(* Title: HOL/Quotient_Examples/Quotient_Int.thy
Author: Cezary Kaliszyk
Author: Christian Urban
Integers based on Quotients, based on an older version by Larry Paulson.
*)
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 default
(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