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src/HOL/Real/Rational.thy

author | nipkow |

Mon, 16 Aug 2004 14:22:27 +0200 | |

changeset 15131 | c69542757a4d |

parent 15013 | 34264f5e4691 |

child 15140 | 322485b816ac |

permissions | -rw-r--r-- |

New theory header syntax.

(* Title: HOL/Library/Rational.thy ID: $Id$ Author: Markus Wenzel, TU Muenchen *) header {* Rational numbers *} theory Rational import Quotient files ("rat_arith.ML") begin subsection {* Fractions *} subsubsection {* The type of fractions *} typedef fraction = "{(a, b) :: int \<times> int | a b. b \<noteq> 0}" proof show "(0, 1) \<in> ?fraction" by simp qed constdefs fract :: "int => int => fraction" "fract a b == Abs_fraction (a, b)" num :: "fraction => int" "num Q == fst (Rep_fraction Q)" den :: "fraction => int" "den Q == snd (Rep_fraction Q)" lemma fract_num [simp]: "b \<noteq> 0 ==> num (fract a b) = a" by (simp add: fract_def num_def fraction_def Abs_fraction_inverse) lemma fract_den [simp]: "b \<noteq> 0 ==> den (fract a b) = b" by (simp add: fract_def den_def fraction_def Abs_fraction_inverse) lemma fraction_cases [case_names fract, cases type: fraction]: "(!!a b. Q = fract a b ==> b \<noteq> 0 ==> C) ==> C" proof - assume r: "!!a b. Q = fract a b ==> b \<noteq> 0 ==> C" obtain a b where "Q = fract a b" and "b \<noteq> 0" by (cases Q) (auto simp add: fract_def fraction_def) thus C by (rule r) qed lemma fraction_induct [case_names fract, induct type: fraction]: "(!!a b. b \<noteq> 0 ==> P (fract a b)) ==> P Q" by (cases Q) simp subsubsection {* Equivalence of fractions *} instance fraction :: eqv .. defs (overloaded) equiv_fraction_def: "Q \<sim> R == num Q * den R = num R * den Q" lemma equiv_fraction_iff [iff]: "b \<noteq> 0 ==> b' \<noteq> 0 ==> (fract a b \<sim> fract a' b') = (a * b' = a' * b)" by (simp add: equiv_fraction_def) instance fraction :: equiv proof fix Q R S :: fraction { show "Q \<sim> Q" proof (induct Q) fix a b :: int assume "b \<noteq> 0" and "b \<noteq> 0" with refl show "fract a b \<sim> fract a b" .. qed next assume "Q \<sim> R" and "R \<sim> S" show "Q \<sim> S" proof (insert prems, induct Q, induct R, induct S) fix a b a' b' a'' b'' :: int assume b: "b \<noteq> 0" and b': "b' \<noteq> 0" and b'': "b'' \<noteq> 0" assume "fract a b \<sim> fract a' b'" hence eq1: "a * b' = a' * b" .. assume "fract a' b' \<sim> fract a'' b''" hence eq2: "a' * b'' = a'' * b'" .. have "a * b'' = a'' * b" proof cases assume "a' = 0" with b' eq1 eq2 have "a = 0 \<and> a'' = 0" by auto thus ?thesis by simp next assume a': "a' \<noteq> 0" from eq1 eq2 have "(a * b') * (a' * b'') = (a' * b) * (a'' * b')" by simp hence "(a * b'') * (a' * b') = (a'' * b) * (a' * b')" by (simp only: mult_ac) with a' b' show ?thesis by simp qed thus "fract a b \<sim> fract a'' b''" .. qed next show "Q \<sim> R ==> R \<sim> Q" proof (induct Q, induct R) fix a b a' b' :: int assume b: "b \<noteq> 0" and b': "b' \<noteq> 0" assume "fract a b \<sim> fract a' b'" hence "a * b' = a' * b" .. hence "a' * b = a * b'" .. thus "fract a' b' \<sim> fract a b" .. qed } qed lemma eq_fraction_iff [iff]: "b \<noteq> 0 ==> b' \<noteq> 0 ==> (\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>) = (a * b' = a' * b)" by (simp add: equiv_fraction_iff quot_equality) subsubsection {* Operations on fractions *} text {* We define the basic arithmetic operations on fractions and demonstrate their ``well-definedness'', i.e.\ congruence with respect to equivalence of fractions. *} instance fraction :: "{zero, one, plus, minus, times, inverse, ord}" .. defs (overloaded) zero_fraction_def: "0 == fract 0 1" one_fraction_def: "1 == fract 1 1" add_fraction_def: "Q + R == fract (num Q * den R + num R * den Q) (den Q * den R)" minus_fraction_def: "-Q == fract (-(num Q)) (den Q)" mult_fraction_def: "Q * R == fract (num Q * num R) (den Q * den R)" inverse_fraction_def: "inverse Q == fract (den Q) (num Q)" le_fraction_def: "Q \<le> R == (num Q * den R) * (den Q * den R) \<le> (num R * den Q) * (den Q * den R)" lemma is_zero_fraction_iff: "b \<noteq> 0 ==> (\<lfloor>fract a b\<rfloor> = \<lfloor>0\<rfloor>) = (a = 0)" by (simp add: zero_fraction_def eq_fraction_iff) theorem add_fraction_cong: "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==> \<lfloor>fract a b + fract c d\<rfloor> = \<lfloor>fract a' b' + fract c' d'\<rfloor>" proof - assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0" assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" .. assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" .. have "\<lfloor>fract (a * d + c * b) (b * d)\<rfloor> = \<lfloor>fract (a' * d' + c' * b') (b' * d')\<rfloor>" proof show "(a * d + c * b) * (b' * d') = (a' * d' + c' * b') * (b * d)" (is "?lhs = ?rhs") proof - have "?lhs = (a * b') * (d * d') + (c * d') * (b * b')" by (simp add: int_distrib mult_ac) also have "... = (a' * b) * (d * d') + (c' * d) * (b * b')" by (simp only: eq1 eq2) also have "... = ?rhs" by (simp add: int_distrib mult_ac) finally show "?lhs = ?rhs" . qed from neq show "b * d \<noteq> 0" by simp from neq show "b' * d' \<noteq> 0" by simp qed with neq show ?thesis by (simp add: add_fraction_def) qed theorem minus_fraction_cong: "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> \<lfloor>-(fract a b)\<rfloor> = \<lfloor>-(fract a' b')\<rfloor>" proof - assume neq: "b \<noteq> 0" "b' \<noteq> 0" assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence "a * b' = a' * b" .. hence "-a * b' = -a' * b" by simp hence "\<lfloor>fract (-a) b\<rfloor> = \<lfloor>fract (-a') b'\<rfloor>" .. with neq show ?thesis by (simp add: minus_fraction_def) qed theorem mult_fraction_cong: "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==> \<lfloor>fract a b * fract c d\<rfloor> = \<lfloor>fract a' b' * fract c' d'\<rfloor>" proof - assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0" assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" .. assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" .. have "\<lfloor>fract (a * c) (b * d)\<rfloor> = \<lfloor>fract (a' * c') (b' * d')\<rfloor>" proof from eq1 eq2 have "(a * b') * (c * d') = (a' * b) * (c' * d)" by simp thus "(a * c) * (b' * d') = (a' * c') * (b * d)" by (simp add: mult_ac) from neq show "b * d \<noteq> 0" by simp from neq show "b' * d' \<noteq> 0" by simp qed with neq show "\<lfloor>fract a b * fract c d\<rfloor> = \<lfloor>fract a' b' * fract c' d'\<rfloor>" by (simp add: mult_fraction_def) qed theorem inverse_fraction_cong: "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor> ==> \<lfloor>fract a' b'\<rfloor> \<noteq> \<lfloor>0\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> \<lfloor>inverse (fract a b)\<rfloor> = \<lfloor>inverse (fract a' b')\<rfloor>" proof - assume neq: "b \<noteq> 0" "b' \<noteq> 0" assume "\<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor>" and "\<lfloor>fract a' b'\<rfloor> \<noteq> \<lfloor>0\<rfloor>" with neq obtain "a \<noteq> 0" and "a' \<noteq> 0" by (simp add: is_zero_fraction_iff) assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence "a * b' = a' * b" .. hence "b * a' = b' * a" by (simp only: mult_ac) hence "\<lfloor>fract b a\<rfloor> = \<lfloor>fract b' a'\<rfloor>" .. with neq show ?thesis by (simp add: inverse_fraction_def) qed theorem le_fraction_cong: "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==> (fract a b \<le> fract c d) = (fract a' b' \<le> fract c' d')" proof - assume neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0" assume "\<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor>" hence eq1: "a * b' = a' * b" .. assume "\<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor>" hence eq2: "c * d' = c' * d" .. let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))" { fix a b c d x :: int assume x: "x \<noteq> 0" have "?le a b c d = ?le (a * x) (b * x) c d" proof - from x have "0 < x * x" by (auto simp add: zero_less_mult_iff) hence "?le a b c d = ((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))" by (simp add: mult_le_cancel_right) also have "... = ?le (a * x) (b * x) c d" by (simp add: mult_ac) finally show ?thesis . qed } note le_factor = this let ?D = "b * d" and ?D' = "b' * d'" from neq have D: "?D \<noteq> 0" by simp from neq have "?D' \<noteq> 0" by simp hence "?le a b c d = ?le (a * ?D') (b * ?D') c d" by (rule le_factor) also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" by (simp add: mult_ac) also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')" by (simp only: eq1 eq2) also have "... = ?le (a' * ?D) (b' * ?D) c' d'" by (simp add: mult_ac) also from D have "... = ?le a' b' c' d'" by (rule le_factor [symmetric]) finally have "?le a b c d = ?le a' b' c' d'" . with neq show ?thesis by (simp add: le_fraction_def) qed subsection {* Rational numbers *} subsubsection {* The type of rational numbers *} typedef (Rat) rat = "UNIV :: fraction quot set" .. lemma RatI [intro, simp]: "Q \<in> Rat" by (simp add: Rat_def) constdefs fraction_of :: "rat => fraction" "fraction_of q == pick (Rep_Rat q)" rat_of :: "fraction => rat" "rat_of Q == Abs_Rat \<lfloor>Q\<rfloor>" theorem rat_of_equality [iff?]: "(rat_of Q = rat_of Q') = (\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor>)" by (simp add: rat_of_def Abs_Rat_inject) lemma rat_of: "\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor> ==> rat_of Q = rat_of Q'" .. constdefs Fract :: "int => int => rat" "Fract a b == rat_of (fract a b)" theorem Fract_inverse: "\<lfloor>fraction_of (Fract a b)\<rfloor> = \<lfloor>fract a b\<rfloor>" by (simp add: fraction_of_def rat_of_def Fract_def Abs_Rat_inverse pick_inverse) theorem Fract_equality [iff?]: "(Fract a b = Fract c d) = (\<lfloor>fract a b\<rfloor> = \<lfloor>fract c d\<rfloor>)" by (simp add: Fract_def rat_of_equality) theorem eq_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> (Fract a b = Fract c d) = (a * d = c * b)" by (simp add: Fract_equality eq_fraction_iff) theorem Rat_cases [case_names Fract, cases type: rat]: "(!!a b. q = Fract a b ==> b \<noteq> 0 ==> C) ==> C" proof - assume r: "!!a b. q = Fract a b ==> b \<noteq> 0 ==> C" obtain x where "q = Abs_Rat x" by (cases q) moreover obtain Q where "x = \<lfloor>Q\<rfloor>" by (cases x) moreover obtain a b where "Q = fract a b" and "b \<noteq> 0" by (cases Q) ultimately have "q = Fract a b" by (simp only: Fract_def rat_of_def) thus ?thesis by (rule r) qed theorem Rat_induct [case_names Fract, induct type: rat]: "(!!a b. b \<noteq> 0 ==> P (Fract a b)) ==> P q" by (cases q) simp subsubsection {* Canonical function definitions *} text {* Note that the unconditional version below is much easier to read. *} theorem rat_cond_function: "(!!q r. P \<lfloor>fraction_of q\<rfloor> \<lfloor>fraction_of r\<rfloor> ==> f q r == g (fraction_of q) (fraction_of r)) ==> (!!a b a' b' c d c' d'. \<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==> P \<lfloor>fract a b\<rfloor> \<lfloor>fract c d\<rfloor> ==> P \<lfloor>fract a' b'\<rfloor> \<lfloor>fract c' d'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==> g (fract a b) (fract c d) = g (fract a' b') (fract c' d')) ==> P \<lfloor>fract a b\<rfloor> \<lfloor>fract c d\<rfloor> ==> f (Fract a b) (Fract c d) = g (fract a b) (fract c d)" (is "PROP ?eq ==> PROP ?cong ==> ?P ==> _") proof - assume eq: "PROP ?eq" and cong: "PROP ?cong" and P: ?P have "f (Abs_Rat \<lfloor>fract a b\<rfloor>) (Abs_Rat \<lfloor>fract c d\<rfloor>) = g (fract a b) (fract c d)" proof (rule quot_cond_function) fix X Y assume "P X Y" with eq show "f (Abs_Rat X) (Abs_Rat Y) == g (pick X) (pick Y)" by (simp add: fraction_of_def pick_inverse Abs_Rat_inverse) next fix Q Q' R R' :: fraction show "\<lfloor>Q\<rfloor> = \<lfloor>Q'\<rfloor> ==> \<lfloor>R\<rfloor> = \<lfloor>R'\<rfloor> ==> P \<lfloor>Q\<rfloor> \<lfloor>R\<rfloor> ==> P \<lfloor>Q'\<rfloor> \<lfloor>R'\<rfloor> ==> g Q R = g Q' R'" by (induct Q, induct Q', induct R, induct R') (rule cong) qed thus ?thesis by (unfold Fract_def rat_of_def) qed theorem rat_function: "(!!q r. f q r == g (fraction_of q) (fraction_of r)) ==> (!!a b a' b' c d c' d'. \<lfloor>fract a b\<rfloor> = \<lfloor>fract a' b'\<rfloor> ==> \<lfloor>fract c d\<rfloor> = \<lfloor>fract c' d'\<rfloor> ==> b \<noteq> 0 ==> b' \<noteq> 0 ==> d \<noteq> 0 ==> d' \<noteq> 0 ==> g (fract a b) (fract c d) = g (fract a' b') (fract c' d')) ==> f (Fract a b) (Fract c d) = g (fract a b) (fract c d)" proof - case rule_context from this TrueI show ?thesis by (rule rat_cond_function) qed subsubsection {* Standard operations on rational numbers *} instance rat :: "{zero, one, plus, minus, times, inverse, ord}" .. defs (overloaded) zero_rat_def: "0 == rat_of 0" one_rat_def: "1 == rat_of 1" add_rat_def: "q + r == rat_of (fraction_of q + fraction_of r)" minus_rat_def: "-q == rat_of (-(fraction_of q))" diff_rat_def: "q - r == q + (-(r::rat))" mult_rat_def: "q * r == rat_of (fraction_of q * fraction_of r)" inverse_rat_def: "inverse q == if q=0 then 0 else rat_of (inverse (fraction_of q))" divide_rat_def: "q / r == q * inverse (r::rat)" le_rat_def: "q \<le> r == fraction_of q \<le> fraction_of r" less_rat_def: "q < r == q \<le> r \<and> q \<noteq> (r::rat)" abs_rat_def: "\<bar>q\<bar> == if q < 0 then -q else (q::rat)" theorem zero_rat: "0 = Fract 0 1" by (simp add: zero_rat_def zero_fraction_def rat_of_def Fract_def) theorem one_rat: "1 = Fract 1 1" by (simp add: one_rat_def one_fraction_def rat_of_def Fract_def) theorem add_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> Fract a b + Fract c d = Fract (a * d + c * b) (b * d)" proof - have "Fract a b + Fract c d = rat_of (fract a b + fract c d)" by (rule rat_function, rule add_rat_def, rule rat_of, rule add_fraction_cong) also assume "b \<noteq> 0" "d \<noteq> 0" hence "fract a b + fract c d = fract (a * d + c * b) (b * d)" by (simp add: add_fraction_def) finally show ?thesis by (unfold Fract_def) qed theorem minus_rat: "b \<noteq> 0 ==> -(Fract a b) = Fract (-a) b" proof - have "-(Fract a b) = rat_of (-(fract a b))" by (rule rat_function, rule minus_rat_def, rule rat_of, rule minus_fraction_cong) also assume "b \<noteq> 0" hence "-(fract a b) = fract (-a) b" by (simp add: minus_fraction_def) finally show ?thesis by (unfold Fract_def) qed theorem diff_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> Fract a b - Fract c d = Fract (a * d - c * b) (b * d)" by (simp add: diff_rat_def add_rat minus_rat) theorem mult_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> Fract a b * Fract c d = Fract (a * c) (b * d)" proof - have "Fract a b * Fract c d = rat_of (fract a b * fract c d)" by (rule rat_function, rule mult_rat_def, rule rat_of, rule mult_fraction_cong) also assume "b \<noteq> 0" "d \<noteq> 0" hence "fract a b * fract c d = fract (a * c) (b * d)" by (simp add: mult_fraction_def) finally show ?thesis by (unfold Fract_def) qed theorem inverse_rat: "Fract a b \<noteq> 0 ==> b \<noteq> 0 ==> inverse (Fract a b) = Fract b a" proof - assume neq: "b \<noteq> 0" and nonzero: "Fract a b \<noteq> 0" hence "\<lfloor>fract a b\<rfloor> \<noteq> \<lfloor>0\<rfloor>" by (simp add: zero_rat eq_rat is_zero_fraction_iff) with _ inverse_fraction_cong [THEN rat_of] have "inverse (Fract a b) = rat_of (inverse (fract a b))" proof (rule rat_cond_function) fix q assume cond: "\<lfloor>fraction_of q\<rfloor> \<noteq> \<lfloor>0\<rfloor>" have "q \<noteq> 0" proof (cases q) fix a b assume "b \<noteq> 0" and "q = Fract a b" from this cond show ?thesis by (simp add: Fract_inverse is_zero_fraction_iff zero_rat eq_rat) qed thus "inverse q == rat_of (inverse (fraction_of q))" by (simp add: inverse_rat_def) qed also from neq nonzero have "inverse (fract a b) = fract b a" by (simp add: inverse_fraction_def) finally show ?thesis by (unfold Fract_def) qed theorem divide_rat: "Fract c d \<noteq> 0 ==> b \<noteq> 0 ==> d \<noteq> 0 ==> Fract a b / Fract c d = Fract (a * d) (b * c)" proof - assume neq: "b \<noteq> 0" "d \<noteq> 0" and nonzero: "Fract c d \<noteq> 0" hence "c \<noteq> 0" by (simp add: zero_rat eq_rat) with neq nonzero show ?thesis by (simp add: divide_rat_def inverse_rat mult_rat) qed theorem le_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> (Fract a b \<le> Fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))" proof - have "(Fract a b \<le> Fract c d) = (fract a b \<le> fract c d)" by (rule rat_function, rule le_rat_def, rule le_fraction_cong) also assume "b \<noteq> 0" "d \<noteq> 0" hence "(fract a b \<le> fract c d) = ((a * d) * (b * d) \<le> (c * b) * (b * d))" by (simp add: le_fraction_def) finally show ?thesis . qed theorem less_rat: "b \<noteq> 0 ==> d \<noteq> 0 ==> (Fract a b < Fract c d) = ((a * d) * (b * d) < (c * b) * (b * d))" by (simp add: less_rat_def le_rat eq_rat order_less_le) theorem abs_rat: "b \<noteq> 0 ==> \<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>" by (simp add: abs_rat_def minus_rat zero_rat less_rat eq_rat) (auto simp add: mult_less_0_iff zero_less_mult_iff order_le_less split: abs_split) subsubsection {* The ordered field of rational numbers *} lemma rat_add_assoc: "(q + r) + s = q + (r + (s::rat))" by (induct q, induct r, induct s) (simp add: add_rat add_ac mult_ac int_distrib) lemma rat_add_0: "0 + q = (q::rat)" by (induct q) (simp add: zero_rat add_rat) lemma rat_left_minus: "(-q) + q = (0::rat)" by (induct q) (simp add: zero_rat minus_rat add_rat eq_rat) instance rat :: field proof fix q r s :: rat show "(q + r) + s = q + (r + s)" by (rule rat_add_assoc) show "q + r = r + q" by (induct q, induct r) (simp add: add_rat add_ac mult_ac) show "0 + q = q" by (induct q) (simp add: zero_rat add_rat) show "(-q) + q = 0" by (rule rat_left_minus) show "q - r = q + (-r)" by (induct q, induct r) (simp add: add_rat minus_rat diff_rat) show "(q * r) * s = q * (r * s)" by (induct q, induct r, induct s) (simp add: mult_rat mult_ac) show "q * r = r * q" by (induct q, induct r) (simp add: mult_rat mult_ac) show "1 * q = q" by (induct q) (simp add: one_rat mult_rat) show "(q + r) * s = q * s + r * s" by (induct q, induct r, induct s) (simp add: add_rat mult_rat eq_rat int_distrib) show "q \<noteq> 0 ==> inverse q * q = 1" by (induct q) (simp add: inverse_rat mult_rat one_rat zero_rat eq_rat) show "q / r = q * inverse r" by (simp add: divide_rat_def) show "0 \<noteq> (1::rat)" by (simp add: zero_rat one_rat eq_rat) qed instance rat :: linorder proof fix q r s :: rat { assume "q \<le> r" and "r \<le> s" show "q \<le> s" proof (insert prems, induct q, induct r, induct s) fix a b c d e f :: int assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract e f" show "Fract a b \<le> Fract e f" proof - from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f" by (auto simp add: zero_less_mult_iff linorder_neq_iff) have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)" proof - from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)" by (simp add: le_rat) with ff show ?thesis by (simp add: mult_le_cancel_right) qed also have "... = (c * f) * (d * f) * (b * b)" by (simp only: mult_ac) also have "... \<le> (e * d) * (d * f) * (b * b)" proof - from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)" by (simp add: le_rat) with bb show ?thesis by (simp add: mult_le_cancel_right) qed finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)" by (simp only: mult_ac) with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)" by (simp add: mult_le_cancel_right) with neq show ?thesis by (simp add: le_rat) qed qed next assume "q \<le> r" and "r \<le> q" show "q = r" proof (insert prems, induct q, induct r) fix a b c d :: int assume neq: "b \<noteq> 0" "d \<noteq> 0" assume 1: "Fract a b \<le> Fract c d" and 2: "Fract c d \<le> Fract a b" show "Fract a b = Fract c d" proof - from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)" by (simp add: le_rat) also have "... \<le> (a * d) * (b * d)" proof - from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)" by (simp add: le_rat) thus ?thesis by (simp only: mult_ac) qed finally have "(a * d) * (b * d) = (c * b) * (b * d)" . moreover from neq have "b * d \<noteq> 0" by simp ultimately have "a * d = c * b" by simp with neq show ?thesis by (simp add: eq_rat) qed qed next show "q \<le> q" by (induct q) (simp add: le_rat) show "(q < r) = (q \<le> r \<and> q \<noteq> r)" by (simp only: less_rat_def) show "q \<le> r \<or> r \<le> q" by (induct q, induct r) (simp add: le_rat mult_ac, arith) } qed instance rat :: ordered_field proof fix q r s :: rat show "q \<le> r ==> s + q \<le> s + r" proof (induct q, induct r, induct s) fix a b c d e f :: int assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" assume le: "Fract a b \<le> Fract c d" show "Fract e f + Fract a b \<le> Fract e f + Fract c d" proof - let ?F = "f * f" from neq have F: "0 < ?F" by (auto simp add: zero_less_mult_iff) from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)" by (simp add: le_rat) with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F" by (simp add: mult_le_cancel_right) with neq show ?thesis by (simp add: add_rat le_rat mult_ac int_distrib) qed qed show "q < r ==> 0 < s ==> s * q < s * r" proof (induct q, induct r, induct s) fix a b c d e f :: int assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" assume le: "Fract a b < Fract c d" assume gt: "0 < Fract e f" show "Fract e f * Fract a b < Fract e f * Fract c d" proof - let ?E = "e * f" and ?F = "f * f" from neq gt have "0 < ?E" by (auto simp add: zero_rat less_rat le_rat order_less_le eq_rat) moreover from neq have "0 < ?F" by (auto simp add: zero_less_mult_iff) moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)" by (simp add: less_rat) ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F" by (simp add: mult_less_cancel_right) with neq show ?thesis by (simp add: less_rat mult_rat mult_ac) qed qed show "\<bar>q\<bar> = (if q < 0 then -q else q)" by (simp only: abs_rat_def) qed instance rat :: division_by_zero proof show "inverse 0 = (0::rat)" by (simp add: inverse_rat_def) qed subsection {* Various Other Results *} lemma minus_rat_cancel [simp]: "b \<noteq> 0 ==> Fract (-a) (-b) = Fract a b" by (simp add: Fract_equality eq_fraction_iff) theorem Rat_induct_pos [case_names Fract, induct type: rat]: assumes step: "!!a b. 0 < b ==> P (Fract a b)" shows "P q" proof (cases q) have step': "!!a b. b < 0 ==> P (Fract a b)" proof - fix a::int and b::int assume b: "b < 0" hence "0 < -b" by simp hence "P (Fract (-a) (-b))" by (rule step) thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b]) qed case (Fract a b) thus "P q" by (force simp add: linorder_neq_iff step step') qed lemma zero_less_Fract_iff: "0 < b ==> (0 < Fract a b) = (0 < a)" by (simp add: zero_rat less_rat order_less_imp_not_eq2 zero_less_mult_iff) lemma Fract_add_one: "n \<noteq> 0 ==> Fract (m + n) n = Fract m n + 1" apply (insert add_rat [of concl: m n 1 1]) apply (simp add: one_rat [symmetric]) done lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k" apply (induct k) apply (simp add: zero_rat) apply (simp add: Fract_add_one) done lemma Fract_of_int_eq: "Fract k 1 = of_int k" proof (cases k rule: int_cases) case (nonneg n) thus ?thesis by (simp add: int_eq_of_nat Fract_of_nat_eq) next case (neg n) hence "Fract k 1 = - (Fract (of_nat (Suc n)) 1)" by (simp only: minus_rat int_eq_of_nat) also have "... = - (of_nat (Suc n))" by (simp only: Fract_of_nat_eq) finally show ?thesis by (simp add: only: prems int_eq_of_nat of_int_minus of_int_of_nat_eq) qed subsection {* Numerals and Arithmetic *} instance rat :: number .. defs (overloaded) rat_number_of_def: "(number_of w :: rat) == of_int (Rep_Bin w)" --{*the type constraint is essential!*} instance rat :: number_ring by (intro_classes, simp add: rat_number_of_def) declare diff_rat_def [symmetric] use "rat_arith.ML" setup rat_arith_setup end