author | blanchet |
Thu, 06 Mar 2014 13:36:48 +0100 | |
changeset 55932 | 68c5104d2204 |
parent 55911 | d00023bd3554 |
child 55945 | e96383acecf9 |
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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header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *} |
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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 {* Definition of integers as a quotient type *} |
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definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" where |
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"intrel = (\<lambda>(x, y) (u, v). x + v = u + y)" |
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lemma intrel_iff [simp]: "intrel (x, y) (u, v) \<longleftrightarrow> x + v = u + y" |
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by (simp add: intrel_def) |
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quotient_type int = "nat \<times> nat" / "intrel" |
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morphisms Rep_Integ Abs_Integ |
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proof (rule equivpI) |
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show "reflp intrel" |
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unfolding reflp_def by auto |
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show "symp intrel" |
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unfolding symp_def by auto |
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show "transp intrel" |
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unfolding transp_def by auto |
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qed |
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lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]: |
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"(!!x y. z = Abs_Integ (x, y) ==> P) ==> P" |
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by (induct z) auto |
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subsection {* Integers form a commutative ring *} |
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instantiation int :: comm_ring_1 |
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begin |
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lift_definition zero_int :: "int" is "(0, 0)" . |
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lift_definition one_int :: "int" is "(1, 0)" . |
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lift_definition plus_int :: "int \<Rightarrow> int \<Rightarrow> int" |
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is "\<lambda>(x, y) (u, v). (x + u, y + v)" |
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by clarsimp |
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lift_definition uminus_int :: "int \<Rightarrow> int" |
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is "\<lambda>(x, y). (y, x)" |
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by clarsimp |
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lift_definition minus_int :: "int \<Rightarrow> int \<Rightarrow> int" |
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is "\<lambda>(x, y) (u, v). (x + v, y + u)" |
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by clarsimp |
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lift_definition times_int :: "int \<Rightarrow> int \<Rightarrow> int" |
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is "\<lambda>(x, y) (u, v). (x*u + y*v, x*v + y*u)" |
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proof (clarsimp) |
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fix s t u v w x y z :: nat |
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assume "s + v = u + t" and "w + z = y + x" |
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hence "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x) |
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= (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)" |
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by simp |
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thus "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)" |
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by (simp add: algebra_simps) |
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qed |
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instance |
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by default (transfer, clarsimp simp: algebra_simps)+ |
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end |
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abbreviation int :: "nat \<Rightarrow> int" where |
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"int \<equiv> of_nat" |
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lemma int_def: "int n = Abs_Integ (n, 0)" |
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by (induct n, simp add: zero_int.abs_eq, |
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simp add: one_int.abs_eq plus_int.abs_eq) |
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lemma int_transfer [transfer_rule]: |
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"(fun_rel (op =) cr_int) (\<lambda>n. (n, 0)) int" |
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unfolding fun_rel_def cr_int_def int_def by simp |
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lemma int_diff_cases: |
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obtains (diff) m n where "z = int m - int n" |
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by transfer clarsimp |
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subsection {* Integers are totally ordered *} |
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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 default (transfer, force)+ |
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end |
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instantiation int :: distrib_lattice |
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begin |
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definition |
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"(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min" |
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definition |
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"(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max" |
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instance |
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by intro_classes |
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(auto simp add: inf_int_def sup_int_def max_min_distrib2) |
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end |
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subsection {* Ordering properties of arithmetic operations *} |
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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{*Strict Monotonicity of Multiplication*} |
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text{*strict, in 1st argument; proof is by induction on k>0*} |
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lemma zmult_zless_mono2_lemma: |
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"(i::int)<j ==> 0<k ==> int k * i < int k * j" |
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apply (induct k) |
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apply simp |
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apply (simp add: distrib_right) |
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apply (case_tac "k=0") |
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apply (simp_all add: add_strict_mono) |
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done |
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lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = int n" |
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apply transfer |
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apply clarsimp |
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apply (rule_tac x="a - b" in exI, simp) |
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done |
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lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = int n" |
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apply transfer |
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apply clarsimp |
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apply (rule_tac x="a - b" in exI, simp) |
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done |
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lemma zmult_zless_mono2: "[| i<j; (0::int) < k |] ==> k*i < k*j" |
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apply (drule zero_less_imp_eq_int) |
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apply (auto simp add: zmult_zless_mono2_lemma) |
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done |
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text{*The integers form an ordered integral domain*} |
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instantiation int :: linordered_idom |
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begin |
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definition |
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zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)" |
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definition |
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zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)" |
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instance proof |
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fix i j k :: int |
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show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j" |
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by (rule zmult_zless_mono2) |
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show "\<bar>i\<bar> = (if i < 0 then -i else i)" |
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by (simp only: zabs_def) |
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show "sgn (i\<Colon>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\<Colon>int) \<le> z" |
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by transfer clarsimp |
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lemma zless_iff_Suc_zadd: |
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"(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))" |
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apply transfer |
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apply auto |
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apply (rename_tac a b c d) |
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apply (rule_tac x="c+b - Suc(a+d)" in exI) |
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apply arith |
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done |
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lemmas int_distrib = |
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distrib_right [of z1 z2 w] |
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distrib_left [of w z1 z2] |
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left_diff_distrib [of z1 z2 w] |
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right_diff_distrib [of w z1 z2] |
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for z1 z2 w :: int |
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subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*} |
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context ring_1 |
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begin |
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|
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lift_definition of_int :: "int \<Rightarrow> 'a" is "\<lambda>(i, j). of_nat i - of_nat j" |
207 |
by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq |
|
208 |
of_nat_add [symmetric] simp del: of_nat_add) |
|
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|
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lemma of_int_0 [simp]: "of_int 0 = 0" |
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by transfer simp |
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|
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lemma of_int_1 [simp]: "of_int 1 = 1" |
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by transfer simp |
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|
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216 |
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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|
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lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)" |
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by (transfer fixing: uminus) clarsimp |
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221 |
|
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222 |
lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z" |
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|
223 |
using of_int_add [of w "- z"] by simp |
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|
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lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z" |
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226 |
by (transfer fixing: times) (clarsimp simp add: algebra_simps of_nat_mult) |
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|
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text{*Collapse nested embeddings*} |
44709 | 229 |
lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n" |
29667 | 230 |
by (induct n) auto |
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231 |
|
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232 |
lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k" |
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233 |
by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric]) |
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234 |
|
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235 |
lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k" |
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236 |
by simp |
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237 |
|
31015 | 238 |
lemma of_int_power: |
239 |
"of_int (z ^ n) = of_int z ^ n" |
|
240 |
by (induct n) simp_all |
|
241 |
||
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242 |
end |
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243 |
|
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244 |
context ring_char_0 |
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245 |
begin |
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246 |
|
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lemma of_int_eq_iff [simp]: |
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248 |
"of_int w = of_int z \<longleftrightarrow> w = z" |
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by transfer (clarsimp simp add: algebra_simps |
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of_nat_add [symmetric] simp del: of_nat_add) |
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251 |
|
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252 |
text{*Special cases where either operand is zero*} |
36424 | 253 |
lemma of_int_eq_0_iff [simp]: |
254 |
"of_int z = 0 \<longleftrightarrow> z = 0" |
|
255 |
using of_int_eq_iff [of z 0] by simp |
|
256 |
||
257 |
lemma of_int_0_eq_iff [simp]: |
|
258 |
"0 = of_int z \<longleftrightarrow> z = 0" |
|
259 |
using of_int_eq_iff [of 0 z] by simp |
|
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260 |
|
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261 |
end |
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262 |
|
36424 | 263 |
context linordered_idom |
264 |
begin |
|
265 |
||
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|
266 |
text{*Every @{text linordered_idom} has characteristic zero.*} |
36424 | 267 |
subclass ring_char_0 .. |
268 |
||
269 |
lemma of_int_le_iff [simp]: |
|
270 |
"of_int w \<le> of_int z \<longleftrightarrow> w \<le> z" |
|
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271 |
by (transfer fixing: less_eq) (clarsimp simp add: algebra_simps |
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272 |
of_nat_add [symmetric] simp del: of_nat_add) |
36424 | 273 |
|
274 |
lemma of_int_less_iff [simp]: |
|
275 |
"of_int w < of_int z \<longleftrightarrow> w < z" |
|
276 |
by (simp add: less_le order_less_le) |
|
277 |
||
278 |
lemma of_int_0_le_iff [simp]: |
|
279 |
"0 \<le> of_int z \<longleftrightarrow> 0 \<le> z" |
|
280 |
using of_int_le_iff [of 0 z] by simp |
|
281 |
||
282 |
lemma of_int_le_0_iff [simp]: |
|
283 |
"of_int z \<le> 0 \<longleftrightarrow> z \<le> 0" |
|
284 |
using of_int_le_iff [of z 0] by simp |
|
285 |
||
286 |
lemma of_int_0_less_iff [simp]: |
|
287 |
"0 < of_int z \<longleftrightarrow> 0 < z" |
|
288 |
using of_int_less_iff [of 0 z] by simp |
|
289 |
||
290 |
lemma of_int_less_0_iff [simp]: |
|
291 |
"of_int z < 0 \<longleftrightarrow> z < 0" |
|
292 |
using of_int_less_iff [of z 0] by simp |
|
293 |
||
294 |
end |
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295 |
|
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lemma of_int_eq_id [simp]: "of_int = id" |
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297 |
proof |
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fix z show "of_int z = id z" |
48045 | 299 |
by (cases z rule: int_diff_cases, simp) |
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300 |
qed |
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301 |
|
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302 |
|
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303 |
instance int :: no_top |
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304 |
apply default |
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305 |
apply (rule_tac x="x + 1" in exI) |
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306 |
apply simp |
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|
307 |
done |
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|
308 |
|
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|
309 |
instance int :: no_bot |
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310 |
apply default |
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311 |
apply (rule_tac x="x - 1" in exI) |
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|
312 |
apply simp |
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|
313 |
done |
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|
314 |
|
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315 |
subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *} |
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316 |
|
48045 | 317 |
lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y" |
318 |
by auto |
|
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319 |
|
44709 | 320 |
lemma nat_int [simp]: "nat (int n) = n" |
48045 | 321 |
by transfer simp |
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322 |
|
44709 | 323 |
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)" |
48045 | 324 |
by transfer clarsimp |
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325 |
|
44709 | 326 |
corollary nat_0_le: "0 \<le> z ==> int (nat z) = z" |
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|
327 |
by simp |
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|
328 |
|
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329 |
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0" |
48045 | 330 |
by transfer clarsimp |
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|
331 |
|
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|
332 |
lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)" |
48045 | 333 |
by transfer (clarsimp, arith) |
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|
334 |
|
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|
335 |
text{*An alternative condition is @{term "0 \<le> w"} *} |
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|
336 |
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)" |
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diff
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|
337 |
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) |
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|
338 |
|
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|
339 |
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)" |
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|
340 |
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) |
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|
341 |
|
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|
342 |
lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)" |
48045 | 343 |
by transfer (clarsimp, arith) |
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|
344 |
|
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diff
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|
345 |
lemma nonneg_eq_int: |
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|
346 |
fixes z :: int |
44709 | 347 |
assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P" |
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|
348 |
shows P |
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|
349 |
using assms by (blast dest: nat_0_le sym) |
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|
350 |
|
54223 | 351 |
lemma nat_eq_iff: |
352 |
"nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)" |
|
48045 | 353 |
by transfer (clarsimp simp add: le_imp_diff_is_add) |
54223 | 354 |
|
355 |
corollary nat_eq_iff2: |
|
356 |
"m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)" |
|
357 |
using nat_eq_iff [of w m] by auto |
|
358 |
||
359 |
lemma nat_0 [simp]: |
|
360 |
"nat 0 = 0" |
|
361 |
by (simp add: nat_eq_iff) |
|
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|
362 |
|
54223 | 363 |
lemma nat_1 [simp]: |
364 |
"nat 1 = Suc 0" |
|
365 |
by (simp add: nat_eq_iff) |
|
366 |
||
367 |
lemma nat_numeral [simp]: |
|
368 |
"nat (numeral k) = numeral k" |
|
369 |
by (simp add: nat_eq_iff) |
|
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|
370 |
|
54223 | 371 |
lemma nat_neg_numeral [simp]: |
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|
372 |
"nat (- numeral k) = 0" |
54223 | 373 |
by simp |
374 |
||
375 |
lemma nat_2: "nat 2 = Suc (Suc 0)" |
|
376 |
by simp |
|
377 |
||
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diff
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|
378 |
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)" |
48045 | 379 |
by transfer (clarsimp, arith) |
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|
380 |
|
44709 | 381 |
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n" |
48045 | 382 |
by transfer (clarsimp simp add: le_diff_conv) |
44707 | 383 |
|
384 |
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y" |
|
48045 | 385 |
by transfer auto |
44707 | 386 |
|
29700 | 387 |
lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0" |
48045 | 388 |
by transfer clarsimp |
29700 | 389 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
390 |
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
391 |
by (auto simp add: nat_eq_iff2) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
392 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
393 |
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
394 |
by (insert zless_nat_conj [of 0], auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
395 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
396 |
lemma nat_add_distrib: |
54223 | 397 |
"0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'" |
48045 | 398 |
by transfer clarsimp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
399 |
|
54223 | 400 |
lemma nat_diff_distrib': |
401 |
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y" |
|
402 |
by transfer clarsimp |
|
403 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
404 |
lemma nat_diff_distrib: |
54223 | 405 |
"0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'" |
406 |
by (rule nat_diff_distrib') auto |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
407 |
|
44709 | 408 |
lemma nat_zminus_int [simp]: "nat (- int n) = 0" |
48045 | 409 |
by transfer simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
410 |
|
53065 | 411 |
lemma le_nat_iff: |
412 |
"k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k" |
|
413 |
by transfer auto |
|
414 |
||
44709 | 415 |
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)" |
48045 | 416 |
by transfer (clarsimp simp add: less_diff_conv) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
417 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
418 |
context ring_1 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
419 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
420 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
421 |
lemma of_nat_nat: "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
|
422 |
by transfer (clarsimp simp add: of_nat_diff) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
423 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
424 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
425 |
|
54249 | 426 |
lemma diff_nat_numeral [simp]: |
427 |
"(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')" |
|
428 |
by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral) |
|
429 |
||
430 |
||
29779 | 431 |
text {* For termination proofs: *} |
432 |
lemma measure_function_int[measure_function]: "is_measure (nat o abs)" .. |
|
433 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
434 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
435 |
subsection{*Lemmas about the Function @{term of_nat} and Orderings*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
436 |
|
44709 | 437 |
lemma negative_zless_0: "- (int (Suc n)) < (0 \<Colon> int)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
438 |
by (simp add: order_less_le del: of_nat_Suc) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
439 |
|
44709 | 440 |
lemma negative_zless [iff]: "- (int (Suc n)) < int m" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
441 |
by (rule negative_zless_0 [THEN order_less_le_trans], simp) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
442 |
|
44709 | 443 |
lemma negative_zle_0: "- int n \<le> 0" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
444 |
by (simp add: minus_le_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
445 |
|
44709 | 446 |
lemma negative_zle [iff]: "- int n \<le> int m" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
447 |
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
448 |
|
44709 | 449 |
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
450 |
by (subst le_minus_iff, simp del: of_nat_Suc) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
451 |
|
44709 | 452 |
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)" |
48045 | 453 |
by transfer simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
454 |
|
44709 | 455 |
lemma not_int_zless_negative [simp]: "~ (int n < - int m)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
456 |
by (simp add: linorder_not_less) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
457 |
|
44709 | 458 |
lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
459 |
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
460 |
|
44709 | 461 |
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
462 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
463 |
have "(w \<le> z) = (0 \<le> z - w)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
464 |
by (simp only: le_diff_eq add_0_left) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
465 |
also have "\<dots> = (\<exists>n. z - w = of_nat n)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
466 |
by (auto elim: zero_le_imp_eq_int) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
467 |
also have "\<dots> = (\<exists>n. z = w + of_nat n)" |
29667 | 468 |
by (simp only: algebra_simps) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
469 |
finally show ?thesis . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
470 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
471 |
|
44709 | 472 |
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
473 |
by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
474 |
|
44709 | 475 |
lemma int_Suc0_eq_1: "int (Suc 0) = 1" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
476 |
by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
477 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
478 |
text{*This version is proved for all ordered rings, not just integers! |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
479 |
It is proved here because attribute @{text arith_split} is not available |
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35032
diff
changeset
|
480 |
in theory @{text Rings}. |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
481 |
But is it really better than just rewriting with @{text abs_if}?*} |
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53652
diff
changeset
|
482 |
lemma abs_split [arith_split, no_atp]: |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
483 |
"P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
484 |
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
485 |
|
44709 | 486 |
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))" |
48045 | 487 |
apply transfer |
488 |
apply clarsimp |
|
489 |
apply (rule_tac x="b - Suc a" in exI, arith) |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
490 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
491 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
492 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
493 |
subsection {* Cases and induction *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
494 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
495 |
text{*Now we replace the case analysis rule by a more conventional one: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
496 |
whether an integer is negative or not.*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
497 |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
498 |
theorem int_cases [case_names nonneg neg, cases type: int]: |
44709 | 499 |
"[|!! n. z = int n ==> P; !! n. z = - (int (Suc n)) ==> P |] ==> P" |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
500 |
apply (cases "z < 0") |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
501 |
apply (blast dest!: negD) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
502 |
apply (simp add: linorder_not_less del: of_nat_Suc) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
503 |
apply auto |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
504 |
apply (blast dest: nat_0_le [THEN sym]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
505 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
506 |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
507 |
theorem int_of_nat_induct [case_names nonneg neg, induct type: int]: |
44709 | 508 |
"[|!! n. P (int n); !!n. P (- (int (Suc n))) |] ==> P z" |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
509 |
by (cases z) auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
510 |
|
47207
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
511 |
lemma nonneg_int_cases: |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
512 |
assumes "0 \<le> k" obtains n where "k = int n" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
513 |
using assms by (rule nonneg_eq_int) |
47207
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
514 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
515 |
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
516 |
-- {* Unfold all @{text let}s involving constants *} |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
517 |
unfolding Let_def .. |
37767 | 518 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
519 |
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
520 |
-- {* Unfold all @{text let}s involving constants *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
521 |
unfolding Let_def .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
522 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
523 |
text {* Unfold @{text min} and @{text max} on numerals. *} |
28958 | 524 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
525 |
lemmas max_number_of [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
526 |
max_def [of "numeral u" "numeral v"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
527 |
max_def [of "numeral u" "- numeral v"] |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
528 |
max_def [of "- numeral u" "numeral v"] |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
529 |
max_def [of "- numeral u" "- numeral v"] for u v |
28958 | 530 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
531 |
lemmas min_number_of [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
532 |
min_def [of "numeral u" "numeral v"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
533 |
min_def [of "numeral u" "- numeral v"] |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
534 |
min_def [of "- numeral u" "numeral v"] |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
535 |
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
|
536 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
537 |
|
28958 | 538 |
subsubsection {* Binary comparisons *} |
539 |
||
540 |
text {* Preliminaries *} |
|
541 |
||
542 |
lemma even_less_0_iff: |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
543 |
"a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)" |
28958 | 544 |
proof - |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
545 |
have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: distrib_right del: one_add_one) |
28958 | 546 |
also have "(1+1)*a < 0 \<longleftrightarrow> a < 0" |
547 |
by (simp add: mult_less_0_iff zero_less_two |
|
548 |
order_less_not_sym [OF zero_less_two]) |
|
549 |
finally show ?thesis . |
|
550 |
qed |
|
551 |
||
552 |
lemma le_imp_0_less: |
|
553 |
assumes le: "0 \<le> z" |
|
554 |
shows "(0::int) < 1 + z" |
|
555 |
proof - |
|
556 |
have "0 \<le> z" by fact |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
557 |
also have "... < z + 1" by (rule less_add_one) |
28958 | 558 |
also have "... = 1 + z" by (simp add: add_ac) |
559 |
finally show "0 < 1 + z" . |
|
560 |
qed |
|
561 |
||
562 |
lemma odd_less_0_iff: |
|
563 |
"(1 + z + z < 0) = (z < (0::int))" |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
564 |
proof (cases z) |
28958 | 565 |
case (nonneg n) |
566 |
thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing |
|
567 |
le_imp_0_less [THEN order_less_imp_le]) |
|
568 |
next |
|
569 |
case (neg n) |
|
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30000
diff
changeset
|
570 |
thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1 |
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30000
diff
changeset
|
571 |
add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric]) |
28958 | 572 |
qed |
573 |
||
574 |
subsubsection {* Comparisons, for Ordered Rings *} |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
575 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
576 |
lemmas double_eq_0_iff = double_zero |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
577 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
578 |
lemma odd_nonzero: |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
579 |
"1 + z + z \<noteq> (0::int)" |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
580 |
proof (cases z) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
581 |
case (nonneg n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
582 |
have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
583 |
thus ?thesis using le_imp_0_less [OF le] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
584 |
by (auto simp add: add_assoc) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
585 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
586 |
case (neg n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
587 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
588 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
589 |
assume eq: "1 + z + z = 0" |
44709 | 590 |
have "(0::int) < 1 + (int n + int n)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
591 |
by (simp add: le_imp_0_less add_increasing) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
592 |
also have "... = - (1 + z + z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
593 |
by (simp add: neg add_assoc [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
594 |
also have "... = 0" by (simp add: eq) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
595 |
finally have "0<0" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
596 |
thus False by blast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
597 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
598 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
599 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
600 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
601 |
subsection {* The Set of Integers *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
602 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
603 |
context ring_1 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
604 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
605 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
606 |
definition Ints :: "'a set" where |
37767 | 607 |
"Ints = range of_int" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
608 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
609 |
notation (xsymbols) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
610 |
Ints ("\<int>") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
611 |
|
35634 | 612 |
lemma Ints_of_int [simp]: "of_int z \<in> \<int>" |
613 |
by (simp add: Ints_def) |
|
614 |
||
615 |
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>" |
|
45533 | 616 |
using Ints_of_int [of "of_nat n"] by simp |
35634 | 617 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
618 |
lemma Ints_0 [simp]: "0 \<in> \<int>" |
45533 | 619 |
using Ints_of_int [of "0"] by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
620 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
621 |
lemma Ints_1 [simp]: "1 \<in> \<int>" |
45533 | 622 |
using Ints_of_int [of "1"] by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
623 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
624 |
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
625 |
apply (auto simp add: Ints_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
626 |
apply (rule range_eqI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
627 |
apply (rule of_int_add [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
628 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
629 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
630 |
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
631 |
apply (auto simp add: Ints_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
632 |
apply (rule range_eqI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
633 |
apply (rule of_int_minus [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
634 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
635 |
|
35634 | 636 |
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>" |
637 |
apply (auto simp add: Ints_def) |
|
638 |
apply (rule range_eqI) |
|
639 |
apply (rule of_int_diff [symmetric]) |
|
640 |
done |
|
641 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
642 |
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
643 |
apply (auto simp add: Ints_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
644 |
apply (rule range_eqI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
645 |
apply (rule of_int_mult [symmetric]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
646 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
647 |
|
35634 | 648 |
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>" |
649 |
by (induct n) simp_all |
|
650 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
651 |
lemma Ints_cases [cases set: Ints]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
652 |
assumes "q \<in> \<int>" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
653 |
obtains (of_int) z where "q = of_int z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
654 |
unfolding Ints_def |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
655 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
656 |
from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
657 |
then obtain z where "q = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
658 |
then show thesis .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
659 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
660 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
661 |
lemma Ints_induct [case_names of_int, induct set: Ints]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
662 |
"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
|
663 |
by (rule Ints_cases) auto |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
664 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
665 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
666 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
667 |
text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
668 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
669 |
lemma Ints_double_eq_0_iff: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
670 |
assumes in_Ints: "a \<in> Ints" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
671 |
shows "(a + a = 0) = (a = (0::'a::ring_char_0))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
672 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
673 |
from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
674 |
then obtain z where a: "a = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
675 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
676 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
677 |
assume "a = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
678 |
thus "a + a = 0" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
679 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
680 |
assume eq: "a + a = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
681 |
hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
682 |
hence "z + z = 0" by (simp only: of_int_eq_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
683 |
hence "z = 0" by (simp only: double_eq_0_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
684 |
thus "a = 0" by (simp add: a) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
685 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
686 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
687 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
688 |
lemma Ints_odd_nonzero: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
689 |
assumes in_Ints: "a \<in> Ints" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
690 |
shows "1 + a + a \<noteq> (0::'a::ring_char_0)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
691 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
692 |
from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
693 |
then obtain z where a: "a = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
694 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
695 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
696 |
assume eq: "1 + a + a = 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
697 |
hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
698 |
hence "1 + z + z = 0" by (simp only: of_int_eq_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
699 |
with odd_nonzero show False by blast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
700 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
701 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
702 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
703 |
lemma Nats_numeral [simp]: "numeral w \<in> Nats" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
704 |
using of_nat_in_Nats [of "numeral w"] by simp |
35634 | 705 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
706 |
lemma Ints_odd_less_0: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
707 |
assumes in_Ints: "a \<in> Ints" |
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34055
diff
changeset
|
708 |
shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
709 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
710 |
from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
711 |
then obtain z where a: "a = of_int z" .. |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
712 |
hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
713 |
by (simp add: a) |
45532
74b17a0881b3
Int.thy: remove duplicate lemmas double_less_0_iff and odd_less_0, use {even,odd}_less_0_iff instead
huffman
parents:
45219
diff
changeset
|
714 |
also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
715 |
also have "... = (a < 0)" by (simp add: a) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
716 |
finally show ?thesis . |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
717 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
718 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
719 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
720 |
subsection {* @{term setsum} and @{term setprod} *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
721 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
722 |
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
723 |
apply (cases "finite A") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
724 |
apply (erule finite_induct, auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
725 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
726 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
727 |
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
728 |
apply (cases "finite A") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
729 |
apply (erule finite_induct, auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
730 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
731 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
732 |
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
733 |
apply (cases "finite A") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
734 |
apply (erule finite_induct, auto simp add: of_nat_mult) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
735 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
736 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
737 |
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
738 |
apply (cases "finite A") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
739 |
apply (erule finite_induct, auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
740 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
741 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
742 |
lemmas int_setsum = of_nat_setsum [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
743 |
lemmas int_setprod = of_nat_setprod [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
744 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
745 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
746 |
text {* Legacy theorems *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
747 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
748 |
lemmas zle_int = of_nat_le_iff [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
749 |
lemmas int_int_eq = of_nat_eq_iff [where 'a=int] |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
750 |
lemmas numeral_1_eq_1 = numeral_One |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
751 |
|
30802 | 752 |
subsection {* Setting up simplification procedures *} |
753 |
||
54249 | 754 |
lemmas of_int_simps = |
755 |
of_int_0 of_int_1 of_int_add of_int_mult |
|
756 |
||
30802 | 757 |
lemmas int_arith_rules = |
54249 | 758 |
numeral_One more_arith_simps of_nat_simps of_int_simps |
30802 | 759 |
|
48891 | 760 |
ML_file "Tools/int_arith.ML" |
30496
7cdcc9dd95cb
vague cleanup in arith proof tools setup: deleted dead code, more proper structures, clearer arrangement
haftmann
parents:
30273
diff
changeset
|
761 |
declaration {* K Int_Arith.setup *} |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
762 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
763 |
simproc_setup fast_arith ("(m::'a::linordered_idom) < n" | |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
764 |
"(m::'a::linordered_idom) <= n" | |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
765 |
"(m::'a::linordered_idom) = n") = |
43595 | 766 |
{* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *} |
767 |
||
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
768 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
769 |
subsection{*More Inequality Reasoning*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
770 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
771 |
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
772 |
by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
773 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
774 |
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
775 |
by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
776 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
777 |
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
778 |
by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
779 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
780 |
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
781 |
by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
782 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
783 |
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
784 |
by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
785 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
786 |
|
28958 | 787 |
subsection{*The functions @{term nat} and @{term int}*} |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
788 |
|
48044
fea6f3060b65
remove unnecessary simp rules involving Abs_Integ
huffman
parents:
47255
diff
changeset
|
789 |
text{*Simplify the term @{term "w + - z"}*} |
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 |
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
792 |
apply (insert zless_nat_conj [of 1 z]) |
47207
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
793 |
apply auto |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
794 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
795 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
796 |
text{*This simplifies expressions of the form @{term "int n = z"} where |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
797 |
z is an integer literal.*} |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
798 |
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
|
799 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
800 |
lemma split_nat [arith_split]: |
44709 | 801 |
"P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
802 |
(is "?P = (?L & ?R)") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
803 |
proof (cases "i < 0") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
804 |
case True thus ?thesis by auto |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
805 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
806 |
case False |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
807 |
have "?P = ?L" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
808 |
proof |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
809 |
assume ?P thus ?L using False by clarsimp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
810 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
811 |
assume ?L thus ?P using False by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
812 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
813 |
with False show ?thesis by simp |
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 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
816 |
context ring_1 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
817 |
begin |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
818 |
|
33056
791a4655cae3
renamed "nitpick_const_xxx" attributes to "nitpick_xxx" and "nitpick_ind_intros" to "nitpick_intros"
blanchet
parents:
32437
diff
changeset
|
819 |
lemma of_int_of_nat [nitpick_simp]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
820 |
"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
|
821 |
proof (cases "k < 0") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
822 |
case True then have "0 \<le> - k" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
823 |
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
|
824 |
with True show ?thesis by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
825 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
826 |
case False then show ?thesis by (simp add: not_less of_nat_nat) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
827 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
828 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
829 |
end |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
830 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
831 |
lemma nat_mult_distrib: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
832 |
fixes z z' :: int |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
833 |
assumes "0 \<le> z" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
834 |
shows "nat (z * z') = nat z * nat z'" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
835 |
proof (cases "0 \<le> z'") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
836 |
case False with assms have "z * z' \<le> 0" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
837 |
by (simp add: not_le mult_le_0_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
838 |
then have "nat (z * z') = 0" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
839 |
moreover from False have "nat z' = 0" by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
840 |
ultimately show ?thesis by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
841 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
842 |
case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
843 |
show ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
844 |
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
|
845 |
(simp only: of_nat_mult of_nat_nat [OF True] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
846 |
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
|
847 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
848 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
849 |
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
850 |
apply (rule trans) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
851 |
apply (rule_tac [2] nat_mult_distrib, auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
852 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
853 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
854 |
lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
855 |
apply (cases "z=0 | w=0") |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
856 |
apply (auto simp add: abs_if nat_mult_distrib [symmetric] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
857 |
nat_mult_distrib_neg [symmetric] mult_less_0_iff) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
858 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
859 |
|
47207
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
860 |
lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)" |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
861 |
apply (rule sym) |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
862 |
apply (simp add: nat_eq_iff) |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
863 |
done |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
864 |
|
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
865 |
lemma diff_nat_eq_if: |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
866 |
"nat z - nat z' = |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
867 |
(if z' < 0 then nat z |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
868 |
else let d = z-z' in |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
869 |
if d < 0 then 0 else nat d)" |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
870 |
by (simp add: Let_def nat_diff_distrib [symmetric]) |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
871 |
|
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
872 |
lemma nat_numeral_diff_1 [simp]: |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
873 |
"numeral v - (1::nat) = nat (numeral v - 1)" |
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
874 |
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
|
875 |
|
25919
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 |
subsection "Induction principles for int" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
878 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
879 |
text{*Well-founded segments of the integers*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
880 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
881 |
definition |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
882 |
int_ge_less_than :: "int => (int * int) set" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
883 |
where |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
884 |
"int_ge_less_than d = {(z',z). d \<le> z' & z' < z}" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
885 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
886 |
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
887 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
888 |
have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
889 |
by (auto simp add: int_ge_less_than_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
890 |
thus ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
891 |
by (rule wf_subset [OF wf_measure]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
892 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
893 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
894 |
text{*This variant looks odd, but is typical of the relations suggested |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
895 |
by RankFinder.*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
896 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
897 |
definition |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
898 |
int_ge_less_than2 :: "int => (int * int) set" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
899 |
where |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
900 |
"int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
901 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
902 |
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
903 |
proof - |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
904 |
have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
905 |
by (auto simp add: int_ge_less_than2_def) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
906 |
thus ?thesis |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
907 |
by (rule wf_subset [OF wf_measure]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
908 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
909 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
910 |
(* `set:int': dummy construction *) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
911 |
theorem int_ge_induct [case_names base step, induct set: int]: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
912 |
fixes i :: int |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
913 |
assumes ge: "k \<le> i" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
914 |
base: "P k" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
915 |
step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
916 |
shows "P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
917 |
proof - |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
918 |
{ fix n |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
919 |
have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
920 |
proof (induct n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
921 |
case 0 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
922 |
hence "i = k" by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
923 |
thus "P i" using base by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
924 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
925 |
case (Suc n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
926 |
then have "n = nat((i - 1) - k)" by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
927 |
moreover |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
928 |
have ki1: "k \<le> i - 1" using Suc.prems by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
929 |
ultimately |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
930 |
have "P (i - 1)" by (rule Suc.hyps) |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
931 |
from step [OF ki1 this] show ?case by simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
932 |
qed |
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 |
with ge show ?thesis by fast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
935 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
936 |
|
25928 | 937 |
(* `set:int': dummy construction *) |
938 |
theorem int_gr_induct [case_names base step, induct set: int]: |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
939 |
assumes gr: "k < (i::int)" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
940 |
base: "P(k+1)" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
941 |
step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
942 |
shows "P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
943 |
apply(rule int_ge_induct[of "k + 1"]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
944 |
using gr apply arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
945 |
apply(rule base) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
946 |
apply (rule step, simp+) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
947 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
948 |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
949 |
theorem int_le_induct [consumes 1, case_names base step]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
950 |
assumes le: "i \<le> (k::int)" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
951 |
base: "P(k)" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
952 |
step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
953 |
shows "P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
954 |
proof - |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
955 |
{ fix n |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
956 |
have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
957 |
proof (induct n) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
958 |
case 0 |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
959 |
hence "i = k" by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
960 |
thus "P i" using base by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
961 |
next |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
962 |
case (Suc n) |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
963 |
hence "n = nat (k - (i + 1))" by arith |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
964 |
moreover |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
965 |
have ki1: "i + 1 \<le> k" using Suc.prems by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
966 |
ultimately |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
967 |
have "P (i + 1)" by(rule Suc.hyps) |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
968 |
from step[OF ki1 this] show ?case by simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
969 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
970 |
} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
971 |
with le show ?thesis by fast |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
972 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
973 |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
974 |
theorem int_less_induct [consumes 1, case_names base step]: |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
975 |
assumes less: "(i::int) < k" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
976 |
base: "P(k - 1)" and |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
977 |
step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
978 |
shows "P i" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
979 |
apply(rule int_le_induct[of _ "k - 1"]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
980 |
using less apply arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
981 |
apply(rule base) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
982 |
apply (rule step, simp+) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
983 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
984 |
|
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
|
985 |
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
|
986 |
fixes k :: int |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
987 |
assumes base: "P k" |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
988 |
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
|
989 |
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
|
990 |
shows "P i" |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
991 |
proof - |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
992 |
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
|
993 |
then show ?thesis |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
994 |
proof |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
995 |
assume "i \<ge> k" |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
996 |
then show ?thesis using base |
36801
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
997 |
by (rule int_ge_induct) (fact step1) |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
998 |
next |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
999 |
assume "i \<le> k" |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1000 |
then show ?thesis using base |
36801
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1001 |
by (rule int_le_induct) (fact step2) |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1002 |
qed |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1003 |
qed |
3560de0fe851
moved int induction lemma to theory Int as int_bidirectional_induct
haftmann
parents:
36749
diff
changeset
|
1004 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1005 |
subsection{*Intermediate value theorems*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1006 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1007 |
lemma int_val_lemma: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1008 |
"(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) --> |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1009 |
f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))" |
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30000
diff
changeset
|
1010 |
unfolding One_nat_def |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1011 |
apply (induct n) |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1012 |
apply simp |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1013 |
apply (intro strip) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1014 |
apply (erule impE, simp) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1015 |
apply (erule_tac x = n in allE, simp) |
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30000
diff
changeset
|
1016 |
apply (case_tac "k = f (Suc n)") |
27106 | 1017 |
apply force |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1018 |
apply (erule impE) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1019 |
apply (simp add: abs_if split add: split_if_asm) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1020 |
apply (blast intro: le_SucI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1021 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1022 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1023 |
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
|
1024 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1025 |
lemma nat_intermed_int_val: |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1026 |
"[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n; |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1027 |
f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1028 |
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1029 |
in int_val_lemma) |
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30000
diff
changeset
|
1030 |
unfolding One_nat_def |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1031 |
apply simp |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1032 |
apply (erule exE) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1033 |
apply (rule_tac x = "i+m" in exI, arith) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1034 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1035 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1036 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1037 |
subsection{*Products and 1, by T. M. Rasmussen*} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1038 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1039 |
lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1040 |
by arith |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1041 |
|
34055 | 1042 |
lemma abs_zmult_eq_1: |
1043 |
assumes mn: "\<bar>m * n\<bar> = 1" |
|
1044 |
shows "\<bar>m\<bar> = (1::int)" |
|
1045 |
proof - |
|
1046 |
have 0: "m \<noteq> 0 & n \<noteq> 0" using mn |
|
1047 |
by auto |
|
1048 |
have "~ (2 \<le> \<bar>m\<bar>)" |
|
1049 |
proof |
|
1050 |
assume "2 \<le> \<bar>m\<bar>" |
|
1051 |
hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>" |
|
1052 |
by (simp add: mult_mono 0) |
|
1053 |
also have "... = \<bar>m*n\<bar>" |
|
1054 |
by (simp add: abs_mult) |
|
1055 |
also have "... = 1" |
|
1056 |
by (simp add: mn) |
|
1057 |
finally have "2*\<bar>n\<bar> \<le> 1" . |
|
1058 |
thus "False" using 0 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1059 |
by arith |
34055 | 1060 |
qed |
1061 |
thus ?thesis using 0 |
|
1062 |
by auto |
|
1063 |
qed |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1064 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1065 |
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1066 |
by (insert abs_zmult_eq_1 [of m n], arith) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1067 |
|
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35634
diff
changeset
|
1068 |
lemma pos_zmult_eq_1_iff: |
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35634
diff
changeset
|
1069 |
assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)" |
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35634
diff
changeset
|
1070 |
proof - |
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35634
diff
changeset
|
1071 |
from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma) |
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35634
diff
changeset
|
1072 |
thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma) |
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35634
diff
changeset
|
1073 |
qed |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1074 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1075 |
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))" |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1076 |
apply (rule iffI) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1077 |
apply (frule pos_zmult_eq_1_iff_lemma) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1078 |
apply (simp add: mult_commute [of m]) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1079 |
apply (frule pos_zmult_eq_1_iff_lemma, auto) |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1080 |
done |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1081 |
|
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1082 |
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)" |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1083 |
proof |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1084 |
assume "finite (UNIV::int set)" |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1085 |
moreover have "inj (\<lambda>i\<Colon>int. 2 * i)" |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1086 |
by (rule injI) simp |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1087 |
ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)" |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1088 |
by (rule finite_UNIV_inj_surj) |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33056
diff
changeset
|
1089 |
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
|
1090 |
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
|
1091 |
qed |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1092 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1093 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1094 |
subsection {* Further theorems on numerals *} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1095 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1096 |
subsubsection{*Special Simplification for Constants*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1097 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1098 |
text{*These distributive laws move literals inside sums and differences.*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1099 |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1100 |
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
|
1101 |
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
|
1102 |
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
|
1103 |
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
|
1104 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1105 |
text{*These are actually for fields, like real: but where else to put them?*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1106 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1107 |
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
|
1108 |
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
|
1109 |
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
|
1110 |
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
|
1111 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1112 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1113 |
text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}. It looks |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1114 |
strange, but then other simprocs simplify the quotient.*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1115 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1116 |
lemmas inverse_eq_divide_numeral [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1117 |
inverse_eq_divide [of "numeral w"] for w |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1118 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1119 |
lemmas inverse_eq_divide_neg_numeral [simp] = |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1120 |
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
|
1121 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1122 |
text {*These laws simplify inequalities, moving unary minus from a term |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1123 |
into the literal.*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1124 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1125 |
lemmas equation_minus_iff_numeral [no_atp] = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1126 |
equation_minus_iff [of "numeral v"] for v |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1127 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1128 |
lemmas minus_equation_iff_numeral [no_atp] = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1129 |
minus_equation_iff [of _ "numeral v"] for v |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1130 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1131 |
lemmas le_minus_iff_numeral [no_atp] = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1132 |
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
|
1133 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1134 |
lemmas minus_le_iff_numeral [no_atp] = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1135 |
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
|
1136 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1137 |
lemmas less_minus_iff_numeral [no_atp] = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1138 |
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
|
1139 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1140 |
lemmas minus_less_iff_numeral [no_atp] = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1141 |
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
|
1142 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1143 |
-- {* FIXME maybe simproc *} |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1144 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1145 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1146 |
text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1147 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1148 |
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
|
1149 |
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
|
1150 |
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
|
1151 |
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
|
1152 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1153 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1154 |
text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1155 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1156 |
lemmas le_divide_eq_numeral1 [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1157 |
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
|
1158 |
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
|
1159 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1160 |
lemmas divide_le_eq_numeral1 [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1161 |
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
|
1162 |
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
|
1163 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1164 |
lemmas less_divide_eq_numeral1 [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1165 |
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
|
1166 |
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
|
1167 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1168 |
lemmas divide_less_eq_numeral1 [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1169 |
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
|
1170 |
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
|
1171 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1172 |
lemmas eq_divide_eq_numeral1 [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1173 |
eq_divide_eq [of _ _ "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1174 |
eq_divide_eq [of _ _ "- numeral w"] for w |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1175 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1176 |
lemmas divide_eq_eq_numeral1 [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1177 |
divide_eq_eq [of _ "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1178 |
divide_eq_eq [of _ "- numeral w"] for w |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1179 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1180 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1181 |
subsubsection{*Optional Simplification Rules Involving Constants*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1182 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1183 |
text{*Simplify quotients that are compared with a literal constant.*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1184 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1185 |
lemmas le_divide_eq_numeral = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1186 |
le_divide_eq [of "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1187 |
le_divide_eq [of "- numeral w"] for w |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1188 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1189 |
lemmas divide_le_eq_numeral = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1190 |
divide_le_eq [of _ _ "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1191 |
divide_le_eq [of _ _ "- numeral w"] for w |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1192 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1193 |
lemmas less_divide_eq_numeral = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1194 |
less_divide_eq [of "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1195 |
less_divide_eq [of "- numeral w"] for w |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1196 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1197 |
lemmas divide_less_eq_numeral = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1198 |
divide_less_eq [of _ _ "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1199 |
divide_less_eq [of _ _ "- numeral w"] for w |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1200 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1201 |
lemmas eq_divide_eq_numeral = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1202 |
eq_divide_eq [of "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1203 |
eq_divide_eq [of "- numeral w"] for w |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1204 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1205 |
lemmas divide_eq_eq_numeral = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1206 |
divide_eq_eq [of _ _ "numeral w"] |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1207 |
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
|
1208 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1209 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1210 |
text{*Not good as automatic simprules because they cause case splits.*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1211 |
lemmas divide_const_simps = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1212 |
le_divide_eq_numeral divide_le_eq_numeral less_divide_eq_numeral |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1213 |
divide_less_eq_numeral eq_divide_eq_numeral divide_eq_eq_numeral |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1214 |
le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1 |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1215 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1216 |
text{*Division By @{text "-1"}*} |
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1217 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1218 |
lemma divide_minus1 [simp]: "(x::'a::field) / -1 = - x" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1219 |
unfolding nonzero_minus_divide_right [OF one_neq_zero, symmetric] |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1220 |
by simp |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1221 |
|
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1222 |
lemma half_gt_zero_iff: |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1223 |
"(0 < r/2) = (0 < (r::'a::linordered_field_inverse_zero))" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1224 |
by auto |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1225 |
|
45607 | 1226 |
lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2] |
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1227 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1228 |
lemma divide_Numeral1: "(x::'a::field) / Numeral1 = x" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1229 |
by (fact divide_numeral_1) |
36719 | 1230 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30496
diff
changeset
|
1231 |
|
33320 | 1232 |
subsection {* The divides relation *} |
1233 |
||
33657 | 1234 |
lemma zdvd_antisym_nonneg: |
1235 |
"0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)" |
|
33320 | 1236 |
apply (simp add: dvd_def, auto) |
33657 | 1237 |
apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff) |
33320 | 1238 |
done |
1239 |
||
33657 | 1240 |
lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a" |
33320 | 1241 |
shows "\<bar>a\<bar> = \<bar>b\<bar>" |
33657 | 1242 |
proof cases |
1243 |
assume "a = 0" with assms show ?thesis by simp |
|
1244 |
next |
|
1245 |
assume "a \<noteq> 0" |
|
33320 | 1246 |
from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast |
1247 |
from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast |
|
1248 |
from k k' have "a = a*k*k'" by simp |
|
1249 |
with mult_cancel_left1[where c="a" and b="k*k'"] |
|
1250 |
have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc) |
|
1251 |
hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff) |
|
1252 |
thus ?thesis using k k' by auto |
|
1253 |
qed |
|
1254 |
||
1255 |
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)" |
|
1256 |
apply (subgoal_tac "m = n + (m - n)") |
|
1257 |
apply (erule ssubst) |
|
1258 |
apply (blast intro: dvd_add, simp) |
|
1259 |
done |
|
1260 |
||
1261 |
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))" |
|
1262 |
apply (rule iffI) |
|
1263 |
apply (erule_tac [2] dvd_add) |
|
1264 |
apply (subgoal_tac "n = (n + k * m) - k * m") |
|
1265 |
apply (erule ssubst) |
|
1266 |
apply (erule dvd_diff) |
|
1267 |
apply(simp_all) |
|
1268 |
done |
|
1269 |
||
1270 |
lemma dvd_imp_le_int: |
|
1271 |
fixes d i :: int |
|
1272 |
assumes "i \<noteq> 0" and "d dvd i" |
|
1273 |
shows "\<bar>d\<bar> \<le> \<bar>i\<bar>" |
|
1274 |
proof - |
|
1275 |
from `d dvd i` obtain k where "i = d * k" .. |
|
1276 |
with `i \<noteq> 0` have "k \<noteq> 0" by auto |
|
1277 |
then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto |
|
1278 |
then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono) |
|
1279 |
with `i = d * k` show ?thesis by (simp add: abs_mult) |
|
1280 |
qed |
|
1281 |
||
1282 |
lemma zdvd_not_zless: |
|
1283 |
fixes m n :: int |
|
1284 |
assumes "0 < m" and "m < n" |
|
1285 |
shows "\<not> n dvd m" |
|
1286 |
proof |
|
1287 |
from assms have "0 < n" by auto |
|
1288 |
assume "n dvd m" then obtain k where k: "m = n * k" .. |
|
1289 |
with `0 < m` have "0 < n * k" by auto |
|
1290 |
with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff) |
|
1291 |
with k `0 < n` `m < n` have "n * k < n * 1" by simp |
|
1292 |
with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto |
|
1293 |
qed |
|
1294 |
||
1295 |
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)" |
|
1296 |
shows "m dvd n" |
|
1297 |
proof- |
|
1298 |
from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast |
|
1299 |
{assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp |
|
1300 |
with h have False by (simp add: mult_assoc)} |
|
1301 |
hence "n = m * h" by blast |
|
1302 |
thus ?thesis by simp |
|
1303 |
qed |
|
1304 |
||
1305 |
theorem zdvd_int: "(x dvd y) = (int x dvd int y)" |
|
1306 |
proof - |
|
1307 |
have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y" |
|
1308 |
proof - |
|
1309 |
fix k |
|
1310 |
assume A: "int y = int x * k" |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1311 |
then show "x dvd y" |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1312 |
proof (cases k) |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1313 |
case (nonneg n) |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1314 |
with A have "y = x * n" by (simp add: of_nat_mult [symmetric]) |
33320 | 1315 |
then show ?thesis .. |
1316 |
next |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1317 |
case (neg n) |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1318 |
with A have "int y = int x * (- int (Suc n))" by simp |
33320 | 1319 |
also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right) |
1320 |
also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric]) |
|
1321 |
finally have "- int (x * Suc n) = int y" .. |
|
1322 |
then show ?thesis by (simp only: negative_eq_positive) auto |
|
1323 |
qed |
|
1324 |
qed |
|
1325 |
then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult) |
|
1326 |
qed |
|
1327 |
||
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1328 |
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)" |
33320 | 1329 |
proof |
1330 |
assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp |
|
1331 |
hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int) |
|
1332 |
hence "nat \<bar>x\<bar> = 1" by simp |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1333 |
thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto |
33320 | 1334 |
next |
1335 |
assume "\<bar>x\<bar>=1" |
|
1336 |
then have "x = 1 \<or> x = -1" by auto |
|
1337 |
then show "x dvd 1" by (auto intro: dvdI) |
|
1338 |
qed |
|
1339 |
||
1340 |
lemma zdvd_mult_cancel1: |
|
1341 |
assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)" |
|
1342 |
proof |
|
1343 |
assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1344 |
by (cases "n >0") (auto simp add: minus_equation_iff) |
33320 | 1345 |
next |
1346 |
assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp |
|
1347 |
from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq) |
|
1348 |
qed |
|
1349 |
||
1350 |
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))" |
|
1351 |
unfolding zdvd_int by (cases "z \<ge> 0") simp_all |
|
1352 |
||
1353 |
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)" |
|
1354 |
unfolding zdvd_int by (cases "z \<ge> 0") simp_all |
|
1355 |
||
1356 |
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)" |
|
1357 |
by (auto simp add: dvd_int_iff) |
|
1358 |
||
33341 | 1359 |
lemma eq_nat_nat_iff: |
1360 |
"0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'" |
|
1361 |
by (auto elim!: nonneg_eq_int) |
|
1362 |
||
1363 |
lemma nat_power_eq: |
|
1364 |
"0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n" |
|
1365 |
by (induct n) (simp_all add: nat_mult_distrib) |
|
1366 |
||
33320 | 1367 |
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)" |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1368 |
apply (cases n) |
33320 | 1369 |
apply (auto simp add: dvd_int_iff) |
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1370 |
apply (cases z) |
33320 | 1371 |
apply (auto simp add: dvd_imp_le) |
1372 |
done |
|
1373 |
||
36749 | 1374 |
lemma zdvd_period: |
1375 |
fixes a d :: int |
|
1376 |
assumes "a dvd d" |
|
1377 |
shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)" |
|
1378 |
proof - |
|
1379 |
from assms obtain k where "d = a * k" by (rule dvdE) |
|
42676
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1380 |
show ?thesis |
8724f20bf69c
proper case_names for int_cases, int_of_nat_induct;
wenzelm
parents:
42411
diff
changeset
|
1381 |
proof |
36749 | 1382 |
assume "a dvd (x + t)" |
1383 |
then obtain l where "x + t = a * l" by (rule dvdE) |
|
1384 |
then have "x = a * l - t" by simp |
|
1385 |
with `d = a * k` show "a dvd x + c * d + t" by simp |
|
1386 |
next |
|
1387 |
assume "a dvd x + c * d + t" |
|
1388 |
then obtain l where "x + c * d + t = a * l" by (rule dvdE) |
|
1389 |
then have "x = a * l - c * d - t" by simp |
|
1390 |
with `d = a * k` show "a dvd (x + t)" by simp |
|
1391 |
qed |
|
1392 |
qed |
|
1393 |
||
33320 | 1394 |
|
46756
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1395 |
subsection {* Finiteness of intervals *} |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1396 |
|
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1397 |
lemma finite_interval_int1 [iff]: "finite {i :: int. a <= i & i <= b}" |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1398 |
proof (cases "a <= b") |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1399 |
case True |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1400 |
from this show ?thesis |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1401 |
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
|
1402 |
case base |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1403 |
have "{i. a <= i & i <= a} = {a}" by auto |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1404 |
from this show ?case by simp |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1405 |
next |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1406 |
case (step b) |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1407 |
from this have "{i. a <= i & i <= b + 1} = {i. a <= i & i <= b} \<union> {b + 1}" by auto |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1408 |
from this step show ?case by simp |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1409 |
qed |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1410 |
next |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1411 |
case False from this show ?thesis |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1412 |
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
|
1413 |
qed |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1414 |
|
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1415 |
lemma finite_interval_int2 [iff]: "finite {i :: int. a <= i & i < b}" |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1416 |
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1417 |
|
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1418 |
lemma finite_interval_int3 [iff]: "finite {i :: int. a < i & i <= b}" |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1419 |
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1420 |
|
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1421 |
lemma finite_interval_int4 [iff]: "finite {i :: int. a < i & i < b}" |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1422 |
by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto |
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1423 |
|
faf62905cd53
adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents:
46027
diff
changeset
|
1424 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1425 |
subsection {* Configuration of the code generator *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1426 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1427 |
text {* Constructors *} |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1428 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1429 |
definition Pos :: "num \<Rightarrow> int" where |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1430 |
[simp, code_abbrev]: "Pos = numeral" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1431 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1432 |
definition Neg :: "num \<Rightarrow> int" where |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1433 |
[simp, code_abbrev]: "Neg n = - (Pos n)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1434 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1435 |
code_datatype "0::int" Pos Neg |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1436 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1437 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
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diff
changeset
|
1438 |
text {* Auxiliary operations *} |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1439 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1440 |
definition dup :: "int \<Rightarrow> int" where |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1441 |
[simp]: "dup k = k + k" |
26507 | 1442 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1443 |
lemma dup_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1444 |
"dup 0 = 0" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1445 |
"dup (Pos n) = Pos (Num.Bit0 n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1446 |
"dup (Neg n) = Neg (Num.Bit0 n)" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1447 |
unfolding Pos_def Neg_def |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1448 |
by (simp_all add: numeral_Bit0) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1449 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1450 |
definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1451 |
[simp]: "sub m n = numeral m - numeral n" |
26507 | 1452 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1453 |
lemma sub_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1454 |
"sub Num.One Num.One = 0" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1455 |
"sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1456 |
"sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1457 |
"sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1458 |
"sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1459 |
"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
|
1460 |
"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
|
1461 |
"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
|
1462 |
"sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1463 |
apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54223
diff
changeset
|
1464 |
apply (simp_all only: algebra_simps minus_diff_eq) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54223
diff
changeset
|
1465 |
apply (simp_all only: add.commute [of _ "- (numeral n + numeral n)"]) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54223
diff
changeset
|
1466 |
apply (simp_all only: minus_add add.assoc left_minus) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54223
diff
changeset
|
1467 |
done |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1468 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1469 |
text {* Implementations *} |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1470 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1471 |
lemma one_int_code [code, code_unfold]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1472 |
"1 = Pos Num.One" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1473 |
by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1474 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1475 |
lemma plus_int_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1476 |
"k + 0 = (k::int)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1477 |
"0 + l = (l::int)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1478 |
"Pos m + Pos n = Pos (m + n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1479 |
"Pos m + Neg n = sub m n" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1480 |
"Neg m + Pos n = sub n m" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1481 |
"Neg m + Neg n = Neg (m + n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1482 |
by simp_all |
26507 | 1483 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1484 |
lemma uminus_int_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1485 |
"uminus 0 = (0::int)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1486 |
"uminus (Pos m) = Neg m" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1487 |
"uminus (Neg m) = Pos m" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1488 |
by simp_all |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1489 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1490 |
lemma minus_int_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1491 |
"k - 0 = (k::int)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1492 |
"0 - l = uminus (l::int)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1493 |
"Pos m - Pos n = sub m n" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1494 |
"Pos m - Neg n = Pos (m + n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1495 |
"Neg m - Pos n = Neg (m + n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1496 |
"Neg m - Neg n = sub n m" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1497 |
by simp_all |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1498 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1499 |
lemma times_int_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1500 |
"k * 0 = (0::int)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1501 |
"0 * l = (0::int)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1502 |
"Pos m * Pos n = Pos (m * n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1503 |
"Pos m * Neg n = Neg (m * n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1504 |
"Neg m * Pos n = Neg (m * n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1505 |
"Neg m * Neg n = Pos (m * n)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1506 |
by simp_all |
26507 | 1507 |
|
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
37887
diff
changeset
|
1508 |
instantiation int :: equal |
26507 | 1509 |
begin |
1510 |
||
37767 | 1511 |
definition |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1512 |
"HOL.equal k l \<longleftrightarrow> k = (l::int)" |
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
37887
diff
changeset
|
1513 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1514 |
instance by default (rule equal_int_def) |
26507 | 1515 |
|
1516 |
end |
|
1517 |
||
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1518 |
lemma equal_int_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1519 |
"HOL.equal 0 (0::int) \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1520 |
"HOL.equal 0 (Pos l) \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1521 |
"HOL.equal 0 (Neg l) \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1522 |
"HOL.equal (Pos k) 0 \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1523 |
"HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1524 |
"HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1525 |
"HOL.equal (Neg k) 0 \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1526 |
"HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1527 |
"HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1528 |
by (auto simp add: equal) |
26507 | 1529 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1530 |
lemma equal_int_refl [code nbe]: |
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
37887
diff
changeset
|
1531 |
"HOL.equal (k::int) k \<longleftrightarrow> True" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1532 |
by (fact equal_refl) |
26507 | 1533 |
|
28562 | 1534 |
lemma less_eq_int_code [code]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1535 |
"0 \<le> (0::int) \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1536 |
"0 \<le> Pos l \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1537 |
"0 \<le> Neg l \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1538 |
"Pos k \<le> 0 \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1539 |
"Pos k \<le> Pos l \<longleftrightarrow> k \<le> l" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1540 |
"Pos k \<le> Neg l \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1541 |
"Neg k \<le> 0 \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1542 |
"Neg k \<le> Pos l \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1543 |
"Neg k \<le> Neg l \<longleftrightarrow> l \<le> k" |
28958 | 1544 |
by simp_all |
26507 | 1545 |
|
28562 | 1546 |
lemma less_int_code [code]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1547 |
"0 < (0::int) \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1548 |
"0 < Pos l \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1549 |
"0 < Neg l \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1550 |
"Pos k < 0 \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1551 |
"Pos k < Pos l \<longleftrightarrow> k < l" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1552 |
"Pos k < Neg l \<longleftrightarrow> False" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1553 |
"Neg k < 0 \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1554 |
"Neg k < Pos l \<longleftrightarrow> True" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1555 |
"Neg k < Neg l \<longleftrightarrow> l < k" |
28958 | 1556 |
by simp_all |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1557 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1558 |
lemma nat_code [code]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1559 |
"nat (Int.Neg k) = 0" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1560 |
"nat 0 = 0" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1561 |
"nat (Int.Pos k) = nat_of_num k" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1562 |
by (simp_all add: nat_of_num_numeral) |
25928 | 1563 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1564 |
lemma (in ring_1) of_int_code [code]: |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1565 |
"of_int (Int.Neg k) = - numeral k" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1566 |
"of_int 0 = 0" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1567 |
"of_int (Int.Pos k) = numeral k" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1568 |
by simp_all |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1569 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1570 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1571 |
text {* Serializer setup *} |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1572 |
|
52435
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
51994
diff
changeset
|
1573 |
code_identifier |
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
51994
diff
changeset
|
1574 |
code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1575 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1576 |
quickcheck_params [default_type = int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1577 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46756
diff
changeset
|
1578 |
hide_const (open) Pos Neg sub dup |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1579 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1580 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1581 |
subsection {* Legacy theorems *} |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1582 |
|
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1583 |
lemmas inj_int = inj_of_nat [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1584 |
lemmas zadd_int = of_nat_add [where 'a=int, symmetric] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1585 |
lemmas int_mult = of_nat_mult [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1586 |
lemmas zmult_int = of_nat_mult [where 'a=int, symmetric] |
45607 | 1587 |
lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n"] for n |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1588 |
lemmas zless_int = of_nat_less_iff [where 'a=int] |
45607 | 1589 |
lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k"] for k |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1590 |
lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1591 |
lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int] |
45607 | 1592 |
lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n"] for n |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1593 |
lemmas int_0 = of_nat_0 [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1594 |
lemmas int_1 = of_nat_1 [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1595 |
lemmas int_Suc = of_nat_Suc [where 'a=int] |
47207
9368aa814518
move lemmas from Nat_Numeral to Int.thy and Num.thy
huffman
parents:
47192
diff
changeset
|
1596 |
lemmas int_numeral = of_nat_numeral [where 'a=int] |
45607 | 1597 |
lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m"] for m |
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1598 |
lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int] |
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1599 |
lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric] |
47255
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47228
diff
changeset
|
1600 |
lemmas zpower_numeral_even = power_numeral_even [where 'a=int] |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47228
diff
changeset
|
1601 |
lemmas zpower_numeral_odd = power_numeral_odd [where 'a=int] |
30960 | 1602 |
|
31015 | 1603 |
lemma zpower_zpower: |
1604 |
"(x ^ y) ^ z = (x ^ (y * z)::int)" |
|
1605 |
by (rule power_mult [symmetric]) |
|
1606 |
||
1607 |
lemma int_power: |
|
1608 |
"int (m ^ n) = int m ^ n" |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54249
diff
changeset
|
1609 |
by (fact of_nat_power) |
31015 | 1610 |
|
1611 |
lemmas zpower_int = int_power [symmetric] |
|
1612 |
||
48045 | 1613 |
text {* De-register @{text "int"} as a quotient type: *} |
1614 |
||
53652
18fbca265e2e
use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents:
53065
diff
changeset
|
1615 |
lifting_update int.lifting |
18fbca265e2e
use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents:
53065
diff
changeset
|
1616 |
lifting_forget int.lifting |
48045 | 1617 |
|
25919
8b1c0d434824
joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents:
diff
changeset
|
1618 |
end |