paulson@3366
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(* Title: HOL/Divides.thy
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paulson@3366
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ID: $Id$
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paulson@3366
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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paulson@6865
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Copyright 1999 University of Cambridge
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huffman@18154
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*)
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paulson@3366
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haftmann@27651
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header {* The division operators div and mod *}
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paulson@3366
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nipkow@15131
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theory Divides
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krauss@26748
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imports Nat Power Product_Type
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haftmann@22993
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uses "~~/src/Provers/Arith/cancel_div_mod.ML"
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nipkow@15131
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begin
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paulson@3366
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haftmann@25942
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subsection {* Syntactic division operations *}
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haftmann@25942
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haftmann@27651
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class div = dvd +
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haftmann@27651
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fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
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haftmann@27651
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and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
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haftmann@21408
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haftmann@25942
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haftmann@27651
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subsection {* Abstract division in commutative semirings. *}
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haftmann@25942
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haftmann@29509
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class semiring_div = comm_semiring_1_cancel + div +
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haftmann@25942
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assumes mod_div_equality: "a div b * b + a mod b = a"
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haftmann@27651
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and div_by_0 [simp]: "a div 0 = 0"
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haftmann@27651
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and div_0 [simp]: "0 div a = 0"
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haftmann@27651
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and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
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haftmann@25942
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begin
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haftmann@25942
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haftmann@26100
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text {* @{const div} and @{const mod} *}
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haftmann@26100
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haftmann@26062
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lemma mod_div_equality2: "b * (a div b) + a mod b = a"
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haftmann@26062
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unfolding mult_commute [of b]
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haftmann@26062
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by (rule mod_div_equality)
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haftmann@26062
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huffman@29400
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lemma mod_div_equality': "a mod b + a div b * b = a"
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huffman@29400
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using mod_div_equality [of a b]
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huffman@29400
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by (simp only: add_ac)
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huffman@29400
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haftmann@26062
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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
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nipkow@29667
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by (simp add: mod_div_equality)
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haftmann@26062
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haftmann@26062
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lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
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nipkow@29667
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by (simp add: mod_div_equality2)
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haftmann@26062
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haftmann@27651
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lemma mod_by_0 [simp]: "a mod 0 = a"
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nipkow@30180
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using mod_div_equality [of a zero] by simp
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haftmann@26100
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haftmann@27651
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lemma mod_0 [simp]: "0 mod a = 0"
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nipkow@30180
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using mod_div_equality [of zero a] div_0 by simp
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haftmann@26062
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haftmann@27651
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lemma div_mult_self2 [simp]:
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haftmann@27651
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assumes "b \<noteq> 0"
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haftmann@27651
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shows "(a + b * c) div b = c + a div b"
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haftmann@27651
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using assms div_mult_self1 [of b a c] by (simp add: mult_commute)
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haftmann@26062
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haftmann@27651
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lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
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haftmann@27651
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proof (cases "b = 0")
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haftmann@27651
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case True then show ?thesis by simp
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haftmann@27651
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next
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haftmann@27651
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case False
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haftmann@27651
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have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
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haftmann@27651
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by (simp add: mod_div_equality)
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haftmann@27651
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also from False div_mult_self1 [of b a c] have
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haftmann@27651
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"\<dots> = (c + a div b) * b + (a + c * b) mod b"
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nipkow@29667
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by (simp add: algebra_simps)
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haftmann@27651
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finally have "a = a div b * b + (a + c * b) mod b"
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haftmann@27651
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by (simp add: add_commute [of a] add_assoc left_distrib)
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haftmann@27651
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then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
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haftmann@27651
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by (simp add: mod_div_equality)
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haftmann@27651
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then show ?thesis by simp
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haftmann@27651
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qed
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haftmann@27651
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haftmann@27651
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lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
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nipkow@29667
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by (simp add: mult_commute [of b])
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haftmann@27651
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haftmann@27651
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lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
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haftmann@27651
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using div_mult_self2 [of b 0 a] by simp
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haftmann@27651
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haftmann@27651
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lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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haftmann@27651
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using div_mult_self1 [of b 0 a] by simp
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haftmann@27651
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haftmann@27651
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lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
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haftmann@27651
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using mod_mult_self2 [of 0 b a] by simp
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haftmann@27651
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haftmann@27651
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lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
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haftmann@27651
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using mod_mult_self1 [of 0 a b] by simp
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haftmann@27651
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haftmann@27651
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lemma div_by_1 [simp]: "a div 1 = a"
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haftmann@27651
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using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
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haftmann@27651
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haftmann@27651
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lemma mod_by_1 [simp]: "a mod 1 = 0"
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haftmann@27651
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proof -
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haftmann@27651
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from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
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haftmann@27651
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then have "a + a mod 1 = a + 0" by simp
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haftmann@27651
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then show ?thesis by (rule add_left_imp_eq)
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haftmann@27651
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qed
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haftmann@27651
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haftmann@27651
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lemma mod_self [simp]: "a mod a = 0"
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haftmann@27651
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using mod_mult_self2_is_0 [of 1] by simp
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haftmann@27651
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haftmann@27651
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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
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haftmann@27651
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using div_mult_self2_is_id [of _ 1] by simp
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haftmann@27651
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haftmann@27676
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lemma div_add_self1 [simp]:
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haftmann@27651
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assumes "b \<noteq> 0"
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haftmann@27651
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shows "(b + a) div b = a div b + 1"
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haftmann@27651
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using assms div_mult_self1 [of b a 1] by (simp add: add_commute)
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haftmann@27651
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haftmann@27676
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lemma div_add_self2 [simp]:
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haftmann@27651
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assumes "b \<noteq> 0"
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haftmann@27651
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shows "(a + b) div b = a div b + 1"
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haftmann@27651
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using assms div_add_self1 [of b a] by (simp add: add_commute)
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haftmann@27651
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haftmann@27676
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lemma mod_add_self1 [simp]:
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haftmann@27651
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"(b + a) mod b = a mod b"
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haftmann@27651
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using mod_mult_self1 [of a 1 b] by (simp add: add_commute)
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haftmann@27651
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haftmann@27676
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lemma mod_add_self2 [simp]:
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haftmann@27651
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"(a + b) mod b = a mod b"
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haftmann@27651
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using mod_mult_self1 [of a 1 b] by simp
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haftmann@27651
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haftmann@27651
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lemma mod_div_decomp:
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haftmann@27651
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fixes a b
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haftmann@27651
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obtains q r where "q = a div b" and "r = a mod b"
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haftmann@27651
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and "a = q * b + r"
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haftmann@27651
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proof -
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haftmann@27651
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from mod_div_equality have "a = a div b * b + a mod b" by simp
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haftmann@27651
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moreover have "a div b = a div b" ..
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haftmann@27651
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moreover have "a mod b = a mod b" ..
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haftmann@27651
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note that ultimately show thesis by blast
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haftmann@27651
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qed
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haftmann@27651
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nipkow@29108
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lemma dvd_eq_mod_eq_0 [code unfold]: "a dvd b \<longleftrightarrow> b mod a = 0"
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haftmann@25942
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proof
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haftmann@25942
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assume "b mod a = 0"
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haftmann@25942
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with mod_div_equality [of b a] have "b div a * a = b" by simp
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haftmann@25942
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then have "b = a * (b div a)" unfolding mult_commute ..
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haftmann@25942
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then have "\<exists>c. b = a * c" ..
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haftmann@25942
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then show "a dvd b" unfolding dvd_def .
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haftmann@25942
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next
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haftmann@25942
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assume "a dvd b"
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haftmann@25942
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then have "\<exists>c. b = a * c" unfolding dvd_def .
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haftmann@25942
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then obtain c where "b = a * c" ..
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haftmann@25942
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then have "b mod a = a * c mod a" by simp
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haftmann@25942
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then have "b mod a = c * a mod a" by (simp add: mult_commute)
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haftmann@27651
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then show "b mod a = 0" by simp
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haftmann@25942
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qed
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haftmann@25942
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huffman@29400
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lemma mod_div_trivial [simp]: "a mod b div b = 0"
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huffman@29400
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proof (cases "b = 0")
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huffman@29400
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assume "b = 0"
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huffman@29400
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thus ?thesis by simp
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huffman@29400
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next
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huffman@29400
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assume "b \<noteq> 0"
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huffman@29400
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hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
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huffman@29400
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by (rule div_mult_self1 [symmetric])
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huffman@29400
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also have "\<dots> = a div b"
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huffman@29400
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by (simp only: mod_div_equality')
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huffman@29400
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also have "\<dots> = a div b + 0"
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huffman@29400
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by simp
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huffman@29400
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finally show ?thesis
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huffman@29400
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by (rule add_left_imp_eq)
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huffman@29400
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qed
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huffman@29400
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huffman@29400
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lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"
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huffman@29400
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proof -
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huffman@29400
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have "a mod b mod b = (a mod b + a div b * b) mod b"
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huffman@29400
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by (simp only: mod_mult_self1)
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huffman@29400
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also have "\<dots> = a mod b"
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huffman@29400
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by (simp only: mod_div_equality')
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huffman@29400
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finally show ?thesis .
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huffman@29400
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qed
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huffman@29400
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nipkow@29862
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lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"
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nipkow@29885
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by (rule dvd_eq_mod_eq_0[THEN iffD1])
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nipkow@29862
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nipkow@29862
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lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
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nipkow@29862
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by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)
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nipkow@29862
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nipkow@29989
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lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"
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nipkow@29989
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apply (cases "a = 0")
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nipkow@29989
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apply simp
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nipkow@29989
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apply (auto simp: dvd_def mult_assoc)
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nipkow@29989
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done
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nipkow@29989
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nipkow@29862
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lemma div_dvd_div[simp]:
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nipkow@29862
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"a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"
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nipkow@29862
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apply (cases "a = 0")
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nipkow@29862
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apply simp
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nipkow@29862
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apply (unfold dvd_def)
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nipkow@29862
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apply auto
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nipkow@29862
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apply(blast intro:mult_assoc[symmetric])
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nipkow@29862
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apply(fastsimp simp add: mult_assoc)
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nipkow@29862
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done
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nipkow@29862
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huffman@30015
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lemma dvd_mod_imp_dvd: "[| k dvd m mod n; k dvd n |] ==> k dvd m"
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huffman@30015
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apply (subgoal_tac "k dvd (m div n) *n + m mod n")
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huffman@30015
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apply (simp add: mod_div_equality)
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huffman@30015
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apply (simp only: dvd_add dvd_mult)
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huffman@30015
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201 |
done
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huffman@30015
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huffman@29400
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text {* Addition respects modular equivalence. *}
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huffman@29400
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huffman@29400
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lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"
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huffman@29400
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proof -
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huffman@29400
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207 |
have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
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huffman@29400
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by (simp only: mod_div_equality)
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huffman@29400
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209 |
also have "\<dots> = (a mod c + b + a div c * c) mod c"
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huffman@29400
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210 |
by (simp only: add_ac)
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huffman@29400
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211 |
also have "\<dots> = (a mod c + b) mod c"
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huffman@29400
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212 |
by (rule mod_mult_self1)
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huffman@29400
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213 |
finally show ?thesis .
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huffman@29400
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214 |
qed
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huffman@29400
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215 |
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huffman@29400
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216 |
lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"
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huffman@29400
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217 |
proof -
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huffman@29400
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218 |
have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
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huffman@29400
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219 |
by (simp only: mod_div_equality)
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huffman@29400
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220 |
also have "\<dots> = (a + b mod c + b div c * c) mod c"
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huffman@29400
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221 |
by (simp only: add_ac)
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huffman@29400
|
222 |
also have "\<dots> = (a + b mod c) mod c"
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huffman@29400
|
223 |
by (rule mod_mult_self1)
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huffman@29400
|
224 |
finally show ?thesis .
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huffman@29400
|
225 |
qed
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huffman@29400
|
226 |
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huffman@29400
|
227 |
lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"
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huffman@29400
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228 |
by (rule trans [OF mod_add_left_eq mod_add_right_eq])
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huffman@29400
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229 |
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huffman@29400
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230 |
lemma mod_add_cong:
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huffman@29400
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231 |
assumes "a mod c = a' mod c"
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huffman@29400
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232 |
assumes "b mod c = b' mod c"
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huffman@29400
|
233 |
shows "(a + b) mod c = (a' + b') mod c"
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huffman@29400
|
234 |
proof -
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huffman@29400
|
235 |
have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
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huffman@29400
|
236 |
unfolding assms ..
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huffman@29400
|
237 |
thus ?thesis
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huffman@29400
|
238 |
by (simp only: mod_add_eq [symmetric])
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huffman@29400
|
239 |
qed
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huffman@29400
|
240 |
|
nipkow@30837
|
241 |
lemma div_add[simp]: "z dvd x \<Longrightarrow> z dvd y
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nipkow@30837
|
242 |
\<Longrightarrow> (x + y) div z = x div z + y div z"
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nipkow@30837
|
243 |
by(cases "z=0", simp, unfold dvd_def, auto simp add: algebra_simps)
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nipkow@30837
|
244 |
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huffman@29400
|
245 |
text {* Multiplication respects modular equivalence. *}
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huffman@29400
|
246 |
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huffman@29400
|
247 |
lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"
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huffman@29400
|
248 |
proof -
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huffman@29400
|
249 |
have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
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huffman@29400
|
250 |
by (simp only: mod_div_equality)
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huffman@29400
|
251 |
also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
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nipkow@29667
|
252 |
by (simp only: algebra_simps)
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huffman@29400
|
253 |
also have "\<dots> = (a mod c * b) mod c"
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huffman@29400
|
254 |
by (rule mod_mult_self1)
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huffman@29400
|
255 |
finally show ?thesis .
|
huffman@29400
|
256 |
qed
|
huffman@29400
|
257 |
|
huffman@29400
|
258 |
lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"
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huffman@29400
|
259 |
proof -
|
huffman@29400
|
260 |
have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
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huffman@29400
|
261 |
by (simp only: mod_div_equality)
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huffman@29400
|
262 |
also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
|
nipkow@29667
|
263 |
by (simp only: algebra_simps)
|
huffman@29400
|
264 |
also have "\<dots> = (a * (b mod c)) mod c"
|
huffman@29400
|
265 |
by (rule mod_mult_self1)
|
huffman@29400
|
266 |
finally show ?thesis .
|
huffman@29400
|
267 |
qed
|
huffman@29400
|
268 |
|
huffman@29400
|
269 |
lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"
|
huffman@29400
|
270 |
by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])
|
huffman@29400
|
271 |
|
huffman@29400
|
272 |
lemma mod_mult_cong:
|
huffman@29400
|
273 |
assumes "a mod c = a' mod c"
|
huffman@29400
|
274 |
assumes "b mod c = b' mod c"
|
huffman@29400
|
275 |
shows "(a * b) mod c = (a' * b') mod c"
|
huffman@29400
|
276 |
proof -
|
huffman@29400
|
277 |
have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
|
huffman@29400
|
278 |
unfolding assms ..
|
huffman@29400
|
279 |
thus ?thesis
|
huffman@29400
|
280 |
by (simp only: mod_mult_eq [symmetric])
|
huffman@29400
|
281 |
qed
|
huffman@29400
|
282 |
|
huffman@29401
|
283 |
lemma mod_mod_cancel:
|
huffman@29401
|
284 |
assumes "c dvd b"
|
huffman@29401
|
285 |
shows "a mod b mod c = a mod c"
|
huffman@29401
|
286 |
proof -
|
huffman@29401
|
287 |
from `c dvd b` obtain k where "b = c * k"
|
huffman@29401
|
288 |
by (rule dvdE)
|
huffman@29401
|
289 |
have "a mod b mod c = a mod (c * k) mod c"
|
huffman@29401
|
290 |
by (simp only: `b = c * k`)
|
huffman@29401
|
291 |
also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
|
huffman@29401
|
292 |
by (simp only: mod_mult_self1)
|
huffman@29401
|
293 |
also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
|
huffman@29401
|
294 |
by (simp only: add_ac mult_ac)
|
huffman@29401
|
295 |
also have "\<dots> = a mod c"
|
huffman@29401
|
296 |
by (simp only: mod_div_equality)
|
huffman@29401
|
297 |
finally show ?thesis .
|
huffman@29401
|
298 |
qed
|
huffman@29401
|
299 |
|
haftmann@25942
|
300 |
end
|
haftmann@25942
|
301 |
|
nipkow@30472
|
302 |
lemma div_mult_div_if_dvd: "(y::'a::{semiring_div,no_zero_divisors}) dvd x \<Longrightarrow>
|
nipkow@30472
|
303 |
z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
|
nipkow@30472
|
304 |
unfolding dvd_def
|
nipkow@30472
|
305 |
apply clarify
|
nipkow@30472
|
306 |
apply (case_tac "y = 0")
|
nipkow@30472
|
307 |
apply simp
|
nipkow@30472
|
308 |
apply (case_tac "z = 0")
|
nipkow@30472
|
309 |
apply simp
|
nipkow@30472
|
310 |
apply (simp add: algebra_simps)
|
nipkow@30472
|
311 |
apply (subst mult_assoc [symmetric])
|
nipkow@30472
|
312 |
apply (simp add: no_zero_divisors)
|
nipkow@30472
|
313 |
done
|
nipkow@30472
|
314 |
|
nipkow@30472
|
315 |
|
nipkow@30472
|
316 |
lemma div_power: "(y::'a::{semiring_div,no_zero_divisors,recpower}) dvd x \<Longrightarrow>
|
nipkow@30472
|
317 |
(x div y)^n = x^n div y^n"
|
nipkow@30472
|
318 |
apply (induct n)
|
nipkow@30472
|
319 |
apply simp
|
nipkow@30472
|
320 |
apply(simp add: div_mult_div_if_dvd dvd_power_same)
|
nipkow@30472
|
321 |
done
|
nipkow@30472
|
322 |
|
huffman@29402
|
323 |
class ring_div = semiring_div + comm_ring_1
|
huffman@29402
|
324 |
begin
|
huffman@29402
|
325 |
|
huffman@29402
|
326 |
text {* Negation respects modular equivalence. *}
|
huffman@29402
|
327 |
|
huffman@29402
|
328 |
lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
|
huffman@29402
|
329 |
proof -
|
huffman@29402
|
330 |
have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
|
huffman@29402
|
331 |
by (simp only: mod_div_equality)
|
huffman@29402
|
332 |
also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
|
huffman@29402
|
333 |
by (simp only: minus_add_distrib minus_mult_left add_ac)
|
huffman@29402
|
334 |
also have "\<dots> = (- (a mod b)) mod b"
|
huffman@29402
|
335 |
by (rule mod_mult_self1)
|
huffman@29402
|
336 |
finally show ?thesis .
|
huffman@29402
|
337 |
qed
|
huffman@29402
|
338 |
|
huffman@29402
|
339 |
lemma mod_minus_cong:
|
huffman@29402
|
340 |
assumes "a mod b = a' mod b"
|
huffman@29402
|
341 |
shows "(- a) mod b = (- a') mod b"
|
huffman@29402
|
342 |
proof -
|
huffman@29402
|
343 |
have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
|
huffman@29402
|
344 |
unfolding assms ..
|
huffman@29402
|
345 |
thus ?thesis
|
huffman@29402
|
346 |
by (simp only: mod_minus_eq [symmetric])
|
huffman@29402
|
347 |
qed
|
huffman@29402
|
348 |
|
huffman@29402
|
349 |
text {* Subtraction respects modular equivalence. *}
|
huffman@29402
|
350 |
|
huffman@29402
|
351 |
lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
|
huffman@29402
|
352 |
unfolding diff_minus
|
huffman@29402
|
353 |
by (intro mod_add_cong mod_minus_cong) simp_all
|
huffman@29402
|
354 |
|
huffman@29402
|
355 |
lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
|
huffman@29402
|
356 |
unfolding diff_minus
|
huffman@29402
|
357 |
by (intro mod_add_cong mod_minus_cong) simp_all
|
huffman@29402
|
358 |
|
huffman@29402
|
359 |
lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
|
huffman@29402
|
360 |
unfolding diff_minus
|
huffman@29402
|
361 |
by (intro mod_add_cong mod_minus_cong) simp_all
|
huffman@29402
|
362 |
|
huffman@29402
|
363 |
lemma mod_diff_cong:
|
huffman@29402
|
364 |
assumes "a mod c = a' mod c"
|
huffman@29402
|
365 |
assumes "b mod c = b' mod c"
|
huffman@29402
|
366 |
shows "(a - b) mod c = (a' - b') mod c"
|
huffman@29402
|
367 |
unfolding diff_minus using assms
|
huffman@29402
|
368 |
by (intro mod_add_cong mod_minus_cong)
|
huffman@29402
|
369 |
|
nipkow@30180
|
370 |
lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
|
nipkow@30180
|
371 |
apply (case_tac "y = 0") apply simp
|
nipkow@30180
|
372 |
apply (auto simp add: dvd_def)
|
nipkow@30180
|
373 |
apply (subgoal_tac "-(y * k) = y * - k")
|
nipkow@30180
|
374 |
apply (erule ssubst)
|
nipkow@30180
|
375 |
apply (erule div_mult_self1_is_id)
|
nipkow@30180
|
376 |
apply simp
|
nipkow@30180
|
377 |
done
|
nipkow@30180
|
378 |
|
nipkow@30180
|
379 |
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
|
nipkow@30180
|
380 |
apply (case_tac "y = 0") apply simp
|
nipkow@30180
|
381 |
apply (auto simp add: dvd_def)
|
nipkow@30180
|
382 |
apply (subgoal_tac "y * k = -y * -k")
|
nipkow@30180
|
383 |
apply (erule ssubst)
|
nipkow@30180
|
384 |
apply (rule div_mult_self1_is_id)
|
nipkow@30180
|
385 |
apply simp
|
nipkow@30180
|
386 |
apply simp
|
nipkow@30180
|
387 |
done
|
nipkow@30180
|
388 |
|
huffman@29402
|
389 |
end
|
huffman@29402
|
390 |
|
haftmann@25942
|
391 |
|
haftmann@26100
|
392 |
subsection {* Division on @{typ nat} *}
|
haftmann@26100
|
393 |
|
haftmann@26100
|
394 |
text {*
|
haftmann@26100
|
395 |
We define @{const div} and @{const mod} on @{typ nat} by means
|
haftmann@26100
|
396 |
of a characteristic relation with two input arguments
|
haftmann@26100
|
397 |
@{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
|
haftmann@26100
|
398 |
@{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
|
haftmann@26100
|
399 |
*}
|
haftmann@26100
|
400 |
|
haftmann@26100
|
401 |
definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
|
haftmann@26100
|
402 |
"divmod_rel m n q r \<longleftrightarrow> m = q * n + r \<and> (if n > 0 then 0 \<le> r \<and> r < n else q = 0)"
|
haftmann@26100
|
403 |
|
haftmann@26100
|
404 |
text {* @{const divmod_rel} is total: *}
|
haftmann@26100
|
405 |
|
haftmann@26100
|
406 |
lemma divmod_rel_ex:
|
haftmann@26100
|
407 |
obtains q r where "divmod_rel m n q r"
|
haftmann@26100
|
408 |
proof (cases "n = 0")
|
haftmann@26100
|
409 |
case True with that show thesis
|
haftmann@26100
|
410 |
by (auto simp add: divmod_rel_def)
|
haftmann@26100
|
411 |
next
|
haftmann@26100
|
412 |
case False
|
haftmann@26100
|
413 |
have "\<exists>q r. m = q * n + r \<and> r < n"
|
haftmann@26100
|
414 |
proof (induct m)
|
haftmann@26100
|
415 |
case 0 with `n \<noteq> 0`
|
haftmann@26100
|
416 |
have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
|
haftmann@26100
|
417 |
then show ?case by blast
|
haftmann@26100
|
418 |
next
|
haftmann@26100
|
419 |
case (Suc m) then obtain q' r'
|
haftmann@26100
|
420 |
where m: "m = q' * n + r'" and n: "r' < n" by auto
|
haftmann@26100
|
421 |
then show ?case proof (cases "Suc r' < n")
|
haftmann@26100
|
422 |
case True
|
haftmann@26100
|
423 |
from m n have "Suc m = q' * n + Suc r'" by simp
|
haftmann@26100
|
424 |
with True show ?thesis by blast
|
haftmann@26100
|
425 |
next
|
haftmann@26100
|
426 |
case False then have "n \<le> Suc r'" by auto
|
haftmann@26100
|
427 |
moreover from n have "Suc r' \<le> n" by auto
|
haftmann@26100
|
428 |
ultimately have "n = Suc r'" by auto
|
haftmann@26100
|
429 |
with m have "Suc m = Suc q' * n + 0" by simp
|
haftmann@26100
|
430 |
with `n \<noteq> 0` show ?thesis by blast
|
haftmann@26100
|
431 |
qed
|
haftmann@26100
|
432 |
qed
|
haftmann@26100
|
433 |
with that show thesis
|
haftmann@26100
|
434 |
using `n \<noteq> 0` by (auto simp add: divmod_rel_def)
|
haftmann@26100
|
435 |
qed
|
haftmann@26100
|
436 |
|
haftmann@26100
|
437 |
text {* @{const divmod_rel} is injective: *}
|
haftmann@26100
|
438 |
|
haftmann@26100
|
439 |
lemma divmod_rel_unique_div:
|
haftmann@26100
|
440 |
assumes "divmod_rel m n q r"
|
haftmann@26100
|
441 |
and "divmod_rel m n q' r'"
|
haftmann@26100
|
442 |
shows "q = q'"
|
haftmann@26100
|
443 |
proof (cases "n = 0")
|
haftmann@26100
|
444 |
case True with assms show ?thesis
|
haftmann@26100
|
445 |
by (simp add: divmod_rel_def)
|
haftmann@26100
|
446 |
next
|
haftmann@26100
|
447 |
case False
|
haftmann@26100
|
448 |
have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
|
haftmann@26100
|
449 |
apply (rule leI)
|
haftmann@26100
|
450 |
apply (subst less_iff_Suc_add)
|
haftmann@26100
|
451 |
apply (auto simp add: add_mult_distrib)
|
haftmann@26100
|
452 |
done
|
haftmann@26100
|
453 |
from `n \<noteq> 0` assms show ?thesis
|
haftmann@26100
|
454 |
by (auto simp add: divmod_rel_def
|
haftmann@26100
|
455 |
intro: order_antisym dest: aux sym)
|
haftmann@26100
|
456 |
qed
|
haftmann@26100
|
457 |
|
haftmann@26100
|
458 |
lemma divmod_rel_unique_mod:
|
haftmann@26100
|
459 |
assumes "divmod_rel m n q r"
|
haftmann@26100
|
460 |
and "divmod_rel m n q' r'"
|
haftmann@26100
|
461 |
shows "r = r'"
|
haftmann@26100
|
462 |
proof -
|
haftmann@26100
|
463 |
from assms have "q = q'" by (rule divmod_rel_unique_div)
|
haftmann@26100
|
464 |
with assms show ?thesis by (simp add: divmod_rel_def)
|
haftmann@26100
|
465 |
qed
|
haftmann@26100
|
466 |
|
haftmann@26100
|
467 |
text {*
|
haftmann@26100
|
468 |
We instantiate divisibility on the natural numbers by
|
haftmann@26100
|
469 |
means of @{const divmod_rel}:
|
haftmann@26100
|
470 |
*}
|
haftmann@25942
|
471 |
|
haftmann@25942
|
472 |
instantiation nat :: semiring_div
|
haftmann@25571
|
473 |
begin
|
haftmann@25571
|
474 |
|
haftmann@26100
|
475 |
definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
|
haftmann@28562
|
476 |
[code del]: "divmod m n = (THE (q, r). divmod_rel m n q r)"
|
haftmann@25571
|
477 |
|
haftmann@26100
|
478 |
definition div_nat where
|
haftmann@26100
|
479 |
"m div n = fst (divmod m n)"
|
haftmann@25942
|
480 |
|
haftmann@26100
|
481 |
definition mod_nat where
|
haftmann@26100
|
482 |
"m mod n = snd (divmod m n)"
|
haftmann@25571
|
483 |
|
haftmann@26100
|
484 |
lemma divmod_div_mod:
|
haftmann@26100
|
485 |
"divmod m n = (m div n, m mod n)"
|
haftmann@26100
|
486 |
unfolding div_nat_def mod_nat_def by simp
|
paulson@14267
|
487 |
|
haftmann@26100
|
488 |
lemma divmod_eq:
|
haftmann@26100
|
489 |
assumes "divmod_rel m n q r"
|
haftmann@26100
|
490 |
shows "divmod m n = (q, r)"
|
haftmann@26100
|
491 |
using assms by (auto simp add: divmod_def
|
haftmann@26100
|
492 |
dest: divmod_rel_unique_div divmod_rel_unique_mod)
|
paulson@14267
|
493 |
|
haftmann@26100
|
494 |
lemma div_eq:
|
haftmann@26100
|
495 |
assumes "divmod_rel m n q r"
|
haftmann@26100
|
496 |
shows "m div n = q"
|
haftmann@26100
|
497 |
using assms by (auto dest: divmod_eq simp add: div_nat_def)
|
paulson@14267
|
498 |
|
haftmann@26100
|
499 |
lemma mod_eq:
|
haftmann@26100
|
500 |
assumes "divmod_rel m n q r"
|
haftmann@26100
|
501 |
shows "m mod n = r"
|
haftmann@26100
|
502 |
using assms by (auto dest: divmod_eq simp add: mod_nat_def)
|
paulson@14267
|
503 |
|
haftmann@26100
|
504 |
lemma divmod_rel: "divmod_rel m n (m div n) (m mod n)"
|
haftmann@26100
|
505 |
proof -
|
haftmann@26100
|
506 |
from divmod_rel_ex
|
haftmann@26100
|
507 |
obtain q r where rel: "divmod_rel m n q r" .
|
haftmann@26100
|
508 |
moreover with div_eq mod_eq have "m div n = q" and "m mod n = r"
|
haftmann@26100
|
509 |
by simp_all
|
haftmann@26100
|
510 |
ultimately show ?thesis by simp
|
haftmann@26100
|
511 |
qed
|
paulson@14267
|
512 |
|
haftmann@26100
|
513 |
lemma divmod_zero:
|
haftmann@26100
|
514 |
"divmod m 0 = (0, m)"
|
haftmann@26100
|
515 |
proof -
|
haftmann@26100
|
516 |
from divmod_rel [of m 0] show ?thesis
|
haftmann@26100
|
517 |
unfolding divmod_div_mod divmod_rel_def by simp
|
haftmann@26100
|
518 |
qed
|
haftmann@25942
|
519 |
|
haftmann@26100
|
520 |
lemma divmod_base:
|
haftmann@26100
|
521 |
assumes "m < n"
|
haftmann@26100
|
522 |
shows "divmod m n = (0, m)"
|
haftmann@26100
|
523 |
proof -
|
haftmann@26100
|
524 |
from divmod_rel [of m n] show ?thesis
|
haftmann@26100
|
525 |
unfolding divmod_div_mod divmod_rel_def
|
haftmann@26100
|
526 |
using assms by (cases "m div n = 0")
|
haftmann@26100
|
527 |
(auto simp add: gr0_conv_Suc [of "m div n"])
|
haftmann@26100
|
528 |
qed
|
haftmann@25942
|
529 |
|
haftmann@26100
|
530 |
lemma divmod_step:
|
haftmann@26100
|
531 |
assumes "0 < n" and "n \<le> m"
|
haftmann@26100
|
532 |
shows "divmod m n = (Suc ((m - n) div n), (m - n) mod n)"
|
haftmann@26100
|
533 |
proof -
|
haftmann@26100
|
534 |
from divmod_rel have divmod_m_n: "divmod_rel m n (m div n) (m mod n)" .
|
haftmann@26100
|
535 |
with assms have m_div_n: "m div n \<ge> 1"
|
haftmann@26100
|
536 |
by (cases "m div n") (auto simp add: divmod_rel_def)
|
huffman@30016
|
537 |
from assms divmod_m_n have "divmod_rel (m - n) n (m div n - Suc 0) (m mod n)"
|
haftmann@26100
|
538 |
by (cases "m div n") (auto simp add: divmod_rel_def)
|
huffman@30016
|
539 |
with divmod_eq have "divmod (m - n) n = (m div n - Suc 0, m mod n)" by simp
|
haftmann@26100
|
540 |
moreover from divmod_div_mod have "divmod (m - n) n = ((m - n) div n, (m - n) mod n)" .
|
haftmann@26100
|
541 |
ultimately have "m div n = Suc ((m - n) div n)"
|
haftmann@26100
|
542 |
and "m mod n = (m - n) mod n" using m_div_n by simp_all
|
haftmann@26100
|
543 |
then show ?thesis using divmod_div_mod by simp
|
haftmann@26100
|
544 |
qed
|
haftmann@26100
|
545 |
|
wenzelm@26300
|
546 |
text {* The ''recursion'' equations for @{const div} and @{const mod} *}
|
haftmann@26100
|
547 |
|
haftmann@26100
|
548 |
lemma div_less [simp]:
|
haftmann@26100
|
549 |
fixes m n :: nat
|
haftmann@26100
|
550 |
assumes "m < n"
|
haftmann@26100
|
551 |
shows "m div n = 0"
|
haftmann@26100
|
552 |
using assms divmod_base divmod_div_mod by simp
|
haftmann@26100
|
553 |
|
haftmann@26100
|
554 |
lemma le_div_geq:
|
haftmann@26100
|
555 |
fixes m n :: nat
|
haftmann@26100
|
556 |
assumes "0 < n" and "n \<le> m"
|
haftmann@26100
|
557 |
shows "m div n = Suc ((m - n) div n)"
|
haftmann@26100
|
558 |
using assms divmod_step divmod_div_mod by simp
|
haftmann@26100
|
559 |
|
haftmann@26100
|
560 |
lemma mod_less [simp]:
|
haftmann@26100
|
561 |
fixes m n :: nat
|
haftmann@26100
|
562 |
assumes "m < n"
|
haftmann@26100
|
563 |
shows "m mod n = m"
|
haftmann@26100
|
564 |
using assms divmod_base divmod_div_mod by simp
|
haftmann@26100
|
565 |
|
haftmann@26100
|
566 |
lemma le_mod_geq:
|
haftmann@26100
|
567 |
fixes m n :: nat
|
haftmann@26100
|
568 |
assumes "n \<le> m"
|
haftmann@26100
|
569 |
shows "m mod n = (m - n) mod n"
|
haftmann@26100
|
570 |
using assms divmod_step divmod_div_mod by (cases "n = 0") simp_all
|
haftmann@25942
|
571 |
|
haftmann@25942
|
572 |
instance proof
|
haftmann@26100
|
573 |
fix m n :: nat show "m div n * n + m mod n = m"
|
haftmann@26100
|
574 |
using divmod_rel [of m n] by (simp add: divmod_rel_def)
|
haftmann@25942
|
575 |
next
|
haftmann@26100
|
576 |
fix n :: nat show "n div 0 = 0"
|
haftmann@26100
|
577 |
using divmod_zero divmod_div_mod [of n 0] by simp
|
haftmann@25942
|
578 |
next
|
haftmann@27651
|
579 |
fix n :: nat show "0 div n = 0"
|
haftmann@27651
|
580 |
using divmod_rel [of 0 n] by (cases n) (simp_all add: divmod_rel_def)
|
haftmann@27651
|
581 |
next
|
haftmann@27651
|
582 |
fix m n q :: nat assume "n \<noteq> 0" then show "(q + m * n) div n = m + q div n"
|
haftmann@25942
|
583 |
by (induct m) (simp_all add: le_div_geq)
|
haftmann@25942
|
584 |
qed
|
haftmann@26100
|
585 |
|
haftmann@25942
|
586 |
end
|
haftmann@25942
|
587 |
|
haftmann@26100
|
588 |
text {* Simproc for cancelling @{const div} and @{const mod} *}
|
haftmann@25942
|
589 |
|
haftmann@27651
|
590 |
(*lemmas mod_div_equality_nat = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard]
|
haftmann@27651
|
591 |
lemmas mod_div_equality2_nat = mod_div_equality2 [of "n\<Colon>nat" m, standard*)
|
haftmann@25942
|
592 |
|
haftmann@25942
|
593 |
ML {*
|
haftmann@25942
|
594 |
structure CancelDivModData =
|
haftmann@25942
|
595 |
struct
|
haftmann@25942
|
596 |
|
haftmann@26100
|
597 |
val div_name = @{const_name div};
|
haftmann@26100
|
598 |
val mod_name = @{const_name mod};
|
haftmann@25942
|
599 |
val mk_binop = HOLogic.mk_binop;
|
haftmann@30494
|
600 |
val mk_sum = Nat_Arith.mk_sum;
|
haftmann@30494
|
601 |
val dest_sum = Nat_Arith.dest_sum;
|
haftmann@25942
|
602 |
|
haftmann@25942
|
603 |
(*logic*)
|
haftmann@25942
|
604 |
|
haftmann@25942
|
605 |
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
|
haftmann@25942
|
606 |
|
haftmann@25942
|
607 |
val trans = trans
|
haftmann@25942
|
608 |
|
haftmann@25942
|
609 |
val prove_eq_sums =
|
haftmann@25942
|
610 |
let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
|
haftmann@30494
|
611 |
in Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac simps) end;
|
haftmann@25942
|
612 |
|
haftmann@25942
|
613 |
end;
|
haftmann@25942
|
614 |
|
haftmann@25942
|
615 |
structure CancelDivMod = CancelDivModFun(CancelDivModData);
|
haftmann@25942
|
616 |
|
wenzelm@28262
|
617 |
val cancel_div_mod_proc = Simplifier.simproc (the_context ())
|
haftmann@26100
|
618 |
"cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
|
haftmann@25942
|
619 |
|
haftmann@25942
|
620 |
Addsimprocs[cancel_div_mod_proc];
|
haftmann@25942
|
621 |
*}
|
haftmann@25942
|
622 |
|
haftmann@26100
|
623 |
text {* code generator setup *}
|
haftmann@26100
|
624 |
|
haftmann@26100
|
625 |
lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else
|
haftmann@26100
|
626 |
let (q, r) = divmod (m - n) n in (Suc q, r))"
|
nipkow@29667
|
627 |
by (simp add: divmod_zero divmod_base divmod_step)
|
haftmann@26100
|
628 |
(simp add: divmod_div_mod)
|
haftmann@26100
|
629 |
|
haftmann@26100
|
630 |
code_modulename SML
|
haftmann@26100
|
631 |
Divides Nat
|
haftmann@26100
|
632 |
|
haftmann@26100
|
633 |
code_modulename OCaml
|
haftmann@26100
|
634 |
Divides Nat
|
haftmann@26100
|
635 |
|
haftmann@26100
|
636 |
code_modulename Haskell
|
haftmann@26100
|
637 |
Divides Nat
|
haftmann@26100
|
638 |
|
haftmann@26100
|
639 |
|
haftmann@26100
|
640 |
subsubsection {* Quotient *}
|
haftmann@26100
|
641 |
|
haftmann@26100
|
642 |
lemma div_geq: "0 < n \<Longrightarrow> \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
|
nipkow@29667
|
643 |
by (simp add: le_div_geq linorder_not_less)
|
haftmann@26100
|
644 |
|
haftmann@26100
|
645 |
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
|
nipkow@29667
|
646 |
by (simp add: div_geq)
|
haftmann@26100
|
647 |
|
haftmann@26100
|
648 |
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
|
nipkow@29667
|
649 |
by simp
|
haftmann@26100
|
650 |
|
haftmann@26100
|
651 |
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
|
nipkow@29667
|
652 |
by simp
|
haftmann@26100
|
653 |
|
haftmann@25942
|
654 |
|
haftmann@25942
|
655 |
subsubsection {* Remainder *}
|
haftmann@25942
|
656 |
|
haftmann@26100
|
657 |
lemma mod_less_divisor [simp]:
|
haftmann@26100
|
658 |
fixes m n :: nat
|
haftmann@26100
|
659 |
assumes "n > 0"
|
haftmann@26100
|
660 |
shows "m mod n < (n::nat)"
|
haftmann@26100
|
661 |
using assms divmod_rel unfolding divmod_rel_def by auto
|
haftmann@25942
|
662 |
|
haftmann@26100
|
663 |
lemma mod_less_eq_dividend [simp]:
|
haftmann@26100
|
664 |
fixes m n :: nat
|
haftmann@26100
|
665 |
shows "m mod n \<le> m"
|
haftmann@26100
|
666 |
proof (rule add_leD2)
|
haftmann@26100
|
667 |
from mod_div_equality have "m div n * n + m mod n = m" .
|
haftmann@26100
|
668 |
then show "m div n * n + m mod n \<le> m" by auto
|
haftmann@26100
|
669 |
qed
|
haftmann@26100
|
670 |
|
haftmann@26100
|
671 |
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
|
nipkow@29667
|
672 |
by (simp add: le_mod_geq linorder_not_less)
|
paulson@14267
|
673 |
|
haftmann@26100
|
674 |
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
|
nipkow@29667
|
675 |
by (simp add: le_mod_geq)
|
haftmann@26100
|
676 |
|
paulson@14267
|
677 |
lemma mod_1 [simp]: "m mod Suc 0 = 0"
|
nipkow@29667
|
678 |
by (induct m) (simp_all add: mod_geq)
|
paulson@14267
|
679 |
|
haftmann@26100
|
680 |
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
|
wenzelm@22718
|
681 |
apply (cases "n = 0", simp)
|
wenzelm@22718
|
682 |
apply (cases "k = 0", simp)
|
wenzelm@22718
|
683 |
apply (induct m rule: nat_less_induct)
|
wenzelm@22718
|
684 |
apply (subst mod_if, simp)
|
wenzelm@22718
|
685 |
apply (simp add: mod_geq diff_mult_distrib)
|
wenzelm@22718
|
686 |
done
|
paulson@14267
|
687 |
|
paulson@14267
|
688 |
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
|
nipkow@29667
|
689 |
by (simp add: mult_commute [of k] mod_mult_distrib)
|
paulson@14267
|
690 |
|
paulson@14267
|
691 |
(* a simple rearrangement of mod_div_equality: *)
|
paulson@14267
|
692 |
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
|
nipkow@29667
|
693 |
by (cut_tac a = m and b = n in mod_div_equality2, arith)
|
paulson@14267
|
694 |
|
nipkow@15439
|
695 |
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
|
wenzelm@22718
|
696 |
apply (drule mod_less_divisor [where m = m])
|
wenzelm@22718
|
697 |
apply simp
|
wenzelm@22718
|
698 |
done
|
paulson@14267
|
699 |
|
haftmann@26100
|
700 |
subsubsection {* Quotient and Remainder *}
|
paulson@14267
|
701 |
|
haftmann@26100
|
702 |
lemma divmod_rel_mult1_eq:
|
haftmann@26100
|
703 |
"[| divmod_rel b c q r; c > 0 |]
|
haftmann@26100
|
704 |
==> divmod_rel (a*b) c (a*q + a*r div c) (a*r mod c)"
|
nipkow@29667
|
705 |
by (auto simp add: split_ifs divmod_rel_def algebra_simps)
|
paulson@14267
|
706 |
|
paulson@14267
|
707 |
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
|
nipkow@25134
|
708 |
apply (cases "c = 0", simp)
|
haftmann@26100
|
709 |
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq])
|
nipkow@25134
|
710 |
done
|
paulson@14267
|
711 |
|
haftmann@26100
|
712 |
lemma divmod_rel_add1_eq:
|
haftmann@26100
|
713 |
"[| divmod_rel a c aq ar; divmod_rel b c bq br; c > 0 |]
|
haftmann@26100
|
714 |
==> divmod_rel (a + b) c (aq + bq + (ar+br) div c) ((ar + br) mod c)"
|
nipkow@29667
|
715 |
by (auto simp add: split_ifs divmod_rel_def algebra_simps)
|
paulson@14267
|
716 |
|
paulson@14267
|
717 |
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
|
paulson@14267
|
718 |
lemma div_add1_eq:
|
nipkow@25134
|
719 |
"(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
|
nipkow@25134
|
720 |
apply (cases "c = 0", simp)
|
haftmann@26100
|
721 |
apply (blast intro: divmod_rel_add1_eq [THEN div_eq] divmod_rel)
|
nipkow@25134
|
722 |
done
|
paulson@14267
|
723 |
|
paulson@14267
|
724 |
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
|
wenzelm@22718
|
725 |
apply (cut_tac m = q and n = c in mod_less_divisor)
|
wenzelm@22718
|
726 |
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
|
wenzelm@22718
|
727 |
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
|
wenzelm@22718
|
728 |
apply (simp add: add_mult_distrib2)
|
wenzelm@22718
|
729 |
done
|
paulson@14267
|
730 |
|
haftmann@26100
|
731 |
lemma divmod_rel_mult2_eq: "[| divmod_rel a b q r; 0 < b; 0 < c |]
|
haftmann@26100
|
732 |
==> divmod_rel a (b*c) (q div c) (b*(q mod c) + r)"
|
nipkow@29667
|
733 |
by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma)
|
paulson@14267
|
734 |
|
paulson@14267
|
735 |
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
|
wenzelm@22718
|
736 |
apply (cases "b = 0", simp)
|
wenzelm@22718
|
737 |
apply (cases "c = 0", simp)
|
haftmann@26100
|
738 |
apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq])
|
wenzelm@22718
|
739 |
done
|
paulson@14267
|
740 |
|
paulson@14267
|
741 |
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
|
wenzelm@22718
|
742 |
apply (cases "b = 0", simp)
|
wenzelm@22718
|
743 |
apply (cases "c = 0", simp)
|
haftmann@26100
|
744 |
apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq])
|
wenzelm@22718
|
745 |
done
|
paulson@14267
|
746 |
|
paulson@14267
|
747 |
|
haftmann@25942
|
748 |
subsubsection{*Cancellation of Common Factors in Division*}
|
paulson@14267
|
749 |
|
paulson@14267
|
750 |
lemma div_mult_mult_lemma:
|
wenzelm@22718
|
751 |
"[| (0::nat) < b; 0 < c |] ==> (c*a) div (c*b) = a div b"
|
nipkow@29667
|
752 |
by (auto simp add: div_mult2_eq)
|
paulson@14267
|
753 |
|
paulson@14267
|
754 |
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
|
wenzelm@22718
|
755 |
apply (cases "b = 0")
|
wenzelm@22718
|
756 |
apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
|
wenzelm@22718
|
757 |
done
|
paulson@14267
|
758 |
|
paulson@14267
|
759 |
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
|
wenzelm@22718
|
760 |
apply (drule div_mult_mult1)
|
wenzelm@22718
|
761 |
apply (auto simp add: mult_commute)
|
wenzelm@22718
|
762 |
done
|
paulson@14267
|
763 |
|
paulson@14267
|
764 |
|
haftmann@25942
|
765 |
subsubsection{*Further Facts about Quotient and Remainder*}
|
paulson@14267
|
766 |
|
paulson@14267
|
767 |
lemma div_1 [simp]: "m div Suc 0 = m"
|
nipkow@29667
|
768 |
by (induct m) (simp_all add: div_geq)
|
paulson@14267
|
769 |
|
paulson@14267
|
770 |
|
paulson@14267
|
771 |
(* Monotonicity of div in first argument *)
|
nipkow@30837
|
772 |
lemma div_le_mono [rule_format]:
|
wenzelm@22718
|
773 |
"\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
|
paulson@14267
|
774 |
apply (case_tac "k=0", simp)
|
paulson@15251
|
775 |
apply (induct "n" rule: nat_less_induct, clarify)
|
paulson@14267
|
776 |
apply (case_tac "n<k")
|
paulson@14267
|
777 |
(* 1 case n<k *)
|
paulson@14267
|
778 |
apply simp
|
paulson@14267
|
779 |
(* 2 case n >= k *)
|
paulson@14267
|
780 |
apply (case_tac "m<k")
|
paulson@14267
|
781 |
(* 2.1 case m<k *)
|
paulson@14267
|
782 |
apply simp
|
paulson@14267
|
783 |
(* 2.2 case m>=k *)
|
nipkow@15439
|
784 |
apply (simp add: div_geq diff_le_mono)
|
paulson@14267
|
785 |
done
|
paulson@14267
|
786 |
|
paulson@14267
|
787 |
(* Antimonotonicity of div in second argument *)
|
paulson@14267
|
788 |
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
|
paulson@14267
|
789 |
apply (subgoal_tac "0<n")
|
wenzelm@22718
|
790 |
prefer 2 apply simp
|
paulson@15251
|
791 |
apply (induct_tac k rule: nat_less_induct)
|
paulson@14267
|
792 |
apply (rename_tac "k")
|
paulson@14267
|
793 |
apply (case_tac "k<n", simp)
|
paulson@14267
|
794 |
apply (subgoal_tac "~ (k<m) ")
|
wenzelm@22718
|
795 |
prefer 2 apply simp
|
paulson@14267
|
796 |
apply (simp add: div_geq)
|
paulson@15251
|
797 |
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
|
paulson@14267
|
798 |
prefer 2
|
paulson@14267
|
799 |
apply (blast intro: div_le_mono diff_le_mono2)
|
paulson@14267
|
800 |
apply (rule le_trans, simp)
|
nipkow@15439
|
801 |
apply (simp)
|
paulson@14267
|
802 |
done
|
paulson@14267
|
803 |
|
paulson@14267
|
804 |
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
|
paulson@14267
|
805 |
apply (case_tac "n=0", simp)
|
paulson@14267
|
806 |
apply (subgoal_tac "m div n \<le> m div 1", simp)
|
paulson@14267
|
807 |
apply (rule div_le_mono2)
|
paulson@14267
|
808 |
apply (simp_all (no_asm_simp))
|
paulson@14267
|
809 |
done
|
paulson@14267
|
810 |
|
wenzelm@22718
|
811 |
(* Similar for "less than" *)
|
paulson@17085
|
812 |
lemma div_less_dividend [rule_format]:
|
paulson@14267
|
813 |
"!!n::nat. 1<n ==> 0 < m --> m div n < m"
|
paulson@15251
|
814 |
apply (induct_tac m rule: nat_less_induct)
|
paulson@14267
|
815 |
apply (rename_tac "m")
|
paulson@14267
|
816 |
apply (case_tac "m<n", simp)
|
paulson@14267
|
817 |
apply (subgoal_tac "0<n")
|
wenzelm@22718
|
818 |
prefer 2 apply simp
|
paulson@14267
|
819 |
apply (simp add: div_geq)
|
paulson@14267
|
820 |
apply (case_tac "n<m")
|
paulson@15251
|
821 |
apply (subgoal_tac "(m-n) div n < (m-n) ")
|
paulson@14267
|
822 |
apply (rule impI less_trans_Suc)+
|
paulson@14267
|
823 |
apply assumption
|
nipkow@15439
|
824 |
apply (simp_all)
|
paulson@14267
|
825 |
done
|
paulson@14267
|
826 |
|
nipkow@30837
|
827 |
lemma nat_div_eq_0 [simp]: "(n::nat) > 0 ==> ((m div n) = 0) = (m < n)"
|
nipkow@30837
|
828 |
by(auto, subst mod_div_equality [of m n, symmetric], auto)
|
nipkow@30837
|
829 |
|
nipkow@30840
|
830 |
lemma nat_div_gt_0 [simp]: "(n::nat) > 0 ==> ((m div n) > 0) = (m >= n)"
|
nipkow@30840
|
831 |
by (subst neq0_conv [symmetric], auto)
|
nipkow@30840
|
832 |
|
paulson@17085
|
833 |
declare div_less_dividend [simp]
|
paulson@17085
|
834 |
|
paulson@14267
|
835 |
text{*A fact for the mutilated chess board*}
|
paulson@14267
|
836 |
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
|
paulson@14267
|
837 |
apply (case_tac "n=0", simp)
|
paulson@15251
|
838 |
apply (induct "m" rule: nat_less_induct)
|
paulson@14267
|
839 |
apply (case_tac "Suc (na) <n")
|
paulson@14267
|
840 |
(* case Suc(na) < n *)
|
paulson@14267
|
841 |
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
|
paulson@14267
|
842 |
(* case n \<le> Suc(na) *)
|
paulson@16796
|
843 |
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
|
nipkow@15439
|
844 |
apply (auto simp add: Suc_diff_le le_mod_geq)
|
paulson@14267
|
845 |
done
|
paulson@14267
|
846 |
|
paulson@14267
|
847 |
|
haftmann@27651
|
848 |
subsubsection {* The Divides Relation *}
|
paulson@24286
|
849 |
|
paulson@14267
|
850 |
lemma dvd_1_left [iff]: "Suc 0 dvd k"
|
wenzelm@22718
|
851 |
unfolding dvd_def by simp
|
paulson@14267
|
852 |
|
paulson@14267
|
853 |
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
|
nipkow@29667
|
854 |
by (simp add: dvd_def)
|
paulson@14267
|
855 |
|
huffman@30016
|
856 |
lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"
|
huffman@30016
|
857 |
by (simp add: dvd_def)
|
huffman@30016
|
858 |
|
paulson@14267
|
859 |
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
|
wenzelm@22718
|
860 |
unfolding dvd_def
|
wenzelm@22718
|
861 |
by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
|
paulson@14267
|
862 |
|
haftmann@23684
|
863 |
text {* @{term "op dvd"} is a partial order *}
|
haftmann@23684
|
864 |
|
wenzelm@30732
|
865 |
interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"
|
haftmann@28823
|
866 |
proof qed (auto intro: dvd_refl dvd_trans dvd_anti_sym)
|
paulson@14267
|
867 |
|
nipkow@29979
|
868 |
lemma nat_dvd_diff[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
|
nipkow@29979
|
869 |
unfolding dvd_def
|
nipkow@29979
|
870 |
by (blast intro: diff_mult_distrib2 [symmetric])
|
paulson@14267
|
871 |
|
paulson@14267
|
872 |
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
|
wenzelm@22718
|
873 |
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
|
wenzelm@22718
|
874 |
apply (blast intro: dvd_add)
|
wenzelm@22718
|
875 |
done
|
paulson@14267
|
876 |
|
paulson@14267
|
877 |
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
|
nipkow@29979
|
878 |
by (drule_tac m = m in nat_dvd_diff, auto)
|
paulson@14267
|
879 |
|
paulson@14267
|
880 |
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
|
wenzelm@22718
|
881 |
apply (rule iffI)
|
wenzelm@22718
|
882 |
apply (erule_tac [2] dvd_add)
|
wenzelm@22718
|
883 |
apply (rule_tac [2] dvd_refl)
|
wenzelm@22718
|
884 |
apply (subgoal_tac "n = (n+k) -k")
|
wenzelm@22718
|
885 |
prefer 2 apply simp
|
wenzelm@22718
|
886 |
apply (erule ssubst)
|
nipkow@29979
|
887 |
apply (erule nat_dvd_diff)
|
wenzelm@22718
|
888 |
apply (rule dvd_refl)
|
wenzelm@22718
|
889 |
done
|
paulson@14267
|
890 |
|
paulson@14267
|
891 |
lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
|
wenzelm@22718
|
892 |
unfolding dvd_def
|
wenzelm@22718
|
893 |
apply (case_tac "n = 0", auto)
|
wenzelm@22718
|
894 |
apply (blast intro: mod_mult_distrib2 [symmetric])
|
wenzelm@22718
|
895 |
done
|
paulson@14267
|
896 |
|
paulson@14267
|
897 |
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
|
nipkow@29667
|
898 |
by (blast intro: dvd_mod_imp_dvd dvd_mod)
|
paulson@14267
|
899 |
|
paulson@14267
|
900 |
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
|
wenzelm@22718
|
901 |
unfolding dvd_def
|
wenzelm@22718
|
902 |
apply (erule exE)
|
wenzelm@22718
|
903 |
apply (simp add: mult_ac)
|
wenzelm@22718
|
904 |
done
|
paulson@14267
|
905 |
|
paulson@14267
|
906 |
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
|
wenzelm@22718
|
907 |
apply auto
|
wenzelm@22718
|
908 |
apply (subgoal_tac "m*n dvd m*1")
|
wenzelm@22718
|
909 |
apply (drule dvd_mult_cancel, auto)
|
wenzelm@22718
|
910 |
done
|
paulson@14267
|
911 |
|
paulson@14267
|
912 |
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
|
wenzelm@22718
|
913 |
apply (subst mult_commute)
|
wenzelm@22718
|
914 |
apply (erule dvd_mult_cancel1)
|
wenzelm@22718
|
915 |
done
|
paulson@14267
|
916 |
|
paulson@14267
|
917 |
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
|
nipkow@30837
|
918 |
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
|
nipkow@30837
|
919 |
|
nipkow@30837
|
920 |
lemma nat_dvd_not_less: "(0::nat) < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
|
nipkow@30837
|
921 |
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
|
paulson@14267
|
922 |
|
paulson@14267
|
923 |
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
|
wenzelm@22718
|
924 |
apply (subgoal_tac "m mod n = 0")
|
wenzelm@22718
|
925 |
apply (simp add: mult_div_cancel)
|
wenzelm@22718
|
926 |
apply (simp only: dvd_eq_mod_eq_0)
|
wenzelm@22718
|
927 |
done
|
paulson@14267
|
928 |
|
nipkow@25162
|
929 |
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
|
wenzelm@22718
|
930 |
by (induct n) auto
|
haftmann@21408
|
931 |
|
haftmann@21408
|
932 |
lemma power_dvd_imp_le: "[|i^m dvd i^n; (1::nat) < i|] ==> m \<le> n"
|
wenzelm@22718
|
933 |
apply (rule power_le_imp_le_exp, assumption)
|
wenzelm@22718
|
934 |
apply (erule dvd_imp_le, simp)
|
wenzelm@22718
|
935 |
done
|
haftmann@21408
|
936 |
|
paulson@14267
|
937 |
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
|
nipkow@29667
|
938 |
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
|
paulson@17084
|
939 |
|
wenzelm@22718
|
940 |
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
|
paulson@14267
|
941 |
|
paulson@14267
|
942 |
(*Loses information, namely we also have r<d provided d is nonzero*)
|
paulson@14267
|
943 |
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
|
haftmann@27651
|
944 |
apply (cut_tac a = m in mod_div_equality)
|
wenzelm@22718
|
945 |
apply (simp only: add_ac)
|
wenzelm@22718
|
946 |
apply (blast intro: sym)
|
wenzelm@22718
|
947 |
done
|
paulson@14267
|
948 |
|
nipkow@13152
|
949 |
lemma split_div:
|
nipkow@13189
|
950 |
"P(n div k :: nat) =
|
nipkow@13189
|
951 |
((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
|
nipkow@13189
|
952 |
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
|
nipkow@13189
|
953 |
proof
|
nipkow@13189
|
954 |
assume P: ?P
|
nipkow@13189
|
955 |
show ?Q
|
nipkow@13189
|
956 |
proof (cases)
|
nipkow@13189
|
957 |
assume "k = 0"
|
haftmann@27651
|
958 |
with P show ?Q by simp
|
nipkow@13189
|
959 |
next
|
nipkow@13189
|
960 |
assume not0: "k \<noteq> 0"
|
nipkow@13189
|
961 |
thus ?Q
|
nipkow@13189
|
962 |
proof (simp, intro allI impI)
|
nipkow@13189
|
963 |
fix i j
|
nipkow@13189
|
964 |
assume n: "n = k*i + j" and j: "j < k"
|
nipkow@13189
|
965 |
show "P i"
|
nipkow@13189
|
966 |
proof (cases)
|
wenzelm@22718
|
967 |
assume "i = 0"
|
wenzelm@22718
|
968 |
with n j P show "P i" by simp
|
nipkow@13189
|
969 |
next
|
wenzelm@22718
|
970 |
assume "i \<noteq> 0"
|
wenzelm@22718
|
971 |
with not0 n j P show "P i" by(simp add:add_ac)
|
nipkow@13189
|
972 |
qed
|
nipkow@13189
|
973 |
qed
|
nipkow@13189
|
974 |
qed
|
nipkow@13189
|
975 |
next
|
nipkow@13189
|
976 |
assume Q: ?Q
|
nipkow@13189
|
977 |
show ?P
|
nipkow@13189
|
978 |
proof (cases)
|
nipkow@13189
|
979 |
assume "k = 0"
|
haftmann@27651
|
980 |
with Q show ?P by simp
|
nipkow@13189
|
981 |
next
|
nipkow@13189
|
982 |
assume not0: "k \<noteq> 0"
|
nipkow@13189
|
983 |
with Q have R: ?R by simp
|
nipkow@13189
|
984 |
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
|
nipkow@13517
|
985 |
show ?P by simp
|
nipkow@13189
|
986 |
qed
|
nipkow@13189
|
987 |
qed
|
nipkow@13189
|
988 |
|
berghofe@13882
|
989 |
lemma split_div_lemma:
|
haftmann@26100
|
990 |
assumes "0 < n"
|
haftmann@26100
|
991 |
shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
|
haftmann@26100
|
992 |
proof
|
haftmann@26100
|
993 |
assume ?rhs
|
haftmann@26100
|
994 |
with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
|
haftmann@26100
|
995 |
then have A: "n * q \<le> m" by simp
|
haftmann@26100
|
996 |
have "n - (m mod n) > 0" using mod_less_divisor assms by auto
|
haftmann@26100
|
997 |
then have "m < m + (n - (m mod n))" by simp
|
haftmann@26100
|
998 |
then have "m < n + (m - (m mod n))" by simp
|
haftmann@26100
|
999 |
with nq have "m < n + n * q" by simp
|
haftmann@26100
|
1000 |
then have B: "m < n * Suc q" by simp
|
haftmann@26100
|
1001 |
from A B show ?lhs ..
|
haftmann@26100
|
1002 |
next
|
haftmann@26100
|
1003 |
assume P: ?lhs
|
haftmann@26100
|
1004 |
then have "divmod_rel m n q (m - n * q)"
|
haftmann@26100
|
1005 |
unfolding divmod_rel_def by (auto simp add: mult_ac)
|
haftmann@26100
|
1006 |
then show ?rhs using divmod_rel by (rule divmod_rel_unique_div)
|
haftmann@26100
|
1007 |
qed
|
berghofe@13882
|
1008 |
|
berghofe@13882
|
1009 |
theorem split_div':
|
berghofe@13882
|
1010 |
"P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
|
paulson@14267
|
1011 |
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
|
berghofe@13882
|
1012 |
apply (case_tac "0 < n")
|
berghofe@13882
|
1013 |
apply (simp only: add: split_div_lemma)
|
haftmann@27651
|
1014 |
apply simp_all
|
berghofe@13882
|
1015 |
done
|
berghofe@13882
|
1016 |
|
nipkow@13189
|
1017 |
lemma split_mod:
|
nipkow@13189
|
1018 |
"P(n mod k :: nat) =
|
nipkow@13189
|
1019 |
((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
|
nipkow@13189
|
1020 |
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
|
nipkow@13189
|
1021 |
proof
|
nipkow@13189
|
1022 |
assume P: ?P
|
nipkow@13189
|
1023 |
show ?Q
|
nipkow@13189
|
1024 |
proof (cases)
|
nipkow@13189
|
1025 |
assume "k = 0"
|
haftmann@27651
|
1026 |
with P show ?Q by simp
|
nipkow@13189
|
1027 |
next
|
nipkow@13189
|
1028 |
assume not0: "k \<noteq> 0"
|
nipkow@13189
|
1029 |
thus ?Q
|
nipkow@13189
|
1030 |
proof (simp, intro allI impI)
|
nipkow@13189
|
1031 |
fix i j
|
nipkow@13189
|
1032 |
assume "n = k*i + j" "j < k"
|
nipkow@13189
|
1033 |
thus "P j" using not0 P by(simp add:add_ac mult_ac)
|
nipkow@13189
|
1034 |
qed
|
nipkow@13189
|
1035 |
qed
|
nipkow@13189
|
1036 |
next
|
nipkow@13189
|
1037 |
assume Q: ?Q
|
nipkow@13189
|
1038 |
show ?P
|
nipkow@13189
|
1039 |
proof (cases)
|
nipkow@13189
|
1040 |
assume "k = 0"
|
haftmann@27651
|
1041 |
with Q show ?P by simp
|
nipkow@13189
|
1042 |
next
|
nipkow@13189
|
1043 |
assume not0: "k \<noteq> 0"
|
nipkow@13189
|
1044 |
with Q have R: ?R by simp
|
nipkow@13189
|
1045 |
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
|
nipkow@13517
|
1046 |
show ?P by simp
|
nipkow@13189
|
1047 |
qed
|
nipkow@13189
|
1048 |
qed
|
nipkow@13189
|
1049 |
|
berghofe@13882
|
1050 |
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
|
berghofe@13882
|
1051 |
apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
|
berghofe@13882
|
1052 |
subst [OF mod_div_equality [of _ n]])
|
berghofe@13882
|
1053 |
apply arith
|
berghofe@13882
|
1054 |
done
|
berghofe@13882
|
1055 |
|
haftmann@22800
|
1056 |
lemma div_mod_equality':
|
haftmann@22800
|
1057 |
fixes m n :: nat
|
haftmann@22800
|
1058 |
shows "m div n * n = m - m mod n"
|
haftmann@22800
|
1059 |
proof -
|
haftmann@22800
|
1060 |
have "m mod n \<le> m mod n" ..
|
haftmann@22800
|
1061 |
from div_mod_equality have
|
haftmann@22800
|
1062 |
"m div n * n + m mod n - m mod n = m - m mod n" by simp
|
haftmann@22800
|
1063 |
with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
|
haftmann@22800
|
1064 |
"m div n * n + (m mod n - m mod n) = m - m mod n"
|
haftmann@22800
|
1065 |
by simp
|
haftmann@22800
|
1066 |
then show ?thesis by simp
|
haftmann@22800
|
1067 |
qed
|
haftmann@22800
|
1068 |
|
haftmann@22800
|
1069 |
|
haftmann@25942
|
1070 |
subsubsection {*An ``induction'' law for modulus arithmetic.*}
|
paulson@14640
|
1071 |
|
paulson@14640
|
1072 |
lemma mod_induct_0:
|
paulson@14640
|
1073 |
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
|
paulson@14640
|
1074 |
and base: "P i" and i: "i<p"
|
paulson@14640
|
1075 |
shows "P 0"
|
paulson@14640
|
1076 |
proof (rule ccontr)
|
paulson@14640
|
1077 |
assume contra: "\<not>(P 0)"
|
paulson@14640
|
1078 |
from i have p: "0<p" by simp
|
paulson@14640
|
1079 |
have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
|
paulson@14640
|
1080 |
proof
|
paulson@14640
|
1081 |
fix k
|
paulson@14640
|
1082 |
show "?A k"
|
paulson@14640
|
1083 |
proof (induct k)
|
paulson@14640
|
1084 |
show "?A 0" by simp -- "by contradiction"
|
paulson@14640
|
1085 |
next
|
paulson@14640
|
1086 |
fix n
|
paulson@14640
|
1087 |
assume ih: "?A n"
|
paulson@14640
|
1088 |
show "?A (Suc n)"
|
paulson@14640
|
1089 |
proof (clarsimp)
|
wenzelm@22718
|
1090 |
assume y: "P (p - Suc n)"
|
wenzelm@22718
|
1091 |
have n: "Suc n < p"
|
wenzelm@22718
|
1092 |
proof (rule ccontr)
|
wenzelm@22718
|
1093 |
assume "\<not>(Suc n < p)"
|
wenzelm@22718
|
1094 |
hence "p - Suc n = 0"
|
wenzelm@22718
|
1095 |
by simp
|
wenzelm@22718
|
1096 |
with y contra show "False"
|
wenzelm@22718
|
1097 |
by simp
|
wenzelm@22718
|
1098 |
qed
|
wenzelm@22718
|
1099 |
hence n2: "Suc (p - Suc n) = p-n" by arith
|
wenzelm@22718
|
1100 |
from p have "p - Suc n < p" by arith
|
wenzelm@22718
|
1101 |
with y step have z: "P ((Suc (p - Suc n)) mod p)"
|
wenzelm@22718
|
1102 |
by blast
|
wenzelm@22718
|
1103 |
show "False"
|
wenzelm@22718
|
1104 |
proof (cases "n=0")
|
wenzelm@22718
|
1105 |
case True
|
wenzelm@22718
|
1106 |
with z n2 contra show ?thesis by simp
|
wenzelm@22718
|
1107 |
next
|
wenzelm@22718
|
1108 |
case False
|
wenzelm@22718
|
1109 |
with p have "p-n < p" by arith
|
wenzelm@22718
|
1110 |
with z n2 False ih show ?thesis by simp
|
wenzelm@22718
|
1111 |
qed
|
paulson@14640
|
1112 |
qed
|
paulson@14640
|
1113 |
qed
|
paulson@14640
|
1114 |
qed
|
paulson@14640
|
1115 |
moreover
|
paulson@14640
|
1116 |
from i obtain k where "0<k \<and> i+k=p"
|
paulson@14640
|
1117 |
by (blast dest: less_imp_add_positive)
|
paulson@14640
|
1118 |
hence "0<k \<and> i=p-k" by auto
|
paulson@14640
|
1119 |
moreover
|
paulson@14640
|
1120 |
note base
|
paulson@14640
|
1121 |
ultimately
|
paulson@14640
|
1122 |
show "False" by blast
|
paulson@14640
|
1123 |
qed
|
paulson@14640
|
1124 |
|
paulson@14640
|
1125 |
lemma mod_induct:
|
paulson@14640
|
1126 |
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
|
paulson@14640
|
1127 |
and base: "P i" and i: "i<p" and j: "j<p"
|
paulson@14640
|
1128 |
shows "P j"
|
paulson@14640
|
1129 |
proof -
|
paulson@14640
|
1130 |
have "\<forall>j<p. P j"
|
paulson@14640
|
1131 |
proof
|
paulson@14640
|
1132 |
fix j
|
paulson@14640
|
1133 |
show "j<p \<longrightarrow> P j" (is "?A j")
|
paulson@14640
|
1134 |
proof (induct j)
|
paulson@14640
|
1135 |
from step base i show "?A 0"
|
wenzelm@22718
|
1136 |
by (auto elim: mod_induct_0)
|
paulson@14640
|
1137 |
next
|
paulson@14640
|
1138 |
fix k
|
paulson@14640
|
1139 |
assume ih: "?A k"
|
paulson@14640
|
1140 |
show "?A (Suc k)"
|
paulson@14640
|
1141 |
proof
|
wenzelm@22718
|
1142 |
assume suc: "Suc k < p"
|
wenzelm@22718
|
1143 |
hence k: "k<p" by simp
|
wenzelm@22718
|
1144 |
with ih have "P k" ..
|
wenzelm@22718
|
1145 |
with step k have "P (Suc k mod p)"
|
wenzelm@22718
|
1146 |
by blast
|
wenzelm@22718
|
1147 |
moreover
|
wenzelm@22718
|
1148 |
from suc have "Suc k mod p = Suc k"
|
wenzelm@22718
|
1149 |
by simp
|
wenzelm@22718
|
1150 |
ultimately
|
wenzelm@22718
|
1151 |
show "P (Suc k)" by simp
|
paulson@14640
|
1152 |
qed
|
paulson@14640
|
1153 |
qed
|
paulson@14640
|
1154 |
qed
|
paulson@14640
|
1155 |
with j show ?thesis by blast
|
paulson@14640
|
1156 |
qed
|
paulson@14640
|
1157 |
|
paulson@3366
|
1158 |
end
|