(* Title: HOL/Quotient_Examples/Quotient_Rat.thy
Author: Cezary Kaliszyk
Rational numbers defined with the quotient package, based on 'HOL/Rat.thy' by Makarius.
*)
theory Quotient_Rat imports Archimedean_Field
"~~/src/HOL/Library/Quotient_Product"
begin
definition
ratrel :: "(int \<times> int) \<Rightarrow> (int \<times> int) \<Rightarrow> bool" (infix "\<approx>" 50)
where
[simp]: "x \<approx> y \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x"
lemma ratrel_equivp:
"part_equivp ratrel"
proof (auto intro!: part_equivpI reflpI sympI transpI exI[of _ "1 :: int"])
fix a b c d e f :: int
assume nz: "d \<noteq> 0" "b \<noteq> 0"
assume y: "a * d = c * b"
assume x: "c * f = e * d"
then have "c * b * f = e * d * b" using nz by simp
then have "a * d * f = e * d * b" using y by simp
then show "a * f = e * b" using nz by simp
qed
quotient_type rat = "int \<times> int" / partial: ratrel
using ratrel_equivp .
instantiation rat :: "{zero, one, plus, uminus, minus, times, ord, abs_if, sgn_if}"
begin
quotient_definition
"0 \<Colon> rat" is "(0\<Colon>int, 1\<Colon>int)" by simp
quotient_definition
"1 \<Colon> rat" is "(1\<Colon>int, 1\<Colon>int)" by simp
fun times_rat_raw where
"times_rat_raw (a :: int, b :: int) (c, d) = (a * c, b * d)"
quotient_definition
"(op *) :: (rat \<Rightarrow> rat \<Rightarrow> rat)" is times_rat_raw by (auto simp add: mult_assoc mult_left_commute)
fun plus_rat_raw where
"plus_rat_raw (a :: int, b :: int) (c, d) = (a * d + c * b, b * d)"
quotient_definition
"(op +) :: (rat \<Rightarrow> rat \<Rightarrow> rat)" is plus_rat_raw
by (auto simp add: mult_commute mult_left_commute int_distrib(2))
fun uminus_rat_raw where
"uminus_rat_raw (a :: int, b :: int) = (-a, b)"
quotient_definition
"(uminus \<Colon> (rat \<Rightarrow> rat))" is "uminus_rat_raw" by fastforce
definition
minus_rat_def: "a - b = a + (-b\<Colon>rat)"
fun le_rat_raw where
"le_rat_raw (a :: int, b) (c, d) \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
quotient_definition
"(op \<le>) :: rat \<Rightarrow> rat \<Rightarrow> bool" is "le_rat_raw"
proof -
{
fix a b c d e f g h :: int
assume "a * f * (b * f) \<le> e * b * (b * f)"
then have le: "a * f * b * f \<le> e * b * b * f" by simp
assume nz: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" "h \<noteq> 0"
then have b2: "b * b > 0"
by (metis linorder_neqE_linordered_idom mult_eq_0_iff not_square_less_zero)
have f2: "f * f > 0" using nz(3)
by (metis linorder_neqE_linordered_idom mult_eq_0_iff not_square_less_zero)
assume eq: "a * d = c * b" "e * h = g * f"
have "a * f * b * f * d * d \<le> e * b * b * f * d * d" using le nz(2)
by (metis linorder_le_cases mult_right_mono mult_right_mono_neg)
then have "c * f * f * d * (b * b) \<le> e * f * d * d * (b * b)" using eq
by (metis (no_types) mult_assoc mult_commute)
then have "c * f * f * d \<le> e * f * d * d" using b2
by (metis leD linorder_le_less_linear mult_strict_right_mono)
then have "c * f * f * d * h * h \<le> e * f * d * d * h * h" using nz(4)
by (metis linorder_le_cases mult_right_mono mult_right_mono_neg)
then have "c * h * (d * h) * (f * f) \<le> g * d * (d * h) * (f * f)" using eq
by (metis (no_types) mult_assoc mult_commute)
then have "c * h * (d * h) \<le> g * d * (d * h)" using f2
by (metis leD linorder_le_less_linear mult_strict_right_mono)
}
then show "\<And>x y xa ya. x \<approx> y \<Longrightarrow> xa \<approx> ya \<Longrightarrow> le_rat_raw x xa = le_rat_raw y ya" by auto
qed
definition
less_rat_def: "(z\<Colon>rat) < w = (z \<le> w \<and> z \<noteq> w)"
definition
rabs_rat_def: "\<bar>i\<Colon>rat\<bar> = (if i < 0 then - i else i)"
definition
sgn_rat_def: "sgn (i\<Colon>rat) = (if i = 0 then 0 else if 0 < i then 1 else - 1)"
instance by intro_classes
(auto simp add: rabs_rat_def sgn_rat_def)
end
definition
Fract_raw :: "int \<Rightarrow> int \<Rightarrow> (int \<times> int)"
where [simp]: "Fract_raw a b = (if b = 0 then (0, 1) else (a, b))"
quotient_definition "Fract :: int \<Rightarrow> int \<Rightarrow> rat" is
Fract_raw by simp
lemmas [simp] = Respects_def
instantiation rat :: comm_ring_1
begin
instance proof
fix a b c :: rat
show "a * b * c = a * (b * c)"
by partiality_descending auto
show "a * b = b * a"
by partiality_descending auto
show "1 * a = a"
by partiality_descending auto
show "a + b + c = a + (b + c)"
by partiality_descending (auto simp add: mult_commute right_distrib)
show "a + b = b + a"
by partiality_descending auto
show "0 + a = a"
by partiality_descending auto
show "- a + a = 0"
by partiality_descending auto
show "a - b = a + - b"
by (simp add: minus_rat_def)
show "(a + b) * c = a * c + b * c"
by partiality_descending (auto simp add: mult_commute right_distrib)
show "(0 :: rat) \<noteq> (1 :: rat)"
by partiality_descending auto
qed
end
lemma add_one_Fract: "1 + Fract (int k) 1 = Fract (1 + int k) 1"
by descending auto
lemma of_nat_rat: "of_nat k = Fract (of_nat k) 1"
apply (induct k)
apply (simp add: zero_rat_def Fract_def)
apply (simp add: add_one_Fract)
done
lemma of_int_rat: "of_int k = Fract k 1"
apply (cases k rule: int_diff_cases)
apply (auto simp add: of_nat_rat minus_rat_def)
apply descending
apply auto
done
instantiation rat :: number_ring
begin
definition
rat_number_of_def: "number_of w = Fract w 1"
instance apply default
unfolding rat_number_of_def of_int_rat ..
end
instantiation rat :: field_inverse_zero begin
fun rat_inverse_raw where
"rat_inverse_raw (a :: int, b :: int) = (if a = 0 then (0, 1) else (b, a))"
quotient_definition
"inverse :: rat \<Rightarrow> rat" is rat_inverse_raw by (force simp add: mult_commute)
definition
divide_rat_def: "q / r = q * inverse (r::rat)"
instance proof
fix q :: rat
assume "q \<noteq> 0"
then show "inverse q * q = 1"
by partiality_descending auto
next
fix q r :: rat
show "q / r = q * inverse r" by (simp add: divide_rat_def)
next
show "inverse 0 = (0::rat)" by partiality_descending auto
qed
end
instantiation rat :: linorder
begin
instance proof
fix q r s :: rat
{
assume "q \<le> r" and "r \<le> s"
then show "q \<le> s"
proof (partiality_descending, auto simp add: mult_assoc[symmetric])
fix a b c d e f :: int
assume nz: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0"
then have d2: "d * d > 0"
by (metis linorder_neqE_linordered_idom mult_eq_0_iff not_square_less_zero)
assume le: "a * d * b * d \<le> c * b * b * d" "c * f * d * f \<le> e * d * d * f"
then have a: "a * d * b * d * f * f \<le> c * b * b * d * f * f" using nz(3)
by (metis linorder_le_cases mult_right_mono mult_right_mono_neg)
have "c * f * d * f * b * b \<le> e * d * d * f * b * b" using nz(1) le
by (metis linorder_le_cases mult_right_mono mult_right_mono_neg)
then have "a * f * b * f * (d * d) \<le> e * b * b * f * (d * d)" using a
by (simp add: algebra_simps)
then show "a * f * b * f \<le> e * b * b * f" using d2
by (metis leD linorder_le_less_linear mult_strict_right_mono)
qed
next
assume "q \<le> r" and "r \<le> q"
then show "q = r"
apply (partiality_descending, auto)
apply (case_tac "b > 0", case_tac [!] "ba > 0")
apply simp_all
done
next
show "q \<le> q" by partiality_descending auto
show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)"
unfolding less_rat_def
by partiality_descending (auto simp add: le_less mult_commute)
show "q \<le> r \<or> r \<le> q"
by partiality_descending (auto simp add: mult_commute linorder_linear)
}
qed
end
instance rat :: archimedean_field
proof
fix q r s :: rat
show "q \<le> r ==> s + q \<le> s + r"
proof (partiality_descending, auto simp add: algebra_simps, simp add: mult_assoc[symmetric])
fix a b c d e :: int
assume "e \<noteq> 0"
then have e2: "e * e > 0"
by (metis linorder_neqE_linordered_idom mult_eq_0_iff not_square_less_zero)
assume "a * b * d * d \<le> b * b * c * d"
then show "a * b * d * d * e * e * e * e \<le> b * b * c * d * e * e * e * e"
using e2 by (metis comm_mult_left_mono mult_commute linorder_le_cases
mult_left_mono_neg)
qed
show "q < r ==> 0 < s ==> s * q < s * r" unfolding less_rat_def
proof (partiality_descending, auto simp add: algebra_simps, simp add: mult_assoc[symmetric])
fix a b c d e f :: int
assume a: "e \<noteq> 0" "f \<noteq> 0" "0 \<le> e * f" "a * b * d * d \<le> b * b * c * d"
have "a * b * d * d * (e * f) \<le> b * b * c * d * (e * f)" using a
by (simp add: mult_right_mono)
then show "a * b * d * d * e * f * f * f \<le> b * b * c * d * e * f * f * f"
by (simp add: mult_assoc[symmetric]) (metis a(3) comm_mult_left_mono
mult_commute mult_left_mono_neg zero_le_mult_iff)
qed
show "\<exists>z. r \<le> of_int z"
unfolding of_int_rat
proof (partiality_descending, auto)
fix a b :: int
assume "b \<noteq> 0"
then have "a * b \<le> (a div b + 1) * b * b"
by (metis mult_commute div_mult_self1_is_id less_int_def linorder_le_cases zdiv_mono1 zdiv_mono1_neg zle_add1_eq_le)
then show "\<exists>z\<Colon>int. a * b \<le> z * b * b" by auto
qed
qed
end