
paulson@3366: (*  Title:      HOL/Divides.thy
paulson@3366:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@6865:     Copyright   1999  University of Cambridge
huffman@18154: *)
paulson@3366: 
haftmann@27651: header {* The division operators div and mod *}
paulson@3366: 
nipkow@15131: theory Divides
huffman@47255: imports Nat_Transfer
nipkow@15131: begin
paulson@3366: 
haftmann@25942: subsection {* Syntactic division operations *}
haftmann@25942: 
haftmann@27651: class div = dvd +
haftmann@27540:   fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
haftmann@27651:     and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
haftmann@27540: 
haftmann@27540: 
haftmann@27651: subsection {* Abstract division in commutative semirings. *}
haftmann@25942: 
haftmann@30930: class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +
haftmann@25942:   assumes mod_div_equality: "a div b * b + a mod b = a"
haftmann@27651:     and div_by_0 [simp]: "a div 0 = 0"
haftmann@27651:     and div_0 [simp]: "0 div a = 0"
haftmann@27651:     and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
haftmann@30930:     and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
haftmann@25942: begin
haftmann@25942: 
haftmann@26100: text {* @{const div} and @{const mod} *}
haftmann@26100: 
haftmann@26062: lemma mod_div_equality2: "b * (a div b) + a mod b = a"
haftmann@57512:   unfolding mult.commute [of b]
haftmann@26062:   by (rule mod_div_equality)
haftmann@26062: 
huffman@29403: lemma mod_div_equality': "a mod b + a div b * b = a"
huffman@29403:   using mod_div_equality [of a b]
haftmann@57514:   by (simp only: ac_simps)
huffman@29403: 
haftmann@26062: lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
haftmann@30934:   by (simp add: mod_div_equality)
haftmann@26062: 
haftmann@26062: lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
haftmann@30934:   by (simp add: mod_div_equality2)
haftmann@26062: 
haftmann@27651: lemma mod_by_0 [simp]: "a mod 0 = a"
haftmann@30934:   using mod_div_equality [of a zero] by simp
haftmann@27651: 
haftmann@27651: lemma mod_0 [simp]: "0 mod a = 0"
haftmann@30934:   using mod_div_equality [of zero a] div_0 by simp
haftmann@27651: 
haftmann@27651: lemma div_mult_self2 [simp]:
haftmann@27651:   assumes "b \<noteq> 0"
haftmann@27651:   shows "(a + b * c) div b = c + a div b"
haftmann@57512:   using assms div_mult_self1 [of b a c] by (simp add: mult.commute)
haftmann@26100: 
haftmann@54221: lemma div_mult_self3 [simp]:
haftmann@54221:   assumes "b \<noteq> 0"
haftmann@54221:   shows "(c * b + a) div b = c + a div b"
haftmann@54221:   using assms by (simp add: add.commute)
haftmann@54221: 
haftmann@54221: lemma div_mult_self4 [simp]:
haftmann@54221:   assumes "b \<noteq> 0"
haftmann@54221:   shows "(b * c + a) div b = c + a div b"
haftmann@54221:   using assms by (simp add: add.commute)
haftmann@54221: 
haftmann@27651: lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
haftmann@27651: proof (cases "b = 0")
haftmann@27651:   case True then show ?thesis by simp
haftmann@27651: next
haftmann@27651:   case False
haftmann@27651:   have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
haftmann@27651:     by (simp add: mod_div_equality)
haftmann@27651:   also from False div_mult_self1 [of b a c] have
haftmann@27651:     "\<dots> = (c + a div b) * b + (a + c * b) mod b"
nipkow@29667:       by (simp add: algebra_simps)
haftmann@27651:   finally have "a = a div b * b + (a + c * b) mod b"
haftmann@57512:     by (simp add: add.commute [of a] add.assoc distrib_right)
haftmann@27651:   then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
haftmann@27651:     by (simp add: mod_div_equality)
haftmann@27651:   then show ?thesis by simp
haftmann@27651: qed
haftmann@27651: 
haftmann@54221: lemma mod_mult_self2 [simp]: 
haftmann@54221:   "(a + b * c) mod b = a mod b"
haftmann@57512:   by (simp add: mult.commute [of b])
haftmann@27651: 
haftmann@54221: lemma mod_mult_self3 [simp]:
haftmann@54221:   "(c * b + a) mod b = a mod b"
haftmann@54221:   by (simp add: add.commute)
haftmann@54221: 
haftmann@54221: lemma mod_mult_self4 [simp]:
haftmann@54221:   "(b * c + a) mod b = a mod b"
haftmann@54221:   by (simp add: add.commute)
haftmann@54221: 
haftmann@27651: lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
haftmann@27651:   using div_mult_self2 [of b 0 a] by simp
haftmann@27651: 
haftmann@27651: lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
haftmann@27651:   using div_mult_self1 [of b 0 a] by simp
haftmann@27651: 
haftmann@27651: lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
haftmann@27651:   using mod_mult_self2 [of 0 b a] by simp
haftmann@27651: 
haftmann@27651: lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
haftmann@27651:   using mod_mult_self1 [of 0 a b] by simp
haftmann@26062: 
haftmann@27651: lemma div_by_1 [simp]: "a div 1 = a"
haftmann@27651:   using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
haftmann@27651: 
haftmann@27651: lemma mod_by_1 [simp]: "a mod 1 = 0"
haftmann@27651: proof -
haftmann@27651:   from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
haftmann@27651:   then have "a + a mod 1 = a + 0" by simp
haftmann@27651:   then show ?thesis by (rule add_left_imp_eq)
haftmann@27651: qed
haftmann@27651: 
haftmann@27651: lemma mod_self [simp]: "a mod a = 0"
haftmann@27651:   using mod_mult_self2_is_0 [of 1] by simp
haftmann@27651: 
haftmann@27651: lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
haftmann@27651:   using div_mult_self2_is_id [of _ 1] by simp
haftmann@27651: 
haftmann@27676: lemma div_add_self1 [simp]:
haftmann@27651:   assumes "b \<noteq> 0"
haftmann@27651:   shows "(b + a) div b = a div b + 1"
haftmann@57512:   using assms div_mult_self1 [of b a 1] by (simp add: add.commute)
haftmann@26062: 
haftmann@27676: lemma div_add_self2 [simp]:
haftmann@27651:   assumes "b \<noteq> 0"
haftmann@27651:   shows "(a + b) div b = a div b + 1"
haftmann@57512:   using assms div_add_self1 [of b a] by (simp add: add.commute)
haftmann@27651: 
haftmann@27676: lemma mod_add_self1 [simp]:
haftmann@27651:   "(b + a) mod b = a mod b"
haftmann@57512:   using mod_mult_self1 [of a 1 b] by (simp add: add.commute)
haftmann@27651: 
haftmann@27676: lemma mod_add_self2 [simp]:
haftmann@27651:   "(a + b) mod b = a mod b"
haftmann@27651:   using mod_mult_self1 [of a 1 b] by simp
haftmann@27651: 
haftmann@27651: lemma mod_div_decomp:
haftmann@27651:   fixes a b
haftmann@27651:   obtains q r where "q = a div b" and "r = a mod b"
haftmann@27651:     and "a = q * b + r"
haftmann@27651: proof -
haftmann@27651:   from mod_div_equality have "a = a div b * b + a mod b" by simp
haftmann@27651:   moreover have "a div b = a div b" ..
haftmann@27651:   moreover have "a mod b = a mod b" ..
haftmann@27651:   note that ultimately show thesis by blast
haftmann@27651: qed
haftmann@27651: 
bulwahn@45231: lemma dvd_eq_mod_eq_0 [code]: "a dvd b \<longleftrightarrow> b mod a = 0"
haftmann@25942: proof
haftmann@25942:   assume "b mod a = 0"
haftmann@25942:   with mod_div_equality [of b a] have "b div a * a = b" by simp
haftmann@57512:   then have "b = a * (b div a)" unfolding mult.commute ..
haftmann@25942:   then have "\<exists>c. b = a * c" ..
haftmann@25942:   then show "a dvd b" unfolding dvd_def .
haftmann@25942: next
haftmann@25942:   assume "a dvd b"
haftmann@25942:   then have "\<exists>c. b = a * c" unfolding dvd_def .
haftmann@25942:   then obtain c where "b = a * c" ..
haftmann@25942:   then have "b mod a = a * c mod a" by simp
haftmann@57512:   then have "b mod a = c * a mod a" by (simp add: mult.commute)
haftmann@27651:   then show "b mod a = 0" by simp
haftmann@25942: qed
haftmann@25942: 
huffman@29403: lemma mod_div_trivial [simp]: "a mod b div b = 0"
huffman@29403: proof (cases "b = 0")
huffman@29403:   assume "b = 0"
huffman@29403:   thus ?thesis by simp
huffman@29403: next
huffman@29403:   assume "b \<noteq> 0"
huffman@29403:   hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
huffman@29403:     by (rule div_mult_self1 [symmetric])
huffman@29403:   also have "\<dots> = a div b"
huffman@29403:     by (simp only: mod_div_equality')
huffman@29403:   also have "\<dots> = a div b + 0"
huffman@29403:     by simp
huffman@29403:   finally show ?thesis
huffman@29403:     by (rule add_left_imp_eq)
huffman@29403: qed
huffman@29403: 
huffman@29403: lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"
huffman@29403: proof -
huffman@29403:   have "a mod b mod b = (a mod b + a div b * b) mod b"
huffman@29403:     by (simp only: mod_mult_self1)
huffman@29403:   also have "\<dots> = a mod b"
huffman@29403:     by (simp only: mod_div_equality')
huffman@29403:   finally show ?thesis .
huffman@29403: qed
huffman@29403: 
nipkow@29925: lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"
nipkow@29948: by (rule dvd_eq_mod_eq_0[THEN iffD1])
nipkow@29925: 
nipkow@29925: lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
nipkow@29925: by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)
nipkow@29925: 
haftmann@33274: lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b"
haftmann@57512: by (drule dvd_div_mult_self) (simp add: mult.commute)
haftmann@33274: 
nipkow@30052: lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"
nipkow@30052: apply (cases "a = 0")
nipkow@30052:  apply simp
haftmann@57512: apply (auto simp: dvd_def mult.assoc)
nipkow@30052: done
nipkow@30052: 
nipkow@29925: lemma div_dvd_div[simp]:
nipkow@29925:   "a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"
nipkow@29925: apply (cases "a = 0")
nipkow@29925:  apply simp
nipkow@29925: apply (unfold dvd_def)
nipkow@29925: apply auto
haftmann@57512:  apply(blast intro:mult.assoc[symmetric])
haftmann@57512: apply(fastforce simp add: mult.assoc)
nipkow@29925: done
nipkow@29925: 
huffman@30078: lemma dvd_mod_imp_dvd: "[| k dvd m mod n;  k dvd n |] ==> k dvd m"
huffman@30078:   apply (subgoal_tac "k dvd (m div n) *n + m mod n")
huffman@30078:    apply (simp add: mod_div_equality)
huffman@30078:   apply (simp only: dvd_add dvd_mult)
huffman@30078:   done
huffman@30078: 
huffman@29403: text {* Addition respects modular equivalence. *}
huffman@29403: 
huffman@29403: lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"
huffman@29403: proof -
huffman@29403:   have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
huffman@29403:     by (simp only: mod_div_equality)
huffman@29403:   also have "\<dots> = (a mod c + b + a div c * c) mod c"
haftmann@57514:     by (simp only: ac_simps)
huffman@29403:   also have "\<dots> = (a mod c + b) mod c"
huffman@29403:     by (rule mod_mult_self1)
huffman@29403:   finally show ?thesis .
huffman@29403: qed
huffman@29403: 
huffman@29403: lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"
huffman@29403: proof -
huffman@29403:   have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
huffman@29403:     by (simp only: mod_div_equality)
huffman@29403:   also have "\<dots> = (a + b mod c + b div c * c) mod c"
haftmann@57514:     by (simp only: ac_simps)
huffman@29403:   also have "\<dots> = (a + b mod c) mod c"
huffman@29403:     by (rule mod_mult_self1)
huffman@29403:   finally show ?thesis .
huffman@29403: qed
huffman@29403: 
huffman@29403: lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"
huffman@29403: by (rule trans [OF mod_add_left_eq mod_add_right_eq])
huffman@29403: 
huffman@29403: lemma mod_add_cong:
huffman@29403:   assumes "a mod c = a' mod c"
huffman@29403:   assumes "b mod c = b' mod c"
huffman@29403:   shows "(a + b) mod c = (a' + b') mod c"
huffman@29403: proof -
huffman@29403:   have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
huffman@29403:     unfolding assms ..
huffman@29403:   thus ?thesis
huffman@29403:     by (simp only: mod_add_eq [symmetric])
huffman@29403: qed
huffman@29403: 
haftmann@30923: lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y
nipkow@30837:   \<Longrightarrow> (x + y) div z = x div z + y div z"
haftmann@30923: by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)
nipkow@30837: 
huffman@29403: text {* Multiplication respects modular equivalence. *}
huffman@29403: 
huffman@29403: lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"
huffman@29403: proof -
huffman@29403:   have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
huffman@29403:     by (simp only: mod_div_equality)
huffman@29403:   also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
nipkow@29667:     by (simp only: algebra_simps)
huffman@29403:   also have "\<dots> = (a mod c * b) mod c"
huffman@29403:     by (rule mod_mult_self1)
huffman@29403:   finally show ?thesis .
huffman@29403: qed
huffman@29403: 
huffman@29403: lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"
huffman@29403: proof -
huffman@29403:   have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
huffman@29403:     by (simp only: mod_div_equality)
huffman@29403:   also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
nipkow@29667:     by (simp only: algebra_simps)
huffman@29403:   also have "\<dots> = (a * (b mod c)) mod c"
huffman@29403:     by (rule mod_mult_self1)
huffman@29403:   finally show ?thesis .
huffman@29403: qed
huffman@29403: 
huffman@29403: lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"
huffman@29403: by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])
huffman@29403: 
huffman@29403: lemma mod_mult_cong:
huffman@29403:   assumes "a mod c = a' mod c"
huffman@29403:   assumes "b mod c = b' mod c"
huffman@29403:   shows "(a * b) mod c = (a' * b') mod c"
huffman@29403: proof -
huffman@29403:   have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
huffman@29403:     unfolding assms ..
huffman@29403:   thus ?thesis
huffman@29403:     by (simp only: mod_mult_eq [symmetric])
huffman@29403: qed
huffman@29403: 
huffman@47164: text {* Exponentiation respects modular equivalence. *}
huffman@47164: 
huffman@47164: lemma power_mod: "(a mod b)^n mod b = a^n mod b"
huffman@47164: apply (induct n, simp_all)
huffman@47164: apply (rule mod_mult_right_eq [THEN trans])
huffman@47164: apply (simp (no_asm_simp))
huffman@47164: apply (rule mod_mult_eq [symmetric])
huffman@47164: done
huffman@47164: 
huffman@29404: lemma mod_mod_cancel:
huffman@29404:   assumes "c dvd b"
huffman@29404:   shows "a mod b mod c = a mod c"
huffman@29404: proof -
huffman@29404:   from `c dvd b` obtain k where "b = c * k"
huffman@29404:     by (rule dvdE)
huffman@29404:   have "a mod b mod c = a mod (c * k) mod c"
huffman@29404:     by (simp only: `b = c * k`)
huffman@29404:   also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
huffman@29404:     by (simp only: mod_mult_self1)
huffman@29404:   also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
haftmann@57514:     by (simp only: ac_simps ac_simps)
huffman@29404:   also have "\<dots> = a mod c"
huffman@29404:     by (simp only: mod_div_equality)
huffman@29404:   finally show ?thesis .
huffman@29404: qed
huffman@29404: 
haftmann@30930: lemma div_mult_div_if_dvd:
haftmann@30930:   "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
haftmann@30930:   apply (cases "y = 0", simp)
haftmann@30930:   apply (cases "z = 0", simp)
haftmann@30930:   apply (auto elim!: dvdE simp add: algebra_simps)
haftmann@57512:   apply (subst mult.assoc [symmetric])
nipkow@30476:   apply (simp add: no_zero_divisors)
haftmann@30930:   done
haftmann@30930: 
haftmann@35367: lemma div_mult_swap:
haftmann@35367:   assumes "c dvd b"
haftmann@35367:   shows "a * (b div c) = (a * b) div c"
haftmann@35367: proof -
haftmann@35367:   from assms have "b div c * (a div 1) = b * a div (c * 1)"
haftmann@35367:     by (simp only: div_mult_div_if_dvd one_dvd)
haftmann@57512:   then show ?thesis by (simp add: mult.commute)
haftmann@35367: qed
haftmann@35367:    
haftmann@30930: lemma div_mult_mult2 [simp]:
haftmann@30930:   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
haftmann@57512:   by (drule div_mult_mult1) (simp add: mult.commute)
haftmann@30930: 
haftmann@30930: lemma div_mult_mult1_if [simp]:
haftmann@30930:   "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
haftmann@30930:   by simp_all
nipkow@30476: 
haftmann@30930: lemma mod_mult_mult1:
haftmann@30930:   "(c * a) mod (c * b) = c * (a mod b)"
haftmann@30930: proof (cases "c = 0")
haftmann@30930:   case True then show ?thesis by simp
haftmann@30930: next
haftmann@30930:   case False
haftmann@30930:   from mod_div_equality
haftmann@30930:   have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
haftmann@30930:   with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
haftmann@30930:     = c * a + c * (a mod b)" by (simp add: algebra_simps)
haftmann@30930:   with mod_div_equality show ?thesis by simp 
haftmann@30930: qed
haftmann@30930:   
haftmann@30930: lemma mod_mult_mult2:
haftmann@30930:   "(a * c) mod (b * c) = (a mod b) * c"
haftmann@57512:   using mod_mult_mult1 [of c a b] by (simp add: mult.commute)
haftmann@30930: 
huffman@47159: lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
huffman@47159:   by (fact mod_mult_mult2 [symmetric])
huffman@47159: 
huffman@47159: lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
huffman@47159:   by (fact mod_mult_mult1 [symmetric])
huffman@47159: 
huffman@31662: lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
huffman@31662:   unfolding dvd_def by (auto simp add: mod_mult_mult1)
huffman@31662: 
huffman@31662: lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
huffman@31662: by (blast intro: dvd_mod_imp_dvd dvd_mod)
huffman@31662: 
haftmann@31009: lemma div_power:
huffman@31661:   "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
nipkow@30476: apply (induct n)
nipkow@30476:  apply simp
nipkow@30476: apply(simp add: div_mult_div_if_dvd dvd_power_same)
nipkow@30476: done
nipkow@30476: 
haftmann@35367: lemma dvd_div_eq_mult:
haftmann@35367:   assumes "a \<noteq> 0" and "a dvd b"  
haftmann@35367:   shows "b div a = c \<longleftrightarrow> b = c * a"
haftmann@35367: proof
haftmann@35367:   assume "b = c * a"
haftmann@35367:   then show "b div a = c" by (simp add: assms)
haftmann@35367: next
haftmann@35367:   assume "b div a = c"
haftmann@35367:   then have "b div a * a = c * a" by simp
haftmann@35367:   moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self)
haftmann@35367:   ultimately show "b = c * a" by simp
haftmann@35367: qed
haftmann@35367:    
haftmann@35367: lemma dvd_div_div_eq_mult:
haftmann@35367:   assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
haftmann@35367:   shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
haftmann@57512:   using assms by (auto simp add: mult.commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)
haftmann@35367: 
huffman@31661: end
huffman@31661: 
haftmann@35673: class ring_div = semiring_div + comm_ring_1
huffman@29405: begin
huffman@29405: 
haftmann@36634: subclass ring_1_no_zero_divisors ..
haftmann@36634: 
huffman@29405: text {* Negation respects modular equivalence. *}
huffman@29405: 
huffman@29405: lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
huffman@29405: proof -
huffman@29405:   have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
huffman@29405:     by (simp only: mod_div_equality)
huffman@29405:   also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
haftmann@57514:     by (simp add: ac_simps)
huffman@29405:   also have "\<dots> = (- (a mod b)) mod b"
huffman@29405:     by (rule mod_mult_self1)
huffman@29405:   finally show ?thesis .
huffman@29405: qed
huffman@29405: 
huffman@29405: lemma mod_minus_cong:
huffman@29405:   assumes "a mod b = a' mod b"
huffman@29405:   shows "(- a) mod b = (- a') mod b"
huffman@29405: proof -
huffman@29405:   have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
huffman@29405:     unfolding assms ..
huffman@29405:   thus ?thesis
huffman@29405:     by (simp only: mod_minus_eq [symmetric])
huffman@29405: qed
huffman@29405: 
huffman@29405: text {* Subtraction respects modular equivalence. *}
huffman@29405: 
haftmann@54230: lemma mod_diff_left_eq:
haftmann@54230:   "(a - b) mod c = (a mod c - b) mod c"
haftmann@54230:   using mod_add_cong [of a c "a mod c" "- b" "- b"] by simp
haftmann@54230: 
haftmann@54230: lemma mod_diff_right_eq:
haftmann@54230:   "(a - b) mod c = (a - b mod c) mod c"
haftmann@54230:   using mod_add_cong [of a c a "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp
haftmann@54230: 
haftmann@54230: lemma mod_diff_eq:
haftmann@54230:   "(a - b) mod c = (a mod c - b mod c) mod c"
haftmann@54230:   using mod_add_cong [of a c "a mod c" "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp
huffman@29405: 
huffman@29405: lemma mod_diff_cong:
huffman@29405:   assumes "a mod c = a' mod c"
huffman@29405:   assumes "b mod c = b' mod c"
huffman@29405:   shows "(a - b) mod c = (a' - b') mod c"
haftmann@54230:   using assms mod_add_cong [of a c a' "- b" "- b'"] mod_minus_cong [of b c "b'"] by simp
huffman@29405: 
nipkow@30180: lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
nipkow@30180: apply (case_tac "y = 0") apply simp
nipkow@30180: apply (auto simp add: dvd_def)
nipkow@30180: apply (subgoal_tac "-(y * k) = y * - k")
thomas@57492:  apply (simp only:)
nipkow@30180:  apply (erule div_mult_self1_is_id)
nipkow@30180: apply simp
nipkow@30180: done
nipkow@30180: 
nipkow@30180: lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
nipkow@30180: apply (case_tac "y = 0") apply simp
nipkow@30180: apply (auto simp add: dvd_def)
nipkow@30180: apply (subgoal_tac "y * k = -y * -k")
thomas@57492:  apply (erule ssubst, rule div_mult_self1_is_id)
nipkow@30180:  apply simp
nipkow@30180: apply simp
nipkow@30180: done
nipkow@30180: 
huffman@47159: lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
huffman@47159:   using div_mult_mult1 [of "- 1" a b]
huffman@47159:   unfolding neg_equal_0_iff_equal by simp
huffman@47159: 
huffman@47159: lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
huffman@47159:   using mod_mult_mult1 [of "- 1" a b] by simp
huffman@47159: 
huffman@47159: lemma div_minus_right: "a div (-b) = (-a) div b"
huffman@47159:   using div_minus_minus [of "-a" b] by simp
huffman@47159: 
huffman@47159: lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
huffman@47159:   using mod_minus_minus [of "-a" b] by simp
huffman@47159: 
huffman@47160: lemma div_minus1_right [simp]: "a div (-1) = -a"
huffman@47160:   using div_minus_right [of a 1] by simp
huffman@47160: 
huffman@47160: lemma mod_minus1_right [simp]: "a mod (-1) = 0"
huffman@47160:   using mod_minus_right [of a 1] by simp
huffman@47160: 
haftmann@54221: lemma minus_mod_self2 [simp]: 
haftmann@54221:   "(a - b) mod b = a mod b"
haftmann@54221:   by (simp add: mod_diff_right_eq)
haftmann@54221: 
haftmann@54221: lemma minus_mod_self1 [simp]: 
haftmann@54221:   "(b - a) mod b = - a mod b"
haftmann@54230:   using mod_add_self2 [of "- a" b] by simp
haftmann@54221: 
huffman@29405: end
huffman@29405: 
haftmann@54226: class semiring_div_parity = semiring_div + semiring_numeral +
haftmann@54226:   assumes parity: "a mod 2 = 0 \<or> a mod 2 = 1"
haftmann@54226: begin
haftmann@54226: 
haftmann@54226: lemma parity_cases [case_names even odd]:
haftmann@54226:   assumes "a mod 2 = 0 \<Longrightarrow> P"
haftmann@54226:   assumes "a mod 2 = 1 \<Longrightarrow> P"
haftmann@54226:   shows P
haftmann@54226:   using assms parity by blast
haftmann@54226: 
haftmann@54226: lemma not_mod_2_eq_0_eq_1 [simp]:
haftmann@54226:   "a mod 2 \<noteq> 0 \<longleftrightarrow> a mod 2 = 1"
haftmann@54226:   by (cases a rule: parity_cases) simp_all
haftmann@54226: 
haftmann@54226: lemma not_mod_2_eq_1_eq_0 [simp]:
haftmann@54226:   "a mod 2 \<noteq> 1 \<longleftrightarrow> a mod 2 = 0"
haftmann@54226:   by (cases a rule: parity_cases) simp_all
haftmann@54226: 
haftmann@54226: end
haftmann@54226: 
haftmann@25942: 
haftmann@53067: subsection {* Generic numeral division with a pragmatic type class *}
haftmann@53067: 
haftmann@53067: text {*
haftmann@53067:   The following type class contains everything necessary to formulate
haftmann@53067:   a division algorithm in ring structures with numerals, restricted
haftmann@53067:   to its positive segments.  This is its primary motiviation, and it
haftmann@53067:   could surely be formulated using a more fine-grained, more algebraic
haftmann@53067:   and less technical class hierarchy.
haftmann@53067: *}
haftmann@53067: 
haftmann@53067: class semiring_numeral_div = linordered_semidom + minus + semiring_div +
haftmann@53067:   assumes diff_invert_add1: "a + b = c \<Longrightarrow> a = c - b"
haftmann@53067:     and le_add_diff_inverse2: "b \<le> a \<Longrightarrow> a - b + b = a"
haftmann@53067:   assumes mult_div_cancel: "b * (a div b) = a - a mod b"
haftmann@53067:     and div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
haftmann@53067:     and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
haftmann@53067:     and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
haftmann@53067:     and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
haftmann@53067:     and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
haftmann@53067:     and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
haftmann@53067:     and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@53067:     and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
haftmann@53067:   assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
haftmann@53067: begin
haftmann@53067: 
haftmann@53067: lemma diff_zero [simp]:
haftmann@53067:   "a - 0 = a"
haftmann@53067:   by (rule diff_invert_add1 [symmetric]) simp
haftmann@53067: 
haftmann@54226: subclass semiring_div_parity
haftmann@54226: proof
haftmann@54226:   fix a
haftmann@54226:   show "a mod 2 = 0 \<or> a mod 2 = 1"
haftmann@54226:   proof (rule ccontr)
haftmann@54226:     assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
haftmann@54226:     then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
haftmann@54226:     have "0 < 2" by simp
haftmann@54226:     with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
haftmann@54226:     with `a mod 2 \<noteq> 0` have "0 < a mod 2" by simp
haftmann@54226:     with discrete have "1 \<le> a mod 2" by simp
haftmann@54226:     with `a mod 2 \<noteq> 1` have "1 < a mod 2" by simp
haftmann@54226:     with discrete have "2 \<le> a mod 2" by simp
haftmann@54226:     with `a mod 2 < 2` show False by simp
haftmann@54226:   qed
haftmann@53067: qed
haftmann@53067: 
haftmann@53067: lemma divmod_digit_1:
haftmann@53067:   assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
haftmann@53067:   shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
haftmann@53067:     and "a mod (2 * b) - b = a mod b" (is "?Q")
haftmann@53067: proof -
haftmann@53067:   from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
haftmann@53067:     by (auto intro: trans)
haftmann@53067:   with `0 < b` have "0 < a div b" by (auto intro: div_positive)
haftmann@53067:   then have [simp]: "1 \<le> a div b" by (simp add: discrete)
haftmann@53067:   with `0 < b` have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
haftmann@53067:   def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067:   have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067:     by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067:   from assms w_exhaust have "w = 1"
haftmann@53067:     by (auto simp add: mod_w) (insert mod_less, auto)
haftmann@53067:   with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
haftmann@53067:   have "2 * (a div (2 * b)) = a div b - w"
haftmann@53067:     by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
haftmann@53067:   with `w = 1` have div: "2 * (a div (2 * b)) = a div b - 1" by simp
haftmann@53067:   then show ?P and ?Q
haftmann@53067:     by (simp_all add: div mod diff_invert_add1 [symmetric] le_add_diff_inverse2)
haftmann@53067: qed
haftmann@53067: 
haftmann@53067: lemma divmod_digit_0:
haftmann@53067:   assumes "0 < b" and "a mod (2 * b) < b"
haftmann@53067:   shows "2 * (a div (2 * b)) = a div b" (is "?P")
haftmann@53067:     and "a mod (2 * b) = a mod b" (is "?Q")
haftmann@53067: proof -
haftmann@53067:   def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067:   have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067:     by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067:   moreover have "b \<le> a mod b + b"
haftmann@53067:   proof -
haftmann@53067:     from `0 < b` pos_mod_sign have "0 \<le> a mod b" by blast
haftmann@53067:     then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
haftmann@53067:     then show ?thesis by simp
haftmann@53067:   qed
haftmann@53067:   moreover note assms w_exhaust
haftmann@53067:   ultimately have "w = 0" by auto
haftmann@53067:   with mod_w have mod: "a mod (2 * b) = a mod b" by simp
haftmann@53067:   have "2 * (a div (2 * b)) = a div b - w"
haftmann@53067:     by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
haftmann@53067:   with `w = 0` have div: "2 * (a div (2 * b)) = a div b" by simp
haftmann@53067:   then show ?P and ?Q
haftmann@53067:     by (simp_all add: div mod)
haftmann@53067: qed
haftmann@53067: 
haftmann@53067: definition divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
haftmann@53067: where
haftmann@53067:   "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@53067: 
haftmann@53067: lemma fst_divmod [simp]:
haftmann@53067:   "fst (divmod m n) = numeral m div numeral n"
haftmann@53067:   by (simp add: divmod_def)
haftmann@53067: 
haftmann@53067: lemma snd_divmod [simp]:
haftmann@53067:   "snd (divmod m n) = numeral m mod numeral n"
haftmann@53067:   by (simp add: divmod_def)
haftmann@53067: 
haftmann@53067: definition divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
haftmann@53067: where
haftmann@53067:   "divmod_step l qr = (let (q, r) = qr
haftmann@53067:     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@53067:     else (2 * q, r))"
haftmann@53067: 
haftmann@53067: text {*
haftmann@53067:   This is a formulation of one step (referring to one digit position)
haftmann@53067:   in school-method division: compare the dividend at the current
haftmann@53070:   digit position with the remainder from previous division steps
haftmann@53067:   and evaluate accordingly.
haftmann@53067: *}
haftmann@53067: 
haftmann@53067: lemma divmod_step_eq [code]:
haftmann@53067:   "divmod_step l (q, r) = (if numeral l \<le> r
haftmann@53067:     then (2 * q + 1, r - numeral l) else (2 * q, r))"
haftmann@53067:   by (simp add: divmod_step_def)
haftmann@53067: 
haftmann@53067: lemma divmod_step_simps [simp]:
haftmann@53067:   "r < numeral l \<Longrightarrow> divmod_step l (q, r) = (2 * q, r)"
haftmann@53067:   "numeral l \<le> r \<Longrightarrow> divmod_step l (q, r) = (2 * q + 1, r - numeral l)"
haftmann@53067:   by (auto simp add: divmod_step_eq not_le)
haftmann@53067: 
haftmann@53067: text {*
haftmann@53067:   This is a formulation of school-method division.
haftmann@53067:   If the divisor is smaller than the dividend, terminate.
haftmann@53067:   If not, shift the dividend to the right until termination
haftmann@53067:   occurs and then reiterate single division steps in the
haftmann@53067:   opposite direction.
haftmann@53067: *}
haftmann@53067: 
haftmann@53067: lemma divmod_divmod_step [code]:
haftmann@53067:   "divmod m n = (if m < n then (0, numeral m)
haftmann@53067:     else divmod_step n (divmod m (Num.Bit0 n)))"
haftmann@53067: proof (cases "m < n")
haftmann@53067:   case True then have "numeral m < numeral n" by simp
haftmann@53067:   then show ?thesis
haftmann@53067:     by (simp add: prod_eq_iff div_less mod_less)
haftmann@53067: next
haftmann@53067:   case False
haftmann@53067:   have "divmod m n =
haftmann@53067:     divmod_step n (numeral m div (2 * numeral n),
haftmann@53067:       numeral m mod (2 * numeral n))"
haftmann@53067:   proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
haftmann@53067:     case True
haftmann@53067:     with divmod_step_simps
haftmann@53067:       have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067:         (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
haftmann@53067:         by blast
haftmann@53067:     moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
haftmann@53067:       have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
haftmann@53067:       and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
haftmann@53067:       by simp_all
haftmann@53067:     ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067:   next
haftmann@53067:     case False then have *: "numeral m mod (2 * numeral n) < numeral n"
haftmann@53067:       by (simp add: not_le)
haftmann@53067:     with divmod_step_simps
haftmann@53067:       have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067:         (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
haftmann@53067:         by blast
haftmann@53067:     moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
haftmann@53067:       have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
haftmann@53067:       and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
haftmann@53067:       by (simp_all only: zero_less_numeral)
haftmann@53067:     ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067:   qed
haftmann@53067:   then have "divmod m n =
haftmann@53067:     divmod_step n (numeral m div numeral (Num.Bit0 n),
haftmann@53067:       numeral m mod numeral (Num.Bit0 n))"
haftmann@53067:     by (simp only: numeral.simps distrib mult_1) 
haftmann@53067:   then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
haftmann@53067:     by (simp add: divmod_def)
haftmann@53067:   with False show ?thesis by simp
haftmann@53067: qed
haftmann@53067: 
haftmann@53069: lemma divmod_cancel [code]:
haftmann@53069:   "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
haftmann@53069:   "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
haftmann@53069: proof -
haftmann@53069:   have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
haftmann@53069:     "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
haftmann@53069:     by (simp_all only: numeral_mult numeral.simps distrib) simp_all
haftmann@53069:   have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
haftmann@53069:   then show ?P and ?Q
haftmann@53069:     by (simp_all add: prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
haftmann@53069:       div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2] add.commute del: numeral_times_numeral)
haftmann@53069:  qed
haftmann@53069: 
haftmann@53067: end
haftmann@53067: 
haftmann@53067: hide_fact (open) diff_invert_add1 le_add_diff_inverse2 diff_zero
haftmann@53067:   -- {* restore simple accesses for more general variants of theorems *}
haftmann@53067: 
haftmann@53067:   
haftmann@26100: subsection {* Division on @{typ nat} *}
haftmann@26100: 
haftmann@26100: text {*
haftmann@26100:   We define @{const div} and @{const mod} on @{typ nat} by means
haftmann@26100:   of a characteristic relation with two input arguments
haftmann@26100:   @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
haftmann@26100:   @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
haftmann@26100: *}
haftmann@26100: 
haftmann@33340: definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
haftmann@33340:   "divmod_nat_rel m n qr \<longleftrightarrow>
haftmann@30923:     m = fst qr * n + snd qr \<and>
haftmann@30923:       (if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"
haftmann@26100: 
haftmann@33340: text {* @{const divmod_nat_rel} is total: *}
haftmann@26100: 
haftmann@33340: lemma divmod_nat_rel_ex:
haftmann@33340:   obtains q r where "divmod_nat_rel m n (q, r)"
haftmann@26100: proof (cases "n = 0")
haftmann@30923:   case True  with that show thesis
haftmann@33340:     by (auto simp add: divmod_nat_rel_def)
haftmann@26100: next
haftmann@26100:   case False
haftmann@26100:   have "\<exists>q r. m = q * n + r \<and> r < n"
haftmann@26100:   proof (induct m)
haftmann@26100:     case 0 with `n \<noteq> 0`
haftmann@26100:     have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
haftmann@26100:     then show ?case by blast
haftmann@26100:   next
haftmann@26100:     case (Suc m) then obtain q' r'
haftmann@26100:       where m: "m = q' * n + r'" and n: "r' < n" by auto
haftmann@26100:     then show ?case proof (cases "Suc r' < n")
haftmann@26100:       case True
haftmann@26100:       from m n have "Suc m = q' * n + Suc r'" by simp
haftmann@26100:       with True show ?thesis by blast
haftmann@26100:     next
haftmann@26100:       case False then have "n \<le> Suc r'" by auto
haftmann@26100:       moreover from n have "Suc r' \<le> n" by auto
haftmann@26100:       ultimately have "n = Suc r'" by auto
haftmann@26100:       with m have "Suc m = Suc q' * n + 0" by simp
haftmann@26100:       with `n \<noteq> 0` show ?thesis by blast
haftmann@26100:     qed
haftmann@26100:   qed
haftmann@26100:   with that show thesis
haftmann@33340:     using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def)
haftmann@26100: qed
haftmann@26100: 
haftmann@33340: text {* @{const divmod_nat_rel} is injective: *}
haftmann@26100: 
haftmann@33340: lemma divmod_nat_rel_unique:
haftmann@33340:   assumes "divmod_nat_rel m n qr"
haftmann@33340:     and "divmod_nat_rel m n qr'"
haftmann@30923:   shows "qr = qr'"
haftmann@26100: proof (cases "n = 0")
haftmann@26100:   case True with assms show ?thesis
haftmann@30923:     by (cases qr, cases qr')
haftmann@33340:       (simp add: divmod_nat_rel_def)
haftmann@26100: next
haftmann@26100:   case False
haftmann@26100:   have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
haftmann@26100:   apply (rule leI)
haftmann@26100:   apply (subst less_iff_Suc_add)
haftmann@26100:   apply (auto simp add: add_mult_distrib)
haftmann@26100:   done
wenzelm@53374:   from `n \<noteq> 0` assms have *: "fst qr = fst qr'"
haftmann@33340:     by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
wenzelm@53374:   with assms have "snd qr = snd qr'"
haftmann@33340:     by (simp add: divmod_nat_rel_def)
wenzelm@53374:   with * show ?thesis by (cases qr, cases qr') simp
haftmann@26100: qed
haftmann@26100: 
haftmann@26100: text {*
haftmann@26100:   We instantiate divisibility on the natural numbers by
haftmann@33340:   means of @{const divmod_nat_rel}:
haftmann@26100: *}
haftmann@25942: 
haftmann@33340: definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
haftmann@37767:   "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
haftmann@30923: 
haftmann@33340: lemma divmod_nat_rel_divmod_nat:
haftmann@33340:   "divmod_nat_rel m n (divmod_nat m n)"
haftmann@30923: proof -
haftmann@33340:   from divmod_nat_rel_ex
haftmann@33340:     obtain qr where rel: "divmod_nat_rel m n qr" .
haftmann@30923:   then show ?thesis
haftmann@33340:   by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
haftmann@30923: qed
haftmann@30923: 
huffman@47135: lemma divmod_nat_unique:
haftmann@33340:   assumes "divmod_nat_rel m n qr" 
haftmann@33340:   shows "divmod_nat m n = qr"
haftmann@33340:   using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
haftmann@26100: 
huffman@46551: instantiation nat :: semiring_div
huffman@46551: begin
huffman@46551: 
haftmann@26100: definition div_nat where
haftmann@33340:   "m div n = fst (divmod_nat m n)"
haftmann@26100: 
huffman@46551: lemma fst_divmod_nat [simp]:
huffman@46551:   "fst (divmod_nat m n) = m div n"
huffman@46551:   by (simp add: div_nat_def)
huffman@46551: 
haftmann@26100: definition mod_nat where
haftmann@33340:   "m mod n = snd (divmod_nat m n)"
haftmann@25571: 
huffman@46551: lemma snd_divmod_nat [simp]:
huffman@46551:   "snd (divmod_nat m n) = m mod n"
huffman@46551:   by (simp add: mod_nat_def)
huffman@46551: 
haftmann@33340: lemma divmod_nat_div_mod:
haftmann@33340:   "divmod_nat m n = (m div n, m mod n)"
huffman@46551:   by (simp add: prod_eq_iff)
haftmann@26100: 
huffman@47135: lemma div_nat_unique:
haftmann@33340:   assumes "divmod_nat_rel m n (q, r)" 
haftmann@26100:   shows "m div n = q"
huffman@47135:   using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
huffman@47135: 
huffman@47135: lemma mod_nat_unique:
haftmann@33340:   assumes "divmod_nat_rel m n (q, r)" 
haftmann@26100:   shows "m mod n = r"
huffman@47135:   using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
haftmann@25571: 
haftmann@33340: lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
huffman@46551:   using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
paulson@14267: 
huffman@47136: lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
huffman@47136:   by (simp add: divmod_nat_unique divmod_nat_rel_def)
huffman@47136: 
huffman@47136: lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
huffman@47136:   by (simp add: divmod_nat_unique divmod_nat_rel_def)
haftmann@25942: 
huffman@47137: lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
huffman@47137:   by (simp add: divmod_nat_unique divmod_nat_rel_def)
haftmann@25942: 
haftmann@33340: lemma divmod_nat_step:
haftmann@26100:   assumes "0 < n" and "n \<le> m"
haftmann@33340:   shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
huffman@47135: proof (rule divmod_nat_unique)
huffman@47134:   have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"
huffman@47134:     by (rule divmod_nat_rel)
huffman@47134:   thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"
huffman@47134:     unfolding divmod_nat_rel_def using assms by auto
haftmann@26100: qed
haftmann@25942: 
wenzelm@26300: text {* The ''recursion'' equations for @{const div} and @{const mod} *}
haftmann@26100: 
haftmann@26100: lemma div_less [simp]:
haftmann@26100:   fixes m n :: nat
haftmann@26100:   assumes "m < n"
haftmann@26100:   shows "m div n = 0"
huffman@46551:   using assms divmod_nat_base by (simp add: prod_eq_iff)
haftmann@25942: 
haftmann@26100: lemma le_div_geq:
haftmann@26100:   fixes m n :: nat
haftmann@26100:   assumes "0 < n" and "n \<le> m"
haftmann@26100:   shows "m div n = Suc ((m - n) div n)"
huffman@46551:   using assms divmod_nat_step by (simp add: prod_eq_iff)
paulson@14267: 
haftmann@26100: lemma mod_less [simp]:
haftmann@26100:   fixes m n :: nat
haftmann@26100:   assumes "m < n"
haftmann@26100:   shows "m mod n = m"
huffman@46551:   using assms divmod_nat_base by (simp add: prod_eq_iff)
haftmann@26100: 
haftmann@26100: lemma le_mod_geq:
haftmann@26100:   fixes m n :: nat
haftmann@26100:   assumes "n \<le> m"
haftmann@26100:   shows "m mod n = (m - n) mod n"
huffman@46551:   using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
paulson@14267: 
huffman@47136: instance proof
huffman@47136:   fix m n :: nat
huffman@47136:   show "m div n * n + m mod n = m"
huffman@47136:     using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
huffman@47136: next
huffman@47136:   fix m n q :: nat
huffman@47136:   assume "n \<noteq> 0"
huffman@47136:   then show "(q + m * n) div n = m + q div n"
huffman@47136:     by (induct m) (simp_all add: le_div_geq)
huffman@47136: next
huffman@47136:   fix m n q :: nat
huffman@47136:   assume "m \<noteq> 0"
huffman@47136:   hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
huffman@47136:     unfolding divmod_nat_rel_def
huffman@47136:     by (auto split: split_if_asm, simp_all add: algebra_simps)
huffman@47136:   moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
huffman@47136:   ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
huffman@47136:   thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
huffman@47136: next
huffman@47136:   fix n :: nat show "n div 0 = 0"
haftmann@33340:     by (simp add: div_nat_def divmod_nat_zero)
huffman@47136: next
huffman@47136:   fix n :: nat show "0 div n = 0"
huffman@47136:     by (simp add: div_nat_def divmod_nat_zero_left)
haftmann@25942: qed
haftmann@26100: 
haftmann@25942: end
paulson@14267: 
haftmann@33361: lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@33361:   let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
blanchet@55414:   by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
haftmann@33361: 
haftmann@26100: text {* Simproc for cancelling @{const div} and @{const mod} *}
haftmann@25942: 
wenzelm@51299: ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
wenzelm@51299: 
haftmann@30934: ML {*
wenzelm@43594: structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
wenzelm@41550: (
haftmann@30934:   val div_name = @{const_name div};
haftmann@30934:   val mod_name = @{const_name mod};
haftmann@30934:   val mk_binop = HOLogic.mk_binop;
huffman@48561:   val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
huffman@48561:   val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
huffman@48561:   fun mk_sum [] = HOLogic.zero
huffman@48561:     | mk_sum [t] = t
huffman@48561:     | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
huffman@48561:   fun dest_sum tm =
huffman@48561:     if HOLogic.is_zero tm then []
huffman@48561:     else
huffman@48561:       (case try HOLogic.dest_Suc tm of
huffman@48561:         SOME t => HOLogic.Suc_zero :: dest_sum t
huffman@48561:       | NONE =>
huffman@48561:           (case try dest_plus tm of
huffman@48561:             SOME (t, u) => dest_sum t @ dest_sum u
huffman@48561:           | NONE => [tm]));
haftmann@25942: 
haftmann@30934:   val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
paulson@14267: 
haftmann@30934:   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
haftmann@57514:     (@{thm add_0_left} :: @{thm add_0_right} :: @{thms ac_simps}))
wenzelm@41550: )
haftmann@25942: *}
haftmann@25942: 
wenzelm@43594: simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}
wenzelm@43594: 
haftmann@26100: 
haftmann@26100: subsubsection {* Quotient *}
haftmann@26100: 
haftmann@26100: lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
nipkow@29667: by (simp add: le_div_geq linorder_not_less)
haftmann@26100: 
haftmann@26100: lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
nipkow@29667: by (simp add: div_geq)
haftmann@26100: 
haftmann@26100: lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
nipkow@29667: by simp
haftmann@26100: 
haftmann@26100: lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
nipkow@29667: by simp
haftmann@26100: 
haftmann@53066: lemma div_positive:
haftmann@53066:   fixes m n :: nat
haftmann@53066:   assumes "n > 0"
haftmann@53066:   assumes "m \<ge> n"
haftmann@53066:   shows "m div n > 0"
haftmann@53066: proof -
haftmann@53066:   from `m \<ge> n` obtain q where "m = n + q"
haftmann@53066:     by (auto simp add: le_iff_add)
haftmann@53066:   with `n > 0` show ?thesis by simp
haftmann@53066: qed
haftmann@53066: 
haftmann@25942: 
haftmann@25942: subsubsection {* Remainder *}
haftmann@25942: 
haftmann@26100: lemma mod_less_divisor [simp]:
haftmann@26100:   fixes m n :: nat
haftmann@26100:   assumes "n > 0"
haftmann@26100:   shows "m mod n < (n::nat)"
haftmann@33340:   using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
paulson@14267: 
haftmann@51173: lemma mod_Suc_le_divisor [simp]:
haftmann@51173:   "m mod Suc n \<le> n"
haftmann@51173:   using mod_less_divisor [of "Suc n" m] by arith
haftmann@51173: 
haftmann@26100: lemma mod_less_eq_dividend [simp]:
haftmann@26100:   fixes m n :: nat
haftmann@26100:   shows "m mod n \<le> m"
haftmann@26100: proof (rule add_leD2)
haftmann@26100:   from mod_div_equality have "m div n * n + m mod n = m" .
haftmann@26100:   then show "m div n * n + m mod n \<le> m" by auto
haftmann@26100: qed
haftmann@26100: 
haftmann@26100: lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
nipkow@29667: by (simp add: le_mod_geq linorder_not_less)
paulson@14267: 
haftmann@26100: lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
nipkow@29667: by (simp add: le_mod_geq)
haftmann@26100: 
paulson@14267: lemma mod_1 [simp]: "m mod Suc 0 = 0"
nipkow@29667: by (induct m) (simp_all add: mod_geq)
paulson@14267: 
paulson@14267: (* a simple rearrangement of mod_div_equality: *)
paulson@14267: lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
huffman@47138:   using mod_div_equality2 [of n m] by arith
paulson@14267: 
nipkow@15439: lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
wenzelm@22718:   apply (drule mod_less_divisor [where m = m])
wenzelm@22718:   apply simp
wenzelm@22718:   done
paulson@14267: 
haftmann@26100: subsubsection {* Quotient and Remainder *}
paulson@14267: 
haftmann@33340: lemma divmod_nat_rel_mult1_eq:
bulwahn@46552:   "divmod_nat_rel b c (q, r)
haftmann@33340:    \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
haftmann@33340: by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267: 
haftmann@30923: lemma div_mult1_eq:
haftmann@30923:   "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
huffman@47135: by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267: 
haftmann@33340: lemma divmod_nat_rel_add1_eq:
bulwahn@46552:   "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
haftmann@33340:    \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
haftmann@33340: by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267: 
paulson@14267: (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267: lemma div_add1_eq:
nipkow@25134:   "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
huffman@47135: by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267: 
paulson@14267: lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@22718:   apply (cut_tac m = q and n = c in mod_less_divisor)
wenzelm@22718:   apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
wenzelm@22718:   apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
wenzelm@22718:   apply (simp add: add_mult_distrib2)
wenzelm@22718:   done
paulson@10559: 
haftmann@33340: lemma divmod_nat_rel_mult2_eq:
bulwahn@46552:   "divmod_nat_rel a b (q, r)
haftmann@33340:    \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
haftmann@57514: by (auto simp add: mult.commute mult.left_commute divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)
paulson@14267: 
blanchet@55085: lemma div_mult2_eq: "a div (b * c) = (a div b) div (c::nat)"
huffman@47135: by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
paulson@14267: 
blanchet@55085: lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"
haftmann@57512: by (auto simp add: mult.commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
paulson@14267: 
paulson@14267: 
huffman@46551: subsubsection {* Further Facts about Quotient and Remainder *}
paulson@14267: 
paulson@14267: lemma div_1 [simp]: "m div Suc 0 = m"
nipkow@29667: by (induct m) (simp_all add: div_geq)
paulson@14267: 
paulson@14267: (* Monotonicity of div in first argument *)
haftmann@30923: lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718:     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267: apply (case_tac "k=0", simp)
paulson@15251: apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267: apply (case_tac "n<k")
paulson@14267: (* 1  case n<k *)
paulson@14267: apply simp
paulson@14267: (* 2  case n >= k *)
paulson@14267: apply (case_tac "m<k")
paulson@14267: (* 2.1  case m<k *)
paulson@14267: apply simp
paulson@14267: (* 2.2  case m>=k *)
nipkow@15439: apply (simp add: div_geq diff_le_mono)
paulson@14267: done
paulson@14267: 
paulson@14267: (* Antimonotonicity of div in second argument *)
paulson@14267: lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267: apply (subgoal_tac "0<n")
wenzelm@22718:  prefer 2 apply simp
paulson@15251: apply (induct_tac k rule: nat_less_induct)
paulson@14267: apply (rename_tac "k")
paulson@14267: apply (case_tac "k<n", simp)
paulson@14267: apply (subgoal_tac "~ (k<m) ")
wenzelm@22718:  prefer 2 apply simp
paulson@14267: apply (simp add: div_geq)
paulson@15251: apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267:  prefer 2
paulson@14267:  apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267: apply (rule le_trans, simp)
nipkow@15439: apply (simp)
paulson@14267: done
paulson@14267: 
paulson@14267: lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267: apply (case_tac "n=0", simp)
paulson@14267: apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267: apply (rule div_le_mono2)
paulson@14267: apply (simp_all (no_asm_simp))
paulson@14267: done
paulson@14267: 
wenzelm@22718: (* Similar for "less than" *)
huffman@47138: lemma div_less_dividend [simp]:
huffman@47138:   "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
huffman@47138: apply (induct m rule: nat_less_induct)
paulson@14267: apply (rename_tac "m")
paulson@14267: apply (case_tac "m<n", simp)
paulson@14267: apply (subgoal_tac "0<n")
wenzelm@22718:  prefer 2 apply simp
paulson@14267: apply (simp add: div_geq)
paulson@14267: apply (case_tac "n<m")
paulson@15251:  apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267:   apply (rule impI less_trans_Suc)+
paulson@14267: apply assumption
nipkow@15439:   apply (simp_all)
paulson@14267: done
paulson@14267: 
paulson@14267: text{*A fact for the mutilated chess board*}
paulson@14267: lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267: apply (case_tac "n=0", simp)
paulson@15251: apply (induct "m" rule: nat_less_induct)
paulson@14267: apply (case_tac "Suc (na) <n")
paulson@14267: (* case Suc(na) < n *)
paulson@14267: apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267: (* case n \<le> Suc(na) *)
paulson@16796: apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439: apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267: done
paulson@14267: 
paulson@14267: lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
nipkow@29667: by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084: 
wenzelm@22718: lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267: 
paulson@14267: (*Loses information, namely we also have r<d provided d is nonzero*)
haftmann@57514: lemma mod_eqD:
haftmann@57514:   fixes m d r q :: nat
haftmann@57514:   assumes "m mod d = r"
haftmann@57514:   shows "\<exists>q. m = r + q * d"
haftmann@57514: proof -
haftmann@57514:   from mod_div_equality obtain q where "q * d + m mod d = m" by blast
haftmann@57514:   with assms have "m = r + q * d" by simp
haftmann@57514:   then show ?thesis ..
haftmann@57514: qed
paulson@14267: 
nipkow@13152: lemma split_div:
nipkow@13189:  "P(n div k :: nat) =
nipkow@13189:  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189:  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189: proof
nipkow@13189:   assume P: ?P
nipkow@13189:   show ?Q
nipkow@13189:   proof (cases)
nipkow@13189:     assume "k = 0"
haftmann@27651:     with P show ?Q by simp
nipkow@13189:   next
nipkow@13189:     assume not0: "k \<noteq> 0"
nipkow@13189:     thus ?Q
nipkow@13189:     proof (simp, intro allI impI)
nipkow@13189:       fix i j
nipkow@13189:       assume n: "n = k*i + j" and j: "j < k"
nipkow@13189:       show "P i"
nipkow@13189:       proof (cases)
wenzelm@22718:         assume "i = 0"
wenzelm@22718:         with n j P show "P i" by simp
nipkow@13189:       next
wenzelm@22718:         assume "i \<noteq> 0"
haftmann@57514:         with not0 n j P show "P i" by(simp add:ac_simps)
nipkow@13189:       qed
nipkow@13189:     qed
nipkow@13189:   qed
nipkow@13189: next
nipkow@13189:   assume Q: ?Q
nipkow@13189:   show ?P
nipkow@13189:   proof (cases)
nipkow@13189:     assume "k = 0"
haftmann@27651:     with Q show ?P by simp
nipkow@13189:   next
nipkow@13189:     assume not0: "k \<noteq> 0"
nipkow@13189:     with Q have R: ?R by simp
nipkow@13189:     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517:     show ?P by simp
nipkow@13189:   qed
nipkow@13189: qed
nipkow@13189: 
berghofe@13882: lemma split_div_lemma:
haftmann@26100:   assumes "0 < n"
haftmann@26100:   shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@26100: proof
haftmann@26100:   assume ?rhs
haftmann@26100:   with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
haftmann@26100:   then have A: "n * q \<le> m" by simp
haftmann@26100:   have "n - (m mod n) > 0" using mod_less_divisor assms by auto
haftmann@26100:   then have "m < m + (n - (m mod n))" by simp
haftmann@26100:   then have "m < n + (m - (m mod n))" by simp
haftmann@26100:   with nq have "m < n + n * q" by simp
haftmann@26100:   then have B: "m < n * Suc q" by simp
haftmann@26100:   from A B show ?lhs ..
haftmann@26100: next
haftmann@26100:   assume P: ?lhs
haftmann@33340:   then have "divmod_nat_rel m n (q, m - n * q)"
haftmann@57514:     unfolding divmod_nat_rel_def by (auto simp add: ac_simps)
haftmann@33340:   with divmod_nat_rel_unique divmod_nat_rel [of m n]
haftmann@30923:   have "(q, m - n * q) = (m div n, m mod n)" by auto
haftmann@30923:   then show ?rhs by simp
haftmann@26100: qed
berghofe@13882: 
berghofe@13882: theorem split_div':
berghofe@13882:   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267:    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
berghofe@13882:   apply (case_tac "0 < n")
berghofe@13882:   apply (simp only: add: split_div_lemma)
haftmann@27651:   apply simp_all
berghofe@13882:   done
berghofe@13882: 
nipkow@13189: lemma split_mod:
nipkow@13189:  "P(n mod k :: nat) =
nipkow@13189:  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189:  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189: proof
nipkow@13189:   assume P: ?P
nipkow@13189:   show ?Q
nipkow@13189:   proof (cases)
nipkow@13189:     assume "k = 0"
haftmann@27651:     with P show ?Q by simp
nipkow@13189:   next
nipkow@13189:     assume not0: "k \<noteq> 0"
nipkow@13189:     thus ?Q
nipkow@13189:     proof (simp, intro allI impI)
nipkow@13189:       fix i j
nipkow@13189:       assume "n = k*i + j" "j < k"
haftmann@57514:       thus "P j" using not0 P by(simp add:ac_simps ac_simps)
nipkow@13189:     qed
nipkow@13189:   qed
nipkow@13189: next
nipkow@13189:   assume Q: ?Q
nipkow@13189:   show ?P
nipkow@13189:   proof (cases)
nipkow@13189:     assume "k = 0"
haftmann@27651:     with Q show ?P by simp
nipkow@13189:   next
nipkow@13189:     assume not0: "k \<noteq> 0"
nipkow@13189:     with Q have R: ?R by simp
nipkow@13189:     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517:     show ?P by simp
nipkow@13189:   qed
nipkow@13189: qed
nipkow@13189: 
berghofe@13882: theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
huffman@47138:   using mod_div_equality [of m n] by arith
huffman@47138: 
huffman@47138: lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"
huffman@47138:   using mod_div_equality [of m n] by arith
huffman@47138: (* FIXME: very similar to mult_div_cancel *)
haftmann@22800: 
noschinl@52398: lemma div_eq_dividend_iff: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1"
noschinl@52398:   apply rule
noschinl@52398:   apply (cases "b = 0")
noschinl@52398:   apply simp_all
noschinl@52398:   apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)
noschinl@52398:   done
noschinl@52398: 
haftmann@22800: 
huffman@46551: subsubsection {* An ``induction'' law for modulus arithmetic. *}
paulson@14640: 
paulson@14640: lemma mod_induct_0:
paulson@14640:   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640:   and base: "P i" and i: "i<p"
paulson@14640:   shows "P 0"
paulson@14640: proof (rule ccontr)
paulson@14640:   assume contra: "\<not>(P 0)"
paulson@14640:   from i have p: "0<p" by simp
paulson@14640:   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640:   proof
paulson@14640:     fix k
paulson@14640:     show "?A k"
paulson@14640:     proof (induct k)
paulson@14640:       show "?A 0" by simp  -- "by contradiction"
paulson@14640:     next
paulson@14640:       fix n
paulson@14640:       assume ih: "?A n"
paulson@14640:       show "?A (Suc n)"
paulson@14640:       proof (clarsimp)
wenzelm@22718:         assume y: "P (p - Suc n)"
wenzelm@22718:         have n: "Suc n < p"
wenzelm@22718:         proof (rule ccontr)
wenzelm@22718:           assume "\<not>(Suc n < p)"
wenzelm@22718:           hence "p - Suc n = 0"
wenzelm@22718:             by simp
wenzelm@22718:           with y contra show "False"
wenzelm@22718:             by simp
wenzelm@22718:         qed
wenzelm@22718:         hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718:         from p have "p - Suc n < p" by arith
wenzelm@22718:         with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718:           by blast
wenzelm@22718:         show "False"
wenzelm@22718:         proof (cases "n=0")
wenzelm@22718:           case True
wenzelm@22718:           with z n2 contra show ?thesis by simp
wenzelm@22718:         next
wenzelm@22718:           case False
wenzelm@22718:           with p have "p-n < p" by arith
wenzelm@22718:           with z n2 False ih show ?thesis by simp
wenzelm@22718:         qed
paulson@14640:       qed
paulson@14640:     qed
paulson@14640:   qed
paulson@14640:   moreover
paulson@14640:   from i obtain k where "0<k \<and> i+k=p"
paulson@14640:     by (blast dest: less_imp_add_positive)
paulson@14640:   hence "0<k \<and> i=p-k" by auto
paulson@14640:   moreover
paulson@14640:   note base
paulson@14640:   ultimately
paulson@14640:   show "False" by blast
paulson@14640: qed
paulson@14640: 
paulson@14640: lemma mod_induct:
paulson@14640:   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640:   and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640:   shows "P j"
paulson@14640: proof -
paulson@14640:   have "\<forall>j<p. P j"
paulson@14640:   proof
paulson@14640:     fix j
paulson@14640:     show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640:     proof (induct j)
paulson@14640:       from step base i show "?A 0"
wenzelm@22718:         by (auto elim: mod_induct_0)
paulson@14640:     next
paulson@14640:       fix k
paulson@14640:       assume ih: "?A k"
paulson@14640:       show "?A (Suc k)"
paulson@14640:       proof
wenzelm@22718:         assume suc: "Suc k < p"
wenzelm@22718:         hence k: "k<p" by simp
wenzelm@22718:         with ih have "P k" ..
wenzelm@22718:         with step k have "P (Suc k mod p)"
wenzelm@22718:           by blast
wenzelm@22718:         moreover
wenzelm@22718:         from suc have "Suc k mod p = Suc k"
wenzelm@22718:           by simp
wenzelm@22718:         ultimately
wenzelm@22718:         show "P (Suc k)" by simp
paulson@14640:       qed
paulson@14640:     qed
paulson@14640:   qed
paulson@14640:   with j show ?thesis by blast
paulson@14640: qed
paulson@14640: 
haftmann@33296: lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
huffman@47138:   by (simp add: numeral_2_eq_2 le_div_geq)
huffman@47138: 
huffman@47138: lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
huffman@47138:   by (simp add: numeral_2_eq_2 le_mod_geq)
haftmann@33296: 
haftmann@33296: lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
huffman@47217: by (simp add: mult_2 [symmetric])
haftmann@33296: 
haftmann@33296: lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
haftmann@33296: proof -
boehmes@35815:   { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
haftmann@33296:   moreover have "m mod 2 < 2" by simp
haftmann@33296:   ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
haftmann@33296:   then show ?thesis by auto
haftmann@33296: qed
haftmann@33296: 
haftmann@33296: text{*These lemmas collapse some needless occurrences of Suc:
haftmann@33296:     at least three Sucs, since two and fewer are rewritten back to Suc again!
haftmann@33296:     We already have some rules to simplify operands smaller than 3.*}
haftmann@33296: 
haftmann@33296: lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
haftmann@33296: by (simp add: Suc3_eq_add_3)
haftmann@33296: 
haftmann@33296: lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
haftmann@33296: by (simp add: Suc3_eq_add_3)
haftmann@33296: 
haftmann@33296: lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
haftmann@33296: by (simp add: Suc3_eq_add_3)
haftmann@33296: 
haftmann@33296: lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
haftmann@33296: by (simp add: Suc3_eq_add_3)
haftmann@33296: 
huffman@47108: lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
huffman@47108: lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
haftmann@33296: 
haftmann@33361: 
haftmann@33361: lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
haftmann@33361: apply (induct "m")
haftmann@33361: apply (simp_all add: mod_Suc)
haftmann@33361: done
haftmann@33361: 
huffman@47108: declare Suc_times_mod_eq [of "numeral w", simp] for w
haftmann@33361: 
huffman@47138: lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
huffman@47138: by (simp add: div_le_mono)
haftmann@33361: 
haftmann@33361: lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
haftmann@33361: by (cases n) simp_all
haftmann@33361: 
boehmes@35815: lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
boehmes@35815: proof -
boehmes@35815:   from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
boehmes@35815:   from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp 
boehmes@35815: qed
haftmann@33361: 
haftmann@33361:   (* Potential use of algebra : Equality modulo n*)
haftmann@33361: lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
haftmann@57514: by (simp add: ac_simps ac_simps)
haftmann@33361: 
haftmann@33361: lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
haftmann@33361: proof -
haftmann@33361:   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
haftmann@33361:   also have "... = Suc m mod n" by (rule mod_mult_self3) 
haftmann@33361:   finally show ?thesis .
haftmann@33361: qed
haftmann@33361: 
haftmann@33361: lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
haftmann@33361: apply (subst mod_Suc [of m]) 
haftmann@33361: apply (subst mod_Suc [of "m mod n"], simp) 
haftmann@33361: done
haftmann@33361: 
huffman@47108: lemma mod_2_not_eq_zero_eq_one_nat:
huffman@47108:   fixes n :: nat
huffman@47108:   shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
huffman@47108:   by simp
huffman@47108: 
haftmann@53067: instance nat :: semiring_numeral_div
haftmann@53067:   by intro_classes (auto intro: div_positive simp add: mult_div_cancel mod_mult2_eq div_mult2_eq)
haftmann@53067: 
haftmann@33361: 
haftmann@33361: subsection {* Division on @{typ int} *}
haftmann@33361: 
haftmann@33361: definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
haftmann@33361:     --{*definition of quotient and remainder*}
huffman@47139:   "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
huffman@47139:     (if 0 < b then 0 \<le> r \<and> r < b else if b < 0 then b < r \<and> r \<le> 0 else q = 0))"
haftmann@33361: 
haftmann@53067: text {*
haftmann@53067:   The following algorithmic devlopment actually echos what has already
haftmann@53067:   been developed in class @{class semiring_numeral_div}.  In the long
haftmann@53067:   run it seems better to derive division on @{typ int} just from
haftmann@53067:   division on @{typ nat} and instantiate @{class semiring_numeral_div}
haftmann@53067:   accordingly.
haftmann@53067: *}
haftmann@53067: 
haftmann@33361: definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
haftmann@33361:     --{*for the division algorithm*}
huffman@47108:     "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
haftmann@33361:                          else (2 * q, r))"
haftmann@33361: 
haftmann@33361: text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
haftmann@33361: function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361:   "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
haftmann@33361:      else adjust b (posDivAlg a (2 * b)))"
haftmann@33361: by auto
haftmann@33361: termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
haftmann@33361:   (auto simp add: mult_2)
haftmann@33361: 
haftmann@33361: text{*algorithm for the case @{text "a<0, b>0"}*}
haftmann@33361: function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361:   "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
haftmann@33361:      else adjust b (negDivAlg a (2 * b)))"
haftmann@33361: by auto
haftmann@33361: termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
haftmann@33361:   (auto simp add: mult_2)
haftmann@33361: 
haftmann@33361: text{*algorithm for the general case @{term "b\<noteq>0"}*}
haftmann@33361: 
haftmann@33361: definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361:     --{*The full division algorithm considers all possible signs for a, b
haftmann@33361:        including the special case @{text "a=0, b<0"} because 
haftmann@33361:        @{term negDivAlg} requires @{term "a<0"}.*}
haftmann@33361:   "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
haftmann@33361:                   else if a = 0 then (0, 0)
huffman@46560:                        else apsnd uminus (negDivAlg (-a) (-b))
haftmann@33361:                else 
haftmann@33361:                   if 0 < b then negDivAlg a b
huffman@46560:                   else apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361: 
haftmann@33361: instantiation int :: Divides.div
haftmann@33361: begin
haftmann@33361: 
huffman@46551: definition div_int where
haftmann@33361:   "a div b = fst (divmod_int a b)"
haftmann@33361: 
huffman@46551: lemma fst_divmod_int [simp]:
huffman@46551:   "fst (divmod_int a b) = a div b"
huffman@46551:   by (simp add: div_int_def)
huffman@46551: 
huffman@46551: definition mod_int where
huffman@46560:   "a mod b = snd (divmod_int a b)"
haftmann@33361: 
huffman@46551: lemma snd_divmod_int [simp]:
huffman@46551:   "snd (divmod_int a b) = a mod b"
huffman@46551:   by (simp add: mod_int_def)
huffman@46551: 
haftmann@33361: instance ..
haftmann@33361: 
paulson@3366: end
haftmann@33361: 
haftmann@33361: lemma divmod_int_mod_div:
haftmann@33361:   "divmod_int p q = (p div q, p mod q)"
huffman@46551:   by (simp add: prod_eq_iff)
haftmann@33361: 
haftmann@33361: text{*
haftmann@33361: Here is the division algorithm in ML:
haftmann@33361: 
haftmann@33361: \begin{verbatim}
haftmann@33361:     fun posDivAlg (a,b) =
haftmann@33361:       if a<b then (0,a)
haftmann@33361:       else let val (q,r) = posDivAlg(a, 2*b)
haftmann@33361:                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
haftmann@33361:            end
haftmann@33361: 
haftmann@33361:     fun negDivAlg (a,b) =
haftmann@33361:       if 0\<le>a+b then (~1,a+b)
haftmann@33361:       else let val (q,r) = negDivAlg(a, 2*b)
haftmann@33361:                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
haftmann@33361:            end;
haftmann@33361: 
haftmann@33361:     fun negateSnd (q,r:int) = (q,~r);
haftmann@33361: 
haftmann@33361:     fun divmod (a,b) = if 0\<le>a then 
haftmann@33361:                           if b>0 then posDivAlg (a,b) 
haftmann@33361:                            else if a=0 then (0,0)
haftmann@33361:                                 else negateSnd (negDivAlg (~a,~b))
haftmann@33361:                        else 
haftmann@33361:                           if 0<b then negDivAlg (a,b)
haftmann@33361:                           else        negateSnd (posDivAlg (~a,~b));
haftmann@33361: \end{verbatim}
haftmann@33361: *}
haftmann@33361: 
haftmann@33361: 
huffman@46551: subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *}
haftmann@33361: 
haftmann@33361: lemma unique_quotient_lemma:
haftmann@33361:      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
haftmann@33361:       ==> q' \<le> (q::int)"
haftmann@33361: apply (subgoal_tac "r' + b * (q'-q) \<le> r")
haftmann@33361:  prefer 2 apply (simp add: right_diff_distrib)
haftmann@33361: apply (subgoal_tac "0 < b * (1 + q - q') ")
haftmann@33361: apply (erule_tac [2] order_le_less_trans)
webertj@49962:  prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361: apply (subgoal_tac "b * q' < b * (1 + q) ")
webertj@49962:  prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361: apply (simp add: mult_less_cancel_left)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma unique_quotient_lemma_neg:
haftmann@33361:      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
haftmann@33361:       ==> q \<le> (q'::int)"
haftmann@33361: by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
haftmann@33361:     auto)
haftmann@33361: 
haftmann@33361: lemma unique_quotient:
bulwahn@46552:      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
haftmann@33361:       ==> q = q'"
haftmann@33361: apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
haftmann@33361: apply (blast intro: order_antisym
haftmann@33361:              dest: order_eq_refl [THEN unique_quotient_lemma] 
haftmann@33361:              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
haftmann@33361: done
haftmann@33361: 
haftmann@33361: 
haftmann@33361: lemma unique_remainder:
bulwahn@46552:      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
haftmann@33361:       ==> r = r'"
haftmann@33361: apply (subgoal_tac "q = q'")
haftmann@33361:  apply (simp add: divmod_int_rel_def)
haftmann@33361: apply (blast intro: unique_quotient)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: 
huffman@46551: subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}
haftmann@33361: 
haftmann@33361: text{*And positive divisors*}
haftmann@33361: 
haftmann@33361: lemma adjust_eq [simp]:
huffman@47108:      "adjust b (q, r) = 
huffman@47108:       (let diff = r - b in  
huffman@47108:         if 0 \<le> diff then (2 * q + 1, diff)   
haftmann@33361:                      else (2*q, r))"
huffman@47108:   by (simp add: Let_def adjust_def)
haftmann@33361: 
haftmann@33361: declare posDivAlg.simps [simp del]
haftmann@33361: 
haftmann@33361: text{*use with a simproc to avoid repeatedly proving the premise*}
haftmann@33361: lemma posDivAlg_eqn:
haftmann@33361:      "0 < b ==>  
haftmann@33361:       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
haftmann@33361: by (rule posDivAlg.simps [THEN trans], simp)
haftmann@33361: 
haftmann@33361: text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
haftmann@33361: theorem posDivAlg_correct:
haftmann@33361:   assumes "0 \<le> a" and "0 < b"
haftmann@33361:   shows "divmod_int_rel a b (posDivAlg a b)"
wenzelm@41550:   using assms
wenzelm@41550:   apply (induct a b rule: posDivAlg.induct)
wenzelm@41550:   apply auto
wenzelm@41550:   apply (simp add: divmod_int_rel_def)
webertj@49962:   apply (subst posDivAlg_eqn, simp add: distrib_left)
wenzelm@41550:   apply (case_tac "a < b")
wenzelm@41550:   apply simp_all
wenzelm@41550:   apply (erule splitE)
haftmann@57514:   apply (auto simp add: distrib_left Let_def ac_simps mult_2_right)
wenzelm@41550:   done
haftmann@33361: 
haftmann@33361: 
huffman@46551: subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}
haftmann@33361: 
haftmann@33361: text{*And positive divisors*}
haftmann@33361: 
haftmann@33361: declare negDivAlg.simps [simp del]
haftmann@33361: 
haftmann@33361: text{*use with a simproc to avoid repeatedly proving the premise*}
haftmann@33361: lemma negDivAlg_eqn:
haftmann@33361:      "0 < b ==>  
haftmann@33361:       negDivAlg a b =       
haftmann@33361:        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
haftmann@33361: by (rule negDivAlg.simps [THEN trans], simp)
haftmann@33361: 
haftmann@33361: (*Correctness of negDivAlg: it computes quotients correctly
haftmann@33361:   It doesn't work if a=0 because the 0/b equals 0, not -1*)
haftmann@33361: lemma negDivAlg_correct:
haftmann@33361:   assumes "a < 0" and "b > 0"
haftmann@33361:   shows "divmod_int_rel a b (negDivAlg a b)"
wenzelm@41550:   using assms
wenzelm@41550:   apply (induct a b rule: negDivAlg.induct)
wenzelm@41550:   apply (auto simp add: linorder_not_le)
wenzelm@41550:   apply (simp add: divmod_int_rel_def)
wenzelm@41550:   apply (subst negDivAlg_eqn, assumption)
wenzelm@41550:   apply (case_tac "a + b < (0\<Colon>int)")
wenzelm@41550:   apply simp_all
wenzelm@41550:   apply (erule splitE)
haftmann@57514:   apply (auto simp add: distrib_left Let_def ac_simps mult_2_right)
wenzelm@41550:   done
haftmann@33361: 
haftmann@33361: 
huffman@46551: subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}
haftmann@33361: 
haftmann@33361: (*the case a=0*)
huffman@47139: lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"
haftmann@33361: by (auto simp add: divmod_int_rel_def linorder_neq_iff)
haftmann@33361: 
haftmann@33361: lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
haftmann@33361: by (subst posDivAlg.simps, auto)
haftmann@33361: 
huffman@47139: lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"
huffman@47139: by (subst posDivAlg.simps, auto)
huffman@47139: 
haftmann@33361: lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
haftmann@33361: by (subst negDivAlg.simps, auto)
haftmann@33361: 
huffman@46560: lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"
huffman@47139: by (auto simp add: divmod_int_rel_def)
huffman@47139: 
huffman@47139: lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"
huffman@47139: apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)
haftmann@33361: by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
haftmann@33361:                     posDivAlg_correct negDivAlg_correct)
haftmann@33361: 
huffman@47141: lemma divmod_int_unique:
huffman@47141:   assumes "divmod_int_rel a b qr" 
huffman@47141:   shows "divmod_int a b = qr"
huffman@47141:   using assms divmod_int_correct [of a b]
huffman@47141:   using unique_quotient [of a b] unique_remainder [of a b]
huffman@47141:   by (metis pair_collapse)
huffman@47141: 
huffman@47141: lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"
huffman@47141:   using divmod_int_correct by (simp add: divmod_int_mod_div)
huffman@47141: 
huffman@47141: lemma div_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a div b = q"
huffman@47141:   by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
huffman@47141: 
huffman@47141: lemma mod_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a mod b = r"
huffman@47141:   by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
huffman@47141: 
huffman@47141: instance int :: ring_div
huffman@47141: proof
huffman@47141:   fix a b :: int
huffman@47141:   show "a div b * b + a mod b = a"
huffman@47141:     using divmod_int_rel_div_mod [of a b]
haftmann@57512:     unfolding divmod_int_rel_def by (simp add: mult.commute)
huffman@47141: next
huffman@47141:   fix a b c :: int
huffman@47141:   assume "b \<noteq> 0"
huffman@47141:   hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"
huffman@47141:     using divmod_int_rel_div_mod [of a b]
huffman@47141:     unfolding divmod_int_rel_def by (auto simp: algebra_simps)
huffman@47141:   thus "(a + c * b) div b = c + a div b"
huffman@47141:     by (rule div_int_unique)
huffman@47141: next
huffman@47141:   fix a b c :: int
huffman@47141:   assume "c \<noteq> 0"
huffman@47141:   hence "\<And>q r. divmod_int_rel a b (q, r)
huffman@47141:     \<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)"
huffman@47141:     unfolding divmod_int_rel_def
huffman@47141:     by - (rule linorder_cases [of 0 b], auto simp: algebra_simps
huffman@47141:       mult_less_0_iff zero_less_mult_iff mult_strict_right_mono
huffman@47141:       mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)
huffman@47141:   hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"
huffman@47141:     using divmod_int_rel_div_mod [of a b] .
huffman@47141:   thus "(c * a) div (c * b) = a div b"
huffman@47141:     by (rule div_int_unique)
huffman@47141: next
huffman@47141:   fix a :: int show "a div 0 = 0"
huffman@47141:     by (rule div_int_unique, simp add: divmod_int_rel_def)
huffman@47141: next
huffman@47141:   fix a :: int show "0 div a = 0"
huffman@47141:     by (rule div_int_unique, auto simp add: divmod_int_rel_def)
huffman@47141: qed
huffman@47141: 
haftmann@33361: text{*Basic laws about division and remainder*}
haftmann@33361: 
haftmann@33361: lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
huffman@47141:   by (fact mod_div_equality2 [symmetric])
haftmann@33361: 
haftmann@33361: text {* Tool setup *}
haftmann@33361: 
huffman@47108: (* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)
huffman@47108: lemmas add_0s = add_0_left add_0_right
huffman@47108: 
haftmann@33361: ML {*
wenzelm@43594: structure Cancel_Div_Mod_Int = Cancel_Div_Mod
wenzelm@41550: (
haftmann@33361:   val div_name = @{const_name div};
haftmann@33361:   val mod_name = @{const_name mod};
haftmann@33361:   val mk_binop = HOLogic.mk_binop;
haftmann@33361:   val mk_sum = Arith_Data.mk_sum HOLogic.intT;
haftmann@33361:   val dest_sum = Arith_Data.dest_sum;
haftmann@33361: 
huffman@47165:   val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
haftmann@33361: 
haftmann@33361:   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
haftmann@57514:     (@{thm diff_conv_add_uminus} :: @{thms add_0s} @ @{thms ac_simps}))
wenzelm@41550: )
haftmann@33361: *}
haftmann@33361: 
wenzelm@43594: simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}
wenzelm@43594: 
huffman@47141: lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
huffman@47141:   using divmod_int_correct [of a b]
huffman@47141:   by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361: 
wenzelm@45607: lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
wenzelm@45607:    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
haftmann@33361: 
huffman@47141: lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
huffman@47141:   using divmod_int_correct [of a b]
huffman@47141:   by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361: 
wenzelm@45607: lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
wenzelm@45607:    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
haftmann@33361: 
haftmann@33361: 
huffman@46551: subsubsection {* General Properties of div and mod *}
haftmann@33361: 
haftmann@33361: lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
huffman@47140: apply (rule div_int_unique)
haftmann@33361: apply (auto simp add: divmod_int_rel_def)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
huffman@47140: apply (rule div_int_unique)
haftmann@33361: apply (auto simp add: divmod_int_rel_def)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
huffman@47140: apply (rule div_int_unique)
haftmann@33361: apply (auto simp add: divmod_int_rel_def)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
haftmann@33361: 
haftmann@33361: lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
huffman@47140: apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361: apply (auto simp add: divmod_int_rel_def)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
huffman@47140: apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361: apply (auto simp add: divmod_int_rel_def)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
huffman@47140: apply (rule_tac q = "-1" in mod_int_unique)
haftmann@33361: apply (auto simp add: divmod_int_rel_def)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: text{*There is no @{text mod_neg_pos_trivial}.*}
haftmann@33361: 
haftmann@33361: 
huffman@46551: subsubsection {* Laws for div and mod with Unary Minus *}
haftmann@33361: 
haftmann@33361: lemma zminus1_lemma:
huffman@47139:      "divmod_int_rel a b (q, r) ==> b \<noteq> 0
haftmann@33361:       ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
haftmann@33361:                           if r=0 then 0 else b-r)"
haftmann@33361: by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
haftmann@33361: 
haftmann@33361: 
haftmann@33361: lemma zdiv_zminus1_eq_if:
haftmann@33361:      "b \<noteq> (0::int)  
haftmann@33361:       ==> (-a) div b =  
haftmann@33361:           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47140: by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])
haftmann@33361: 
haftmann@33361: lemma zmod_zminus1_eq_if:
haftmann@33361:      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
haftmann@33361: apply (case_tac "b = 0", simp)
huffman@47140: apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma zmod_zminus1_not_zero:
haftmann@33361:   fixes k l :: int
haftmann@33361:   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361:   unfolding zmod_zminus1_eq_if by auto
haftmann@33361: 
haftmann@33361: lemma zdiv_zminus2_eq_if:
haftmann@33361:      "b \<noteq> (0::int)  
haftmann@33361:       ==> a div (-b) =  
haftmann@33361:           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47159: by (simp add: zdiv_zminus1_eq_if div_minus_right)
haftmann@33361: 
haftmann@33361: lemma zmod_zminus2_eq_if:
haftmann@33361:      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
huffman@47159: by (simp add: zmod_zminus1_eq_if mod_minus_right)
haftmann@33361: 
haftmann@33361: lemma zmod_zminus2_not_zero:
haftmann@33361:   fixes k l :: int
haftmann@33361:   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361:   unfolding zmod_zminus2_eq_if by auto 
haftmann@33361: 
haftmann@33361: 
huffman@46551: subsubsection {* Computation of Division and Remainder *}
haftmann@33361: 
haftmann@33361: lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@33361: by (simp add: div_int_def divmod_int_def)
haftmann@33361: 
haftmann@33361: lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@33361: by (simp add: mod_int_def divmod_int_def)
haftmann@33361: 
haftmann@33361: text{*a positive, b positive *}
haftmann@33361: 
haftmann@33361: lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
haftmann@33361: by (simp add: div_int_def divmod_int_def)
haftmann@33361: 
haftmann@33361: lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
haftmann@33361: by (simp add: mod_int_def divmod_int_def)
haftmann@33361: 
haftmann@33361: text{*a negative, b positive *}
haftmann@33361: 
haftmann@33361: lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
haftmann@33361: by (simp add: div_int_def divmod_int_def)
haftmann@33361: 
haftmann@33361: lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
haftmann@33361: by (simp add: mod_int_def divmod_int_def)
haftmann@33361: 
haftmann@33361: text{*a positive, b negative *}
haftmann@33361: 
haftmann@33361: lemma div_pos_neg:
huffman@46560:      "[| 0 < a;  b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"
haftmann@33361: by (simp add: div_int_def divmod_int_def)
haftmann@33361: 
haftmann@33361: lemma mod_pos_neg:
huffman@46560:      "[| 0 < a;  b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"
haftmann@33361: by (simp add: mod_int_def divmod_int_def)
haftmann@33361: 
haftmann@33361: text{*a negative, b negative *}
haftmann@33361: 
haftmann@33361: lemma div_neg_neg:
huffman@46560:      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361: by (simp add: div_int_def divmod_int_def)
haftmann@33361: 
haftmann@33361: lemma mod_neg_neg:
huffman@46560:      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361: by (simp add: mod_int_def divmod_int_def)
haftmann@33361: 
haftmann@33361: text {*Simplify expresions in which div and mod combine numerical constants*}
haftmann@33361: 
huffman@45530: lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
huffman@47140:   by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)
huffman@45530: 
huffman@45530: lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
huffman@47140:   by (rule div_int_unique [of a b q r],
bulwahn@46552:     simp add: divmod_int_rel_def)
huffman@45530: 
huffman@45530: lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
huffman@47140:   by (rule mod_int_unique [of a b q r],
bulwahn@46552:     simp add: divmod_int_rel_def)
huffman@45530: 
huffman@45530: lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
huffman@47140:   by (rule mod_int_unique [of a b q r],
bulwahn@46552:     simp add: divmod_int_rel_def)
huffman@45530: 
haftmann@53069: text {*
haftmann@53069:   numeral simprocs -- high chance that these can be replaced
haftmann@53069:   by divmod algorithm from @{class semiring_numeral_div}
haftmann@53069: *}
haftmann@53069: 
haftmann@33361: ML {*
haftmann@33361: local
huffman@45530:   val mk_number = HOLogic.mk_number HOLogic.intT
huffman@45530:   val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"}
huffman@45530:   val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"}
huffman@45530:   val zero = @{term "0 :: int"}
huffman@45530:   val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}
huffman@45530:   val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}
haftmann@54489:   val simps = @{thms arith_simps} @ @{thms rel_simps} @ [@{thm numeral_1_eq_1 [symmetric]}]
haftmann@54489:   fun prove ctxt goal = (writeln "prove"; Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)
haftmann@54489:     (K (ALLGOALS (full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps simps)))));
wenzelm@51717:   fun binary_proc proc ctxt ct =
haftmann@33361:     (case Thm.term_of ct of
haftmann@33361:       _ $ t $ u =>
haftmann@33361:       (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
wenzelm@51717:         SOME args => proc ctxt args
haftmann@33361:       | NONE => NONE)
haftmann@33361:     | _ => NONE);
haftmann@33361: in
huffman@45530:   fun divmod_proc posrule negrule =
huffman@45530:     binary_proc (fn ctxt => fn ((a, t), (b, u)) =>
huffman@45530:       if b = 0 then NONE else let
huffman@45530:         val (q, r) = pairself mk_number (Integer.div_mod a b)
huffman@45530:         val goal1 = HOLogic.mk_eq (t, plus $ (times $ u $ q) $ r)
huffman@45530:         val (goal2, goal3, rule) = if b > 0
huffman@45530:           then (le $ zero $ r, less $ r $ u, posrule RS eq_reflection)
huffman@45530:           else (le $ r $ zero, less $ u $ r, negrule RS eq_reflection)
huffman@45530:       in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)
haftmann@33361: end
haftmann@33361: *}
haftmann@33361: 
huffman@47108: simproc_setup binary_int_div
huffman@47108:   ("numeral m div numeral n :: int" |
haftmann@54489:    "numeral m div - numeral n :: int" |
haftmann@54489:    "- numeral m div numeral n :: int" |
haftmann@54489:    "- numeral m div - numeral n :: int") =
huffman@45530:   {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}
haftmann@33361: 
huffman@47108: simproc_setup binary_int_mod
huffman@47108:   ("numeral m mod numeral n :: int" |
haftmann@54489:    "numeral m mod - numeral n :: int" |
haftmann@54489:    "- numeral m mod numeral n :: int" |
haftmann@54489:    "- numeral m mod - numeral n :: int") =
huffman@45530:   {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}
haftmann@33361: 
huffman@47108: lemmas posDivAlg_eqn_numeral [simp] =
huffman@47108:     posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w
huffman@47108: 
huffman@47108: lemmas negDivAlg_eqn_numeral [simp] =
haftmann@54489:     negDivAlg_eqn [of "numeral v" "- numeral w", OF zero_less_numeral] for v w
haftmann@33361: 
haftmann@33361: 
haftmann@55172: text {* Special-case simplification: @{text "\<plusminus>1 div z"} and @{text "\<plusminus>1 mod z"} *}
haftmann@55172: 
haftmann@55172: lemma [simp]:
haftmann@55172:   shows div_one_bit0: "1 div numeral (Num.Bit0 v) = (0 :: int)"
haftmann@55172:     and mod_one_bit0: "1 mod numeral (Num.Bit0 v) = (1 :: int)"
wenzelm@55439:     and div_one_bit1: "1 div numeral (Num.Bit1 v) = (0 :: int)"
wenzelm@55439:     and mod_one_bit1: "1 mod numeral (Num.Bit1 v) = (1 :: int)"
wenzelm@55439:     and div_one_neg_numeral: "1 div - numeral v = (- 1 :: int)"
wenzelm@55439:     and mod_one_neg_numeral: "1 mod - numeral v = (1 :: int) - numeral v"
haftmann@55172:   by (simp_all del: arith_special
haftmann@55172:     add: div_pos_pos mod_pos_pos div_pos_neg mod_pos_neg posDivAlg_eqn)
wenzelm@55439: 
haftmann@55172: lemma [simp]:
haftmann@55172:   shows div_neg_one_numeral: "- 1 div numeral v = (- 1 :: int)"
haftmann@55172:     and mod_neg_one_numeral: "- 1 mod numeral v = numeral v - (1 :: int)"
haftmann@55172:     and div_neg_one_neg_bit0: "- 1 div - numeral (Num.Bit0 v) = (0 :: int)"
haftmann@55172:     and mod_neg_one_neb_bit0: "- 1 mod - numeral (Num.Bit0 v) = (- 1 :: int)"
haftmann@55172:     and div_neg_one_neg_bit1: "- 1 div - numeral (Num.Bit1 v) = (0 :: int)"
haftmann@55172:     and mod_neg_one_neb_bit1: "- 1 mod - numeral (Num.Bit1 v) = (- 1 :: int)"
haftmann@55172:   by (simp_all add: div_eq_minus1 zmod_minus1)
haftmann@33361: 
haftmann@33361: 
huffman@46551: subsubsection {* Monotonicity in the First Argument (Dividend) *}
haftmann@33361: 
haftmann@33361: lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
haftmann@33361: apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361: apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361: apply (rule unique_quotient_lemma)
haftmann@33361: apply (erule subst)
haftmann@33361: apply (erule subst, simp_all)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
haftmann@33361: apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361: apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361: apply (rule unique_quotient_lemma_neg)
haftmann@33361: apply (erule subst)
haftmann@33361: apply (erule subst, simp_all)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: 
huffman@46551: subsubsection {* Monotonicity in the Second Argument (Divisor) *}
haftmann@33361: 
haftmann@33361: lemma q_pos_lemma:
haftmann@33361:      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
haftmann@33361: apply (subgoal_tac "0 < b'* (q' + 1) ")
haftmann@33361:  apply (simp add: zero_less_mult_iff)
webertj@49962: apply (simp add: distrib_left)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma zdiv_mono2_lemma:
haftmann@33361:      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
haftmann@33361:          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
haftmann@33361:       ==> q \<le> (q'::int)"
haftmann@33361: apply (frule q_pos_lemma, assumption+) 
haftmann@33361: apply (subgoal_tac "b*q < b* (q' + 1) ")
haftmann@33361:  apply (simp add: mult_less_cancel_left)
haftmann@33361: apply (subgoal_tac "b*q = r' - r + b'*q'")
haftmann@33361:  prefer 2 apply simp
webertj@49962: apply (simp (no_asm_simp) add: distrib_left)
haftmann@57512: apply (subst add.commute, rule add_less_le_mono, arith)
haftmann@33361: apply (rule mult_right_mono, auto)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma zdiv_mono2:
haftmann@33361:      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
haftmann@33361: apply (subgoal_tac "b \<noteq> 0")
haftmann@33361:  prefer 2 apply arith
haftmann@33361: apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361: apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361: apply (rule zdiv_mono2_lemma)
haftmann@33361: apply (erule subst)
haftmann@33361: apply (erule subst, simp_all)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma q_neg_lemma:
haftmann@33361:      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
haftmann@33361: apply (subgoal_tac "b'*q' < 0")
haftmann@33361:  apply (simp add: mult_less_0_iff, arith)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma zdiv_mono2_neg_lemma:
haftmann@33361:      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
haftmann@33361:          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
haftmann@33361:       ==> q' \<le> (q::int)"
haftmann@33361: apply (frule q_neg_lemma, assumption+) 
haftmann@33361: apply (subgoal_tac "b*q' < b* (q + 1) ")
haftmann@33361:  apply (simp add: mult_less_cancel_left)
webertj@49962: apply (simp add: distrib_left)
haftmann@33361: apply (subgoal_tac "b*q' \<le> b'*q'")
haftmann@33361:  prefer 2 apply (simp add: mult_right_mono_neg, arith)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma zdiv_mono2_neg:
haftmann@33361:      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
haftmann@33361: apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361: apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361: apply (rule zdiv_mono2_neg_lemma)
haftmann@33361: apply (erule subst)
haftmann@33361: apply (erule subst, simp_all)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: 
huffman@46551: subsubsection {* More Algebraic Laws for div and mod *}
haftmann@33361: 
haftmann@33361: text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
haftmann@33361: 
haftmann@33361: lemma zmult1_lemma:
bulwahn@46552:      "[| divmod_int_rel b c (q, r) |]  
haftmann@33361:       ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
haftmann@57514: by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left ac_simps)
haftmann@33361: 
haftmann@33361: lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
haftmann@33361: apply (case_tac "c = 0", simp)
huffman@47140: apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])
haftmann@33361: done
haftmann@33361: 
haftmann@33361: text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
haftmann@33361: 
haftmann@33361: lemma zadd1_lemma:
bulwahn@46552:      "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]  
haftmann@33361:       ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
webertj@49962: by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)
haftmann@33361: 
haftmann@33361: (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@33361: lemma zdiv_zadd1_eq:
haftmann@33361:      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@33361: apply (case_tac "c = 0", simp)
huffman@47140: apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma posDivAlg_div_mod:
haftmann@33361:   assumes "k \<ge> 0"
haftmann@33361:   and "l \<ge> 0"
haftmann@33361:   shows "posDivAlg k l = (k div l, k mod l)"
haftmann@33361: proof (cases "l = 0")
haftmann@33361:   case True then show ?thesis by (simp add: posDivAlg.simps)
haftmann@33361: next
haftmann@33361:   case False with assms posDivAlg_correct
haftmann@33361:     have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
haftmann@33361:     by simp
huffman@47140:   from div_int_unique [OF this] mod_int_unique [OF this]
haftmann@33361:   show ?thesis by simp
haftmann@33361: qed
haftmann@33361: 
haftmann@33361: lemma negDivAlg_div_mod:
haftmann@33361:   assumes "k < 0"
haftmann@33361:   and "l > 0"
haftmann@33361:   shows "negDivAlg k l = (k div l, k mod l)"
haftmann@33361: proof -
haftmann@33361:   from assms have "l \<noteq> 0" by simp
haftmann@33361:   from assms negDivAlg_correct
haftmann@33361:     have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
haftmann@33361:     by simp
huffman@47140:   from div_int_unique [OF this] mod_int_unique [OF this]
haftmann@33361:   show ?thesis by simp
haftmann@33361: qed
haftmann@33361: 
haftmann@33361: lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
haftmann@33361: by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@33361: 
haftmann@33361: (* REVISIT: should this be generalized to all semiring_div types? *)
haftmann@33361: lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
haftmann@33361: 
huffman@47108: lemma zmod_zdiv_equality':
huffman@47108:   "(m\<Colon>int) mod n = m - (m div n) * n"
huffman@47141:   using mod_div_equality [of m n] by arith
huffman@47108: 
haftmann@33361: 
blanchet@55085: subsubsection {* Proving  @{term "a div (b * c) = (a div b) div c"} *}
haftmann@33361: 
haftmann@33361: (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
haftmann@33361:   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
haftmann@33361:   to cause particular problems.*)
haftmann@33361: 
haftmann@33361: text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
haftmann@33361: 
blanchet@55085: lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b * c < b * (q mod c) + r"
haftmann@33361: apply (subgoal_tac "b * (c - q mod c) < r * 1")
haftmann@33361:  apply (simp add: algebra_simps)
haftmann@33361: apply (rule order_le_less_trans)
haftmann@33361:  apply (erule_tac [2] mult_strict_right_mono)
haftmann@33361:  apply (rule mult_left_mono_neg)
huffman@35216:   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
haftmann@33361:  apply (simp)
haftmann@33361: apply (simp)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma zmult2_lemma_aux2:
haftmann@33361:      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
haftmann@33361: apply (subgoal_tac "b * (q mod c) \<le> 0")
haftmann@33361:  apply arith
haftmann@33361: apply (simp add: mult_le_0_iff)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
haftmann@33361: apply (subgoal_tac "0 \<le> b * (q mod c) ")
haftmann@33361: apply arith
haftmann@33361: apply (simp add: zero_le_mult_iff)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
haftmann@33361: apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
haftmann@33361:  apply (simp add: right_diff_distrib)
haftmann@33361: apply (rule order_less_le_trans)
haftmann@33361:  apply (erule mult_strict_right_mono)
haftmann@33361:  apply (rule_tac [2] mult_left_mono)
haftmann@33361:   apply simp
huffman@35216:  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
haftmann@33361: apply simp
haftmann@33361: done
haftmann@33361: 
bulwahn@46552: lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  
haftmann@33361:       ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
haftmann@57514: by (auto simp add: mult.assoc divmod_int_rel_def linorder_neq_iff
webertj@49962:                    zero_less_mult_iff distrib_left [symmetric] 
huffman@47139:                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)
haftmann@33361: 
haftmann@53068: lemma zdiv_zmult2_eq:
haftmann@53068:   fixes a b c :: int
haftmann@53068:   shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
haftmann@33361: apply (case_tac "b = 0", simp)
haftmann@53068: apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma zmod_zmult2_eq:
haftmann@53068:   fixes a b c :: int
haftmann@53068:   shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@33361: apply (case_tac "b = 0", simp)
haftmann@53068: apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])
haftmann@33361: done
haftmann@33361: 
huffman@47108: lemma div_pos_geq:
huffman@47108:   fixes k l :: int
huffman@47108:   assumes "0 < l" and "l \<le> k"
huffman@47108:   shows "k div l = (k - l) div l + 1"
huffman@47108: proof -
huffman@47108:   have "k = (k - l) + l" by simp
huffman@47108:   then obtain j where k: "k = j + l" ..
huffman@47108:   with assms show ?thesis by simp
huffman@47108: qed
huffman@47108: 
huffman@47108: lemma mod_pos_geq:
huffman@47108:   fixes k l :: int
huffman@47108:   assumes "0 < l" and "l \<le> k"
huffman@47108:   shows "k mod l = (k - l) mod l"
huffman@47108: proof -
huffman@47108:   have "k = (k - l) + l" by simp
huffman@47108:   then obtain j where k: "k = j + l" ..
huffman@47108:   with assms show ?thesis by simp
huffman@47108: qed
huffman@47108: 
haftmann@33361: 
huffman@46551: subsubsection {* Splitting Rules for div and mod *}
haftmann@33361: 
haftmann@33361: text{*The proofs of the two lemmas below are essentially identical*}
haftmann@33361: 
haftmann@33361: lemma split_pos_lemma:
haftmann@33361:  "0<k ==> 
haftmann@33361:     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
haftmann@33361: apply (rule iffI, clarify)
haftmann@33361:  apply (erule_tac P="P ?x ?y" in rev_mp)  
haftmann@33361:  apply (subst mod_add_eq) 
haftmann@33361:  apply (subst zdiv_zadd1_eq) 
haftmann@33361:  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
haftmann@33361: txt{*converse direction*}
haftmann@33361: apply (drule_tac x = "n div k" in spec) 
haftmann@33361: apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma split_neg_lemma:
haftmann@33361:  "k<0 ==>
haftmann@33361:     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
haftmann@33361: apply (rule iffI, clarify)
haftmann@33361:  apply (erule_tac P="P ?x ?y" in rev_mp)  
haftmann@33361:  apply (subst mod_add_eq) 
haftmann@33361:  apply (subst zdiv_zadd1_eq) 
haftmann@33361:  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
haftmann@33361: txt{*converse direction*}
haftmann@33361: apply (drule_tac x = "n div k" in spec) 
haftmann@33361: apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma split_zdiv:
haftmann@33361:  "P(n div k :: int) =
haftmann@33361:   ((k = 0 --> P 0) & 
haftmann@33361:    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
haftmann@33361:    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
haftmann@33361: apply (case_tac "k=0", simp)
haftmann@33361: apply (simp only: linorder_neq_iff)
haftmann@33361: apply (erule disjE) 
haftmann@33361:  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
haftmann@33361:                       split_neg_lemma [of concl: "%x y. P x"])
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma split_zmod:
haftmann@33361:  "P(n mod k :: int) =
haftmann@33361:   ((k = 0 --> P n) & 
haftmann@33361:    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
haftmann@33361:    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
haftmann@33361: apply (case_tac "k=0", simp)
haftmann@33361: apply (simp only: linorder_neq_iff)
haftmann@33361: apply (erule disjE) 
haftmann@33361:  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
haftmann@33361:                       split_neg_lemma [of concl: "%x y. P y"])
haftmann@33361: done
haftmann@33361: 
webertj@33730: text {* Enable (lin)arith to deal with @{const div} and @{const mod}
webertj@33730:   when these are applied to some constant that is of the form
huffman@47108:   @{term "numeral k"}: *}
huffman@47108: declare split_zdiv [of _ _ "numeral k", arith_split] for k
huffman@47108: declare split_zmod [of _ _ "numeral k", arith_split] for k
haftmann@33361: 
haftmann@33361: 
huffman@47166: subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}
huffman@47166: 
huffman@47166: lemma pos_divmod_int_rel_mult_2:
huffman@47166:   assumes "0 \<le> b"
huffman@47166:   assumes "divmod_int_rel a b (q, r)"
huffman@47166:   shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"
huffman@47166:   using assms unfolding divmod_int_rel_def by auto
huffman@47166: 
haftmann@54489: declaration {* K (Lin_Arith.add_simps @{thms uminus_numeral_One}) *}
haftmann@54489: 
huffman@47166: lemma neg_divmod_int_rel_mult_2:
huffman@47166:   assumes "b \<le> 0"
huffman@47166:   assumes "divmod_int_rel (a + 1) b (q, r)"
huffman@47166:   shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"
huffman@47166:   using assms unfolding divmod_int_rel_def by auto
haftmann@33361: 
haftmann@33361: text{*computing div by shifting *}
haftmann@33361: 
haftmann@33361: lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
huffman@47166:   using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]
huffman@47166:   by (rule div_int_unique)
haftmann@33361: 
boehmes@35815: lemma neg_zdiv_mult_2: 
boehmes@35815:   assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
huffman@47166:   using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]
huffman@47166:   by (rule div_int_unique)
haftmann@33361: 
huffman@47108: (* FIXME: add rules for negative numerals *)
huffman@47108: lemma zdiv_numeral_Bit0 [simp]:
huffman@47108:   "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
huffman@47108:     numeral v div (numeral w :: int)"
huffman@47108:   unfolding numeral.simps unfolding mult_2 [symmetric]
huffman@47108:   by (rule div_mult_mult1, simp)
huffman@47108: 
huffman@47108: lemma zdiv_numeral_Bit1 [simp]:
huffman@47108:   "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  
huffman@47108:     (numeral v div (numeral w :: int))"
huffman@47108:   unfolding numeral.simps
haftmann@57512:   unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108:   by (rule pos_zdiv_mult_2, simp)
haftmann@33361: 
haftmann@33361: lemma pos_zmod_mult_2:
haftmann@33361:   fixes a b :: int
haftmann@33361:   assumes "0 \<le> a"
haftmann@33361:   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
huffman@47166:   using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
huffman@47166:   by (rule mod_int_unique)
haftmann@33361: 
haftmann@33361: lemma neg_zmod_mult_2:
haftmann@33361:   fixes a b :: int
haftmann@33361:   assumes "a \<le> 0"
haftmann@33361:   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
huffman@47166:   using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
huffman@47166:   by (rule mod_int_unique)
haftmann@33361: 
huffman@47108: (* FIXME: add rules for negative numerals *)
huffman@47108: lemma zmod_numeral_Bit0 [simp]:
huffman@47108:   "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =  
huffman@47108:     (2::int) * (numeral v mod numeral w)"
huffman@47108:   unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
huffman@47108:   unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
huffman@47108: 
huffman@47108: lemma zmod_numeral_Bit1 [simp]:
huffman@47108:   "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
huffman@47108:     2 * (numeral v mod numeral w) + (1::int)"
huffman@47108:   unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
haftmann@57512:   unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108:   by (rule pos_zmod_mult_2, simp)
haftmann@33361: 
nipkow@39489: lemma zdiv_eq_0_iff:
nipkow@39489:  "(i::int) div k = 0 \<longleftrightarrow> k=0 \<or> 0\<le>i \<and> i<k \<or> i\<le>0 \<and> k<i" (is "?L = ?R")
nipkow@39489: proof
nipkow@39489:   assume ?L
nipkow@39489:   have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
nipkow@39489:   with `?L` show ?R by blast
nipkow@39489: next
nipkow@39489:   assume ?R thus ?L
nipkow@39489:     by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
nipkow@39489: qed
nipkow@39489: 
nipkow@39489: 
huffman@46551: subsubsection {* Quotients of Signs *}
haftmann@33361: 
haftmann@33361: lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
haftmann@33361: apply (subgoal_tac "a div b \<le> -1", force)
haftmann@33361: apply (rule order_trans)
haftmann@33361: apply (rule_tac a' = "-1" in zdiv_mono1)
haftmann@33361: apply (auto simp add: div_eq_minus1)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
haftmann@33361: by (drule zdiv_mono1_neg, auto)
haftmann@33361: 
haftmann@33361: lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
haftmann@33361: by (drule zdiv_mono1, auto)
haftmann@33361: 
nipkow@33804: text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}
nipkow@33804: conditional upon the sign of @{text a} or @{text b}. There are many more.
nipkow@33804: They should all be simp rules unless that causes too much search. *}
nipkow@33804: 
haftmann@33361: lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
haftmann@33361: apply auto
haftmann@33361: apply (drule_tac [2] zdiv_mono1)
haftmann@33361: apply (auto simp add: linorder_neq_iff)
haftmann@33361: apply (simp (no_asm_use) add: linorder_not_less [symmetric])
haftmann@33361: apply (blast intro: div_neg_pos_less0)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma neg_imp_zdiv_nonneg_iff:
nipkow@33804:   "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
huffman@47159: apply (subst div_minus_minus [symmetric])
haftmann@33361: apply (subst pos_imp_zdiv_nonneg_iff, auto)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
haftmann@33361: lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
haftmann@33361: by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
haftmann@33361: 
nipkow@39489: lemma pos_imp_zdiv_pos_iff:
nipkow@39489:   "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
nipkow@39489: using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
nipkow@39489: by arith
nipkow@39489: 
haftmann@33361: (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
haftmann@33361: lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
haftmann@33361: by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
haftmann@33361: 
nipkow@33804: lemma nonneg1_imp_zdiv_pos_iff:
nipkow@33804:   "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
nipkow@33804: apply rule
nipkow@33804:  apply rule
nipkow@33804:   using div_pos_pos_trivial[of a b]apply arith
nipkow@33804:  apply(cases "b=0")apply simp
nipkow@33804:  using div_nonneg_neg_le0[of a b]apply arith
nipkow@33804: using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
nipkow@33804: done
nipkow@33804: 
nipkow@39489: lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
nipkow@39489: apply (rule split_zmod[THEN iffD2])
nipkow@44890: apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
nipkow@39489: done
nipkow@39489: 
nipkow@39489: 
haftmann@33361: subsubsection {* The Divides Relation *}
haftmann@33361: 
huffman@47268: lemma dvd_neg_numeral_left [simp]:
huffman@47268:   fixes y :: "'a::comm_ring_1"
haftmann@54489:   shows "(- numeral k) dvd y \<longleftrightarrow> (numeral k) dvd y"
haftmann@54489:   by (fact minus_dvd_iff)
huffman@47268: 
huffman@47268: lemma dvd_neg_numeral_right [simp]:
huffman@47268:   fixes x :: "'a::comm_ring_1"
haftmann@54489:   shows "x dvd (- numeral k) \<longleftrightarrow> x dvd (numeral k)"
haftmann@54489:   by (fact dvd_minus_iff)
haftmann@33361: 
huffman@47108: lemmas dvd_eq_mod_eq_0_numeral [simp] =
huffman@47108:   dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y
huffman@47108: 
huffman@47108: 
huffman@47108: subsubsection {* Further properties *}
huffman@47108: 
haftmann@33361: lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
haftmann@33361:   using zmod_zdiv_equality[where a="m" and b="n"]
huffman@47142:   by (simp add: algebra_simps) (* FIXME: generalize *)
haftmann@33361: 
haftmann@33361: lemma zdiv_int: "int (a div b) = (int a) div (int b)"
haftmann@33361: apply (subst split_div, auto)
haftmann@33361: apply (subst split_zdiv, auto)
haftmann@33361: apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)
haftmann@33361: apply (auto simp add: divmod_int_rel_def of_nat_mult)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
haftmann@33361: apply (subst split_mod, auto)
haftmann@33361: apply (subst split_zmod, auto)
haftmann@33361: apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
haftmann@33361:        in unique_remainder)
haftmann@33361: apply (auto simp add: divmod_int_rel_def of_nat_mult)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
haftmann@33361: by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
haftmann@33361: 
haftmann@33361: text{*Suggested by Matthias Daum*}
haftmann@33361: lemma int_power_div_base:
haftmann@33361:      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
haftmann@33361: apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
haftmann@33361:  apply (erule ssubst)
haftmann@33361:  apply (simp only: power_add)
haftmann@33361:  apply simp_all
haftmann@33361: done
haftmann@33361: 
haftmann@33361: text {* by Brian Huffman *}
haftmann@33361: lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
haftmann@33361: by (rule mod_minus_eq [symmetric])
haftmann@33361: 
haftmann@33361: lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
haftmann@33361: by (rule mod_diff_left_eq [symmetric])
haftmann@33361: 
haftmann@33361: lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
haftmann@33361: by (rule mod_diff_right_eq [symmetric])
haftmann@33361: 
haftmann@33361: lemmas zmod_simps =
haftmann@33361:   mod_add_left_eq  [symmetric]
haftmann@33361:   mod_add_right_eq [symmetric]
huffman@47142:   mod_mult_right_eq[symmetric]
haftmann@33361:   mod_mult_left_eq [symmetric]
huffman@47164:   power_mod
haftmann@33361:   zminus_zmod zdiff_zmod_left zdiff_zmod_right
haftmann@33361: 
haftmann@33361: text {* Distributive laws for function @{text nat}. *}
haftmann@33361: 
haftmann@33361: lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
haftmann@33361: apply (rule linorder_cases [of y 0])
haftmann@33361: apply (simp add: div_nonneg_neg_le0)
haftmann@33361: apply simp
haftmann@33361: apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
haftmann@33361: lemma nat_mod_distrib:
haftmann@33361:   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
haftmann@33361: apply (case_tac "y = 0", simp)
haftmann@33361: apply (simp add: nat_eq_iff zmod_int)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: text  {* transfer setup *}
haftmann@33361: 
haftmann@33361: lemma transfer_nat_int_functions:
haftmann@33361:     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
haftmann@33361:     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
haftmann@33361:   by (auto simp add: nat_div_distrib nat_mod_distrib)
haftmann@33361: 
haftmann@33361: lemma transfer_nat_int_function_closures:
haftmann@33361:     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
haftmann@33361:     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
haftmann@33361:   apply (cases "y = 0")
haftmann@33361:   apply (auto simp add: pos_imp_zdiv_nonneg_iff)
haftmann@33361:   apply (cases "y = 0")
haftmann@33361:   apply auto
haftmann@33361: done
haftmann@33361: 
haftmann@35644: declare transfer_morphism_nat_int [transfer add return:
haftmann@33361:   transfer_nat_int_functions
haftmann@33361:   transfer_nat_int_function_closures
haftmann@33361: ]
haftmann@33361: 
haftmann@33361: lemma transfer_int_nat_functions:
haftmann@33361:     "(int x) div (int y) = int (x div y)"
haftmann@33361:     "(int x) mod (int y) = int (x mod y)"
haftmann@33361:   by (auto simp add: zdiv_int zmod_int)
haftmann@33361: 
haftmann@33361: lemma transfer_int_nat_function_closures:
haftmann@33361:     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
haftmann@33361:     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
haftmann@33361:   by (simp_all only: is_nat_def transfer_nat_int_function_closures)
haftmann@33361: 
haftmann@35644: declare transfer_morphism_int_nat [transfer add return:
haftmann@33361:   transfer_int_nat_functions
haftmann@33361:   transfer_int_nat_function_closures
haftmann@33361: ]
haftmann@33361: 
haftmann@33361: text{*Suggested by Matthias Daum*}
haftmann@33361: lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
haftmann@33361: apply (subgoal_tac "nat x div nat k < nat x")
nipkow@34225:  apply (simp add: nat_div_distrib [symmetric])
haftmann@33361: apply (rule Divides.div_less_dividend, simp_all)
haftmann@33361: done
haftmann@33361: 
haftmann@33361: lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
haftmann@33361: proof
haftmann@33361:   assume H: "x mod n = y mod n"
haftmann@33361:   hence "x mod n - y mod n = 0" by simp
haftmann@33361:   hence "(x mod n - y mod n) mod n = 0" by simp 
haftmann@33361:   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
haftmann@33361:   thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
haftmann@33361: next
haftmann@33361:   assume H: "n dvd x - y"
haftmann@33361:   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
haftmann@33361:   hence "x = n*k + y" by simp
haftmann@33361:   hence "x mod n = (n*k + y) mod n" by simp
haftmann@33361:   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
haftmann@33361: qed
haftmann@33361: 
haftmann@33361: lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
haftmann@33361:   shows "\<exists>q. x = y + n * q"
haftmann@33361: proof-
haftmann@33361:   from xy have th: "int x - int y = int (x - y)" by simp 
haftmann@33361:   from xyn have "int x mod int n = int y mod int n" 
huffman@46551:     by (simp add: zmod_int [symmetric])
haftmann@33361:   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
haftmann@33361:   hence "n dvd x - y" by (simp add: th zdvd_int)
haftmann@33361:   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
haftmann@33361: qed
haftmann@33361: 
haftmann@33361: lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
haftmann@33361:   (is "?lhs = ?rhs")
haftmann@33361: proof
haftmann@33361:   assume H: "x mod n = y mod n"
haftmann@33361:   {assume xy: "x \<le> y"
haftmann@33361:     from H have th: "y mod n = x mod n" by simp
haftmann@33361:     from nat_mod_eq_lemma[OF th xy] have ?rhs 
haftmann@33361:       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
haftmann@33361:   moreover
haftmann@33361:   {assume xy: "y \<le> x"
haftmann@33361:     from nat_mod_eq_lemma[OF H xy] have ?rhs 
haftmann@33361:       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
haftmann@33361:   ultimately  show ?rhs using linear[of x y] by blast  
haftmann@33361: next
haftmann@33361:   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
haftmann@33361:   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
haftmann@33361:   thus  ?lhs by simp
haftmann@33361: qed
haftmann@33361: 
haftmann@53067: text {*
haftmann@53067:   This re-embedding of natural division on integers goes back to the
haftmann@53067:   time when numerals had been signed numerals.  It should
haftmann@53070:   now be replaced by the algorithm developed in @{class semiring_numeral_div}.  
haftmann@53067: *}
haftmann@53067: 
huffman@47108: lemma div_nat_numeral [simp]:
huffman@47108:   "(numeral v :: nat) div numeral v' = nat (numeral v div numeral v')"
haftmann@33361:   by (simp add: nat_div_distrib)
haftmann@33361: 
huffman@47108: lemma one_div_nat_numeral [simp]:
huffman@47108:   "Suc 0 div numeral v' = nat (1 div numeral v')"
huffman@47108:   by (subst nat_div_distrib, simp_all)
huffman@47108: 
huffman@47108: lemma mod_nat_numeral [simp]:
huffman@47108:   "(numeral v :: nat) mod numeral v' = nat (numeral v mod numeral v')"
haftmann@33361:   by (simp add: nat_mod_distrib)
haftmann@33361: 
huffman@47108: lemma one_mod_nat_numeral [simp]:
huffman@47108:   "Suc 0 mod numeral v' = nat (1 mod numeral v')"
huffman@47108:   by (subst nat_mod_distrib) simp_all
huffman@47108: 
haftmann@53067: instance int :: semiring_numeral_div
haftmann@53068:   by intro_classes (auto intro: zmod_le_nonneg_dividend
haftmann@53068:     simp add: zmult_div_cancel
haftmann@53068:     pos_imp_zdiv_pos_iff div_pos_pos_trivial mod_pos_pos_trivial
haftmann@53068:     zmod_zmult2_eq zdiv_zmult2_eq)
haftmann@53067: 
huffman@47108: 
huffman@47108: subsubsection {* Tools setup *}
huffman@47108: 
huffman@47108: text {* Nitpick *}
blanchet@34126: 
blanchet@41792: lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality'
blanchet@34126: 
haftmann@35673: 
haftmann@33361: subsubsection {* Code generation *}
haftmann@33361: 
haftmann@53069: definition divmod_abs :: "int \<Rightarrow> int \<Rightarrow> int \<times> int"
haftmann@53069: where
haftmann@53069:   "divmod_abs k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
haftmann@53069: 
haftmann@53069: lemma fst_divmod_abs [simp]:
haftmann@53069:   "fst (divmod_abs k l) = \<bar>k\<bar> div \<bar>l\<bar>"
haftmann@53069:   by (simp add: divmod_abs_def)
haftmann@53069: 
haftmann@53069: lemma snd_divmod_abs [simp]:
haftmann@53069:   "snd (divmod_abs k l) = \<bar>k\<bar> mod \<bar>l\<bar>"
haftmann@53069:   by (simp add: divmod_abs_def)
haftmann@53069: 
haftmann@53069: lemma divmod_abs_code [code]:
haftmann@53069:   "divmod_abs (Int.Pos k) (Int.Pos l) = divmod k l"
haftmann@53069:   "divmod_abs (Int.Neg k) (Int.Neg l) = divmod k l"
haftmann@53069:   "divmod_abs (Int.Neg k) (Int.Pos l) = divmod k l"
haftmann@53069:   "divmod_abs (Int.Pos k) (Int.Neg l) = divmod k l"
haftmann@53069:   "divmod_abs j 0 = (0, \<bar>j\<bar>)"
haftmann@53069:   "divmod_abs 0 j = (0, 0)"
haftmann@53069:   by (simp_all add: prod_eq_iff)
haftmann@53069: 
haftmann@53069: lemma divmod_int_divmod_abs:
haftmann@53069:   "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
haftmann@33361:   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
haftmann@53069:     then divmod_abs k l
haftmann@53069:     else (let (r, s) = divmod_abs k l in
huffman@47108:        if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
haftmann@33361: proof -
haftmann@33361:   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
haftmann@33361:   show ?thesis
haftmann@53069:     by (simp add: prod_eq_iff split_def Let_def)
haftmann@33361:       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
haftmann@33361:       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
haftmann@33361: qed
haftmann@33361: 
haftmann@53069: lemma divmod_int_code [code]:
haftmann@53069:   "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
haftmann@33361:   apsnd ((op *) (sgn l)) (if sgn k = sgn l
haftmann@53069:     then divmod_abs k l
haftmann@53069:     else (let (r, s) = divmod_abs k l in
haftmann@33361:       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
haftmann@33361: proof -
haftmann@33361:   have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
haftmann@33361:     by (auto simp add: not_less sgn_if)
haftmann@53069:   then show ?thesis by (simp add: divmod_int_divmod_abs)
haftmann@33361: qed
haftmann@33361: 
haftmann@53069: hide_const (open) divmod_abs
haftmann@53069: 
haftmann@52435: code_identifier
haftmann@52435:   code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364: 
haftmann@33361: end
haftmann@52435: 




