--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/Quot/Examples/Quotient_Int.thy Wed Apr 28 17:05:20 2010 +0200
@@ -0,0 +1,379 @@
+(* Title: HOL/Quotient_Examples/Quotient_Int.thy
+ Author: Cezary Kaliszyk
+ Author: Christian Urban
+
+Integers based on Quotients.
+*)
+theory Quotient_Int
+imports Quotient_Product Nat
+begin
+
+fun
+ intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" (infix "\<approx>" 50)
+where
+ "intrel (x, y) (u, v) = (x + v = u + y)"
+
+quotient_type int = "nat \<times> nat" / intrel
+ by (auto simp add: equivp_def expand_fun_eq)
+
+instantiation int :: "{zero, one, plus, uminus, minus, times, ord, abs, sgn}"
+begin
+
+quotient_definition
+ "0 \<Colon> int" is "(0\<Colon>nat, 0\<Colon>nat)"
+
+quotient_definition
+ "1 \<Colon> int" is "(1\<Colon>nat, 0\<Colon>nat)"
+
+fun
+ plus_int_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
+where
+ "plus_int_raw (x, y) (u, v) = (x + u, y + v)"
+
+quotient_definition
+ "(op +) \<Colon> (int \<Rightarrow> int \<Rightarrow> int)" is "plus_int_raw"
+
+fun
+ uminus_int_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
+where
+ "uminus_int_raw (x, y) = (y, x)"
+
+quotient_definition
+ "(uminus \<Colon> (int \<Rightarrow> int))" is "uminus_int_raw"
+
+definition
+ minus_int_def [code del]: "z - w = z + (-w\<Colon>int)"
+
+fun
+ times_int_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
+where
+ "times_int_raw (x, y) (u, v) = (x*u + y*v, x*v + y*u)"
+
+quotient_definition
+ "(op *) :: (int \<Rightarrow> int \<Rightarrow> int)" is "times_int_raw"
+
+fun
+ le_int_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool"
+where
+ "le_int_raw (x, y) (u, v) = (x+v \<le> u+y)"
+
+quotient_definition
+ le_int_def: "(op \<le>) :: int \<Rightarrow> int \<Rightarrow> bool" is "le_int_raw"
+
+definition
+ less_int_def [code del]: "(z\<Colon>int) < w = (z \<le> w \<and> z \<noteq> w)"
+
+definition
+ zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"
+
+definition
+ zsgn_def: "sgn (i\<Colon>int) = (if i = 0 then 0 else if 0 < i then 1 else - 1)"
+
+instance ..
+
+end
+
+lemma [quot_respect]:
+ shows "(op \<approx> ===> op \<approx> ===> op \<approx>) plus_int_raw plus_int_raw"
+ and "(op \<approx> ===> op \<approx>) uminus_int_raw uminus_int_raw"
+ and "(op \<approx> ===> op \<approx> ===> op =) le_int_raw le_int_raw"
+ by auto
+
+lemma times_int_raw_fst:
+ assumes a: "x \<approx> z"
+ shows "times_int_raw x y \<approx> times_int_raw z y"
+ using a
+ apply(cases x, cases y, cases z)
+ apply(auto simp add: times_int_raw.simps intrel.simps)
+ apply(rename_tac u v w x y z)
+ apply(subgoal_tac "u*w + z*w = y*w + v*w & u*x + z*x = y*x + v*x")
+ apply(simp add: mult_ac)
+ apply(simp add: add_mult_distrib [symmetric])
+ done
+
+lemma times_int_raw_snd:
+ assumes a: "x \<approx> z"
+ shows "times_int_raw y x \<approx> times_int_raw y z"
+ using a
+ apply(cases x, cases y, cases z)
+ apply(auto simp add: times_int_raw.simps intrel.simps)
+ apply(rename_tac u v w x y z)
+ apply(subgoal_tac "u*w + z*w = y*w + v*w & u*x + z*x = y*x + v*x")
+ apply(simp add: mult_ac)
+ apply(simp add: add_mult_distrib [symmetric])
+ done
+
+lemma [quot_respect]:
+ shows "(op \<approx> ===> op \<approx> ===> op \<approx>) times_int_raw times_int_raw"
+ apply(simp only: fun_rel_def)
+ apply(rule allI | rule impI)+
+ apply(rule equivp_transp[OF int_equivp])
+ apply(rule times_int_raw_fst)
+ apply(assumption)
+ apply(rule times_int_raw_snd)
+ apply(assumption)
+ done
+
+lemma plus_assoc_raw:
+ shows "plus_int_raw (plus_int_raw i j) k \<approx> plus_int_raw i (plus_int_raw j k)"
+ by (cases i, cases j, cases k) (simp)
+
+lemma plus_sym_raw:
+ shows "plus_int_raw i j \<approx> plus_int_raw j i"
+ by (cases i, cases j) (simp)
+
+lemma plus_zero_raw:
+ shows "plus_int_raw (0, 0) i \<approx> i"
+ by (cases i) (simp)
+
+lemma plus_minus_zero_raw:
+ shows "plus_int_raw (uminus_int_raw i) i \<approx> (0, 0)"
+ by (cases i) (simp)
+
+lemma times_assoc_raw:
+ shows "times_int_raw (times_int_raw i j) k \<approx> times_int_raw i (times_int_raw j k)"
+ by (cases i, cases j, cases k)
+ (simp add: algebra_simps)
+
+lemma times_sym_raw:
+ shows "times_int_raw i j \<approx> times_int_raw j i"
+ by (cases i, cases j) (simp add: algebra_simps)
+
+lemma times_one_raw:
+ shows "times_int_raw (1, 0) i \<approx> i"
+ by (cases i) (simp)
+
+lemma times_plus_comm_raw:
+ shows "times_int_raw (plus_int_raw i j) k \<approx> plus_int_raw (times_int_raw i k) (times_int_raw j k)"
+by (cases i, cases j, cases k)
+ (simp add: algebra_simps)
+
+lemma one_zero_distinct:
+ shows "\<not> (0, 0) \<approx> ((1::nat), (0::nat))"
+ by simp
+
+text{* The integers form a @{text comm_ring_1}*}
+
+instance int :: comm_ring_1
+proof
+ fix i j k :: int
+ show "(i + j) + k = i + (j + k)"
+ by (lifting plus_assoc_raw)
+ show "i + j = j + i"
+ by (lifting plus_sym_raw)
+ show "0 + i = (i::int)"
+ by (lifting plus_zero_raw)
+ show "- i + i = 0"
+ by (lifting plus_minus_zero_raw)
+ show "i - j = i + - j"
+ by (simp add: minus_int_def)
+ show "(i * j) * k = i * (j * k)"
+ by (lifting times_assoc_raw)
+ show "i * j = j * i"
+ by (lifting times_sym_raw)
+ show "1 * i = i"
+ by (lifting times_one_raw)
+ show "(i + j) * k = i * k + j * k"
+ by (lifting times_plus_comm_raw)
+ show "0 \<noteq> (1::int)"
+ by (lifting one_zero_distinct)
+qed
+
+lemma plus_int_raw_rsp_aux:
+ assumes a: "a \<approx> b" "c \<approx> d"
+ shows "plus_int_raw a c \<approx> plus_int_raw b d"
+ using a
+ by (cases a, cases b, cases c, cases d)
+ (simp)
+
+lemma add_abs_int:
+ "(abs_int (x,y)) + (abs_int (u,v)) =
+ (abs_int (x + u, y + v))"
+ apply(simp add: plus_int_def id_simps)
+ apply(fold plus_int_raw.simps)
+ apply(rule Quotient_rel_abs[OF Quotient_int])
+ apply(rule plus_int_raw_rsp_aux)
+ apply(simp_all add: rep_abs_rsp_left[OF Quotient_int])
+ done
+
+definition int_of_nat_raw:
+ "int_of_nat_raw m = (m :: nat, 0 :: nat)"
+
+quotient_definition
+ "int_of_nat :: nat \<Rightarrow> int" is "int_of_nat_raw"
+
+lemma[quot_respect]:
+ shows "(op = ===> op \<approx>) int_of_nat_raw int_of_nat_raw"
+ by (simp add: equivp_reflp[OF int_equivp])
+
+lemma int_of_nat:
+ shows "of_nat m = int_of_nat m"
+ by (induct m)
+ (simp_all add: zero_int_def one_int_def int_of_nat_def int_of_nat_raw add_abs_int)
+
+lemma le_antisym_raw:
+ shows "le_int_raw i j \<Longrightarrow> le_int_raw j i \<Longrightarrow> i \<approx> j"
+ by (cases i, cases j) (simp)
+
+lemma le_refl_raw:
+ shows "le_int_raw i i"
+ by (cases i) (simp)
+
+lemma le_trans_raw:
+ shows "le_int_raw i j \<Longrightarrow> le_int_raw j k \<Longrightarrow> le_int_raw i k"
+ by (cases i, cases j, cases k) (simp)
+
+lemma le_cases_raw:
+ shows "le_int_raw i j \<or> le_int_raw j i"
+ by (cases i, cases j)
+ (simp add: linorder_linear)
+
+instance int :: linorder
+proof
+ fix i j k :: int
+ show antisym: "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"
+ by (lifting le_antisym_raw)
+ show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)"
+ by (auto simp add: less_int_def dest: antisym)
+ show "i \<le> i"
+ by (lifting le_refl_raw)
+ show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
+ by (lifting le_trans_raw)
+ show "i \<le> j \<or> j \<le> i"
+ by (lifting le_cases_raw)
+qed
+
+instantiation int :: distrib_lattice
+begin
+
+definition
+ "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
+
+definition
+ "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
+
+instance
+ by intro_classes
+ (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
+
+end
+
+lemma le_plus_int_raw:
+ shows "le_int_raw i j \<Longrightarrow> le_int_raw (plus_int_raw k i) (plus_int_raw k j)"
+ by (cases i, cases j, cases k) (simp)
+
+instance int :: ordered_cancel_ab_semigroup_add
+proof
+ fix i j k :: int
+ show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
+ by (lifting le_plus_int_raw)
+qed
+
+abbreviation
+ "less_int_raw i j \<equiv> le_int_raw i j \<and> \<not>(i \<approx> j)"
+
+lemma zmult_zless_mono2_lemma:
+ fixes i j::int
+ and k::nat
+ shows "i < j \<Longrightarrow> 0 < k \<Longrightarrow> of_nat k * i < of_nat k * j"
+ apply(induct "k")
+ apply(simp)
+ apply(case_tac "k = 0")
+ apply(simp_all add: left_distrib add_strict_mono)
+ done
+
+lemma zero_le_imp_eq_int_raw:
+ fixes k::"(nat \<times> nat)"
+ shows "less_int_raw (0, 0) k \<Longrightarrow> (\<exists>n > 0. k \<approx> int_of_nat_raw n)"
+ apply(cases k)
+ apply(simp add:int_of_nat_raw)
+ apply(auto)
+ apply(rule_tac i="b" and j="a" in less_Suc_induct)
+ apply(auto)
+ done
+
+lemma zero_le_imp_eq_int:
+ fixes k::int
+ shows "0 < k \<Longrightarrow> \<exists>n > 0. k = of_nat n"
+ unfolding less_int_def int_of_nat
+ by (lifting zero_le_imp_eq_int_raw)
+
+lemma zmult_zless_mono2:
+ fixes i j k::int
+ assumes a: "i < j" "0 < k"
+ shows "k * i < k * j"
+ using a
+ by (drule_tac zero_le_imp_eq_int) (auto simp add: zmult_zless_mono2_lemma)
+
+text{*The integers form an ordered integral domain*}
+instance int :: linordered_idom
+proof
+ fix i j k :: int
+ show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
+ by (rule zmult_zless_mono2)
+ show "\<bar>i\<bar> = (if i < 0 then -i else i)"
+ by (simp only: zabs_def)
+ show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
+ by (simp only: zsgn_def)
+qed
+
+lemmas int_distrib =
+ left_distrib [of "z1::int" "z2" "w", standard]
+ right_distrib [of "w::int" "z1" "z2", standard]
+ left_diff_distrib [of "z1::int" "z2" "w", standard]
+ right_diff_distrib [of "w::int" "z1" "z2", standard]
+ minus_add_distrib[of "z1::int" "z2", standard]
+
+lemma int_induct_raw:
+ assumes a: "P (0::nat, 0)"
+ and b: "\<And>i. P i \<Longrightarrow> P (plus_int_raw i (1, 0))"
+ and c: "\<And>i. P i \<Longrightarrow> P (plus_int_raw i (uminus_int_raw (1, 0)))"
+ shows "P x"
+proof (cases x, clarify)
+ fix a b
+ show "P (a, b)"
+ proof (induct a arbitrary: b rule: Nat.induct)
+ case zero
+ show "P (0, b)" using assms by (induct b) auto
+ next
+ case (Suc n)
+ then show ?case using assms by auto
+ qed
+qed
+
+lemma int_induct:
+ fixes x :: int
+ assumes a: "P 0"
+ and b: "\<And>i. P i \<Longrightarrow> P (i + 1)"
+ and c: "\<And>i. P i \<Longrightarrow> P (i - 1)"
+ shows "P x"
+ using a b c unfolding minus_int_def
+ by (lifting int_induct_raw)
+
+text {* Magnitide of an Integer, as a Natural Number: @{term nat} *}
+
+definition
+ "int_to_nat_raw \<equiv> \<lambda>(x, y).x - (y::nat)"
+
+quotient_definition
+ "int_to_nat::int \<Rightarrow> nat"
+is
+ "int_to_nat_raw"
+
+lemma [quot_respect]:
+ shows "(intrel ===> op =) int_to_nat_raw int_to_nat_raw"
+ by (auto iff: int_to_nat_raw_def)
+
+lemma nat_le_eq_zle_raw:
+ assumes a: "less_int_raw (0, 0) w \<or> le_int_raw (0, 0) z"
+ shows "(int_to_nat_raw w \<le> int_to_nat_raw z) = (le_int_raw w z)"
+ using a
+ by (cases w, cases z) (auto simp add: int_to_nat_raw_def)
+
+lemma nat_le_eq_zle:
+ fixes w z::"int"
+ shows "0 < w \<or> 0 \<le> z \<Longrightarrow> (int_to_nat w \<le> int_to_nat z) = (w \<le> z)"
+ unfolding less_int_def
+ by (lifting nat_le_eq_zle_raw)
+
+end