (z ^ n) = of_int z ^ n"
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  by (induct n) simp_all
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end
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context ring_char_0
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begin
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lemma of_int_eq_iff [simp]:
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   "of_int w = of_int z \<longleftrightarrow> w = z"
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  by transfer (clarsimp simp add: algebra_simps
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    of_nat_add [symmetric] simp del: of_nat_add)
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text\<open>Special cases where either operand is zero\<close>
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lemma of_int_eq_0_iff [simp]:
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  "of_int z = 0 \<longleftrightarrow> z = 0"
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  using of_int_eq_iff [of z 0] by simp
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lemma of_int_0_eq_iff [simp]:
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  "0 = of_int z \<longleftrightarrow> z = 0"
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  using of_int_eq_iff [of 0 z] by simp
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61234
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lemma of_int_eq_1_iff [iff]:
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   "of_int z = 1 \<longleftrightarrow> z = 1"
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  using of_int_eq_iff [of z 1] by simp
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end
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context linordered_idom
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begin
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text\<open>Every @{text linordered_idom} has characteristic zero.\<close>
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subclass ring_char_0 ..
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lemma of_int_le_iff [simp]:
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  "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
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  by (transfer fixing: less_eq) (clarsimp simp add: algebra_simps
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    of_nat_add [symmetric] simp del: of_nat_add)
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lemma of_int_less_iff [simp]:
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  "of_int w < of_int z \<longleftrightarrow> w < z"
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  by (simp add: less_le order_less_le)
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lemma of_int_0_le_iff [simp]:
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  "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
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  using of_int_le_iff [of 0 z] by simp
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lemma of_int_le_0_iff [simp]:
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  "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
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  using of_int_le_iff [of z 0] by simp
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lemma of_int_0_less_iff [simp]:
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  "0 < of_int z \<longleftrightarrow> 0 < z"
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  using of_int_less_iff [of 0 z] by simp
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lemma of_int_less_0_iff [simp]:
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  "of_int z < 0 \<longleftrightarrow> z < 0"
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   299
  using of_int_less_iff [of z 0] by simp
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   300
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   301
lemma of_int_1_le_iff [simp]:
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   302
  "1 \<le> of_int z \<longleftrightarrow> 1 \<le> z"
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   303
  using of_int_le_iff [of 1 z] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   304
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   305
lemma of_int_le_1_iff [simp]:
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   306
  "of_int z \<le> 1 \<longleftrightarrow> z \<le> 1"
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   307
  using of_int_le_iff [of z 1] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   308
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   309
lemma of_int_1_less_iff [simp]:
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   310
  "1 < of_int z \<longleftrightarrow> 1 < z"
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   311
  using of_int_less_iff [of 1 z] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   312
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   313
lemma of_int_less_1_iff [simp]:
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   314
  "of_int z < 1 \<longleftrightarrow> z < 1"
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   315
  using of_int_less_iff [of z 1] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   316
36424
f3f389fc7974 got rid of [simplified]
haftmann
parents: 36409
diff changeset
   317
end
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   318
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   319
text \<open>Comparisons involving @{term of_int}.\<close>
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   320
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   321
lemma of_int_eq_numeral_iff [iff]:
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
   322
   "of_int z = (numeral n :: 'a::ring_char_0)
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   323
   \<longleftrightarrow> z = numeral n"
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   324
  using of_int_eq_iff by fastforce
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   325
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
   326
lemma of_int_le_numeral_iff [simp]:
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
   327
   "of_int z \<le> (numeral n :: 'a::linordered_idom)
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   328
   \<longleftrightarrow> z \<le> numeral n"
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   329
  using of_int_le_iff [of z "numeral n"] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   330
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
   331
lemma of_int_numeral_le_iff [simp]:
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   332
   "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   333
  using of_int_le_iff [of "numeral n"] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   334
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
   335
lemma of_int_less_numeral_iff [simp]:
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
   336
   "of_int z < (numeral n :: 'a::linordered_idom)
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   337
   \<longleftrightarrow> z < numeral n"
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   338
  using of_int_less_iff [of z "numeral n"] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   339
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
   340
lemma of_int_numeral_less_iff [simp]:
61234
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   341
   "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   342
  using of_int_less_iff [of "numeral n" z] by simp
a9e6052188fa New lemmas
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
   343
56889
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56525
diff changeset
   344
lemma of_nat_less_of_int_iff:
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56525
diff changeset
   345
  "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56525
diff changeset
   346
  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
   347
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   348
lemma of_int_eq_id [simp]: "of_int = id"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   349
proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   350
  fix z show "of_int z = id z"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   351
    by (cases z rule: int_diff_cases, simp)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   352
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   353
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   354
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   355
instance int :: no_top
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 61144
diff changeset
   356
  apply standard
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   357
  apply (rule_tac x="x + 1" in exI)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   358
  apply simp
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   359
  done
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   360
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   361
instance int :: no_bot
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 61144
diff changeset
   362
  apply standard
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   363
  apply (rule_tac x="x - 1" in exI)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   364
  apply simp
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   365
  done
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51185
diff changeset
   366
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   367
subsection \<open>Magnitude of an Integer, as a Natural Number: @{text nat}\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   368
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   369
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
   370
  by auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   371
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   372
lemma nat_int [simp]: "nat (int n) = n"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   373
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   374
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   375
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
   376
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   377
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   378
corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   379
by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   380
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   381
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   382
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   383
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   384
lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   385
  by transfer (clarsimp, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   386
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   387
text\<open>An alternative condition is @{term "0 \<le> w"}\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   388
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   389
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
   390
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   391
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   392
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
   393
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   394
lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   395
  by transfer (clarsimp, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   396
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   397
lemma nonneg_eq_int:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   398
  fixes z :: int
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   399
  assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   400
  shows P
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   401
  using assms by (blast dest: nat_0_le sym)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   402
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   403
lemma nat_eq_iff:
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   404
  "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
   405
  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
   406
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   407
corollary nat_eq_iff2:
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   408
  "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   409
  using nat_eq_iff [of w m] by auto
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   410
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   411
lemma nat_0 [simp]:
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   412
  "nat 0 = 0"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   413
  by (simp add: nat_eq_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   414
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   415
lemma nat_1 [simp]:
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   416
  "nat 1 = Suc 0"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   417
  by (simp add: nat_eq_iff)
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   418
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   419
lemma nat_numeral [simp]:
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   420
  "nat (numeral k) = numeral k"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   421
  by (simp add: nat_eq_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   422
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   423
lemma nat_neg_numeral [simp]:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   424
  "nat (- numeral k) = 0"
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   425
  by simp
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   426
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   427
lemma nat_2: "nat 2 = Suc (Suc 0)"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   428
  by simp
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   429
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   430
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   431
  by transfer (clarsimp, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   432
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   433
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
   434
  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
   435
487ae6317f7b move lemmas nat_le_iff and nat_mono into Int.thy
huffman
parents: 44695
diff changeset
   436
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
   437
  by transfer auto
44707
487ae6317f7b move lemmas nat_le_iff and nat_mono into Int.thy
huffman
parents: 44695
diff changeset
   438
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
   439
lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   440
  by transfer clarsimp
29700
22faf21db3df added some simp rules
nipkow
parents: 29668
diff changeset
   441
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   442
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   443
by (auto simp add: nat_eq_iff2)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   444
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   445
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   446
by (insert zless_nat_conj [of 0], auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   447
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   448
lemma nat_add_distrib:
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   449
  "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
   450
  by transfer clarsimp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   451
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   452
lemma nat_diff_distrib':
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   453
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   454
  by transfer clarsimp
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   455
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   456
lemma nat_diff_distrib:
54223
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   457
  "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
85705ba18add restructed
haftmann
parents: 54147
diff changeset
   458
  by (rule nat_diff_distrib') auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   459
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   460
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   461
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   462
53065
de1816a7293e added lemma
haftmann
parents: 52435
diff changeset
   463
lemma le_nat_iff:
de1816a7293e added lemma
haftmann
parents: 52435
diff changeset
   464
  "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
de1816a7293e added lemma
haftmann
parents: 52435
diff changeset
   465
  by transfer auto
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   466
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   467
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   468
  by transfer (clarsimp simp add: less_diff_conv)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   469
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   470
context ring_1
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   471
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   472
61609
77b453bd616f Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents: 61552
diff changeset
   473
lemma 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
   474
  by transfer (clarsimp simp add: of_nat_diff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   475
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   476
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   477
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   478
lemma diff_nat_numeral [simp]:
54249
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   479
  "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   480
  by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   481
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   482
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   483
text \<open>For termination proofs:\<close>
29779
2786b348c376 declare "nat o abs" as default measure for int
krauss
parents: 29700
diff changeset
   484
lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
2786b348c376 declare "nat o abs" as default measure for int
krauss
parents: 29700
diff changeset
   485
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   486
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   487
subsection\<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   488
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
   489
lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   490
by (simp add: order_less_le del: of_nat_Suc)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   491
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   492
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   493
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   494
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   495
lemma negative_zle_0: "- int n \<le> 0"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   496
by (simp add: minus_le_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   497
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   498
lemma negative_zle [iff]: "- int n \<le> int m"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   499
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   500
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   501
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   502
by (subst le_minus_iff, simp del: of_nat_Suc)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   503
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   504
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   505
  by transfer simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   506
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   507
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   508
by (simp add: linorder_not_less)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   509
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   510
lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   511
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   512
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   513
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   514
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   515
  have "(w \<le> z) = (0 \<le> z - w)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   516
    by (simp only: le_diff_eq add_0_left)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   517
  also have "\<dots> = (\<exists>n. z - w = of_nat n)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   518
    by (auto elim: zero_le_imp_eq_int)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   519
  also have "\<dots> = (\<exists>n. z = w + of_nat n)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29046
diff changeset
   520
    by (simp only: algebra_simps)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   521
  finally show ?thesis .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   522
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   523
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   524
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   525
by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   526
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   527
text\<open>This version is proved for all ordered rings, not just integers!
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   528
      It is proved here because attribute @{text arith_split} is not available
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 35032
diff changeset
   529
      in theory @{text Rings}.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   530
      But is it really better than just rewriting with @{text abs_if}?\<close>
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 53652
diff changeset
   531
lemma abs_split [arith_split, no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34055
diff changeset
   532
     "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   533
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   534
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   535
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
48045
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   536
apply transfer
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   537
apply clarsimp
fbf77fdf9ae4 convert Int.thy to use lifting and transfer
huffman
parents: 48044
diff changeset
   538
apply (rule_tac x="b - Suc a" in exI, arith)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   539
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   540
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   541
subsection \<open>Cases and induction\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   542
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   543
text\<open>Now we replace the case analysis rule by a more conventional one:
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   544
whether an integer is negative or not.\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   545
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   546
text\<open>This version is symmetric in the two subgoals.\<close>
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   547
theorem int_cases2 [case_names nonneg nonpos, cases type: int]:
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   548
  "\<lbrakk>!! n. z = int n \<Longrightarrow> P;  !! n. z = - (int n) \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   549
apply (cases "z < 0")
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   550
apply (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   551
done
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   552
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   553
text\<open>This is the default, with a negative case.\<close>
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   554
theorem int_cases [case_names nonneg neg, cases type: int]:
59613
7103019278f0 The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents: 59582
diff changeset
   555
  "[|!! n. z = int n ==> P;  !! n. z = - (int (Suc n)) ==> P |] ==> P"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   556
apply (cases "z < 0")
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   557
apply (blast dest!: negD)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   558
apply (simp add: linorder_not_less del: of_nat_Suc)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   559
apply auto
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   560
apply (blast dest: nat_0_le [THEN sym])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   561
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   562
60868
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   563
lemma int_cases3 [case_names zero pos neg]:
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   564
  fixes k :: int
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   565
  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
   566
    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
   567
  shows "P"
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   568
proof (cases k "0::int" rule: linorder_cases)
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   569
  case equal with assms(1) show P by simp
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   570
next
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   571
  case greater
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   572
  then have "nat k > 0" by simp
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   573
  moreover from this have "k = int (nat k)" by auto
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   574
  ultimately show P using assms(2) by blast
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   575
next
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   576
  case less
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   577
  then have "nat (- k) > 0" by simp
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   578
  moreover from this have "k = - int (nat (- k))" by auto
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   579
  ultimately show P using assms(3) by blast
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   580
qed
dd18c33c001e direct bootstrap of integer division from natural division
haftmann
parents: 60758
diff changeset
   581
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   582
theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   583
     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   584
  by (cases z) auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   585
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   586
lemma nonneg_int_cases:
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   587
  assumes "0 \<le> k" obtains n where "k = int n"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   588
  using assms by (rule nonneg_eq_int)
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   589
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   590
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   591
  -- \<open>Unfold all @{text let}s involving constants\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   592
  by (fact Let_numeral) -- \<open>FIXME drop\<close>
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36811
diff changeset
   593
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   594
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   595
  -- \<open>Unfold all @{text let}s involving constants\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   596
  by (fact Let_neg_numeral) -- \<open>FIXME drop\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   597
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   598
text \<open>Unfold @{text min} and @{text max} on numerals.\<close>
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   599
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   600
lemmas max_number_of [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   601
  max_def [of "numeral u" "numeral v"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   602
  max_def [of "numeral u" "- numeral v"]
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   603
  max_def [of "- numeral u" "numeral v"]
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   604
  max_def [of "- numeral u" "- numeral v"] for u v
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   605
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   606
lemmas min_number_of [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   607
  min_def [of "numeral u" "numeral v"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   608
  min_def [of "numeral u" "- numeral v"]
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   609
  min_def [of "- numeral u" "numeral v"]
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
   610
  min_def [of "- numeral u" "- numeral v"] for u v
26075
815f3ccc0b45 added lemma lists {normalize,succ,pred,minus,add,mult}_bin_simps
huffman
parents: 26072
diff changeset
   611
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   612
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   613
subsubsection \<open>Binary comparisons\<close>
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   614
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   615
text \<open>Preliminaries\<close>
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   616
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   617
lemma le_imp_0_less:
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   618
  assumes le: "0 \<le> z"
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   619
  shows "(0::int) < 1 + z"
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   620
proof -
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   621
  have "0 \<le> z" by fact
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   622
  also have "... < z + 1" by (rule less_add_one)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   623
  also have "... = 1 + z" by (simp add: ac_simps)
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   624
  finally show "0 < 1 + z" .
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   625
qed
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   626
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   627
lemma odd_less_0_iff:
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   628
  "(1 + z + z < 0) = (z < (0::int))"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   629
proof (cases z)
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   630
  case (nonneg n)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56889
diff changeset
   631
  thus ?thesis by (simp add: linorder_not_less add.assoc add_increasing
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   632
                             le_imp_0_less [THEN order_less_imp_le])
28958
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   633
next
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   634
  case (neg n)
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30000
diff changeset
   635
  thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30000
diff changeset
   636
    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
   637
qed
74c60b78969c cleaned up subsection headings;
huffman
parents: 28952
diff changeset
   638
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   639
subsubsection \<open>Comparisons, for Ordered Rings\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   640
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   641
lemmas double_eq_0_iff = double_zero
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   642
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   643
lemma odd_nonzero:
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
   644
  "1 + z + z \<noteq> (0::int)"
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
   645
proof (cases z)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   646
  case (nonneg n)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   647
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   648
  thus ?thesis using  le_imp_0_less [OF le]
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   649
    by (auto simp add: add.assoc)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   650
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   651
  case (neg n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   652
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   653
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   654
    assume eq: "1 + z + z = 0"
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   655
    have "(0::int) < 1 + (int n + int n)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   656
      by (simp add: le_imp_0_less add_increasing)
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   657
    also have "... = - (1 + z + z)"
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   658
      by (simp add: neg add.assoc [symmetric])
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   659
    also have "... = 0" by (simp add: eq)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   660
    finally have "0<0" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   661
    thus False by blast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   662
  qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   663
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   664
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
   665
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   666
subsection \<open>The Set of Integers\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   667
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   668
context ring_1
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   669
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   670
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60868
diff changeset
   671
definition Ints :: "'a set"  ("\<int>")
b72a990adfe2 prefer symbols;
wenzelm
parents: 60868
diff changeset
   672
  where "\<int> = range of_int"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   673
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   674
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   675
  by (simp add: Ints_def)
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   676
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   677
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
45533
af3690f6bd79 simplify some proofs
huffman
parents: 45532
diff changeset
   678
  using Ints_of_int [of "of_nat n"] by simp
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   679
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   680
lemma Ints_0 [simp]: "0 \<in> \<int>"
45533
af3690f6bd79 simplify some proofs
huffman
parents: 45532
diff changeset
   681
  using Ints_of_int [of "0"] by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   682
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   683
lemma Ints_1 [simp]: "1 \<in> \<int>"
45533
af3690f6bd79 simplify some proofs
huffman
parents: 45532
diff changeset
   684
  using Ints_of_int [of "1"] by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   685
61552
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61531
diff changeset
   686
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
   687
  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
   688
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   689
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   690
apply (auto simp add: Ints_def)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   691
apply (rule range_eqI)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   692
apply (rule of_int_add [symmetric])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   693
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   694
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   695
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   696
apply (auto simp add: Ints_def)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   697
apply (rule range_eqI)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   698
apply (rule of_int_minus [symmetric])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   699
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   700
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   701
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   702
apply (auto simp add: Ints_def)
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   703
apply (rule range_eqI)
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   704
apply (rule of_int_diff [symmetric])
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   705
done
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   706
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   707
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   708
apply (auto simp add: Ints_def)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   709
apply (rule range_eqI)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   710
apply (rule of_int_mult [symmetric])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   711
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   712
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   713
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   714
by (induct n) simp_all
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   715
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   716
lemma Ints_cases [cases set: Ints]:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   717
  assumes "q \<in> \<int>"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   718
  obtains (of_int) z where "q = of_int z"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   719
  unfolding Ints_def
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   720
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   721
  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
   722
  then obtain z where "q = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   723
  then show thesis ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   724
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   725
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   726
lemma Ints_induct [case_names of_int, induct set: Ints]:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   727
  "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
   728
  by (rule Ints_cases) auto
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   729
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   730
lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   731
  unfolding Nats_def Ints_def
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   732
  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
   733
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   734
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
   735
proof (intro subsetI equalityI)
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   736
  fix x :: 'a assume "x \<in> {of_int n |n. n \<ge> 0}"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   737
  then obtain n where "x = of_int n" "n \<ge> 0" by (auto elim!: Ints_cases)
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   738
  hence "x = of_nat (nat n)" by (subst of_nat_nat) simp_all
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   739
  thus "x \<in> \<nat>" by simp
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   740
next
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   741
  fix x :: 'a assume "x \<in> \<nat>"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   742
  then obtain n where "x = of_nat n" by (auto elim!: Nats_cases)
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   743
  hence "x = of_int (int n)" by simp
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   744
  also have "int n \<ge> 0" by simp
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   745
  hence "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   746
  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
   747
qed
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   748
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   749
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   750
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   751
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
   752
proof (intro subsetI equalityI)
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   753
  fix x :: 'a assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   754
  then obtain n where "x = of_int n" "n \<ge> 0" by (auto elim!: Ints_cases)
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   755
  hence "x = of_nat (nat n)" by (subst of_nat_nat) simp_all
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   756
  thus "x \<in> \<nat>" by simp
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   757
qed (auto elim!: Nats_cases)
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   758
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61234
diff changeset
   759
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   760
text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   761
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   762
lemma Ints_double_eq_0_iff:
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60868
diff changeset
   763
  assumes in_Ints: "a \<in> \<int>"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   764
  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   765
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   766
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   767
  then obtain z where a: "a = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   768
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   769
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   770
    assume "a = 0"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   771
    thus "a + a = 0" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   772
  next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   773
    assume eq: "a + a = 0"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   774
    hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   775
    hence "z + z = 0" by (simp only: of_int_eq_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   776
    hence "z = 0" by (simp only: double_eq_0_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   777
    thus "a = 0" by (simp add: a)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   778
  qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   779
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   780
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   781
lemma Ints_odd_nonzero:
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60868
diff changeset
   782
  assumes in_Ints: "a \<in> \<int>"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   783
  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   784
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   785
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   786
  then obtain z where a: "a = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   787
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   788
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   789
    assume eq: "1 + a + a = 0"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   790
    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   791
    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   792
    with odd_nonzero show False by blast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   793
  qed
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   794
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   795
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60868
diff changeset
   796
lemma Nats_numeral [simp]: "numeral w \<in> \<nat>"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   797
  using of_nat_in_Nats [of "numeral w"] by simp
35634
6fdfe37b84d6 add more simp rules for Ints
huffman
parents: 35216
diff changeset
   798
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   799
lemma Ints_odd_less_0:
61070
b72a990adfe2 prefer symbols;
wenzelm
parents: 60868
diff changeset
   800
  assumes in_Ints: "a \<in> \<int>"
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34055
diff changeset
   801
  shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   802
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   803
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   804
  then obtain z where a: "a = of_int z" ..
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   805
  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   806
    by (simp add: a)
45532
74b17a0881b3 Int.thy: remove duplicate lemmas double_less_0_iff and odd_less_0, use {even,odd}_less_0_iff instead
huffman
parents: 45219
diff changeset
   807
  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   808
  also have "... = (a < 0)" by (simp add: a)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   809
  finally show ?thesis .
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   810
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   811
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   812
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   813
subsection \<open>@{term setsum} and @{term setprod}\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   814
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
   815
lemma of_nat_setsum [simp]: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   816
  apply (cases "finite A")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   817
  apply (erule finite_induct, auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   818
  done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   819
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
   820
lemma of_int_setsum [simp]: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   821
  apply (cases "finite A")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   822
  apply (erule finite_induct, auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   823
  done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   824
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
   825
lemma of_nat_setprod [simp]: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   826
  apply (cases "finite A")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   827
  apply (erule finite_induct, auto simp add: of_nat_mult)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   828
  done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   829
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
   830
lemma of_int_setprod [simp]: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   831
  apply (cases "finite A")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   832
  apply (erule finite_induct, auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   833
  done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   834
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   835
lemmas int_setsum = of_nat_setsum [where 'a=int]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   836
lemmas int_setprod = of_nat_setprod [where 'a=int]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   837
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   838
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   839
text \<open>Legacy theorems\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   840
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   841
lemmas zle_int = of_nat_le_iff [where 'a=int]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   842
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   843
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   844
subsection \<open>Setting up simplification procedures\<close>
30802
f9e9e800d27e simplify theorem references
huffman
parents: 30796
diff changeset
   845
54249
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   846
lemmas of_int_simps =
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   847
  of_int_0 of_int_1 of_int_add of_int_mult
ce00f2fef556 streamlined setup of linear arithmetic
haftmann
parents: 54230
diff changeset
   848
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 48066
diff changeset
   849
ML_file "Tools/int_arith.ML"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   850
declaration \<open>K Int_Arith.setup\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   851
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   852
simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
61144
5e94dfead1c2 simplified simproc programming interfaces;
wenzelm
parents: 61076
diff changeset
   853
  "(m::'a::linordered_idom) \<le> n" |
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   854
  "(m::'a::linordered_idom) = n") =
61144
5e94dfead1c2 simplified simproc programming interfaces;
wenzelm
parents: 61076
diff changeset
   855
  \<open>K Lin_Arith.simproc\<close>
43595
7ae4a23b5be6 modernized some simproc setup;
wenzelm
parents: 43531
diff changeset
   856
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   857
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   858
subsection\<open>More Inequality Reasoning\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   859
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   860
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   861
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   862
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   863
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   864
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   865
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   866
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   867
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   868
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   869
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   870
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   871
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   872
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   873
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   874
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   875
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   876
subsection\<open>The functions @{term nat} and @{term int}\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   877
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   878
text\<open>Simplify the term @{term "w + - z"}\<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
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   881
  using zless_nat_conj [of 1 z] by auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   882
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   883
text\<open>This simplifies expressions of the form @{term "int n = z"} where
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   884
      z is an integer literal.\<close>
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
   885
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   886
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   887
lemma split_nat [arith_split]:
44709
79f10d9e63c1 introduce abbreviation 'int' earlier in Int.thy
huffman
parents: 44707
diff changeset
   888
  "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   889
  (is "?P = (?L & ?R)")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   890
proof (cases "i < 0")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   891
  case True thus ?thesis by auto
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   892
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   893
  case False
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   894
  have "?P = ?L"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   895
  proof
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   896
    assume ?P thus ?L using False by clarsimp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   897
  next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   898
    assume ?L thus ?P using False by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   899
  qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   900
  with False show ?thesis by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   901
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   902
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   903
lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   904
  by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   905
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   906
lemma nat_int_add: "nat (int a + int b) = a + b"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   907
  by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58889
diff changeset
   908
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   909
context ring_1
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   910
begin
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   911
33056
791a4655cae3 renamed "nitpick_const_xxx" attributes to "nitpick_xxx" and "nitpick_ind_intros" to "nitpick_intros"
blanchet
parents: 32437
diff changeset
   912
lemma of_int_of_nat [nitpick_simp]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   913
  "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
   914
proof (cases "k < 0")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   915
  case True then have "0 \<le> - k" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   916
  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
   917
  with True show ?thesis by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   918
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   919
  case False then show ?thesis by (simp add: not_less of_nat_nat)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   920
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   921
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   922
end
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   923
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   924
lemma nat_mult_distrib:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   925
  fixes z z' :: int
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   926
  assumes "0 \<le> z"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   927
  shows "nat (z * z') = nat z * nat z'"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   928
proof (cases "0 \<le> z'")
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   929
  case False with assms have "z * z' \<le> 0"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   930
    by (simp add: not_le mult_le_0_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   931
  then have "nat (z * z') = 0" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   932
  moreover from False have "nat z' = 0" by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   933
  ultimately show ?thesis by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   934
next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   935
  case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   936
  show ?thesis
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   937
    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
   938
      (simp only: of_nat_mult of_nat_nat [OF True]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   939
         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
   940
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   941
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   942
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   943
apply (rule trans)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   944
apply (rule_tac [2] nat_mult_distrib, auto)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   945
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   946
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   947
lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   948
apply (cases "z=0 | w=0")
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   949
apply (auto simp add: abs_if nat_mult_distrib [symmetric]
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   950
                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   951
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   952
60570
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   953
lemma int_in_range_abs [simp]:
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   954
  "int n \<in> range abs"
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   955
proof (rule range_eqI)
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   956
  show "int n = \<bar>int n\<bar>"
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   957
    by simp
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   958
qed
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   959
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   960
lemma range_abs_Nats [simp]:
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   961
  "range abs = (\<nat> :: int set)"
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   962
proof -
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   963
  have "\<bar>k\<bar> \<in> \<nat>" for k :: int
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   964
    by (cases k) simp_all
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   965
  moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   966
    using that by induct simp
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   967
  ultimately show ?thesis by blast
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 61169
diff changeset
   968
qed
60570
7ed2cde6806d more theorems
haftmann
parents: 60162
diff changeset
   969
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   970
lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   971
apply (rule sym)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   972
apply (simp add: nat_eq_iff)
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   973
done
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   974
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   975
lemma diff_nat_eq_if:
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   976
     "nat z - nat z' =
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   977
        (if z' < 0 then nat z
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   978
         else let d = z-z' in
47207
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   979
              if d < 0 then 0 else nat d)"
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   980
by (simp add: Let_def nat_diff_distrib [symmetric])
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   981
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   982
lemma nat_numeral_diff_1 [simp]:
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   983
  "numeral v - (1::nat) = nat (numeral v - 1)"
9368aa814518 move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents: 47192
diff changeset
   984
  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
   985
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   986
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   987
subsection "Induction principles for int"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   988
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
   989
text\<open>Well-founded segments of the integers\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   990
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   991
definition
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   992
  int_ge_less_than  ::  "int => (int * int) set"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   993
where
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   994
  "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   995
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   996
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   997
proof -
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   998
  have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
   999
    by (auto simp add: int_ge_less_than_def)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1000
  thus ?thesis
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1001
    by (rule wf_subset [OF wf_measure])
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1002
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1003
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1004
text\<open>This variant looks odd, but is typical of the relations suggested
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1005
by RankFinder.\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1006
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1007
definition
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1008
  int_ge_less_than2 ::  "int => (int * int) set"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1009
where
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1010
  "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1011
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1012
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1013
proof -
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1014
  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1015
    by (auto simp add: int_ge_less_than2_def)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1016
  thus ?thesis
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1017
    by (rule wf_subset [OF wf_measure])
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1018
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1019
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1020
(* `set:int': dummy construction *)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1021
theorem int_ge_induct [case_names base step, induct set: int]:
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1022
  fixes i :: int
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1023
  assumes ge: "k \<le> i" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1024
    base: "P k" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1025
    step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1026
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1027
proof -
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1028
  { fix n
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1029
    have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1030
    proof (induct n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1031
      case 0
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1032
      hence "i = k" by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1033
      thus "P i" using base by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1034
    next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1035
      case (Suc n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1036
      then have "n = nat((i - 1) - k)" by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1037
      moreover
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1038
      have ki1: "k \<le> i - 1" using Suc.prems by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1039
      ultimately
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1040
      have "P (i - 1)" by (rule Suc.hyps)
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1041
      from step [OF ki1 this] show ?case by simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1042
    qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1043
  }
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1044
  with ge show ?thesis by fast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1045
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1046
25928
042e877d9841 tuned code setup
haftmann
parents: 25919
diff changeset
  1047
(* `set:int': dummy construction *)
042e877d9841 tuned code setup
haftmann
parents: 25919
diff changeset
  1048
theorem int_gr_induct [case_names base step, induct set: int]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1049
  assumes gr: "k < (i::int)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1050
        base: "P(k+1)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1051
        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1052
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1053
apply(rule int_ge_induct[of "k + 1"])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1054
  using gr apply arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1055
 apply(rule base)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1056
apply (rule step, simp+)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1057
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1058
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1059
theorem int_le_induct [consumes 1, case_names base step]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1060
  assumes le: "i \<le> (k::int)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1061
        base: "P(k)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1062
        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1063
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1064
proof -
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1065
  { fix n
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1066
    have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1067
    proof (induct n)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1068
      case 0
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1069
      hence "i = k" by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1070
      thus "P i" using base by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1071
    next
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1072
      case (Suc n)
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1073
      hence "n = nat (k - (i + 1))" by arith
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1074
      moreover
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1075
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1076
      ultimately
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1077
      have "P (i + 1)" by(rule Suc.hyps)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1078
      from step[OF ki1 this] show ?case by simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1079
    qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1080
  }
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1081
  with le show ?thesis by fast
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1082
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1083
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1084
theorem int_less_induct [consumes 1, case_names base step]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1085
  assumes less: "(i::int) < k" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1086
        base: "P(k - 1)" and
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1087
        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1088
  shows "P i"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1089
apply(rule int_le_induct[of _ "k - 1"])
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1090
  using less apply arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1091
 apply(rule base)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1092
apply (rule step, simp+)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1093
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1094
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
  1095
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
  1096
  fixes k :: int
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1097
  assumes base: "P k"
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1098
    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
  1099
    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
  1100
  shows "P i"
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1101
proof -
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1102
  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
  1103
  then show ?thesis
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1104
  proof
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1105
    assume "i \<ge> k"
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1106
    then show ?thesis using base
36801
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1107
      by (rule int_ge_induct) (fact step1)
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1108
  next
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1109
    assume "i \<le> k"
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1110
    then show ?thesis using base
36801
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1111
      by (rule int_le_induct) (fact step2)
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1112
  qed
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1113
qed
3560de0fe851 moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents: 36749
diff changeset
  1114
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1115
subsection\<open>Intermediate value theorems\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1116
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1117
lemma int_val_lemma:
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1118
     "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1119
      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30000
diff changeset
  1120
unfolding One_nat_def
42676
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1121
apply (induct n)
8724f20bf69c proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents: 42411
diff changeset
  1122
apply simp
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1123
apply (intro strip)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1124
apply (erule impE, simp)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1125
apply (erule_tac x = n in allE, simp)
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30000
diff changeset
  1126
apply (case_tac "k = f (Suc n)")
27106
ff27dc6e7d05 removed some dubious code lemmas
haftmann
parents: 26975
diff changeset
  1127
apply force
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1128
apply (erule impE)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1129
 apply (simp add: abs_if split add: split_if_asm)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1130
apply (blast intro: le_SucI)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1131
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1132
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1133
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1134
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1135
lemma nat_intermed_int_val:
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1136
     "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1137
         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1138
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1139
       in int_val_lemma)
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30000
diff changeset
  1140
unfolding One_nat_def
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1141
apply simp
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1142
apply (erule exE)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1143
apply (rule_tac x = "i+m" in exI, arith)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1144
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1145
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1146
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1147
subsection\<open>Products and 1, by T. M. Rasmussen\<close>
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1148
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1149
lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1150
by arith
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1151
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1152
lemma abs_zmult_eq_1:
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1153
  assumes mn: "\<bar>m * n\<bar> = 1"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1154
  shows "\<bar>m\<bar> = (1::int)"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1155
proof -
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1156
  have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1157
    by auto
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1158
  have "~ (2 \<le> \<bar>m\<bar>)"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1159
  proof
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1160
    assume "2 \<le> \<bar>m\<bar>"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1161
    hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1162
      by (simp add: mult_mono 0)
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1163
    also have "... = \<bar>m*n\<bar>"
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1164
      by (simp add: abs_mult)
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1165
    also have "... = 1"
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1166
      by (simp add: mn)
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1167
    finally have "2*\<bar>n\<bar> \<le> 1" .
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1168
    thus "False" using 0
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1169
      by arith
34055
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1170
  qed
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1171
  thus ?thesis using 0
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1172
    by auto
fdf294ee08b2 streamlined proofs
paulson
parents: 33657
diff changeset
  1173
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1174
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1175
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1176
by (insert abs_zmult_eq_1 [of m n], arith)
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1177
35815
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1178
lemma pos_zmult_eq_1_iff:
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1179
  assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1180
proof -
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1181
  from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1182
  thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
10e723e54076 tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents: 35634
diff changeset
  1183
qed
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1184
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1185
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1186
apply (rule iffI)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1187
 apply (frule pos_zmult_eq_1_iff_lemma)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1188
 apply (simp add: mult.commute [of m])
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1189
 apply (frule pos_zmult_eq_1_iff_lemma, auto)
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1190
done
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1191
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1192
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1193
proof
33296
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1194
  assume "finite (UNIV::int set)"
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
  1195
  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
  1196
    by (rule injI) simp
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
  1197
  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
  1198
    by (rule finite_UNIV_inj_surj)
a3924d1069e5 moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents: 33056
diff changeset
  1199
  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
  1200
  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
  1201
qed
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1202
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff changeset
  1203
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1204
subsection \<open>Further theorems on numerals\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1205
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1206
subsubsection\<open>Special Simplification for Constants\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1207
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1208
text\<open>These distributive laws move literals inside sums and differences.\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1209
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1210
lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 48891
diff changeset
  1211
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1212
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1213
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1214
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1215
text\<open>These are actually for fields, like real: but where else to put them?\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1216
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1217
lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1218
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1219
lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1220
lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1221
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1222
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1223
text \<open>Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1224
  strange, but then other simprocs simplify the quotient.\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1225
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1226
lemmas inverse_eq_divide_numeral [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1227
  inverse_eq_divide [of "numeral w"] for w
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1228
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1229
lemmas inverse_eq_divide_neg_numeral [simp] =
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1230
  inverse_eq_divide [of "- numeral w"] for w
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1231
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1232
text \<open>These laws simplify inequalities, moving unary minus from a term
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1233
into the literal.\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1234
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1235
lemmas equation_minus_iff_numeral [no_atp] =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1236
  equation_minus_iff [of "numeral v"] for v
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1237
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1238
lemmas minus_equation_iff_numeral [no_atp] =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1239
  minus_equation_iff [of _ "numeral v"] for v
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1240
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1241
lemmas le_minus_iff_numeral [no_atp] =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1242
  le_minus_iff [of "numeral v"] for v
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1243
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1244
lemmas minus_le_iff_numeral [no_atp] =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1245
  minus_le_iff [of _ "numeral v"] for v
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1246
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1247
lemmas less_minus_iff_numeral [no_atp] =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1248
  less_minus_iff [of "numeral v"] for v
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1249
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1250
lemmas minus_less_iff_numeral [no_atp] =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1251
  minus_less_iff [of _ "numeral v"] for v
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1252
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1253
-- \<open>FIXME maybe simproc\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1254
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1255
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1256
text \<open>Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"})\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1257
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1258
lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1259
lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1260
lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1261
lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1262
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1263
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1264
text \<open>Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="})\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1265
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1266
lemmas le_divide_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1267
  pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1268
  neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1269
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1270
lemmas divide_le_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1271
  pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1272
  neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1273
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1274
lemmas less_divide_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1275
  pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1276
  neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1277
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1278
lemmas divide_less_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1279
  pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1280
  neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1281
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1282
lemmas eq_divide_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1283
  eq_divide_eq [of _ _ "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1284
  eq_divide_eq [of _ _ "- numeral w"] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1285
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1286
lemmas divide_eq_eq_numeral1 [simp] =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1287
  divide_eq_eq [of _ "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1288
  divide_eq_eq [of _ "- numeral w"] for w
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1289
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1290
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1291
subsubsection\<open>Optional Simplification Rules Involving Constants\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1292
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1293
text\<open>Simplify quotients that are compared with a literal constant.\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1294
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1295
lemmas le_divide_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1296
  le_divide_eq [of "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1297
  le_divide_eq [of "- numeral w"] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1298
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1299
lemmas divide_le_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1300
  divide_le_eq [of _ _ "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1301
  divide_le_eq [of _ _ "- numeral w"] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1302
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1303
lemmas less_divide_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1304
  less_divide_eq [of "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1305
  less_divide_eq [of "- numeral w"] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1306
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1307
lemmas divide_less_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1308
  divide_less_eq [of _ _ "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1309
  divide_less_eq [of _ _ "- numeral w"] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1310
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1311
lemmas eq_divide_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1312
  eq_divide_eq [of "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1313
  eq_divide_eq [of "- numeral w"] for w
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1314
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1315
lemmas divide_eq_eq_numeral =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1316
  divide_eq_eq [of _ _ "numeral w"]
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54249
diff changeset
  1317
  divide_eq_eq [of _ _ "- numeral w"] for w
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1318
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1319
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1320
text\<open>Not good as automatic simprules because they cause case splits.\<close>
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1321
lemmas divide_const_simps =
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1322
  le_divide_eq_numeral divide_le_eq_numeral less_divide_eq_numeral
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46756
diff changeset
  1323
  divide_less_eq_numeral eq_divide_eq_numeral divide_eq_eq_numeral
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1324
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1325
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30496
diff changeset
  1326
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1327
subsection \<open>The divides relation\<close>
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1328
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1329
lemma zdvd_antisym_nonneg:
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1330
    "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1331
  apply (simp add: dvd_def, auto)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56889
diff changeset
  1332
  apply (auto simp add: mult.assoc zero_le_mult_iff zmult_eq_1_iff)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1333
  done
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1334
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1335
lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a"
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1336
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1337
proof cases
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1338
  assume "a = 0" with assms show ?thesis by simp
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1339
next
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33523
diff changeset
  1340
  assume "a \<noteq> 0"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1341
  from \<open>a dvd b\<close> obtain k where k:"b = a*k" unfolding dvd_def by blast
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1342
  from \<open>b dvd a\<close> obtain k' where k':"a = b*k'" unfolding dvd_def by blast
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1343
  from k k' have "a = a*k*k'" by simp
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1344
  with mult_cancel_left1[where c="a" and b="k*k'"]
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60570
diff changeset
  1345
  have kk':"k*k' = 1" using \<open>a\<noteq>0\<close> by (simp add: mult.assoc)
33320
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1346
  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1347
  thus ?thesis using k k' by auto
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1348
qed
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1349
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1350
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
  1351
  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
  1352
73998ef6ea91 moved some dvd [int] facts to Int
haftmann
parents: 33296
diff changeset
  1353
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
58649
a62065b5e1e2 generalized and consolidated some theorems concerning divisibility
haftmann
parents: 58512
diff changeset
  1354
  using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)
33320