author | haftmann |
Fri, 20 Oct 2017 20:57:55 +0200 | |
changeset 66888 | 930abfdf8727 |
parent 66886 | 960509bfd47e |
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permissions | -rw-r--r-- |
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(* Title: HOL/Int.thy |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Author: Tobias Nipkow, Florian Haftmann, TU Muenchen |
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*) |
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section \<open>The Integers as Equivalence Classes over Pairs of Natural Numbers\<close> |
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theory Int |
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imports Equiv_Relations Power Quotient Fun_Def |
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begin |
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subsection \<open>Definition of integers as a quotient type\<close> |
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definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" |
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where "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)" |
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lemma intrel_iff [simp]: "intrel (x, y) (u, v) \<longleftrightarrow> x + v = u + y" |
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by (simp add: intrel_def) |
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quotient_type int = "nat \<times> nat" / "intrel" |
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morphisms Rep_Integ Abs_Integ |
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proof (rule equivpI) |
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show "reflp intrel" by (auto simp: reflp_def) |
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show "symp intrel" by (auto simp: symp_def) |
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show "transp intrel" by (auto simp: transp_def) |
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qed |
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lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]: |
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"(\<And>x y. z = Abs_Integ (x, y) \<Longrightarrow> P) \<Longrightarrow> P" |
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by (induct z) auto |
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subsection \<open>Integers form a commutative ring\<close> |
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instantiation int :: comm_ring_1 |
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begin |
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lift_definition zero_int :: "int" is "(0, 0)" . |
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lift_definition one_int :: "int" is "(1, 0)" . |
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lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int" |
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is "\<lambda>(x, y) (u, v). (x + u, y + v)" |
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by clarsimp |
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lift_definition uminus_int :: "int \<Rightarrow> int" |
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is "\<lambda>(x, y). (y, x)" |
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by clarsimp |
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lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int" |
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is "\<lambda>(x, y) (u, v). (x + v, y + u)" |
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by clarsimp |
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lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int" |
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is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)" |
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proof (clarsimp) |
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fix s t u v w x y z :: nat |
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assume "s + v = u + t" and "w + z = y + x" |
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then have "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x) = |
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(u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)" |
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by simp |
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then show "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)" |
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by (simp add: algebra_simps) |
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qed |
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instance |
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by standard (transfer; clarsimp simp: algebra_simps)+ |
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end |
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abbreviation int :: "nat \<Rightarrow> int" |
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where "int \<equiv> of_nat" |
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lemma int_def: "int n = Abs_Integ (n, 0)" |
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by (induct n) (simp add: zero_int.abs_eq, simp add: one_int.abs_eq plus_int.abs_eq) |
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lemma int_transfer [transfer_rule]: "(rel_fun (op =) pcr_int) (\<lambda>n. (n, 0)) int" |
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by (simp add: rel_fun_def int.pcr_cr_eq cr_int_def int_def) |
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lemma int_diff_cases: obtains (diff) m n where "z = int m - int n" |
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by transfer clarsimp |
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subsection \<open>Integers are totally ordered\<close> |
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instantiation int :: linorder |
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begin |
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lift_definition less_eq_int :: "int \<Rightarrow> int \<Rightarrow> bool" |
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is "\<lambda>(x, y) (u, v). x + v \<le> u + y" |
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by auto |
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lift_definition less_int :: "int \<Rightarrow> int \<Rightarrow> bool" |
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is "\<lambda>(x, y) (u, v). x + v < u + y" |
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by auto |
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instance |
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by standard (transfer, force)+ |
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end |
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instantiation int :: distrib_lattice |
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begin |
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definition "(inf :: int \<Rightarrow> int \<Rightarrow> int) = min" |
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definition "(sup :: int \<Rightarrow> int \<Rightarrow> int) = max" |
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instance |
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by standard (auto simp add: inf_int_def sup_int_def max_min_distrib2) |
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end |
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subsection \<open>Ordering properties of arithmetic operations\<close> |
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instance int :: ordered_cancel_ab_semigroup_add |
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proof |
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fix i j k :: int |
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show "i \<le> j \<Longrightarrow> k + i \<le> k + j" |
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by transfer clarsimp |
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qed |
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text \<open>Strict Monotonicity of Multiplication.\<close> |
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text \<open>Strict, in 1st argument; proof is by induction on \<open>k > 0\<close>.\<close> |
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lemma zmult_zless_mono2_lemma: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> int k * i < int k * j" |
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for i j :: int |
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proof (induct k) |
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case 0 |
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then show ?case by simp |
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next |
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case (Suc k) |
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then show ?case |
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by (cases "k = 0") (simp_all add: distrib_right add_strict_mono) |
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qed |
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lemma zero_le_imp_eq_int: "0 \<le> k \<Longrightarrow> \<exists>n. k = int n" |
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for k :: int |
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apply transfer |
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apply clarsimp |
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apply (rule_tac x="a - b" in exI) |
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apply simp |
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done |
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lemma zero_less_imp_eq_int: "0 < k \<Longrightarrow> \<exists>n>0. k = int n" |
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for k :: int |
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apply transfer |
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apply clarsimp |
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apply (rule_tac x="a - b" in exI) |
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apply simp |
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done |
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lemma zmult_zless_mono2: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j" |
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for i j k :: int |
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by (drule zero_less_imp_eq_int) (auto simp add: zmult_zless_mono2_lemma) |
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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 zabs_def: "\<bar>i::int\<bar> = (if i < 0 then - i else i)" |
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definition zsgn_def: "sgn (i::int) = (if i = 0 then 0 else if 0 < i then 1 else - 1)" |
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instance |
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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 \<le> z" |
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for w z :: int |
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by transfer clarsimp |
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lemma zless_iff_Suc_zadd: "w < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))" |
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for w z :: int |
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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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lemma zabs_less_one_iff [simp]: "\<bar>z\<bar> < 1 \<longleftrightarrow> z = 0" (is "?lhs \<longleftrightarrow> ?rhs") |
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for z :: int |
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proof |
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assume ?rhs |
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then show ?lhs by simp |
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next |
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assume ?lhs |
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with zless_imp_add1_zle [of "\<bar>z\<bar>" 1] have "\<bar>z\<bar> + 1 \<le> 1" by simp |
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then have "\<bar>z\<bar> \<le> 0" by simp |
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then show ?rhs by simp |
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qed |
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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 \<open>ring_1\<close>: \<open>of_int\<close>\<close> |
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context ring_1 |
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begin |
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lift_definition of_int :: "int \<Rightarrow> 'a" |
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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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|
234 |
by (transfer fixing: uminus) clarsimp |
25919
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joined theories IntDef, Numeral, IntArith to theory Int
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parents:
diff
changeset
|
235 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
236 |
lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54223
diff
changeset
|
237 |
using of_int_add [of w "- z"] by simp |
25919
8b1c0d434824
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haftmann
parents:
diff
changeset
|
238 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
239 |
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z" |
63652 | 240 |
by (transfer fixing: times) (clarsimp simp add: algebra_simps) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
241 |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
242 |
lemma mult_of_int_commute: "of_int x * y = y * of_int x" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
243 |
by (transfer fixing: times) (auto simp: algebra_simps mult_of_nat_commute) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
244 |
|
63652 | 245 |
text \<open>Collapse nested embeddings.\<close> |
44709 | 246 |
lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n" |
63652 | 247 |
by (induct n) auto |
25919
8b1c0d434824
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haftmann
parents:
diff
changeset
|
248 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
249 |
lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
250 |
by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric]) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
251 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
252 |
lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
253 |
by simp |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
254 |
|
63652 | 255 |
lemma of_int_power [simp]: "of_int (z ^ n) = of_int z ^ n" |
31015 | 256 |
by (induct n) simp_all |
257 |
||
66816
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
258 |
lemma of_int_of_bool [simp]: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
259 |
"of_int (of_bool P) = of_bool P" |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
260 |
by auto |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
261 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
262 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
263 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
264 |
context ring_char_0 |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
265 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
266 |
|
63652 | 267 |
lemma of_int_eq_iff [simp]: "of_int w = of_int z \<longleftrightarrow> w = z" |
268 |
by transfer (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
269 |
|
63652 | 270 |
text \<open>Special cases where either operand is zero.\<close> |
271 |
lemma of_int_eq_0_iff [simp]: "of_int z = 0 \<longleftrightarrow> z = 0" |
|
36424 | 272 |
using of_int_eq_iff [of z 0] by simp |
273 |
||
63652 | 274 |
lemma of_int_0_eq_iff [simp]: "0 = of_int z \<longleftrightarrow> z = 0" |
36424 | 275 |
using of_int_eq_iff [of 0 z] by simp |
25919
8b1c0d434824
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haftmann
parents:
diff
changeset
|
276 |
|
63652 | 277 |
lemma of_int_eq_1_iff [iff]: "of_int z = 1 \<longleftrightarrow> z = 1" |
61234 | 278 |
using of_int_eq_iff [of z 1] by simp |
279 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
280 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
281 |
|
36424 | 282 |
context linordered_idom |
283 |
begin |
|
284 |
||
63652 | 285 |
text \<open>Every \<open>linordered_idom\<close> has characteristic zero.\<close> |
36424 | 286 |
subclass ring_char_0 .. |
287 |
||
63652 | 288 |
lemma of_int_le_iff [simp]: "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z" |
289 |
by (transfer fixing: less_eq) |
|
290 |
(clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add) |
|
36424 | 291 |
|
63652 | 292 |
lemma of_int_less_iff [simp]: "of_int w < of_int z \<longleftrightarrow> w < z" |
36424 | 293 |
by (simp add: less_le order_less_le) |
294 |
||
63652 | 295 |
lemma of_int_0_le_iff [simp]: "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z" |
36424 | 296 |
using of_int_le_iff [of 0 z] by simp |
297 |
||
63652 | 298 |
lemma of_int_le_0_iff [simp]: "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0" |
36424 | 299 |
using of_int_le_iff [of z 0] by simp |
300 |
||
63652 | 301 |
lemma of_int_0_less_iff [simp]: "0 < of_int z \<longleftrightarrow> 0 < z" |
36424 | 302 |
using of_int_less_iff [of 0 z] by simp |
303 |
||
63652 | 304 |
lemma of_int_less_0_iff [simp]: "of_int z < 0 \<longleftrightarrow> z < 0" |
36424 | 305 |
using of_int_less_iff [of z 0] by simp |
306 |
||
63652 | 307 |
lemma of_int_1_le_iff [simp]: "1 \<le> of_int z \<longleftrightarrow> 1 \<le> z" |
61234 | 308 |
using of_int_le_iff [of 1 z] by simp |
309 |
||
63652 | 310 |
lemma of_int_le_1_iff [simp]: "of_int z \<le> 1 \<longleftrightarrow> z \<le> 1" |
61234 | 311 |
using of_int_le_iff [of z 1] by simp |
312 |
||
63652 | 313 |
lemma of_int_1_less_iff [simp]: "1 < of_int z \<longleftrightarrow> 1 < z" |
61234 | 314 |
using of_int_less_iff [of 1 z] by simp |
315 |
||
63652 | 316 |
lemma of_int_less_1_iff [simp]: "of_int z < 1 \<longleftrightarrow> z < 1" |
61234 | 317 |
using of_int_less_iff [of z 1] by simp |
318 |
||
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61944
diff
changeset
|
319 |
lemma of_int_pos: "z > 0 \<Longrightarrow> of_int z > 0" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61944
diff
changeset
|
320 |
by simp |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61944
diff
changeset
|
321 |
|
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61944
diff
changeset
|
322 |
lemma of_int_nonneg: "z \<ge> 0 \<Longrightarrow> of_int z \<ge> 0" |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61944
diff
changeset
|
323 |
by simp |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61944
diff
changeset
|
324 |
|
63652 | 325 |
lemma of_int_abs [simp]: "of_int \<bar>x\<bar> = \<bar>of_int x\<bar>" |
62347 | 326 |
by (auto simp add: abs_if) |
327 |
||
328 |
lemma of_int_lessD: |
|
329 |
assumes "\<bar>of_int n\<bar> < x" |
|
330 |
shows "n = 0 \<or> x > 1" |
|
331 |
proof (cases "n = 0") |
|
63652 | 332 |
case True |
333 |
then show ?thesis by simp |
|
62347 | 334 |
next |
335 |
case False |
|
336 |
then have "\<bar>n\<bar> \<noteq> 0" by simp |
|
337 |
then have "\<bar>n\<bar> > 0" by simp |
|
338 |
then have "\<bar>n\<bar> \<ge> 1" |
|
339 |
using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp |
|
340 |
then have "\<bar>of_int n\<bar> \<ge> 1" |
|
341 |
unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp |
|
342 |
then have "1 < x" using assms by (rule le_less_trans) |
|
343 |
then show ?thesis .. |
|
344 |
qed |
|
345 |
||
346 |
lemma of_int_leD: |
|
347 |
assumes "\<bar>of_int n\<bar> \<le> x" |
|
348 |
shows "n = 0 \<or> 1 \<le> x" |
|
349 |
proof (cases "n = 0") |
|
63652 | 350 |
case True |
351 |
then show ?thesis by simp |
|
62347 | 352 |
next |
353 |
case False |
|
354 |
then have "\<bar>n\<bar> \<noteq> 0" by simp |
|
355 |
then have "\<bar>n\<bar> > 0" by simp |
|
356 |
then have "\<bar>n\<bar> \<ge> 1" |
|
357 |
using zless_imp_add1_zle [of 0 "\<bar>n\<bar>"] by simp |
|
358 |
then have "\<bar>of_int n\<bar> \<ge> 1" |
|
359 |
unfolding of_int_1_le_iff [of "\<bar>n\<bar>", symmetric] by simp |
|
360 |
then have "1 \<le> x" using assms by (rule order_trans) |
|
361 |
then show ?thesis .. |
|
362 |
qed |
|
363 |
||
36424 | 364 |
end |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
365 |
|
61234 | 366 |
text \<open>Comparisons involving @{term of_int}.\<close> |
367 |
||
63652 | 368 |
lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) \<longleftrightarrow> z = numeral n" |
61234 | 369 |
using of_int_eq_iff by fastforce |
370 |
||
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
|
371 |
lemma of_int_le_numeral_iff [simp]: |
63652 | 372 |
"of_int z \<le> (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z \<le> numeral n" |
61234 | 373 |
using of_int_le_iff [of z "numeral n"] by simp |
374 |
||
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
|
375 |
lemma of_int_numeral_le_iff [simp]: |
63652 | 376 |
"(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z" |
61234 | 377 |
using of_int_le_iff [of "numeral n"] by simp |
378 |
||
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
|
379 |
lemma of_int_less_numeral_iff [simp]: |
63652 | 380 |
"of_int z < (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z < numeral n" |
61234 | 381 |
using of_int_less_iff [of z "numeral n"] by simp |
382 |
||
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
|
383 |
lemma of_int_numeral_less_iff [simp]: |
63652 | 384 |
"(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z" |
61234 | 385 |
using of_int_less_iff [of "numeral n" z] by simp |
386 |
||
63652 | 387 |
lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x" |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56525
diff
changeset
|
388 |
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
|
389 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
390 |
lemma of_int_eq_id [simp]: "of_int = id" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
391 |
proof |
63652 | 392 |
show "of_int z = id z" for z |
393 |
by (cases z rule: int_diff_cases) simp |
|
25919
8b1c0d434824
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haftmann
parents:
diff
changeset
|
394 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
395 |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51185
diff
changeset
|
396 |
instance int :: no_top |
61169 | 397 |
apply standard |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51185
diff
changeset
|
398 |
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
|
399 |
apply simp |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51185
diff
changeset
|
400 |
done |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51185
diff
changeset
|
401 |
|
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51185
diff
changeset
|
402 |
instance int :: no_bot |
61169 | 403 |
apply standard |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51185
diff
changeset
|
404 |
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
|
405 |
apply simp |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51185
diff
changeset
|
406 |
done |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51185
diff
changeset
|
407 |
|
63652 | 408 |
|
61799 | 409 |
subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
410 |
|
48045 | 411 |
lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y" |
412 |
by auto |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
413 |
|
44709 | 414 |
lemma nat_int [simp]: "nat (int n) = n" |
48045 | 415 |
by transfer simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
416 |
|
44709 | 417 |
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)" |
48045 | 418 |
by transfer clarsimp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
419 |
|
63652 | 420 |
lemma nat_0_le: "0 \<le> z \<Longrightarrow> int (nat z) = z" |
421 |
by simp |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
422 |
|
63652 | 423 |
lemma nat_le_0 [simp]: "z \<le> 0 \<Longrightarrow> nat z = 0" |
48045 | 424 |
by transfer clarsimp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
425 |
|
63652 | 426 |
lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> nat w \<le> nat z \<longleftrightarrow> w \<le> z" |
48045 | 427 |
by transfer (clarsimp, arith) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
428 |
|
63652 | 429 |
text \<open>An alternative condition is @{term "0 \<le> w"}.\<close> |
430 |
lemma nat_mono_iff: "0 < z \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z" |
|
431 |
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
|
432 |
|
63652 | 433 |
lemma nat_less_eq_zless: "0 \<le> w \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z" |
434 |
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
|
435 |
|
63652 | 436 |
lemma zless_nat_conj [simp]: "nat w < nat z \<longleftrightarrow> 0 < z \<and> w < z" |
48045 | 437 |
by transfer (clarsimp, arith) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
438 |
|
64714
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
439 |
lemma nonneg_int_cases: |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
440 |
assumes "0 \<le> k" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
441 |
obtains n where "k = int n" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
442 |
proof - |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
443 |
from assms have "k = int (nat k)" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
444 |
by simp |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
445 |
then show thesis |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
446 |
by (rule that) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
447 |
qed |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
448 |
|
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
449 |
lemma pos_int_cases: |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
450 |
assumes "0 < k" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
451 |
obtains n where "k = int n" and "n > 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
452 |
proof - |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
453 |
from assms have "0 \<le> k" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
454 |
by simp |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
455 |
then obtain n where "k = int n" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
456 |
by (rule nonneg_int_cases) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
457 |
moreover have "n > 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
458 |
using \<open>k = int n\<close> assms by simp |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
459 |
ultimately show thesis |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
460 |
by (rule that) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
461 |
qed |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
462 |
|
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
463 |
lemma nonpos_int_cases: |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
464 |
assumes "k \<le> 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
465 |
obtains n where "k = - int n" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
466 |
proof - |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
467 |
from assms have "- k \<ge> 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
468 |
by simp |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
469 |
then obtain n where "- k = int n" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
470 |
by (rule nonneg_int_cases) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
471 |
then have "k = - int n" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
472 |
by simp |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
473 |
then show thesis |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
474 |
by (rule that) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
475 |
qed |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
476 |
|
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
477 |
lemma neg_int_cases: |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
478 |
assumes "k < 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
479 |
obtains n where "k = - int n" and "n > 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
480 |
proof - |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
481 |
from assms have "- k > 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
482 |
by simp |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
483 |
then obtain n where "- k = int n" and "- k > 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
484 |
by (blast elim: pos_int_cases) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
485 |
then have "k = - int n" and "n > 0" |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
486 |
by simp_all |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
487 |
then show thesis |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
488 |
by (rule that) |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
489 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
490 |
|
63652 | 491 |
lemma nat_eq_iff: "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)" |
48045 | 492 |
by transfer (clarsimp simp add: le_imp_diff_is_add) |
60162 | 493 |
|
63652 | 494 |
lemma nat_eq_iff2: "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)" |
54223 | 495 |
using nat_eq_iff [of w m] by auto |
496 |
||
63652 | 497 |
lemma nat_0 [simp]: "nat 0 = 0" |
54223 | 498 |
by (simp add: nat_eq_iff) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
499 |
|
63652 | 500 |
lemma nat_1 [simp]: "nat 1 = Suc 0" |
54223 | 501 |
by (simp add: nat_eq_iff) |
502 |
||
63652 | 503 |
lemma nat_numeral [simp]: "nat (numeral k) = numeral k" |
54223 | 504 |
by (simp add: nat_eq_iff) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
505 |
|
63652 | 506 |
lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0" |
54223 | 507 |
by simp |
508 |
||
509 |
lemma nat_2: "nat 2 = Suc (Suc 0)" |
|
510 |
by simp |
|
60162 | 511 |
|
63652 | 512 |
lemma nat_less_iff: "0 \<le> w \<Longrightarrow> nat w < m \<longleftrightarrow> w < of_nat m" |
48045 | 513 |
by transfer (clarsimp, arith) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
514 |
|
44709 | 515 |
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n" |
48045 | 516 |
by transfer (clarsimp simp add: le_diff_conv) |
44707 | 517 |
|
518 |
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y" |
|
48045 | 519 |
by transfer auto |
44707 | 520 |
|
63652 | 521 |
lemma nat_0_iff[simp]: "nat i = 0 \<longleftrightarrow> i \<le> 0" |
522 |
for i :: int |
|
48045 | 523 |
by transfer clarsimp |
29700 | 524 |
|
63652 | 525 |
lemma int_eq_iff: "of_nat m = z \<longleftrightarrow> m = nat z \<and> 0 \<le> z" |
526 |
by (auto simp add: nat_eq_iff2) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
527 |
|
63652 | 528 |
lemma zero_less_nat_eq [simp]: "0 < nat z \<longleftrightarrow> 0 < z" |
529 |
using zless_nat_conj [of 0] by auto |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
530 |
|
63652 | 531 |
lemma nat_add_distrib: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'" |
48045 | 532 |
by transfer clarsimp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
533 |
|
63652 | 534 |
lemma nat_diff_distrib': "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y" |
54223 | 535 |
by transfer clarsimp |
60162 | 536 |
|
63652 | 537 |
lemma nat_diff_distrib: "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'" |
54223 | 538 |
by (rule nat_diff_distrib') auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
539 |
|
44709 | 540 |
lemma nat_zminus_int [simp]: "nat (- int n) = 0" |
48045 | 541 |
by transfer simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
542 |
|
63652 | 543 |
lemma le_nat_iff: "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k" |
53065 | 544 |
by transfer auto |
60162 | 545 |
|
63652 | 546 |
lemma zless_nat_eq_int_zless: "m < nat z \<longleftrightarrow> int m < z" |
48045 | 547 |
by transfer (clarsimp simp add: less_diff_conv) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
548 |
|
63652 | 549 |
lemma (in ring_1) of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z" |
48066
c6783c9b87bf
transfer method now handles transfer rules for compound terms, e.g. locale-defined constants with hidden parameters
huffman
parents:
48045
diff
changeset
|
550 |
by transfer (clarsimp simp add: of_nat_diff) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
551 |
|
63652 | 552 |
lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')" |
54249 | 553 |
by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral) |
554 |
||
66886 | 555 |
lemma nat_abs_triangle_ineq: |
556 |
"nat \<bar>k + l\<bar> \<le> nat \<bar>k\<bar> + nat \<bar>l\<bar>" |
|
557 |
by (simp add: nat_add_distrib [symmetric] nat_le_eq_zle abs_triangle_ineq) |
|
558 |
||
66816
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
559 |
lemma nat_of_bool [simp]: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
560 |
"nat (of_bool P) = of_bool P" |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
561 |
by auto |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
562 |
|
66836 | 563 |
lemma split_nat [arith_split]: "P (nat i) \<longleftrightarrow> ((\<forall>n. i = int n \<longrightarrow> P n) \<and> (i < 0 \<longrightarrow> P 0))" |
564 |
(is "?P = (?L \<and> ?R)") |
|
565 |
for i :: int |
|
566 |
proof (cases "i < 0") |
|
567 |
case True |
|
568 |
then show ?thesis |
|
569 |
by auto |
|
570 |
next |
|
571 |
case False |
|
572 |
have "?P = ?L" |
|
573 |
proof |
|
574 |
assume ?P |
|
575 |
then show ?L using False by auto |
|
576 |
next |
|
577 |
assume ?L |
|
578 |
moreover from False have "int (nat i) = i" |
|
579 |
by (simp add: not_less) |
|
580 |
ultimately show ?P |
|
581 |
by simp |
|
582 |
qed |
|
583 |
with False show ?thesis by simp |
|
584 |
qed |
|
585 |
||
586 |
lemma all_nat: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x\<ge>0. P (nat x))" |
|
587 |
by (auto split: split_nat) |
|
588 |
||
589 |
lemma ex_nat: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> P (nat x))" |
|
590 |
proof |
|
591 |
assume "\<exists>x. P x" |
|
592 |
then obtain x where "P x" .. |
|
593 |
then have "int x \<ge> 0 \<and> P (nat (int x))" by simp |
|
594 |
then show "\<exists>x\<ge>0. P (nat x)" .. |
|
595 |
next |
|
596 |
assume "\<exists>x\<ge>0. P (nat x)" |
|
597 |
then show "\<exists>x. P x" by auto |
|
598 |
qed |
|
599 |
||
54249 | 600 |
|
60758 | 601 |
text \<open>For termination proofs:\<close> |
63652 | 602 |
lemma measure_function_int[measure_function]: "is_measure (nat \<circ> abs)" .. |
29779 | 603 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
604 |
|
63652 | 605 |
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
|
606 |
|
61076 | 607 |
lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)" |
63652 | 608 |
by (simp add: order_less_le del: of_nat_Suc) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
609 |
|
44709 | 610 |
lemma negative_zless [iff]: "- (int (Suc n)) < int m" |
63652 | 611 |
by (rule negative_zless_0 [THEN order_less_le_trans], simp) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
612 |
|
44709 | 613 |
lemma negative_zle_0: "- int n \<le> 0" |
63652 | 614 |
by (simp add: minus_le_iff) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
615 |
|
44709 | 616 |
lemma negative_zle [iff]: "- int n \<le> int m" |
63652 | 617 |
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff]) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
618 |
|
63652 | 619 |
lemma not_zle_0_negative [simp]: "\<not> 0 \<le> - int (Suc n)" |
620 |
by (subst le_minus_iff) (simp del: of_nat_Suc) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
621 |
|
63652 | 622 |
lemma int_zle_neg: "int n \<le> - int m \<longleftrightarrow> n = 0 \<and> m = 0" |
48045 | 623 |
by transfer simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
624 |
|
63652 | 625 |
lemma not_int_zless_negative [simp]: "\<not> int n < - int m" |
626 |
by (simp add: linorder_not_less) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
627 |
|
63652 | 628 |
lemma negative_eq_positive [simp]: "- int n = of_nat m \<longleftrightarrow> n = 0 \<and> m = 0" |
629 |
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
630 |
|
63652 | 631 |
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)" |
632 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
62348 | 633 |
proof |
63652 | 634 |
assume ?rhs |
635 |
then show ?lhs by auto |
|
62348 | 636 |
next |
63652 | 637 |
assume ?lhs |
62348 | 638 |
then have "0 \<le> z - w" by simp |
639 |
then obtain n where "z - w = int n" |
|
640 |
using zero_le_imp_eq_int [of "z - w"] by blast |
|
63652 | 641 |
then have "z = w + int n" by simp |
642 |
then show ?rhs .. |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
643 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
644 |
|
44709 | 645 |
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z" |
63652 | 646 |
by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
647 |
|
63652 | 648 |
text \<open> |
649 |
This version is proved for all ordered rings, not just integers! |
|
650 |
It is proved here because attribute \<open>arith_split\<close> is not available |
|
651 |
in theory \<open>Rings\<close>. |
|
652 |
But is it really better than just rewriting with \<open>abs_if\<close>? |
|
653 |
\<close> |
|
654 |
lemma abs_split [arith_split, no_atp]: "P \<bar>a\<bar> \<longleftrightarrow> (0 \<le> a \<longrightarrow> P a) \<and> (a < 0 \<longrightarrow> P (- a))" |
|
655 |
for a :: "'a::linordered_idom" |
|
656 |
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
657 |
|
44709 | 658 |
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))" |
63652 | 659 |
apply transfer |
660 |
apply clarsimp |
|
661 |
apply (rule_tac x="b - Suc a" in exI) |
|
662 |
apply arith |
|
663 |
done |
|
664 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
665 |
|
60758 | 666 |
subsection \<open>Cases and induction\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
667 |
|
63652 | 668 |
text \<open> |
669 |
Now we replace the case analysis rule by a more conventional one: |
|
670 |
whether an integer is negative or not. |
|
671 |
\<close> |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
672 |
|
63652 | 673 |
text \<open>This version is symmetric in the two subgoals.\<close> |
674 |
lemma int_cases2 [case_names nonneg nonpos, cases type: int]: |
|
675 |
"(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int n) \<Longrightarrow> P) \<Longrightarrow> P" |
|
676 |
by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym]) |
|
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59582
diff
changeset
|
677 |
|
63652 | 678 |
text \<open>This is the default, with a negative case.\<close> |
679 |
lemma int_cases [case_names nonneg neg, cases type: int]: |
|
680 |
"(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int (Suc n)) \<Longrightarrow> P) \<Longrightarrow> P" |
|
681 |
apply (cases "z < 0") |
|
682 |
apply (blast dest!: negD) |
|
683 |
apply (simp add: linorder_not_less del: of_nat_Suc) |
|
684 |
apply auto |
|
685 |
apply (blast dest: nat_0_le [THEN sym]) |
|
686 |
done |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
687 |
|
60868
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
688 |
lemma int_cases3 [case_names zero pos neg]: |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
689 |
fixes k :: int |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
690 |
assumes "k = 0 \<Longrightarrow> P" and "\<And>n. k = int n \<Longrightarrow> n > 0 \<Longrightarrow> P" |
61204 | 691 |
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
|
692 |
shows "P" |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
693 |
proof (cases k "0::int" rule: linorder_cases) |
63652 | 694 |
case equal |
695 |
with assms(1) show P by simp |
|
60868
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
696 |
next |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
697 |
case greater |
63539 | 698 |
then have *: "nat k > 0" by simp |
699 |
moreover from * have "k = int (nat k)" by auto |
|
60868
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
700 |
ultimately show P using assms(2) by blast |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
701 |
next |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
702 |
case less |
63539 | 703 |
then have *: "nat (- k) > 0" by simp |
704 |
moreover from * have "k = - int (nat (- k))" by auto |
|
60868
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
705 |
ultimately show P using assms(3) by blast |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
706 |
qed |
dd18c33c001e
direct bootstrap of integer division from natural division
haftmann
parents:
60758
diff
changeset
|
707 |
|
63652 | 708 |
lemma int_of_nat_induct [case_names nonneg neg, induct type: int]: |
709 |
"(\<And>n. P (int n)) \<Longrightarrow> (\<And>n. P (- (int (Suc n)))) \<Longrightarrow> P z" |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
710 |
by (cases z) auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
711 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
712 |
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)" |
61799 | 713 |
\<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close> |
714 |
by (fact Let_numeral) \<comment> \<open>FIXME drop\<close> |
|
37767 | 715 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
716 |
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)" |
61799 | 717 |
\<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close> |
718 |
by (fact Let_neg_numeral) \<comment> \<open>FIXME drop\<close> |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
719 |
|
66816
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
720 |
lemma sgn_mult_dvd_iff [simp]: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
721 |
"sgn r * l dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
722 |
by (cases r rule: int_cases3) auto |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
723 |
|
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
724 |
lemma mult_sgn_dvd_iff [simp]: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
725 |
"l * sgn r dvd k \<longleftrightarrow> l dvd k \<and> (r = 0 \<longrightarrow> k = 0)" for k l r :: int |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
726 |
using sgn_mult_dvd_iff [of r l k] by (simp add: ac_simps) |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
727 |
|
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
728 |
lemma dvd_sgn_mult_iff [simp]: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
729 |
"l dvd sgn r * k \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
730 |
by (cases r rule: int_cases3) simp_all |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
731 |
|
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
732 |
lemma dvd_mult_sgn_iff [simp]: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
733 |
"l dvd k * sgn r \<longleftrightarrow> l dvd k \<or> r = 0" for k l r :: int |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
734 |
using dvd_sgn_mult_iff [of l r k] by (simp add: ac_simps) |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
735 |
|
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
736 |
lemma int_sgnE: |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
737 |
fixes k :: int |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
738 |
obtains n and l where "k = sgn l * int n" |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
739 |
proof - |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
740 |
have "k = sgn k * int (nat \<bar>k\<bar>)" |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
741 |
by (simp add: sgn_mult_abs) |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
742 |
then show ?thesis .. |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
743 |
qed |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
744 |
|
61799 | 745 |
text \<open>Unfold \<open>min\<close> and \<open>max\<close> on numerals.\<close> |
28958 | 746 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
747 |
lemmas max_number_of [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
748 |
max_def [of "numeral u" "numeral v"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
749 |
max_def [of "numeral u" "- numeral v"] |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
750 |
max_def [of "- numeral u" "numeral v"] |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
751 |
max_def [of "- numeral u" "- numeral v"] for u v |
28958 | 752 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
753 |
lemmas min_number_of [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
754 |
min_def [of "numeral u" "numeral v"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
755 |
min_def [of "numeral u" "- numeral v"] |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
756 |
min_def [of "- numeral u" "numeral v"] |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
757 |
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
|
758 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
759 |
|
60758 | 760 |
subsubsection \<open>Binary comparisons\<close> |
28958 | 761 |
|
60758 | 762 |
text \<open>Preliminaries\<close> |
28958 | 763 |
|
60162 | 764 |
lemma le_imp_0_less: |
63652 | 765 |
fixes z :: int |
28958 | 766 |
assumes le: "0 \<le> z" |
63652 | 767 |
shows "0 < 1 + z" |
28958 | 768 |
proof - |
769 |
have "0 \<le> z" by fact |
|
63652 | 770 |
also have "\<dots> < z + 1" by (rule less_add_one) |
771 |
also have "\<dots> = 1 + z" by (simp add: ac_simps) |
|
28958 | 772 |
finally show "0 < 1 + z" . |
773 |
qed |
|
774 |
||
63652 | 775 |
lemma odd_less_0_iff: "1 + z + z < 0 \<longleftrightarrow> z < 0" |
776 |
for z :: int |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
777 |
proof (cases z) |
28958 | 778 |
case (nonneg n) |
63652 | 779 |
then show ?thesis |
780 |
by (simp add: linorder_not_less add.assoc add_increasing le_imp_0_less [THEN order_less_imp_le]) |
|
28958 | 781 |
next |
782 |
case (neg n) |
|
63652 | 783 |
then show ?thesis |
784 |
by (simp del: of_nat_Suc of_nat_add of_nat_1 |
|
785 |
add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric]) |
|
28958 | 786 |
qed |
787 |
||
63652 | 788 |
|
60758 | 789 |
subsubsection \<open>Comparisons, for Ordered Rings\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
790 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
791 |
lemmas double_eq_0_iff = double_zero |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
792 |
|
63652 | 793 |
lemma odd_nonzero: "1 + z + z \<noteq> 0" |
794 |
for z :: int |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
795 |
proof (cases z) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
796 |
case (nonneg n) |
63652 | 797 |
have le: "0 \<le> z + z" |
798 |
by (simp add: nonneg add_increasing) |
|
799 |
then show ?thesis |
|
800 |
using le_imp_0_less [OF le] by (auto simp: add.assoc) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
801 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
802 |
case (neg n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
803 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
804 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
805 |
assume eq: "1 + z + z = 0" |
63652 | 806 |
have "0 < 1 + (int n + int n)" |
60162 | 807 |
by (simp add: le_imp_0_less add_increasing) |
63652 | 808 |
also have "\<dots> = - (1 + z + z)" |
60162 | 809 |
by (simp add: neg add.assoc [symmetric]) |
63652 | 810 |
also have "\<dots> = 0" by (simp add: eq) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
811 |
finally have "0<0" .. |
63652 | 812 |
then show False by blast |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
813 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
814 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
815 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
816 |
|
60758 | 817 |
subsection \<open>The Set of Integers\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
818 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
819 |
context ring_1 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
820 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
821 |
|
61070 | 822 |
definition Ints :: "'a set" ("\<int>") |
823 |
where "\<int> = range of_int" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
824 |
|
35634 | 825 |
lemma Ints_of_int [simp]: "of_int z \<in> \<int>" |
826 |
by (simp add: Ints_def) |
|
827 |
||
828 |
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>" |
|
45533 | 829 |
using Ints_of_int [of "of_nat n"] by simp |
35634 | 830 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
831 |
lemma Ints_0 [simp]: "0 \<in> \<int>" |
45533 | 832 |
using Ints_of_int [of "0"] by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
833 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
834 |
lemma Ints_1 [simp]: "1 \<in> \<int>" |
45533 | 835 |
using Ints_of_int [of "1"] by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
836 |
|
61552
980dd46a03fb
Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents:
61531
diff
changeset
|
837 |
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
|
838 |
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
|
839 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
840 |
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>" |
63652 | 841 |
apply (auto simp add: Ints_def) |
842 |
apply (rule range_eqI) |
|
843 |
apply (rule of_int_add [symmetric]) |
|
844 |
done |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
845 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
846 |
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>" |
63652 | 847 |
apply (auto simp add: Ints_def) |
848 |
apply (rule range_eqI) |
|
849 |
apply (rule of_int_minus [symmetric]) |
|
850 |
done |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
851 |
|
35634 | 852 |
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>" |
63652 | 853 |
apply (auto simp add: Ints_def) |
854 |
apply (rule range_eqI) |
|
855 |
apply (rule of_int_diff [symmetric]) |
|
856 |
done |
|
35634 | 857 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
858 |
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>" |
63652 | 859 |
apply (auto simp add: Ints_def) |
860 |
apply (rule range_eqI) |
|
861 |
apply (rule of_int_mult [symmetric]) |
|
862 |
done |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
863 |
|
35634 | 864 |
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>" |
63652 | 865 |
by (induct n) simp_all |
35634 | 866 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
867 |
lemma Ints_cases [cases set: Ints]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
868 |
assumes "q \<in> \<int>" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
869 |
obtains (of_int) z where "q = of_int z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
870 |
unfolding Ints_def |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
871 |
proof - |
60758 | 872 |
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
|
873 |
then obtain z where "q = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
874 |
then show thesis .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
875 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
876 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
877 |
lemma Ints_induct [case_names of_int, induct set: Ints]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
878 |
"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
|
879 |
by (rule Ints_cases) auto |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
880 |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
881 |
lemma Nats_subset_Ints: "\<nat> \<subseteq> \<int>" |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
882 |
unfolding Nats_def Ints_def |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
883 |
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
|
884 |
|
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
885 |
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
|
886 |
proof (intro subsetI equalityI) |
63652 | 887 |
fix x :: 'a |
888 |
assume "x \<in> {of_int n |n. n \<ge> 0}" |
|
889 |
then obtain n where "x = of_int n" "n \<ge> 0" |
|
890 |
by (auto elim!: Ints_cases) |
|
891 |
then have "x = of_nat (nat n)" |
|
892 |
by (subst of_nat_nat) simp_all |
|
893 |
then show "x \<in> \<nat>" |
|
894 |
by simp |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
895 |
next |
63652 | 896 |
fix x :: 'a |
897 |
assume "x \<in> \<nat>" |
|
898 |
then obtain n where "x = of_nat n" |
|
899 |
by (auto elim!: Nats_cases) |
|
900 |
then have "x = of_int (int n)" by simp |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
901 |
also have "int n \<ge> 0" by simp |
63652 | 902 |
then have "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast |
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
903 |
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
|
904 |
qed |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
905 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
906 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
907 |
|
64758
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
908 |
lemma (in linordered_idom) Ints_abs [simp]: |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
909 |
shows "a \<in> \<int> \<Longrightarrow> abs a \<in> \<int>" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
910 |
by (auto simp: abs_if) |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
911 |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
912 |
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
|
913 |
proof (intro subsetI equalityI) |
63652 | 914 |
fix x :: 'a |
915 |
assume "x \<in> {n \<in> \<int>. n \<ge> 0}" |
|
916 |
then obtain n where "x = of_int n" "n \<ge> 0" |
|
917 |
by (auto elim!: Ints_cases) |
|
918 |
then have "x = of_nat (nat n)" |
|
919 |
by (subst of_nat_nat) simp_all |
|
920 |
then show "x \<in> \<nat>" |
|
921 |
by simp |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
922 |
qed (auto elim!: Nats_cases) |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
923 |
|
64849 | 924 |
lemma (in idom_divide) of_int_divide_in_Ints: |
925 |
"of_int a div of_int b \<in> \<int>" if "b dvd a" |
|
926 |
proof - |
|
927 |
from that obtain c where "a = b * c" .. |
|
928 |
then show ?thesis |
|
929 |
by (cases "of_int b = 0") simp_all |
|
930 |
qed |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61234
diff
changeset
|
931 |
|
60758 | 932 |
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
|
933 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
934 |
lemma Ints_double_eq_0_iff: |
63652 | 935 |
fixes a :: "'a::ring_char_0" |
61070 | 936 |
assumes in_Ints: "a \<in> \<int>" |
63652 | 937 |
shows "a + a = 0 \<longleftrightarrow> a = 0" |
938 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
939 |
proof - |
63652 | 940 |
from in_Ints have "a \<in> range of_int" |
941 |
unfolding Ints_def [symmetric] . |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
942 |
then obtain z where a: "a = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
943 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
944 |
proof |
63652 | 945 |
assume ?rhs |
946 |
then show ?lhs by simp |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
947 |
next |
63652 | 948 |
assume ?lhs |
949 |
with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp |
|
950 |
then have "z + z = 0" by (simp only: of_int_eq_iff) |
|
951 |
then have "z = 0" by (simp only: double_eq_0_iff) |
|
952 |
with a show ?rhs by simp |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
953 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
954 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
955 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
956 |
lemma Ints_odd_nonzero: |
63652 | 957 |
fixes a :: "'a::ring_char_0" |
61070 | 958 |
assumes in_Ints: "a \<in> \<int>" |
63652 | 959 |
shows "1 + a + a \<noteq> 0" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
960 |
proof - |
63652 | 961 |
from in_Ints have "a \<in> range of_int" |
962 |
unfolding Ints_def [symmetric] . |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
963 |
then obtain z where a: "a = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
964 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
965 |
proof |
63652 | 966 |
assume "1 + a + a = 0" |
967 |
with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp |
|
968 |
then have "1 + z + z = 0" by (simp only: of_int_eq_iff) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
969 |
with odd_nonzero show False by blast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
970 |
qed |
60162 | 971 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
972 |
|
61070 | 973 |
lemma Nats_numeral [simp]: "numeral w \<in> \<nat>" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
974 |
using of_nat_in_Nats [of "numeral w"] by simp |
35634 | 975 |
|
60162 | 976 |
lemma Ints_odd_less_0: |
63652 | 977 |
fixes a :: "'a::linordered_idom" |
61070 | 978 |
assumes in_Ints: "a \<in> \<int>" |
63652 | 979 |
shows "1 + a + a < 0 \<longleftrightarrow> a < 0" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
980 |
proof - |
63652 | 981 |
from in_Ints have "a \<in> range of_int" |
982 |
unfolding Ints_def [symmetric] . |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
983 |
then obtain z where a: "a = of_int z" .. |
63652 | 984 |
with a have "1 + a + a < 0 \<longleftrightarrow> of_int (1 + z + z) < (of_int 0 :: 'a)" |
985 |
by simp |
|
986 |
also have "\<dots> \<longleftrightarrow> z < 0" |
|
987 |
by (simp only: of_int_less_iff odd_less_0_iff) |
|
988 |
also have "\<dots> \<longleftrightarrow> a < 0" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
989 |
by (simp add: a) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
990 |
finally show ?thesis . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
991 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
992 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
993 |
|
64272 | 994 |
subsection \<open>@{term sum} and @{term prod}\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
995 |
|
64267 | 996 |
lemma of_nat_sum [simp]: "of_nat (sum f A) = (\<Sum>x\<in>A. of_nat(f x))" |
63652 | 997 |
by (induct A rule: infinite_finite_induct) auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
998 |
|
64267 | 999 |
lemma of_int_sum [simp]: "of_int (sum f A) = (\<Sum>x\<in>A. of_int(f x))" |
63652 | 1000 |
by (induct A rule: infinite_finite_induct) auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1001 |
|
64272 | 1002 |
lemma of_nat_prod [simp]: "of_nat (prod f A) = (\<Prod>x\<in>A. of_nat(f x))" |
63652 | 1003 |
by (induct A rule: infinite_finite_induct) auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1004 |
|
64272 | 1005 |
lemma of_int_prod [simp]: "of_int (prod f A) = (\<Prod>x\<in>A. of_int(f x))" |
63652 | 1006 |
by (induct A rule: infinite_finite_induct) auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1007 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1008 |
|
60758 | 1009 |
text \<open>Legacy theorems\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1010 |
|
64714
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
1011 |
lemmas int_sum = of_nat_sum [where 'a=int] |
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
1012 |
lemmas int_prod = of_nat_prod [where 'a=int] |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1013 |
lemmas zle_int = of_nat_le_iff [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1014 |
lemmas int_int_eq = of_nat_eq_iff [where 'a=int] |
64714
53bab28983f1
complete set of cases rules for integers known to be (non-)positive/negative;
haftmann
parents:
64272
diff
changeset
|
1015 |
lemmas nonneg_eq_int = nonneg_int_cases |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1016 |
|
63652 | 1017 |
|
60758 | 1018 |
subsection \<open>Setting up simplification procedures\<close> |
30802 | 1019 |
|
54249 | 1020 |
lemmas of_int_simps = |
1021 |
of_int_0 of_int_1 of_int_add of_int_mult |
|
1022 |
||
48891 | 1023 |
ML_file "Tools/int_arith.ML" |
60758 | 1024 |
declaration \<open>K Int_Arith.setup\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1025 |
|
63652 | 1026 |
simproc_setup fast_arith |
1027 |
("(m::'a::linordered_idom) < n" | |
|
1028 |
"(m::'a::linordered_idom) \<le> n" | |
|
1029 |
"(m::'a::linordered_idom) = n") = |
|
61144 | 1030 |
\<open>K Lin_Arith.simproc\<close> |
43595 | 1031 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1032 |
|
60758 | 1033 |
subsection\<open>More Inequality Reasoning\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1034 |
|
63652 | 1035 |
lemma zless_add1_eq: "w < z + 1 \<longleftrightarrow> w < z \<or> w = z" |
1036 |
for w z :: int |
|
1037 |
by arith |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1038 |
|
63652 | 1039 |
lemma add1_zle_eq: "w + 1 \<le> z \<longleftrightarrow> w < z" |
1040 |
for w z :: int |
|
1041 |
by arith |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1042 |
|
63652 | 1043 |
lemma zle_diff1_eq [simp]: "w \<le> z - 1 \<longleftrightarrow> w < z" |
1044 |
for w z :: int |
|
1045 |
by arith |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1046 |
|
63652 | 1047 |
lemma zle_add1_eq_le [simp]: "w < z + 1 \<longleftrightarrow> w \<le> z" |
1048 |
for w z :: int |
|
1049 |
by arith |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1050 |
|
63652 | 1051 |
lemma int_one_le_iff_zero_less: "1 \<le> z \<longleftrightarrow> 0 < z" |
1052 |
for z :: int |
|
1053 |
by arith |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1054 |
|
64758
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1055 |
lemma Ints_nonzero_abs_ge1: |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1056 |
fixes x:: "'a :: linordered_idom" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1057 |
assumes "x \<in> Ints" "x \<noteq> 0" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1058 |
shows "1 \<le> abs x" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1059 |
proof (rule Ints_cases [OF \<open>x \<in> Ints\<close>]) |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1060 |
fix z::int |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1061 |
assume "x = of_int z" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1062 |
with \<open>x \<noteq> 0\<close> |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1063 |
show "1 \<le> \<bar>x\<bar>" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1064 |
apply (auto simp add: abs_if) |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1065 |
by (metis diff_0 of_int_1 of_int_le_iff of_int_minus zle_diff1_eq) |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1066 |
qed |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1067 |
|
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1068 |
lemma Ints_nonzero_abs_less1: |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1069 |
fixes x:: "'a :: linordered_idom" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1070 |
shows "\<lbrakk>x \<in> Ints; abs x < 1\<rbrakk> \<Longrightarrow> x = 0" |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1071 |
using Ints_nonzero_abs_ge1 [of x] by auto |
3b33d2fc5fc0
A few new lemmas and needed adaptations
paulson <lp15@cam.ac.uk>
parents:
64714
diff
changeset
|
1072 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1073 |
|
63652 | 1074 |
subsection \<open>The functions @{term nat} and @{term int}\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1075 |
|
63652 | 1076 |
text \<open>Simplify the term @{term "w + - z"}.\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1077 |
|
63652 | 1078 |
lemma one_less_nat_eq [simp]: "Suc 0 < nat z \<longleftrightarrow> 1 < z" |
60162 | 1079 |
using zless_nat_conj [of 1 z] by auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1080 |
|
63652 | 1081 |
text \<open> |
1082 |
This simplifies expressions of the form @{term "int n = z"} where |
|
1083 |
\<open>z\<close> is an integer literal. |
|
1084 |
\<close> |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1085 |
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
|
1086 |
|
59000 | 1087 |
lemma nat_abs_int_diff: "nat \<bar>int a - int b\<bar> = (if a \<le> b then b - a else a - b)" |
1088 |
by auto |
|
1089 |
||
1090 |
lemma nat_int_add: "nat (int a + int b) = a + b" |
|
1091 |
by auto |
|
1092 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1093 |
context ring_1 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1094 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1095 |
|
33056
791a4655cae3
renamed "nitpick_const_xxx" attributes to "nitpick_xxx" and "nitpick_ind_intros" to "nitpick_intros"
blanchet
parents:
32437
diff
changeset
|
1096 |
lemma of_int_of_nat [nitpick_simp]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1097 |
"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
|
1098 |
proof (cases "k < 0") |
63652 | 1099 |
case True |
1100 |
then have "0 \<le> - k" by simp |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1101 |
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
|
1102 |
with True show ?thesis by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1103 |
next |
63652 | 1104 |
case False |
1105 |
then show ?thesis by (simp add: not_less) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1106 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1107 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1108 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1109 |
|
64014 | 1110 |
lemma transfer_rule_of_int: |
1111 |
fixes R :: "'a::ring_1 \<Rightarrow> 'b::ring_1 \<Rightarrow> bool" |
|
1112 |
assumes [transfer_rule]: "R 0 0" "R 1 1" |
|
1113 |
"rel_fun R (rel_fun R R) plus plus" |
|
1114 |
"rel_fun R R uminus uminus" |
|
1115 |
shows "rel_fun HOL.eq R of_int of_int" |
|
1116 |
proof - |
|
1117 |
note transfer_rule_of_nat [transfer_rule] |
|
1118 |
have [transfer_rule]: "rel_fun HOL.eq R of_nat of_nat" |
|
1119 |
by transfer_prover |
|
1120 |
show ?thesis |
|
1121 |
by (unfold of_int_of_nat [abs_def]) transfer_prover |
|
1122 |
qed |
|
1123 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1124 |
lemma nat_mult_distrib: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1125 |
fixes z z' :: int |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1126 |
assumes "0 \<le> z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1127 |
shows "nat (z * z') = nat z * nat z'" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1128 |
proof (cases "0 \<le> z'") |
63652 | 1129 |
case False |
1130 |
with assms have "z * z' \<le> 0" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1131 |
by (simp add: not_le mult_le_0_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1132 |
then have "nat (z * z') = 0" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1133 |
moreover from False have "nat z' = 0" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1134 |
ultimately show ?thesis by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1135 |
next |
63652 | 1136 |
case True |
1137 |
with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1138 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1139 |
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
|
1140 |
(simp only: of_nat_mult of_nat_nat [OF True] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1141 |
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
|
1142 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1143 |
|
63652 | 1144 |
lemma nat_mult_distrib_neg: "z \<le> 0 \<Longrightarrow> nat (z * z') = nat (- z) * nat (- z')" |
1145 |
for z z' :: int |
|
1146 |
apply (rule trans) |
|
1147 |
apply (rule_tac [2] nat_mult_distrib) |
|
1148 |
apply auto |
|
1149 |
done |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1150 |
|
61944 | 1151 |
lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>" |
63652 | 1152 |
by (cases "z = 0 \<or> w = 0") |
1153 |
(auto simp add: abs_if nat_mult_distrib [symmetric] |
|
1154 |
nat_mult_distrib_neg [symmetric] mult_less_0_iff) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1155 |
|
63652 | 1156 |
lemma int_in_range_abs [simp]: "int n \<in> range abs" |
60570 | 1157 |
proof (rule range_eqI) |
63652 | 1158 |
show "int n = \<bar>int n\<bar>" by simp |
60570 | 1159 |
qed |
1160 |
||
63652 | 1161 |
lemma range_abs_Nats [simp]: "range abs = (\<nat> :: int set)" |
60570 | 1162 |
proof - |
1163 |
have "\<bar>k\<bar> \<in> \<nat>" for k :: int |
|
1164 |
by (cases k) simp_all |
|
1165 |
moreover have "k \<in> range abs" if "k \<in> \<nat>" for k :: int |
|
1166 |
using that by induct simp |
|
1167 |
ultimately show ?thesis by blast |
|
61204 | 1168 |
qed |
60570 | 1169 |
|
63652 | 1170 |
lemma Suc_nat_eq_nat_zadd1: "0 \<le> z \<Longrightarrow> Suc (nat z) = nat (1 + z)" |
1171 |
for z :: int |
|
1172 |
by (rule sym) (simp add: nat_eq_iff) |
|
47207
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
1173 |
|
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
1174 |
lemma diff_nat_eq_if: |
63652 | 1175 |
"nat z - nat z' = |
1176 |
(if z' < 0 then nat z |
|
1177 |
else |
|
1178 |
let d = z - z' |
|
1179 |
in if d < 0 then 0 else nat d)" |
|
1180 |
by (simp add: Let_def nat_diff_distrib [symmetric]) |
|
47207
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
1181 |
|
63652 | 1182 |
lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)" |
47207
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
1183 |
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
|
1184 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1185 |
|
63652 | 1186 |
subsection \<open>Induction principles for int\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1187 |
|
63652 | 1188 |
text \<open>Well-founded segments of the integers.\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1189 |
|
63652 | 1190 |
definition int_ge_less_than :: "int \<Rightarrow> (int \<times> int) set" |
1191 |
where "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1192 |
|
63652 | 1193 |
lemma wf_int_ge_less_than: "wf (int_ge_less_than d)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1194 |
proof - |
63652 | 1195 |
have "int_ge_less_than d \<subseteq> measure (\<lambda>z. nat (z - d))" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1196 |
by (auto simp add: int_ge_less_than_def) |
63652 | 1197 |
then show ?thesis |
60162 | 1198 |
by (rule wf_subset [OF wf_measure]) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1199 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1200 |
|
63652 | 1201 |
text \<open> |
1202 |
This variant looks odd, but is typical of the relations suggested |
|
1203 |
by RankFinder.\<close> |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1204 |
|
63652 | 1205 |
definition int_ge_less_than2 :: "int \<Rightarrow> (int \<times> int) set" |
1206 |
where "int_ge_less_than2 d = {(z',z). d \<le> z \<and> z' < z}" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1207 |
|
63652 | 1208 |
lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1209 |
proof - |
63652 | 1210 |
have "int_ge_less_than2 d \<subseteq> measure (\<lambda>z. nat (1 + z - d))" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1211 |
by (auto simp add: int_ge_less_than2_def) |
63652 | 1212 |
then show ?thesis |
60162 | 1213 |
by (rule wf_subset [OF wf_measure]) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1214 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1215 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1216 |
(* `set:int': dummy construction *) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1217 |
theorem int_ge_induct [case_names base step, induct set: int]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1218 |
fixes i :: int |
63652 | 1219 |
assumes ge: "k \<le> i" |
1220 |
and base: "P k" |
|
1221 |
and step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1222 |
shows "P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1223 |
proof - |
63652 | 1224 |
have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" for n |
1225 |
proof (induct n) |
|
1226 |
case 0 |
|
1227 |
then have "i = k" by arith |
|
1228 |
with base show "P i" by simp |
|
1229 |
next |
|
1230 |
case (Suc n) |
|
1231 |
then have "n = nat ((i - 1) - k)" by arith |
|
1232 |
moreover have k: "k \<le> i - 1" using Suc.prems by arith |
|
1233 |
ultimately have "P (i - 1)" by (rule Suc.hyps) |
|
1234 |
from step [OF k this] show ?case by simp |
|
1235 |
qed |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1236 |
with ge show ?thesis by fast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1237 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1238 |
|
25928 | 1239 |
(* `set:int': dummy construction *) |
1240 |
theorem int_gr_induct [case_names base step, induct set: int]: |
|
63652 | 1241 |
fixes i k :: int |
1242 |
assumes gr: "k < i" |
|
1243 |
and base: "P (k + 1)" |
|
1244 |
and step: "\<And>i. k < i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1245 |
shows "P i" |
63652 | 1246 |
apply (rule int_ge_induct[of "k + 1"]) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1247 |
using gr apply arith |
63652 | 1248 |
apply (rule base) |
1249 |
apply (rule step) |
|
1250 |
apply simp_all |
|
1251 |
done |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1252 |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1253 |
theorem int_le_induct [consumes 1, case_names base step]: |
63652 | 1254 |
fixes i k :: int |
1255 |
assumes le: "i \<le> k" |
|
1256 |
and base: "P k" |
|
1257 |
and step: "\<And>i. i \<le> k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1258 |
shows "P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1259 |
proof - |
63652 | 1260 |
have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" for n |
1261 |
proof (induct n) |
|
1262 |
case 0 |
|
1263 |
then have "i = k" by arith |
|
1264 |
with base show "P i" by simp |
|
1265 |
next |
|
1266 |
case (Suc n) |
|
1267 |
then have "n = nat (k - (i + 1))" by arith |
|
1268 |
moreover have k: "i + 1 \<le> k" using Suc.prems by arith |
|
1269 |
ultimately have "P (i + 1)" by (rule Suc.hyps) |
|
1270 |
from step[OF k this] show ?case by simp |
|
1271 |
qed |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1272 |
with le show ?thesis by fast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1273 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1274 |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1275 |
theorem int_less_induct [consumes 1, case_names base step]: |
63652 | 1276 |
fixes i k :: int |
1277 |
assumes less: "i < k" |
|
1278 |
and base: "P (k - 1)" |
|
1279 |
and step: "\<And>i. i < k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)" |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1280 |
shows "P i" |
63652 | 1281 |
apply (rule int_le_induct[of _ "k - 1"]) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1282 |
using less apply arith |
63652 | 1283 |
apply (rule base) |
1284 |
apply (rule step) |
|
1285 |
apply simp_all |
|
1286 |
done |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1287 |
|
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
|
1288 |
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
|
1289 |
fixes k :: int |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1290 |
assumes base: "P k" |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1291 |
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
|
1292 |
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
|
1293 |
shows "P i" |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1294 |
proof - |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1295 |
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
|
1296 |
then show ?thesis |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1297 |
proof |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1298 |
assume "i \<ge> k" |
63652 | 1299 |
then show ?thesis |
1300 |
using base by (rule int_ge_induct) (fact step1) |
|
36801
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1301 |
next |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1302 |
assume "i \<le> k" |
63652 | 1303 |
then show ?thesis |
1304 |
using base by (rule int_le_induct) (fact step2) |
|
36801
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1305 |
qed |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1306 |
qed |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1307 |
|
63652 | 1308 |
|
1309 |
subsection \<open>Intermediate value theorems\<close> |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1310 |
|
63652 | 1311 |
lemma int_val_lemma: "(\<forall>i<n. \<bar>f (i + 1) - f i\<bar> \<le> 1) \<longrightarrow> f 0 \<le> k \<longrightarrow> k \<le> f n \<longrightarrow> (\<exists>i \<le> n. f i = k)" |
1312 |
for n :: nat and k :: int |
|
1313 |
unfolding One_nat_def |
|
1314 |
apply (induct n) |
|
1315 |
apply simp |
|
1316 |
apply (intro strip) |
|
1317 |
apply (erule impE) |
|
1318 |
apply simp |
|
1319 |
apply (erule_tac x = n in allE) |
|
1320 |
apply simp |
|
1321 |
apply (case_tac "k = f (Suc n)") |
|
1322 |
apply force |
|
1323 |
apply (erule impE) |
|
1324 |
apply (simp add: abs_if split: if_split_asm) |
|
1325 |
apply (blast intro: le_SucI) |
|
1326 |
done |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1327 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1328 |
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
|
1329 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1330 |
lemma nat_intermed_int_val: |
63652 | 1331 |
"\<forall>i. m \<le> i \<and> i < n \<longrightarrow> \<bar>f (i + 1) - f i\<bar> \<le> 1 \<Longrightarrow> m < n \<Longrightarrow> |
1332 |
f m \<le> k \<Longrightarrow> k \<le> f n \<Longrightarrow> \<exists>i. m \<le> i \<and> i \<le> n \<and> f i = k" |
|
1333 |
for f :: "nat \<Rightarrow> int" and k :: int |
|
1334 |
apply (cut_tac n = "n-m" and f = "\<lambda>i. f (i + m)" and k = k in int_val_lemma) |
|
1335 |
unfolding One_nat_def |
|
1336 |
apply simp |
|
1337 |
apply (erule exE) |
|
1338 |
apply (rule_tac x = "i+m" in exI) |
|
1339 |
apply arith |
|
1340 |
done |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1341 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1342 |
|
63652 | 1343 |
subsection \<open>Products and 1, by T. M. Rasmussen\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1344 |
|
34055 | 1345 |
lemma abs_zmult_eq_1: |
63652 | 1346 |
fixes m n :: int |
34055 | 1347 |
assumes mn: "\<bar>m * n\<bar> = 1" |
63652 | 1348 |
shows "\<bar>m\<bar> = 1" |
34055 | 1349 |
proof - |
63652 | 1350 |
from mn have 0: "m \<noteq> 0" "n \<noteq> 0" by auto |
1351 |
have "\<not> 2 \<le> \<bar>m\<bar>" |
|
34055 | 1352 |
proof |
1353 |
assume "2 \<le> \<bar>m\<bar>" |
|
63652 | 1354 |
then have "2 * \<bar>n\<bar> \<le> \<bar>m\<bar> * \<bar>n\<bar>" by (simp add: mult_mono 0) |
1355 |
also have "\<dots> = \<bar>m * n\<bar>" by (simp add: abs_mult) |
|
1356 |
also from mn have "\<dots> = 1" by simp |
|
1357 |
finally have "2 * \<bar>n\<bar> \<le> 1" . |
|
1358 |
with 0 show "False" by arith |
|
34055 | 1359 |
qed |
63652 | 1360 |
with 0 show ?thesis by auto |
34055 | 1361 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1362 |
|
63652 | 1363 |
lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 \<Longrightarrow> m = 1 \<or> m = - 1" |
1364 |
for m n :: int |
|
1365 |
using abs_zmult_eq_1 [of m n] by arith |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1366 |
|
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35634
diff
changeset
|
1367 |
lemma pos_zmult_eq_1_iff: |
63652 | 1368 |
fixes m n :: int |
1369 |
assumes "0 < m" |
|
1370 |
shows "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1" |
|
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35634
diff
changeset
|
1371 |
proof - |
63652 | 1372 |
from assms have "m * n = 1 \<Longrightarrow> m = 1" |
1373 |
by (auto dest: pos_zmult_eq_1_iff_lemma) |
|
1374 |
then show ?thesis |
|
1375 |
by (auto dest: pos_zmult_eq_1_iff_lemma) |
|
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35634
diff
changeset
|
1376 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1377 |
|
63652 | 1378 |
lemma zmult_eq_1_iff: "m * n = 1 \<longleftrightarrow> (m = 1 \<and> n = 1) \<or> (m = - 1 \<and> n = - 1)" |
1379 |
for m n :: int |
|
1380 |
apply (rule iffI) |
|
1381 |
apply (frule pos_zmult_eq_1_iff_lemma) |
|
1382 |
apply (simp add: mult.commute [of m]) |
|
1383 |
apply (frule pos_zmult_eq_1_iff_lemma) |
|
1384 |
apply auto |
|
1385 |
done |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1386 |
|
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1387 |
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1388 |
proof |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1389 |
assume "finite (UNIV::int set)" |
61076 | 1390 |
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
|
1391 |
by (rule injI) simp |
61076 | 1392 |
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
|
1393 |
by (rule finite_UNIV_inj_surj) |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1394 |
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
|
1395 |
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
|
1396 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1397 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1398 |
|
60758 | 1399 |
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
|
1400 |
|
63652 | 1401 |
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
|
1402 |
|
63652 | 1403 |
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
|
1404 |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1405 |
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
|
1406 |
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
|
1407 |
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
|
1408 |
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
|
1409 |
|
63652 | 1410 |
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
|
1411 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1412 |
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
|
1413 |
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
|
1414 |
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
|
1415 |
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
|
1416 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1417 |
|
61799 | 1418 |
text \<open>Replaces \<open>inverse #nn\<close> by \<open>1/#nn\<close>. It looks |
60758 | 1419 |
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
|
1420 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1421 |
lemmas inverse_eq_divide_numeral [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1422 |
inverse_eq_divide [of "numeral w"] for w |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1423 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1424 |
lemmas inverse_eq_divide_neg_numeral [simp] = |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1425 |
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
|
1426 |
|
60758 | 1427 |
text \<open>These laws simplify inequalities, moving unary minus from a term |
63652 | 1428 |
into the literal.\<close> |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1429 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1430 |
lemmas equation_minus_iff_numeral [no_atp] = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1431 |
equation_minus_iff [of "numeral v"] for v |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1432 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1433 |
lemmas minus_equation_iff_numeral [no_atp] = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1434 |
minus_equation_iff [of _ "numeral v"] for v |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1435 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1436 |
lemmas le_minus_iff_numeral [no_atp] = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1437 |
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
|
1438 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1439 |
lemmas minus_le_iff_numeral [no_atp] = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1440 |
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
|
1441 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1442 |
lemmas less_minus_iff_numeral [no_atp] = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1443 |
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
|
1444 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1445 |
lemmas minus_less_iff_numeral [no_atp] = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1446 |
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
|
1447 |
|
63652 | 1448 |
(* FIXME maybe simproc *) |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1449 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1450 |
|
61799 | 1451 |
text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close> |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1452 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1453 |
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
|
1454 |
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
|
1455 |
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
|
1456 |
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
|
1457 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1458 |
|
61799 | 1459 |
text \<open>Multiplying out constant divisors in comparisons (\<open><\<close>, \<open>\<le>\<close> and \<open>=\<close>)\<close> |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1460 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1461 |
named_theorems divide_const_simps "simplification rules to simplify comparisons involving constant divisors" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1462 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1463 |
lemmas le_divide_eq_numeral1 [simp,divide_const_simps] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1464 |
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
|
1465 |
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
|
1466 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1467 |
lemmas divide_le_eq_numeral1 [simp,divide_const_simps] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1468 |
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
|
1469 |
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
|
1470 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1471 |
lemmas less_divide_eq_numeral1 [simp,divide_const_simps] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1472 |
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
|
1473 |
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
|
1474 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1475 |
lemmas divide_less_eq_numeral1 [simp,divide_const_simps] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1476 |
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
|
1477 |
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
|
1478 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1479 |
lemmas eq_divide_eq_numeral1 [simp,divide_const_simps] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1480 |
eq_divide_eq [of _ _ "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1481 |
eq_divide_eq [of _ _ "- numeral w"] for w |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1482 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1483 |
lemmas divide_eq_eq_numeral1 [simp,divide_const_simps] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1484 |
divide_eq_eq [of _ "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1485 |
divide_eq_eq [of _ "- numeral w"] for w |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1486 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1487 |
|
63652 | 1488 |
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
|
1489 |
|
63652 | 1490 |
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
|
1491 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1492 |
lemmas le_divide_eq_numeral [divide_const_simps] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1493 |
le_divide_eq [of "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1494 |
le_divide_eq [of "- numeral w"] for w |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1495 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1496 |
lemmas divide_le_eq_numeral [divide_const_simps] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1497 |
divide_le_eq [of _ _ "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1498 |
divide_le_eq [of _ _ "- numeral w"] for w |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1499 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1500 |
lemmas less_divide_eq_numeral [divide_const_simps] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1501 |
less_divide_eq [of "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1502 |
less_divide_eq [of "- numeral w"] for w |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1503 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1504 |
lemmas divide_less_eq_numeral [divide_const_simps] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1505 |
divide_less_eq [of _ _ "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1506 |
divide_less_eq [of _ _ "- numeral w"] for w |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1507 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1508 |
lemmas eq_divide_eq_numeral [divide_const_simps] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1509 |
eq_divide_eq [of "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1510 |
eq_divide_eq [of "- numeral w"] for w |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1511 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1512 |
lemmas divide_eq_eq_numeral [divide_const_simps] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1513 |
divide_eq_eq [of _ _ "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1514 |
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
|
1515 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1516 |
|
63652 | 1517 |
text \<open>Not good as automatic simprules because they cause case splits.\<close> |
1518 |
lemmas [divide_const_simps] = |
|
1519 |
le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1520 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1521 |
|
60758 | 1522 |
subsection \<open>The divides relation\<close> |
33320 | 1523 |
|
63652 | 1524 |
lemma zdvd_antisym_nonneg: "0 \<le> m \<Longrightarrow> 0 \<le> n \<Longrightarrow> m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n" |
1525 |
for m n :: int |
|
1526 |
by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff) |
|
33320 | 1527 |
|
63652 | 1528 |
lemma zdvd_antisym_abs: |
1529 |
fixes a b :: int |
|
1530 |
assumes "a dvd b" and "b dvd a" |
|
33320 | 1531 |
shows "\<bar>a\<bar> = \<bar>b\<bar>" |
63652 | 1532 |
proof (cases "a = 0") |
1533 |
case True |
|
1534 |
with assms show ?thesis by simp |
|
33657 | 1535 |
next |
63652 | 1536 |
case False |
1537 |
from \<open>a dvd b\<close> obtain k where k: "b = a * k" |
|
1538 |
unfolding dvd_def by blast |
|
1539 |
from \<open>b dvd a\<close> obtain k' where k': "a = b * k'" |
|
1540 |
unfolding dvd_def by blast |
|
1541 |
from k k' have "a = a * k * k'" by simp |
|
1542 |
with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1" |
|
1543 |
using \<open>a \<noteq> 0\<close> by (simp add: mult.assoc) |
|
1544 |
then have "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" |
|
1545 |
by (simp add: zmult_eq_1_iff) |
|
1546 |
with k k' show ?thesis by auto |
|
33320 | 1547 |
qed |
1548 |
||
63652 | 1549 |
lemma zdvd_zdiffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> k dvd m" |
1550 |
for k m n :: int |
|
60162 | 1551 |
using dvd_add_right_iff [of k "- n" m] by simp |
33320 | 1552 |
|
63652 | 1553 |
lemma zdvd_reduce: "k dvd n + k * m \<longleftrightarrow> k dvd n" |
1554 |
for k m n :: int |
|
58649
a62065b5e1e2
generalized and consolidated some theorems concerning divisibility
haftmann
parents:
58512
diff
changeset
|
1555 |
using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps) |
33320 | 1556 |
|
1557 |
lemma dvd_imp_le_int: |
|
1558 |
fixes d i :: int |
|
1559 |
assumes "i \<noteq> 0" and "d dvd i" |
|
1560 |
shows "\<bar>d\<bar> \<le> \<bar>i\<bar>" |
|
1561 |
proof - |
|
60758 | 1562 |
from \<open>d dvd i\<close> obtain k where "i = d * k" .. |
1563 |
with \<open>i \<noteq> 0\<close> have "k \<noteq> 0" by auto |
|
33320 | 1564 |
then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto |
1565 |
then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono) |
|
60758 | 1566 |
with \<open>i = d * k\<close> show ?thesis by (simp add: abs_mult) |
33320 | 1567 |
qed |
1568 |
||
1569 |
lemma zdvd_not_zless: |
|
1570 |
fixes m n :: int |
|
1571 |
assumes "0 < m" and "m < n" |
|
1572 |
shows "\<not> n dvd m" |
|
1573 |
proof |
|
1574 |
from assms have "0 < n" by auto |
|
1575 |
assume "n dvd m" then obtain k where k: "m = n * k" .. |
|
60758 | 1576 |
with \<open>0 < m\<close> have "0 < n * k" by auto |
1577 |
with \<open>0 < n\<close> have "0 < k" by (simp add: zero_less_mult_iff) |
|
1578 |
with k \<open>0 < n\<close> \<open>m < n\<close> have "n * k < n * 1" by simp |
|
1579 |
with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto |
|
33320 | 1580 |
qed |
1581 |
||
63652 | 1582 |
lemma zdvd_mult_cancel: |
1583 |
fixes k m n :: int |
|
1584 |
assumes d: "k * m dvd k * n" |
|
1585 |
and "k \<noteq> 0" |
|
33320 | 1586 |
shows "m dvd n" |
63652 | 1587 |
proof - |
1588 |
from d obtain h where h: "k * n = k * m * h" |
|
1589 |
unfolding dvd_def by blast |
|
1590 |
have "n = m * h" |
|
1591 |
proof (rule ccontr) |
|
1592 |
assume "\<not> ?thesis" |
|
1593 |
with \<open>k \<noteq> 0\<close> have "k * n \<noteq> k * (m * h)" by simp |
|
1594 |
with h show False |
|
1595 |
by (simp add: mult.assoc) |
|
1596 |
qed |
|
1597 |
then show ?thesis by simp |
|
33320 | 1598 |
qed |
1599 |
||
63652 | 1600 |
theorem zdvd_int: "x dvd y \<longleftrightarrow> int x dvd int y" |
33320 | 1601 |
proof - |
63652 | 1602 |
have "x dvd y" if "int y = int x * k" for k |
1603 |
proof (cases k) |
|
1604 |
case (nonneg n) |
|
1605 |
with that have "y = x * n" |
|
1606 |
by (simp del: of_nat_mult add: of_nat_mult [symmetric]) |
|
1607 |
then show ?thesis .. |
|
1608 |
next |
|
1609 |
case (neg n) |
|
1610 |
with that have "int y = int x * (- int (Suc n))" |
|
1611 |
by simp |
|
1612 |
also have "\<dots> = - (int x * int (Suc n))" |
|
1613 |
by (simp only: mult_minus_right) |
|
1614 |
also have "\<dots> = - int (x * Suc n)" |
|
1615 |
by (simp only: of_nat_mult [symmetric]) |
|
1616 |
finally have "- int (x * Suc n) = int y" .. |
|
1617 |
then show ?thesis |
|
1618 |
by (simp only: negative_eq_positive) auto |
|
33320 | 1619 |
qed |
63652 | 1620 |
then show ?thesis |
1621 |
by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult) |
|
33320 | 1622 |
qed |
1623 |
||
63652 | 1624 |
lemma zdvd1_eq[simp]: "x dvd 1 \<longleftrightarrow> \<bar>x\<bar> = 1" |
1625 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
1626 |
for x :: int |
|
33320 | 1627 |
proof |
63652 | 1628 |
assume ?lhs |
1629 |
then have "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp |
|
1630 |
then have "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int) |
|
1631 |
then have "nat \<bar>x\<bar> = 1" by simp |
|
1632 |
then show ?rhs by (cases "x < 0") auto |
|
33320 | 1633 |
next |
63652 | 1634 |
assume ?rhs |
1635 |
then have "x = 1 \<or> x = - 1" by auto |
|
1636 |
then show ?lhs by (auto intro: dvdI) |
|
33320 | 1637 |
qed |
1638 |
||
60162 | 1639 |
lemma zdvd_mult_cancel1: |
63652 | 1640 |
fixes m :: int |
1641 |
assumes mp: "m \<noteq> 0" |
|
1642 |
shows "m * n dvd m \<longleftrightarrow> \<bar>n\<bar> = 1" |
|
1643 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
33320 | 1644 |
proof |
63652 | 1645 |
assume ?rhs |
1646 |
then show ?lhs |
|
1647 |
by (cases "n > 0") (auto simp add: minus_equation_iff) |
|
33320 | 1648 |
next |
63652 | 1649 |
assume ?lhs |
1650 |
then have "m * n dvd m * 1" by simp |
|
1651 |
from zdvd_mult_cancel[OF this mp] show ?rhs |
|
1652 |
by (simp only: zdvd1_eq) |
|
33320 | 1653 |
qed |
1654 |
||
63652 | 1655 |
lemma int_dvd_iff: "int m dvd z \<longleftrightarrow> m dvd nat \<bar>z\<bar>" |
1656 |
by (cases "z \<ge> 0") (simp_all add: zdvd_int) |
|
33320 | 1657 |
|
63652 | 1658 |
lemma dvd_int_iff: "z dvd int m \<longleftrightarrow> nat \<bar>z\<bar> dvd m" |
1659 |
by (cases "z \<ge> 0") (simp_all add: zdvd_int) |
|
58650 | 1660 |
|
63652 | 1661 |
lemma dvd_int_unfold_dvd_nat: "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>" |
1662 |
by (simp add: dvd_int_iff [symmetric]) |
|
1663 |
||
1664 |
lemma nat_dvd_iff: "nat z dvd m \<longleftrightarrow> (if 0 \<le> z then z dvd int m else m = 0)" |
|
33320 | 1665 |
by (auto simp add: dvd_int_iff) |
1666 |
||
63652 | 1667 |
lemma eq_nat_nat_iff: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'" |
33341 | 1668 |
by (auto elim!: nonneg_eq_int) |
1669 |
||
63652 | 1670 |
lemma nat_power_eq: "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n" |
33341 | 1671 |
by (induct n) (simp_all add: nat_mult_distrib) |
1672 |
||
63652 | 1673 |
lemma zdvd_imp_le: "z dvd n \<Longrightarrow> 0 < n \<Longrightarrow> z \<le> n" |
1674 |
for n z :: int |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1675 |
apply (cases n) |
63652 | 1676 |
apply (auto simp add: dvd_int_iff) |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1677 |
apply (cases z) |
63652 | 1678 |
apply (auto simp add: dvd_imp_le) |
33320 | 1679 |
done |
1680 |
||
36749 | 1681 |
lemma zdvd_period: |
1682 |
fixes a d :: int |
|
1683 |
assumes "a dvd d" |
|
1684 |
shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)" |
|
63652 | 1685 |
(is "?lhs \<longleftrightarrow> ?rhs") |
36749 | 1686 |
proof - |
66816
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
1687 |
from assms have "a dvd (x + t) \<longleftrightarrow> a dvd ((x + t) + c * d)" |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
1688 |
by (simp add: dvd_add_left_iff) |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
1689 |
then show ?thesis |
212a3334e7da
more fundamental definition of div and mod on int
haftmann
parents:
66035
diff
changeset
|
1690 |
by (simp add: ac_simps) |
36749 | 1691 |
qed |
1692 |
||
33320 | 1693 |
|
60758 | 1694 |
subsection \<open>Finiteness of intervals\<close> |
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1695 |
|
63652 | 1696 |
lemma finite_interval_int1 [iff]: "finite {i :: int. a \<le> i \<and> i \<le> b}" |
1697 |
proof (cases "a \<le> b") |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1698 |
case True |
63652 | 1699 |
then show ?thesis |
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1700 |
proof (induct b rule: int_ge_induct) |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1701 |
case base |
63652 | 1702 |
have "{i. a \<le> i \<and> i \<le> a} = {a}" by auto |
1703 |
then show ?case by simp |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1704 |
next |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1705 |
case (step b) |
63652 | 1706 |
then have "{i. a \<le> i \<and> i \<le> b + 1} = {i. a \<le> i \<and> i \<le> b} \<union> {b + 1}" by auto |
1707 |
with step show ?case by simp |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1708 |
qed |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1709 |
next |
63652 | 1710 |
case False |
1711 |
then show ?thesis |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1712 |
by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans) |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1713 |
qed |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1714 |
|
63652 | 1715 |
lemma finite_interval_int2 [iff]: "finite {i :: int. a \<le> i \<and> i < b}" |
1716 |
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1717 |
|
63652 | 1718 |
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i \<and> i \<le> b}" |
1719 |
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1720 |
|
63652 | 1721 |
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i \<and> i < b}" |
1722 |
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1723 |
|
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1724 |
|
60758 | 1725 |
subsection \<open>Configuration of the code generator\<close> |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1726 |
|
60758 | 1727 |
text \<open>Constructors\<close> |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1728 |
|
63652 | 1729 |
definition Pos :: "num \<Rightarrow> int" |
1730 |
where [simp, code_abbrev]: "Pos = numeral" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1731 |
|
63652 | 1732 |
definition Neg :: "num \<Rightarrow> int" |
1733 |
where [simp, code_abbrev]: "Neg n = - (Pos n)" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1734 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1735 |
code_datatype "0::int" Pos Neg |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1736 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1737 |
|
63652 | 1738 |
text \<open>Auxiliary operations.\<close> |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1739 |
|
63652 | 1740 |
definition dup :: "int \<Rightarrow> int" |
1741 |
where [simp]: "dup k = k + k" |
|
26507 | 1742 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1743 |
lemma dup_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1744 |
"dup 0 = 0" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1745 |
"dup (Pos n) = Pos (Num.Bit0 n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1746 |
"dup (Neg n) = Neg (Num.Bit0 n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1747 |
by (simp_all add: numeral_Bit0) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1748 |
|
63652 | 1749 |
definition sub :: "num \<Rightarrow> num \<Rightarrow> int" |
1750 |
where [simp]: "sub m n = numeral m - numeral n" |
|
26507 | 1751 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1752 |
lemma sub_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1753 |
"sub Num.One Num.One = 0" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1754 |
"sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1755 |
"sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1756 |
"sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1757 |
"sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1758 |
"sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1759 |
"sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1760 |
"sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1761 |
"sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1" |
66035
de6cd60b1226
replace non-arithmetic terms by fresh variables before replaying linear-arithmetic proofs: avoid failed proof replays due to an overambitious simpset which may cause proof replay to diverge from the pre-computed proof trace
boehmes
parents:
64996
diff
changeset
|
1762 |
by (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM) |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1763 |
|
63652 | 1764 |
text \<open>Implementations.\<close> |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1765 |
|
64996 | 1766 |
lemma one_int_code [code]: "1 = Pos Num.One" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1767 |
by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1768 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1769 |
lemma plus_int_code [code]: |
63652 | 1770 |
"k + 0 = k" |
1771 |
"0 + l = l" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1772 |
"Pos m + Pos n = Pos (m + n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1773 |
"Pos m + Neg n = sub m n" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1774 |
"Neg m + Pos n = sub n m" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1775 |
"Neg m + Neg n = Neg (m + n)" |
63652 | 1776 |
for k l :: int |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1777 |
by simp_all |
26507 | 1778 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1779 |
lemma uminus_int_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1780 |
"uminus 0 = (0::int)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1781 |
"uminus (Pos m) = Neg m" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1782 |
"uminus (Neg m) = Pos m" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1783 |
by simp_all |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1784 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1785 |
lemma minus_int_code [code]: |
63652 | 1786 |
"k - 0 = k" |
1787 |
"0 - l = uminus l" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1788 |
"Pos m - Pos n = sub m n" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1789 |
"Pos m - Neg n = Pos (m + n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1790 |
"Neg m - Pos n = Neg (m + n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1791 |
"Neg m - Neg n = sub n m" |
63652 | 1792 |
for k l :: int |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1793 |
by simp_all |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1794 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1795 |
lemma times_int_code [code]: |
63652 | 1796 |
"k * 0 = 0" |
1797 |
"0 * l = 0" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1798 |
"Pos m * Pos n = Pos (m * n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1799 |
"Pos m * Neg n = Neg (m * n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1800 |
"Neg m * Pos n = Neg (m * n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1801 |
"Neg m * Neg n = Pos (m * n)" |
63652 | 1802 |
for k l :: int |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1803 |
by simp_all |
26507 | 1804 |
|
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
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|
1805 |
instantiation int :: equal |
26507 | 1806 |
begin |
1807 |
||
63652 | 1808 |
definition "HOL.equal k l \<longleftrightarrow> k = (l::int)" |
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|
1809 |
|
61169 | 1810 |
instance |
1811 |
by standard (rule equal_int_def) |
|
26507 | 1812 |
|
1813 |
end |
|
1814 |
||
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|
1815 |
lemma equal_int_code [code]: |
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|
1816 |
"HOL.equal 0 (0::int) \<longleftrightarrow> True" |
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1817 |
"HOL.equal 0 (Pos l) \<longleftrightarrow> False" |
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|
1818 |
"HOL.equal 0 (Neg l) \<longleftrightarrow> False" |
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|
1819 |
"HOL.equal (Pos k) 0 \<longleftrightarrow> False" |
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|
1820 |
"HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l" |
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|
1821 |
"HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False" |
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|
1822 |
"HOL.equal (Neg k) 0 \<longleftrightarrow> False" |
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|
1823 |
"HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False" |
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|
1824 |
"HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l" |
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|
1825 |
by (auto simp add: equal) |
26507 | 1826 |
|
63652 | 1827 |
lemma equal_int_refl [code nbe]: "HOL.equal k k \<longleftrightarrow> True" |
1828 |
for k :: int |
|
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|
1829 |
by (fact equal_refl) |
26507 | 1830 |
|
28562 | 1831 |
lemma less_eq_int_code [code]: |
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|
1832 |
"0 \<le> (0::int) \<longleftrightarrow> True" |
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|
1833 |
"0 \<le> Pos l \<longleftrightarrow> True" |
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|
1834 |
"0 \<le> Neg l \<longleftrightarrow> False" |
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|
1835 |
"Pos k \<le> 0 \<longleftrightarrow> False" |
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|
1836 |
"Pos k \<le> Pos l \<longleftrightarrow> k \<le> l" |
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|
1837 |
"Pos k \<le> Neg l \<longleftrightarrow> False" |
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|
1838 |
"Neg k \<le> 0 \<longleftrightarrow> True" |
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|
1839 |
"Neg k \<le> Pos l \<longleftrightarrow> True" |
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|
1840 |
"Neg k \<le> Neg l \<longleftrightarrow> l \<le> k" |
28958 | 1841 |
by simp_all |
26507 | 1842 |
|
28562 | 1843 |
lemma less_int_code [code]: |
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|
1844 |
"0 < (0::int) \<longleftrightarrow> False" |
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|
1845 |
"0 < Pos l \<longleftrightarrow> True" |
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|
1846 |
"0 < Neg l \<longleftrightarrow> False" |
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|
1847 |
"Pos k < 0 \<longleftrightarrow> False" |
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|
1848 |
"Pos k < Pos l \<longleftrightarrow> k < l" |
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|
1849 |
"Pos k < Neg l \<longleftrightarrow> False" |
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|
1850 |
"Neg k < 0 \<longleftrightarrow> True" |
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|
1851 |
"Neg k < Pos l \<longleftrightarrow> True" |
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|
1852 |
"Neg k < Neg l \<longleftrightarrow> l < k" |
28958 | 1853 |
by simp_all |
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|
1854 |
|
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|
1855 |
lemma nat_code [code]: |
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|
1856 |
"nat (Int.Neg k) = 0" |
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|
1857 |
"nat 0 = 0" |
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|
1858 |
"nat (Int.Pos k) = nat_of_num k" |
54489
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eliminiated neg_numeral in favour of - (numeral _)
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parents:
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|
1859 |
by (simp_all add: nat_of_num_numeral) |
25928 | 1860 |
|
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|
1861 |
lemma (in ring_1) of_int_code [code]: |
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eliminiated neg_numeral in favour of - (numeral _)
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parents:
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|
1862 |
"of_int (Int.Neg k) = - numeral k" |
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|
1863 |
"of_int 0 = 0" |
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|
1864 |
"of_int (Int.Pos k) = numeral k" |
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|
1865 |
by simp_all |
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|
1866 |
|
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|
1867 |
|
63652 | 1868 |
text \<open>Serializer setup.\<close> |
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parents:
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|
1869 |
|
52435
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migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
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parents:
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diff
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|
1870 |
code_identifier |
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migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
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parents:
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diff
changeset
|
1871 |
code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith |
25919
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parents:
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|
1872 |
|
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parents:
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changeset
|
1873 |
quickcheck_params [default_type = int] |
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parents:
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changeset
|
1874 |
|
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|
1875 |
hide_const (open) Pos Neg sub dup |
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parents:
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changeset
|
1876 |
|
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parents:
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changeset
|
1877 |
|
61799 | 1878 |
text \<open>De-register \<open>int\<close> as a quotient type:\<close> |
48045 | 1879 |
|
53652
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use lifting_forget for deregistering numeric types as a quotient type
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parents:
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changeset
|
1880 |
lifting_update int.lifting |
18fbca265e2e
use lifting_forget for deregistering numeric types as a quotient type
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parents:
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diff
changeset
|
1881 |
lifting_forget int.lifting |
48045 | 1882 |
|
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parents:
diff
changeset
|
1883 |
end |