# HG changeset patch # User haftmann # Date 1233929746 -3600 # Node ID 2cf979ed69b88ba22a7e403bebb471d0f5d06337 # Parent ab8c54355f2e669406c548e0d0ac462ad80b2f21# Parent 0ab754d13ccd48f759aebd6b5c646b8a9e279df1 merged diff -r ab8c54355f2e -r 2cf979ed69b8 NEWS --- a/NEWS Fri Feb 06 14:36:58 2009 +0100 +++ b/NEWS Fri Feb 06 15:15:46 2009 +0100 @@ -196,14 +196,17 @@ * Auxiliary class "itself" has disappeared -- classes without any parameter are treated as expected by the 'class' command. -* Theory "Reflection" now resides in HOL/Library. Common reflection examples -(Cooper, MIR, Ferrack, Approximation) now in distinct session directory -HOL/Reflection. Here Approximation provides the new proof method -"approximation". It proves formulas on real values by using interval arithmetic. +* Leibnitz's Series for Pi and the arcus tangens and logarithm series. + +* Common decision procedures (Cooper, MIR, Ferrack, Approximation, Dense_Linear_Order) +now in directory HOL/Decision_Procs. + +* Theory HOL/Decisioin_Procs/Approximation.thy provides the new proof method +"approximation". It proves formulas on real values by using interval arithmetic. In the formulas are also the transcendental functions sin, cos, tan, atan, ln, -exp and the constant pi are allowed. For examples see -src/HOL/ex/ApproximationEx.thy. To reach this the Leibnitz's Series for Pi and -the arcus tangens and logarithm series is now proved in Isabelle. +exp and the constant pi are allowed. For examples see HOL/ex/ApproximationEx.thy. + +* Theory "Reflection" now resides in HOL/Library. * Entry point to Word library now simply named "Word". INCOMPATIBILITY. @@ -212,7 +215,6 @@ src/HOL/Library/RType.thy ~> src/HOL/Typerep.thy src/HOL/Library/Code_Message.thy ~> src/HOL/ - src/HOL/Library/Dense_Linear_Order.thy ~> src/HOL/ src/HOL/Library/GCD.thy ~> src/HOL/ src/HOL/Library/Order_Relation.thy ~> src/HOL/ src/HOL/Library/Parity.thy ~> src/HOL/ diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Decision_Procs/Approximation.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Decision_Procs/Approximation.thy Fri Feb 06 15:15:46 2009 +0100 @@ -0,0 +1,2507 @@ +(* Title: HOL/Reflection/Approximation.thy + * Author: Johannes Hölzl 2008 / 2009 + *) +header {* Prove unequations about real numbers by computation *} +theory Approximation +imports Complex_Main Float Reflection Dense_Linear_Order Efficient_Nat +begin + +section "Horner Scheme" + +subsection {* Define auxiliary helper @{text horner} function *} + +fun horner :: "(nat \ nat) \ (nat \ nat \ nat) \ nat \ nat \ nat \ real \ real" where +"horner F G 0 i k x = 0" | +"horner F G (Suc n) i k x = 1 / real k - x * horner F G n (F i) (G i k) x" + +lemma horner_schema': fixes x :: real and a :: "nat \ real" + shows "a 0 - x * (\ i=0.. i=0..i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)" by auto + show ?thesis unfolding setsum_right_distrib shift_pow real_diff_def setsum_negf[symmetric] setsum_head_upt_Suc[OF zero_less_Suc] + setsum_reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\ n. (-1)^n *a n * x^n"] by auto +qed + +lemma horner_schema: fixes f :: "nat \ nat" and G :: "nat \ nat \ nat" and F :: "nat \ nat" + assumes f_Suc: "\n. f (Suc n) = G ((F^n) s) (f n)" + shows "horner F G n ((F^j') s) (f j') x = (\ j = 0..< n. -1^j * (1 / real (f (j' + j))) * x^j)" +proof (induct n arbitrary: i k j') + case (Suc n) + + show ?case unfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc] + using horner_schema'[of "\ j. 1 / real (f (j' + j))"] by auto +qed auto + +lemma horner_bounds': + assumes "0 \ Ifloat x" and f_Suc: "\n. f (Suc n) = G ((F^n) s) (f n)" + and lb_0: "\ i k x. lb 0 i k x = 0" + and lb_Suc: "\ n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) - x * (ub n (F i) (G i k) x)" + and ub_0: "\ i k x. ub 0 i k x = 0" + and ub_Suc: "\ n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) - x * (lb n (F i) (G i k) x)" + shows "Ifloat (lb n ((F^j') s) (f j') x) \ horner F G n ((F^j') s) (f j') (Ifloat x) \ + horner F G n ((F^j') s) (f j') (Ifloat x) \ Ifloat (ub n ((F^j') s) (f j') x)" + (is "?lb n j' \ ?horner n j' \ ?horner n j' \ ?ub n j'") +proof (induct n arbitrary: j') + case 0 thus ?case unfolding lb_0 ub_0 horner.simps by auto +next + case (Suc n) + have "?lb (Suc n) j' \ ?horner (Suc n) j'" unfolding lb_Suc ub_Suc horner.simps Ifloat_sub diff_def + proof (rule add_mono) + show "Ifloat (lapprox_rat prec 1 (int (f j'))) \ 1 / real (f j')" using lapprox_rat[of prec 1 "int (f j')"] by auto + from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct2] `0 \ Ifloat x` + show "- Ifloat (x * ub n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) x) \ - (Ifloat x * horner F G n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) (Ifloat x))" + unfolding Ifloat_mult neg_le_iff_le by (rule mult_left_mono) + qed + moreover have "?horner (Suc n) j' \ ?ub (Suc n) j'" unfolding ub_Suc ub_Suc horner.simps Ifloat_sub diff_def + proof (rule add_mono) + show "1 / real (f j') \ Ifloat (rapprox_rat prec 1 (int (f j')))" using rapprox_rat[of 1 "int (f j')" prec] by auto + from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct1] `0 \ Ifloat x` + show "- (Ifloat x * horner F G n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) (Ifloat x)) \ + - Ifloat (x * lb n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) x)" + unfolding Ifloat_mult neg_le_iff_le by (rule mult_left_mono) + qed + ultimately show ?case by blast +qed + +subsection "Theorems for floating point functions implementing the horner scheme" + +text {* + +Here @{term_type "f :: nat \ nat"} is the sequence defining the Taylor series, the coefficients are +all alternating and reciprocs. We use @{term G} and @{term F} to describe the computation of @{term f}. + +*} + +lemma horner_bounds: fixes F :: "nat \ nat" and G :: "nat \ nat \ nat" + assumes "0 \ Ifloat x" and f_Suc: "\n. f (Suc n) = G ((F^n) s) (f n)" + and lb_0: "\ i k x. lb 0 i k x = 0" + and lb_Suc: "\ n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) - x * (ub n (F i) (G i k) x)" + and ub_0: "\ i k x. ub 0 i k x = 0" + and ub_Suc: "\ n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) - x * (lb n (F i) (G i k) x)" + shows "Ifloat (lb n ((F^j') s) (f j') x) \ (\j=0..j=0.. Ifloat (ub n ((F^j') s) (f j') x)" (is "?ub") +proof - + have "?lb \ ?ub" + using horner_bounds'[where lb=lb, OF `0 \ Ifloat x` f_Suc lb_0 lb_Suc ub_0 ub_Suc] + unfolding horner_schema[where f=f, OF f_Suc] . + thus "?lb" and "?ub" by auto +qed + +lemma horner_bounds_nonpos: fixes F :: "nat \ nat" and G :: "nat \ nat \ nat" + assumes "Ifloat x \ 0" and f_Suc: "\n. f (Suc n) = G ((F^n) s) (f n)" + and lb_0: "\ i k x. lb 0 i k x = 0" + and lb_Suc: "\ n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) + x * (ub n (F i) (G i k) x)" + and ub_0: "\ i k x. ub 0 i k x = 0" + and ub_Suc: "\ n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) + x * (lb n (F i) (G i k) x)" + shows "Ifloat (lb n ((F^j') s) (f j') x) \ (\j=0..j=0.. Ifloat (ub n ((F^j') s) (f j') x)" (is "?ub") +proof - + { fix x y z :: float have "x - y * z = x + - y * z" + by (cases x, cases y, cases z, simp add: plus_float.simps minus_float.simps uminus_float.simps times_float.simps algebra_simps) + } note diff_mult_minus = this + + { fix x :: float have "- (- x) = x" by (cases x, auto simp add: uminus_float.simps) } note minus_minus = this + + have move_minus: "Ifloat (-x) = -1 * Ifloat x" by auto + + have sum_eq: "(\j=0..j = 0.. {0 ..< n}" + show "1 / real (f (j' + j)) * Ifloat x ^ j = -1 ^ j * (1 / real (f (j' + j))) * Ifloat (- x) ^ j" + unfolding move_minus power_mult_distrib real_mult_assoc[symmetric] + unfolding real_mult_commute unfolding real_mult_assoc[of "-1^j", symmetric] power_mult_distrib[symmetric] + by auto + qed + + have "0 \ Ifloat (-x)" using assms by auto + from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec + and lb="\ n i k x. lb n i k (-x)" and ub="\ n i k x. ub n i k (-x)", unfolded lb_Suc ub_Suc diff_mult_minus, + OF this f_Suc lb_0 refl ub_0 refl] + show "?lb" and "?ub" unfolding minus_minus sum_eq + by auto +qed + +subsection {* Selectors for next even or odd number *} + +text {* + +The horner scheme computes alternating series. To get the upper and lower bounds we need to +guarantee to access a even or odd member. To do this we use @{term get_odd} and @{term get_even}. + +*} + +definition get_odd :: "nat \ nat" where + "get_odd n = (if odd n then n else (Suc n))" + +definition get_even :: "nat \ nat" where + "get_even n = (if even n then n else (Suc n))" + +lemma get_odd[simp]: "odd (get_odd n)" unfolding get_odd_def by (cases "odd n", auto) +lemma get_even[simp]: "even (get_even n)" unfolding get_even_def by (cases "even n", auto) +lemma get_odd_ex: "\ k. Suc k = get_odd n \ odd (Suc k)" +proof (cases "odd n") + case True hence "0 < n" by (rule odd_pos) + from gr0_implies_Suc[OF this] obtain k where "Suc k = n" by auto + thus ?thesis unfolding get_odd_def if_P[OF True] using True[unfolded `Suc k = n`[symmetric]] by blast +next + case False hence "odd (Suc n)" by auto + thus ?thesis unfolding get_odd_def if_not_P[OF False] by blast +qed + +lemma get_even_double: "\i. get_even n = 2 * i" using get_even[unfolded even_mult_two_ex] . +lemma get_odd_double: "\i. get_odd n = 2 * i + 1" using get_odd[unfolded odd_Suc_mult_two_ex] by auto + +section "Power function" + +definition float_power_bnds :: "nat \ float \ float \ float * float" where +"float_power_bnds n l u = (if odd n \ 0 < l then (l ^ n, u ^ n) + else if u < 0 then (u ^ n, l ^ n) + else (0, (max (-l) u) ^ n))" + +lemma float_power_bnds: assumes "(l1, u1) = float_power_bnds n l u" and "x \ {Ifloat l .. Ifloat u}" + shows "x^n \ {Ifloat l1..Ifloat u1}" +proof (cases "even n") + case True + show ?thesis + proof (cases "0 < l") + case True hence "odd n \ 0 < l" and "0 \ Ifloat l" unfolding less_float_def by auto + have u1: "u1 = u ^ n" and l1: "l1 = l ^ n" using assms unfolding float_power_bnds_def if_P[OF `odd n \ 0 < l`] by auto + have "Ifloat l^n \ x^n" and "x^n \ Ifloat u^n " using `0 \ Ifloat l` and assms unfolding atLeastAtMost_iff using power_mono[of "Ifloat l" x] power_mono[of x "Ifloat u"] by auto + thus ?thesis using assms `0 < l` unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto + next + case False hence P: "\ (odd n \ 0 < l)" using `even n` by auto + show ?thesis + proof (cases "u < 0") + case True hence "0 \ - Ifloat u" and "- Ifloat u \ - x" and "0 \ - x" and "-x \ - Ifloat l" using assms unfolding less_float_def by auto + hence "Ifloat u^n \ x^n" and "x^n \ Ifloat l^n" using power_mono[of "-x" "-Ifloat l" n] power_mono[of "-Ifloat u" "-x" n] + unfolding power_minus_even[OF `even n`] by auto + moreover have u1: "u1 = l ^ n" and l1: "l1 = u ^ n" using assms unfolding float_power_bnds_def if_not_P[OF P] if_P[OF True] by auto + ultimately show ?thesis using float_power by auto + next + case False + have "\x\ \ Ifloat (max (-l) u)" + proof (cases "-l \ u") + case True thus ?thesis unfolding max_def if_P[OF True] using assms unfolding le_float_def by auto + next + case False thus ?thesis unfolding max_def if_not_P[OF False] using assms unfolding le_float_def by auto + qed + hence x_abs: "\x\ \ \Ifloat (max (-l) u)\" by auto + have u1: "u1 = (max (-l) u) ^ n" and l1: "l1 = 0" using assms unfolding float_power_bnds_def if_not_P[OF P] if_not_P[OF False] by auto + show ?thesis unfolding atLeastAtMost_iff l1 u1 float_power using zero_le_even_power[OF `even n`] power_mono_even[OF `even n` x_abs] by auto + qed + qed +next + case False hence "odd n \ 0 < l" by auto + have u1: "u1 = u ^ n" and l1: "l1 = l ^ n" using assms unfolding float_power_bnds_def if_P[OF `odd n \ 0 < l`] by auto + have "Ifloat l^n \ x^n" and "x^n \ Ifloat u^n " using assms unfolding atLeastAtMost_iff using power_mono_odd[OF False] by auto + thus ?thesis unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto +qed + +lemma bnds_power: "\ x l u. (l1, u1) = float_power_bnds n l u \ x \ {Ifloat l .. Ifloat u} \ Ifloat l1 \ x^n \ x^n \ Ifloat u1" + using float_power_bnds by auto + +section "Square root" + +text {* + +The square root computation is implemented as newton iteration. As first first step we use the +nearest power of two greater than the square root. + +*} + +fun sqrt_iteration :: "nat \ nat \ float \ float" where +"sqrt_iteration prec 0 (Float m e) = Float 1 ((e + bitlen m) div 2 + 1)" | +"sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x + in Float 1 -1 * (y + float_divr prec x y))" + +definition ub_sqrt :: "nat \ float \ float option" where +"ub_sqrt prec x = (if 0 < x then Some (sqrt_iteration prec prec x) else if x < 0 then None else Some 0)" + +definition lb_sqrt :: "nat \ float \ float option" where +"lb_sqrt prec x = (if 0 < x then Some (float_divl prec x (sqrt_iteration prec prec x)) else if x < 0 then None else Some 0)" + +lemma sqrt_ub_pos_pos_1: + assumes "sqrt x < b" and "0 < b" and "0 < x" + shows "sqrt x < (b + x / b)/2" +proof - + from assms have "0 < (b - sqrt x) ^ 2 " by simp + also have "\ = b ^ 2 - 2 * b * sqrt x + (sqrt x) ^ 2" by algebra + also have "\ = b ^ 2 - 2 * b * sqrt x + x" using assms by (simp add: real_sqrt_pow2) + finally have "0 < b ^ 2 - 2 * b * sqrt x + x" by assumption + hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms + by (simp add: field_simps power2_eq_square) + thus ?thesis by (simp add: field_simps) +qed + +lemma sqrt_iteration_bound: assumes "0 < Ifloat x" + shows "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec n x)" +proof (induct n) + case 0 + show ?case + proof (cases x) + case (Float m e) + hence "0 < m" using float_pos_m_pos[unfolded less_float_def] assms by auto + hence "0 < sqrt (real m)" by auto + + have int_nat_bl: "int (nat (bitlen m)) = bitlen m" using bitlen_ge0 by auto + + have "Ifloat x = (real m / 2^nat (bitlen m)) * pow2 (e + int (nat (bitlen m)))" + unfolding pow2_add pow2_int Float Ifloat.simps by auto + also have "\ < 1 * pow2 (e + int (nat (bitlen m)))" + proof (rule mult_strict_right_mono, auto) + show "real m < 2^nat (bitlen m)" using bitlen_bounds[OF `0 < m`, THEN conjunct2] + unfolding real_of_int_less_iff[of m, symmetric] by auto + qed + finally have "sqrt (Ifloat x) < sqrt (pow2 (e + bitlen m))" unfolding int_nat_bl by auto + also have "\ \ pow2 ((e + bitlen m) div 2 + 1)" + proof - + let ?E = "e + bitlen m" + have E_mod_pow: "pow2 (?E mod 2) < 4" + proof (cases "?E mod 2 = 1") + case True thus ?thesis by auto + next + case False + have "0 \ ?E mod 2" by auto + have "?E mod 2 < 2" by auto + from this[THEN zless_imp_add1_zle] + have "?E mod 2 \ 0" using False by auto + from xt1(5)[OF `0 \ ?E mod 2` this] + show ?thesis by auto + qed + hence "sqrt (pow2 (?E mod 2)) < sqrt (2 * 2)" by auto + hence E_mod_pow: "sqrt (pow2 (?E mod 2)) < 2" unfolding real_sqrt_abs2 by auto + + have E_eq: "pow2 ?E = pow2 (?E div 2 + ?E div 2 + ?E mod 2)" by auto + have "sqrt (pow2 ?E) = sqrt (pow2 (?E div 2) * pow2 (?E div 2) * pow2 (?E mod 2))" + unfolding E_eq unfolding pow2_add .. + also have "\ = pow2 (?E div 2) * sqrt (pow2 (?E mod 2))" + unfolding real_sqrt_mult[of _ "pow2 (?E mod 2)"] real_sqrt_abs2 by auto + also have "\ < pow2 (?E div 2) * 2" + by (rule mult_strict_left_mono, auto intro: E_mod_pow) + also have "\ = pow2 (?E div 2 + 1)" unfolding zadd_commute[of _ 1] pow2_add1 by auto + finally show ?thesis by auto + qed + finally show ?thesis + unfolding Float sqrt_iteration.simps Ifloat.simps by auto + qed +next + case (Suc n) + let ?b = "sqrt_iteration prec n x" + have "0 < sqrt (Ifloat x)" using `0 < Ifloat x` by auto + also have "\ < Ifloat ?b" using Suc . + finally have "sqrt (Ifloat x) < (Ifloat ?b + Ifloat x / Ifloat ?b)/2" using sqrt_ub_pos_pos_1[OF Suc _ `0 < Ifloat x`] by auto + also have "\ \ (Ifloat ?b + Ifloat (float_divr prec x ?b))/2" by (rule divide_right_mono, auto simp add: float_divr) + also have "\ = Ifloat (Float 1 -1) * (Ifloat ?b + Ifloat (float_divr prec x ?b))" by auto + finally show ?case unfolding sqrt_iteration.simps Let_def Ifloat_mult Ifloat_add right_distrib . +qed + +lemma sqrt_iteration_lower_bound: assumes "0 < Ifloat x" + shows "0 < Ifloat (sqrt_iteration prec n x)" (is "0 < ?sqrt") +proof - + have "0 < sqrt (Ifloat x)" using assms by auto + also have "\ < ?sqrt" using sqrt_iteration_bound[OF assms] . + finally show ?thesis . +qed + +lemma lb_sqrt_lower_bound: assumes "0 \ Ifloat x" + shows "0 \ Ifloat (the (lb_sqrt prec x))" +proof (cases "0 < x") + case True hence "0 < Ifloat x" and "0 \ x" using `0 \ Ifloat x` unfolding less_float_def le_float_def by auto + hence "0 < sqrt_iteration prec prec x" unfolding less_float_def using sqrt_iteration_lower_bound by auto + hence "0 \ Ifloat (float_divl prec x (sqrt_iteration prec prec x))" using float_divl_lower_bound[OF `0 \ x`] unfolding le_float_def by auto + thus ?thesis unfolding lb_sqrt_def using True by auto +next + case False with `0 \ Ifloat x` have "Ifloat x = 0" unfolding less_float_def by auto + thus ?thesis unfolding lb_sqrt_def less_float_def by auto +qed + +lemma lb_sqrt_upper_bound: assumes "0 \ Ifloat x" + shows "Ifloat (the (lb_sqrt prec x)) \ sqrt (Ifloat x)" +proof (cases "0 < x") + case True hence "0 < Ifloat x" and "0 \ Ifloat x" unfolding less_float_def by auto + hence sqrt_gt0: "0 < sqrt (Ifloat x)" by auto + hence sqrt_ub: "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto + + have "Ifloat (float_divl prec x (sqrt_iteration prec prec x)) \ Ifloat x / Ifloat (sqrt_iteration prec prec x)" by (rule float_divl) + also have "\ < Ifloat x / sqrt (Ifloat x)" + by (rule divide_strict_left_mono[OF sqrt_ub `0 < Ifloat x` mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]]) + also have "\ = sqrt (Ifloat x)" unfolding inverse_eq_iff_eq[of _ "sqrt (Ifloat x)", symmetric] sqrt_divide_self_eq[OF `0 \ Ifloat x`, symmetric] by auto + finally show ?thesis unfolding lb_sqrt_def if_P[OF `0 < x`] by auto +next + case False with `0 \ Ifloat x` + have "\ x < 0" unfolding less_float_def le_float_def by auto + show ?thesis unfolding lb_sqrt_def if_not_P[OF False] if_not_P[OF `\ x < 0`] using assms by auto +qed + +lemma lb_sqrt: assumes "Some y = lb_sqrt prec x" + shows "Ifloat y \ sqrt (Ifloat x)" and "0 \ Ifloat x" +proof - + show "0 \ Ifloat x" + proof (rule ccontr) + assume "\ 0 \ Ifloat x" + hence "lb_sqrt prec x = None" unfolding lb_sqrt_def less_float_def by auto + thus False using assms by auto + qed + from lb_sqrt_upper_bound[OF this, of prec] + show "Ifloat y \ sqrt (Ifloat x)" unfolding assms[symmetric] by auto +qed + +lemma ub_sqrt_lower_bound: assumes "0 \ Ifloat x" + shows "sqrt (Ifloat x) \ Ifloat (the (ub_sqrt prec x))" +proof (cases "0 < x") + case True hence "0 < Ifloat x" unfolding less_float_def by auto + hence "0 < sqrt (Ifloat x)" by auto + hence "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto + thus ?thesis unfolding ub_sqrt_def if_P[OF `0 < x`] by auto +next + case False with `0 \ Ifloat x` + have "Ifloat x = 0" unfolding less_float_def le_float_def by auto + thus ?thesis unfolding ub_sqrt_def less_float_def le_float_def by auto +qed + +lemma ub_sqrt: assumes "Some y = ub_sqrt prec x" + shows "sqrt (Ifloat x) \ Ifloat y" and "0 \ Ifloat x" +proof - + show "0 \ Ifloat x" + proof (rule ccontr) + assume "\ 0 \ Ifloat x" + hence "ub_sqrt prec x = None" unfolding ub_sqrt_def less_float_def by auto + thus False using assms by auto + qed + from ub_sqrt_lower_bound[OF this, of prec] + show "sqrt (Ifloat x) \ Ifloat y" unfolding assms[symmetric] by auto +qed + +lemma bnds_sqrt: "\ x lx ux. (Some l, Some u) = (lb_sqrt prec lx, ub_sqrt prec ux) \ x \ {Ifloat lx .. Ifloat ux} \ Ifloat l \ sqrt x \ sqrt x \ Ifloat u" +proof (rule allI, rule allI, rule allI, rule impI) + fix x lx ux + assume "(Some l, Some u) = (lb_sqrt prec lx, ub_sqrt prec ux) \ x \ {Ifloat lx .. Ifloat ux}" + hence l: "Some l = lb_sqrt prec lx " and u: "Some u = ub_sqrt prec ux" and x: "x \ {Ifloat lx .. Ifloat ux}" by auto + + have "Ifloat lx \ x" and "x \ Ifloat ux" using x by auto + + from lb_sqrt(1)[OF l] real_sqrt_le_mono[OF `Ifloat lx \ x`] + have "Ifloat l \ sqrt x" by (rule order_trans) + moreover + from real_sqrt_le_mono[OF `x \ Ifloat ux`] ub_sqrt(1)[OF u] + have "sqrt x \ Ifloat u" by (rule order_trans) + ultimately show "Ifloat l \ sqrt x \ sqrt x \ Ifloat u" .. +qed + +section "Arcus tangens and \" + +subsection "Compute arcus tangens series" + +text {* + +As first step we implement the computation of the arcus tangens series. This is only valid in the range +@{term "{-1 :: real .. 1}"}. This is used to compute \ and then the entire arcus tangens. + +*} + +fun ub_arctan_horner :: "nat \ nat \ nat \ float \ float" +and lb_arctan_horner :: "nat \ nat \ nat \ float \ float" where + "ub_arctan_horner prec 0 k x = 0" +| "ub_arctan_horner prec (Suc n) k x = + (rapprox_rat prec 1 (int k)) - x * (lb_arctan_horner prec n (k + 2) x)" +| "lb_arctan_horner prec 0 k x = 0" +| "lb_arctan_horner prec (Suc n) k x = + (lapprox_rat prec 1 (int k)) - x * (ub_arctan_horner prec n (k + 2) x)" + +lemma arctan_0_1_bounds': assumes "0 \ Ifloat x" "Ifloat x \ 1" and "even n" + shows "arctan (Ifloat x) \ {Ifloat (x * lb_arctan_horner prec n 1 (x * x)) .. Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x * x))}" +proof - + let "?c i" = "-1^i * (1 / real (i * 2 + 1) * Ifloat x ^ (i * 2 + 1))" + let "?S n" = "\ i=0.. Ifloat (x * x)" by auto + from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto + + have "arctan (Ifloat x) \ { ?S n .. ?S (Suc n) }" + proof (cases "Ifloat x = 0") + case False + hence "0 < Ifloat x" using `0 \ Ifloat x` by auto + hence prem: "0 < 1 / real (0 * 2 + (1::nat)) * Ifloat x ^ (0 * 2 + 1)" by auto + + have "\ Ifloat x \ \ 1" using `0 \ Ifloat x` `Ifloat x \ 1` by auto + from mp[OF summable_Leibniz(2)[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] prem, THEN spec, of m, unfolded `2 * m = n`] + show ?thesis unfolding arctan_series[OF `\ Ifloat x \ \ 1`] Suc_plus1 . + qed auto + note arctan_bounds = this[unfolded atLeastAtMost_iff] + + have F: "\n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto + + note bounds = horner_bounds[where s=1 and f="\i. 2 * i + 1" and j'=0 + and lb="\n i k x. lb_arctan_horner prec n k x" + and ub="\n i k x. ub_arctan_horner prec n k x", + OF `0 \ Ifloat (x*x)` F lb_arctan_horner.simps ub_arctan_horner.simps] + + { have "Ifloat (x * lb_arctan_horner prec n 1 (x*x)) \ ?S n" + using bounds(1) `0 \ Ifloat x` + unfolding Ifloat_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric] + unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "Ifloat x"] + by (auto intro!: mult_left_mono) + also have "\ \ arctan (Ifloat x)" using arctan_bounds .. + finally have "Ifloat (x * lb_arctan_horner prec n 1 (x*x)) \ arctan (Ifloat x)" . } + moreover + { have "arctan (Ifloat x) \ ?S (Suc n)" using arctan_bounds .. + also have "\ \ Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x*x))" + using bounds(2)[of "Suc n"] `0 \ Ifloat x` + unfolding Ifloat_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric] + unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "Ifloat x"] + by (auto intro!: mult_left_mono) + finally have "arctan (Ifloat x) \ Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . } + ultimately show ?thesis by auto +qed + +lemma arctan_0_1_bounds: assumes "0 \ Ifloat x" "Ifloat x \ 1" + shows "arctan (Ifloat x) \ {Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) .. Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))}" +proof (cases "even n") + case True + obtain n' where "Suc n' = get_odd n" and "odd (Suc n')" using get_odd_ex by auto + hence "even n'" unfolding even_nat_Suc by auto + have "arctan (Ifloat x) \ Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))" + unfolding `Suc n' = get_odd n`[symmetric] using arctan_0_1_bounds'[OF `0 \ Ifloat x` `Ifloat x \ 1` `even n'`] by auto + moreover + have "Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \ arctan (Ifloat x)" + unfolding get_even_def if_P[OF True] using arctan_0_1_bounds'[OF `0 \ Ifloat x` `Ifloat x \ 1` `even n`] by auto + ultimately show ?thesis by auto +next + case False hence "0 < n" by (rule odd_pos) + from gr0_implies_Suc[OF this] obtain n' where "n = Suc n'" .. + from False[unfolded this even_nat_Suc] + have "even n'" and "even (Suc (Suc n'))" by auto + have "get_odd n = Suc n'" unfolding get_odd_def if_P[OF False] using `n = Suc n'` . + + have "arctan (Ifloat x) \ Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))" + unfolding `get_odd n = Suc n'` using arctan_0_1_bounds'[OF `0 \ Ifloat x` `Ifloat x \ 1` `even n'`] by auto + moreover + have "Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \ arctan (Ifloat x)" + unfolding get_even_def if_not_P[OF False] unfolding `n = Suc n'` using arctan_0_1_bounds'[OF `0 \ Ifloat x` `Ifloat x \ 1` `even (Suc (Suc n'))`] by auto + ultimately show ?thesis by auto +qed + +subsection "Compute \" + +definition ub_pi :: "nat \ float" where + "ub_pi prec = (let A = rapprox_rat prec 1 5 ; + B = lapprox_rat prec 1 239 + in ((Float 1 2) * ((Float 1 2) * A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (A * A)) - + B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1 (B * B)))))" + +definition lb_pi :: "nat \ float" where + "lb_pi prec = (let A = lapprox_rat prec 1 5 ; + B = rapprox_rat prec 1 239 + in ((Float 1 2) * ((Float 1 2) * A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (A * A)) - + B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 (B * B)))))" + +lemma pi_boundaries: "pi \ {Ifloat (lb_pi n) .. Ifloat (ub_pi n)}" +proof - + have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))" unfolding machin[symmetric] by auto + + { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \ k" and "0 < k" and "1 \ k" by auto + let ?k = "rapprox_rat prec 1 k" + have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto + + have "0 \ Ifloat ?k" by (rule order_trans[OF _ rapprox_rat], auto simp add: `0 \ k`) + have "Ifloat ?k \ 1" unfolding rapprox_rat.simps(2)[OF zero_le_one `0 < k`] + by (rule rapprox_posrat_le1, auto simp add: `0 < k` `1 \ k`) + + have "1 / real k \ Ifloat ?k" using rapprox_rat[where x=1 and y=k] by auto + hence "arctan (1 / real k) \ arctan (Ifloat ?k)" by (rule arctan_monotone') + also have "\ \ Ifloat (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))" + using arctan_0_1_bounds[OF `0 \ Ifloat ?k` `Ifloat ?k \ 1`] by auto + finally have "arctan (1 / (real k)) \ Ifloat (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))" . + } note ub_arctan = this + + { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \ k" and "0 < k" by auto + let ?k = "lapprox_rat prec 1 k" + have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto + have "1 / real k \ 1" using `1 < k` by auto + + have "\n. 0 \ Ifloat ?k" using lapprox_rat_bottom[where x=1 and y=k, OF zero_le_one `0 < k`] by (auto simp add: `1 div k = 0`) + have "\n. Ifloat ?k \ 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / real k \ 1`) + + have "Ifloat ?k \ 1 / real k" using lapprox_rat[where x=1 and y=k] by auto + + have "Ifloat (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \ arctan (Ifloat ?k)" + using arctan_0_1_bounds[OF `0 \ Ifloat ?k` `Ifloat ?k \ 1`] by auto + also have "\ \ arctan (1 / real k)" using `Ifloat ?k \ 1 / real k` by (rule arctan_monotone') + finally have "Ifloat (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \ arctan (1 / (real k))" . + } note lb_arctan = this + + have "pi \ Ifloat (ub_pi n)" + unfolding ub_pi_def machin_pi Let_def Ifloat_mult Ifloat_sub unfolding Float_num + using lb_arctan[of 239] ub_arctan[of 5] + by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps) + moreover + have "Ifloat (lb_pi n) \ pi" + unfolding lb_pi_def machin_pi Let_def Ifloat_mult Ifloat_sub Float_num + using lb_arctan[of 5] ub_arctan[of 239] + by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps) + ultimately show ?thesis by auto +qed + +subsection "Compute arcus tangens in the entire domain" + +function lb_arctan :: "nat \ float \ float" and ub_arctan :: "nat \ float \ float" where + "lb_arctan prec x = (let ub_horner = \ x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x) ; + lb_horner = \ x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x) + in (if x < 0 then - ub_arctan prec (-x) else + if x \ Float 1 -1 then lb_horner x else + if x \ Float 1 1 then Float 1 1 * lb_horner (float_divl prec x (1 + the (ub_sqrt prec (1 + x * x)))) + else (let inv = float_divr prec 1 x + in if inv > 1 then 0 + else lb_pi prec * Float 1 -1 - ub_horner inv)))" + +| "ub_arctan prec x = (let lb_horner = \ x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x) ; + ub_horner = \ x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x) + in (if x < 0 then - lb_arctan prec (-x) else + if x \ Float 1 -1 then ub_horner x else + if x \ Float 1 1 then let y = float_divr prec x (1 + the (lb_sqrt prec (1 + x * x))) + in if y > 1 then ub_pi prec * Float 1 -1 + else Float 1 1 * ub_horner y + else ub_pi prec * Float 1 -1 - lb_horner (float_divl prec 1 x)))" +by pat_completeness auto +termination by (relation "measure (\ v. let (prec, x) = sum_case id id v in (if x < 0 then 1 else 0))", auto simp add: less_float_def) + +declare ub_arctan_horner.simps[simp del] +declare lb_arctan_horner.simps[simp del] + +lemma lb_arctan_bound': assumes "0 \ Ifloat x" + shows "Ifloat (lb_arctan prec x) \ arctan (Ifloat x)" +proof - + have "\ x < 0" and "0 \ x" unfolding less_float_def le_float_def using `0 \ Ifloat x` by auto + let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)" + and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)" + + show ?thesis + proof (cases "x \ Float 1 -1") + case True hence "Ifloat x \ 1" unfolding le_float_def Float_num by auto + show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\ x < 0`] if_P[OF True] + using arctan_0_1_bounds[OF `0 \ Ifloat x` `Ifloat x \ 1`] by auto + next + case False hence "0 < Ifloat x" unfolding le_float_def Float_num by auto + let ?R = "1 + sqrt (1 + Ifloat x * Ifloat x)" + let ?fR = "1 + the (ub_sqrt prec (1 + x * x))" + let ?DIV = "float_divl prec x ?fR" + + have sqr_ge0: "0 \ 1 + Ifloat x * Ifloat x" using sum_power2_ge_zero[of 1 "Ifloat x", unfolded numeral_2_eq_2] by auto + hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg) + + have "sqrt (Ifloat (1 + x * x)) \ Ifloat (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0) + hence "?R \ Ifloat ?fR" by auto + hence "0 < ?fR" and "0 < Ifloat ?fR" unfolding less_float_def using `0 < ?R` by auto + + have monotone: "Ifloat (float_divl prec x ?fR) \ Ifloat x / ?R" + proof - + have "Ifloat ?DIV \ Ifloat x / Ifloat ?fR" by (rule float_divl) + also have "\ \ Ifloat x / ?R" by (rule divide_left_mono[OF `?R \ Ifloat ?fR` `0 \ Ifloat x` mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 `?R \ Ifloat ?fR`] divisor_gt0]]) + finally show ?thesis . + qed + + show ?thesis + proof (cases "x \ Float 1 1") + case True + + have "Ifloat x \ sqrt (Ifloat (1 + x * x))" using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto + also have "\ \ Ifloat (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0) + finally have "Ifloat x \ Ifloat ?fR" by auto + moreover have "Ifloat ?DIV \ Ifloat x / Ifloat ?fR" by (rule float_divl) + ultimately have "Ifloat ?DIV \ 1" unfolding divide_le_eq_1_pos[OF `0 < Ifloat ?fR`, symmetric] by auto + + have "0 \ Ifloat ?DIV" using float_divl_lower_bound[OF `0 \ x` `0 < ?fR`] unfolding le_float_def by auto + + have "Ifloat (Float 1 1 * ?lb_horner ?DIV) \ 2 * arctan (Ifloat (float_divl prec x ?fR))" unfolding Ifloat_mult[of "Float 1 1"] Float_num + using arctan_0_1_bounds[OF `0 \ Ifloat ?DIV` `Ifloat ?DIV \ 1`] by auto + also have "\ \ 2 * arctan (Ifloat x / ?R)" + using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono) + also have "2 * arctan (Ifloat x / ?R) = arctan (Ifloat x)" using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 realpow_0 real_mult_1 . + finally show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\ x < 0`] if_not_P[OF `\ x \ Float 1 -1`] if_P[OF True] . + next + case False + hence "2 < Ifloat x" unfolding le_float_def Float_num by auto + hence "1 \ Ifloat x" by auto + + let "?invx" = "float_divr prec 1 x" + have "0 \ arctan (Ifloat x)" using arctan_monotone'[OF `0 \ Ifloat x`] using arctan_tan[of 0, unfolded tan_zero] by auto + + show ?thesis + proof (cases "1 < ?invx") + case True + show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\ x < 0`] if_not_P[OF `\ x \ Float 1 -1`] if_not_P[OF False] if_P[OF True] + using `0 \ arctan (Ifloat x)` by auto + next + case False + hence "Ifloat ?invx \ 1" unfolding less_float_def by auto + have "0 \ Ifloat ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \ Ifloat x`) + + have "1 / Ifloat x \ 0" and "0 < 1 / Ifloat x" using `0 < Ifloat x` by auto + + have "arctan (1 / Ifloat x) \ arctan (Ifloat ?invx)" unfolding Ifloat_1[symmetric] by (rule arctan_monotone', rule float_divr) + also have "\ \ Ifloat (?ub_horner ?invx)" using arctan_0_1_bounds[OF `0 \ Ifloat ?invx` `Ifloat ?invx \ 1`] by auto + finally have "pi / 2 - Ifloat (?ub_horner ?invx) \ arctan (Ifloat x)" + using `0 \ arctan (Ifloat x)` arctan_inverse[OF `1 / Ifloat x \ 0`] + unfolding real_sgn_pos[OF `0 < 1 / Ifloat x`] le_diff_eq by auto + moreover + have "Ifloat (lb_pi prec * Float 1 -1) \ pi / 2" unfolding Ifloat_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto + ultimately + show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\ x < 0`] if_not_P[OF `\ x \ Float 1 -1`] if_not_P[OF `\ x \ Float 1 1`] if_not_P[OF False] + by auto + qed + qed + qed +qed + +lemma ub_arctan_bound': assumes "0 \ Ifloat x" + shows "arctan (Ifloat x) \ Ifloat (ub_arctan prec x)" +proof - + have "\ x < 0" and "0 \ x" unfolding less_float_def le_float_def using `0 \ Ifloat x` by auto + + let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)" + and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)" + + show ?thesis + proof (cases "x \ Float 1 -1") + case True hence "Ifloat x \ 1" unfolding le_float_def Float_num by auto + show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\ x < 0`] if_P[OF True] + using arctan_0_1_bounds[OF `0 \ Ifloat x` `Ifloat x \ 1`] by auto + next + case False hence "0 < Ifloat x" unfolding le_float_def Float_num by auto + let ?R = "1 + sqrt (1 + Ifloat x * Ifloat x)" + let ?fR = "1 + the (lb_sqrt prec (1 + x * x))" + let ?DIV = "float_divr prec x ?fR" + + have sqr_ge0: "0 \ 1 + Ifloat x * Ifloat x" using sum_power2_ge_zero[of 1 "Ifloat x", unfolded numeral_2_eq_2] by auto + hence "0 \ Ifloat (1 + x*x)" by auto + + hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg) + + have "Ifloat (the (lb_sqrt prec (1 + x * x))) \ sqrt (Ifloat (1 + x * x))" by (rule lb_sqrt_upper_bound, auto simp add: sqr_ge0) + hence "Ifloat ?fR \ ?R" by auto + have "0 < Ifloat ?fR" unfolding Ifloat_add Ifloat_1 by (rule order_less_le_trans[OF zero_less_one], auto simp add: lb_sqrt_lower_bound[OF `0 \ Ifloat (1 + x*x)`]) + + have monotone: "Ifloat x / ?R \ Ifloat (float_divr prec x ?fR)" + proof - + from divide_left_mono[OF `Ifloat ?fR \ ?R` `0 \ Ifloat x` mult_pos_pos[OF divisor_gt0 `0 < Ifloat ?fR`]] + have "Ifloat x / ?R \ Ifloat x / Ifloat ?fR" . + also have "\ \ Ifloat ?DIV" by (rule float_divr) + finally show ?thesis . + qed + + show ?thesis + proof (cases "x \ Float 1 1") + case True + show ?thesis + proof (cases "?DIV > 1") + case True + have "pi / 2 \ Ifloat (ub_pi prec * Float 1 -1)" unfolding Ifloat_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto + from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le] + show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\ x < 0`] if_not_P[OF `\ x \ Float 1 -1`] if_P[OF `x \ Float 1 1`] if_P[OF True] . + next + case False + hence "Ifloat ?DIV \ 1" unfolding less_float_def by auto + + have "0 \ Ifloat x / ?R" using `0 \ Ifloat x` `0 < ?R` unfolding real_0_le_divide_iff by auto + hence "0 \ Ifloat ?DIV" using monotone by (rule order_trans) + + have "arctan (Ifloat x) = 2 * arctan (Ifloat x / ?R)" using arctan_half unfolding numeral_2_eq_2 power_Suc2 realpow_0 real_mult_1 . + also have "\ \ 2 * arctan (Ifloat ?DIV)" + using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono) + also have "\ \ Ifloat (Float 1 1 * ?ub_horner ?DIV)" unfolding Ifloat_mult[of "Float 1 1"] Float_num + using arctan_0_1_bounds[OF `0 \ Ifloat ?DIV` `Ifloat ?DIV \ 1`] by auto + finally show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\ x < 0`] if_not_P[OF `\ x \ Float 1 -1`] if_P[OF `x \ Float 1 1`] if_not_P[OF False] . + qed + next + case False + hence "2 < Ifloat x" unfolding le_float_def Float_num by auto + hence "1 \ Ifloat x" by auto + hence "0 < Ifloat x" by auto + hence "0 < x" unfolding less_float_def by auto + + let "?invx" = "float_divl prec 1 x" + have "0 \ arctan (Ifloat x)" using arctan_monotone'[OF `0 \ Ifloat x`] using arctan_tan[of 0, unfolded tan_zero] by auto + + have "Ifloat ?invx \ 1" unfolding less_float_def by (rule order_trans[OF float_divl], auto simp add: `1 \ Ifloat x` divide_le_eq_1_pos[OF `0 < Ifloat x`]) + have "0 \ Ifloat ?invx" unfolding Ifloat_0[symmetric] by (rule float_divl_lower_bound[unfolded le_float_def], auto simp add: `0 < x`) + + have "1 / Ifloat x \ 0" and "0 < 1 / Ifloat x" using `0 < Ifloat x` by auto + + have "Ifloat (?lb_horner ?invx) \ arctan (Ifloat ?invx)" using arctan_0_1_bounds[OF `0 \ Ifloat ?invx` `Ifloat ?invx \ 1`] by auto + also have "\ \ arctan (1 / Ifloat x)" unfolding Ifloat_1[symmetric] by (rule arctan_monotone', rule float_divl) + finally have "arctan (Ifloat x) \ pi / 2 - Ifloat (?lb_horner ?invx)" + using `0 \ arctan (Ifloat x)` arctan_inverse[OF `1 / Ifloat x \ 0`] + unfolding real_sgn_pos[OF `0 < 1 / Ifloat x`] le_diff_eq by auto + moreover + have "pi / 2 \ Ifloat (ub_pi prec * Float 1 -1)" unfolding Ifloat_mult Float_num times_divide_eq_right mult_1_right using pi_boundaries by auto + ultimately + show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\ x < 0`] if_not_P[OF `\ x \ Float 1 -1`] if_not_P[OF `\ x \ Float 1 1`] if_not_P[OF False] + by auto + qed + qed +qed + +lemma arctan_boundaries: + "arctan (Ifloat x) \ {Ifloat (lb_arctan prec x) .. Ifloat (ub_arctan prec x)}" +proof (cases "0 \ x") + case True hence "0 \ Ifloat x" unfolding le_float_def by auto + show ?thesis using ub_arctan_bound'[OF `0 \ Ifloat x`] lb_arctan_bound'[OF `0 \ Ifloat x`] unfolding atLeastAtMost_iff by auto +next + let ?mx = "-x" + case False hence "x < 0" and "0 \ Ifloat ?mx" unfolding le_float_def less_float_def by auto + hence bounds: "Ifloat (lb_arctan prec ?mx) \ arctan (Ifloat ?mx) \ arctan (Ifloat ?mx) \ Ifloat (ub_arctan prec ?mx)" + using ub_arctan_bound'[OF `0 \ Ifloat ?mx`] lb_arctan_bound'[OF `0 \ Ifloat ?mx`] by auto + show ?thesis unfolding Ifloat_minus arctan_minus lb_arctan.simps[where x=x] ub_arctan.simps[where x=x] Let_def if_P[OF `x < 0`] + unfolding atLeastAtMost_iff using bounds[unfolded Ifloat_minus arctan_minus] by auto +qed + +lemma bnds_arctan: "\ x lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \ x \ {Ifloat lx .. Ifloat ux} \ Ifloat l \ arctan x \ arctan x \ Ifloat u" +proof (rule allI, rule allI, rule allI, rule impI) + fix x lx ux + assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \ x \ {Ifloat lx .. Ifloat ux}" + hence l: "lb_arctan prec lx = l " and u: "ub_arctan prec ux = u" and x: "x \ {Ifloat lx .. Ifloat ux}" by auto + + { from arctan_boundaries[of lx prec, unfolded l] + have "Ifloat l \ arctan (Ifloat lx)" by (auto simp del: lb_arctan.simps) + also have "\ \ arctan x" using x by (auto intro: arctan_monotone') + finally have "Ifloat l \ arctan x" . + } moreover + { have "arctan x \ arctan (Ifloat ux)" using x by (auto intro: arctan_monotone') + also have "\ \ Ifloat u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps) + finally have "arctan x \ Ifloat u" . + } ultimately show "Ifloat l \ arctan x \ arctan x \ Ifloat u" .. +qed + +section "Sinus and Cosinus" + +subsection "Compute the cosinus and sinus series" + +fun ub_sin_cos_aux :: "nat \ nat \ nat \ nat \ float \ float" +and lb_sin_cos_aux :: "nat \ nat \ nat \ nat \ float \ float" where + "ub_sin_cos_aux prec 0 i k x = 0" +| "ub_sin_cos_aux prec (Suc n) i k x = + (rapprox_rat prec 1 (int k)) - x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)" +| "lb_sin_cos_aux prec 0 i k x = 0" +| "lb_sin_cos_aux prec (Suc n) i k x = + (lapprox_rat prec 1 (int k)) - x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)" + +lemma cos_aux: + shows "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \ (\ i=0.. i=0.. Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub") +proof - + have "0 \ Ifloat (x * x)" unfolding Ifloat_mult by auto + let "?f n" = "fact (2 * n)" + + { fix n + have F: "\m. ((\i. i + 2) ^ n) m = m + 2 * n" by (induct n arbitrary: m, auto) + have "?f (Suc n) = ?f n * ((\i. i + 2) ^ n) 1 * (((\i. i + 2) ^ n) 1 + 1)" + unfolding F by auto } note f_eq = this + + from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0, + OF `0 \ Ifloat (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps] + show "?lb" and "?ub" by (auto simp add: power_mult power2_eq_square[of "Ifloat x"]) +qed + +lemma cos_boundaries: assumes "0 \ Ifloat x" and "Ifloat x \ pi / 2" + shows "cos (Ifloat x) \ {Ifloat (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. Ifloat (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}" +proof (cases "Ifloat x = 0") + case False hence "Ifloat x \ 0" by auto + hence "0 < x" and "0 < Ifloat x" using `0 \ Ifloat x` unfolding less_float_def by auto + have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0 + using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto + + { fix x n have "(\ i=0.. i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum") + proof - + have "?sum = ?sum + (\ j = 0 ..< n. 0)" by auto + also have "\ = + (\ j = 0 ..< n. -1 ^ ((2 * j) div 2) / (real (fact (2 * j))) * x ^(2 * j)) + (\ j = 0 ..< n. 0)" by auto + also have "\ = (\ i = 0 ..< 2 * n. if even i then -1 ^ (i div 2) / (real (fact i)) * x ^ i else 0)" + unfolding sum_split_even_odd .. + also have "\ = (\ i = 0 ..< 2 * n. (if even i then -1 ^ (i div 2) / (real (fact i)) else 0) * x ^ i)" + by (rule setsum_cong2) auto + finally show ?thesis by assumption + qed } note morph_to_if_power = this + + + { fix n :: nat assume "0 < n" + hence "0 < 2 * n" by auto + obtain t where "0 < t" and "t < Ifloat x" and + cos_eq: "cos (Ifloat x) = (\ i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * (Ifloat x) ^ i) + + (cos (t + 1/2 * real (2 * n) * pi) / real (fact (2*n))) * (Ifloat x)^(2*n)" + (is "_ = ?SUM + ?rest / ?fact * ?pow") + using Maclaurin_cos_expansion2[OF `0 < Ifloat x` `0 < 2 * n`] by auto + + have "cos t * -1^n = cos t * cos (real n * pi) + sin t * sin (real n * pi)" by auto + also have "\ = cos (t + real n * pi)" using cos_add by auto + also have "\ = ?rest" by auto + finally have "cos t * -1^n = ?rest" . + moreover + have "t \ pi / 2" using `t < Ifloat x` and `Ifloat x \ pi / 2` by auto + hence "0 \ cos t" using `0 < t` and cos_ge_zero by auto + ultimately have even: "even n \ 0 \ ?rest" and odd: "odd n \ 0 \ - ?rest " by auto + + have "0 < ?fact" by auto + have "0 < ?pow" using `0 < Ifloat x` by auto + + { + assume "even n" + have "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \ ?SUM" + unfolding morph_to_if_power[symmetric] using cos_aux by auto + also have "\ \ cos (Ifloat x)" + proof - + from even[OF `even n`] `0 < ?fact` `0 < ?pow` + have "0 \ (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le) + thus ?thesis unfolding cos_eq by auto + qed + finally have "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \ cos (Ifloat x)" . + } note lb = this + + { + assume "odd n" + have "cos (Ifloat x) \ ?SUM" + proof - + from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`] + have "0 \ (- ?rest) / ?fact * ?pow" + by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le) + thus ?thesis unfolding cos_eq by auto + qed + also have "\ \ Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" + unfolding morph_to_if_power[symmetric] using cos_aux by auto + finally have "cos (Ifloat x) \ Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" . + } note ub = this and lb + } note ub = this(1) and lb = this(2) + + have "cos (Ifloat x) \ Ifloat (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] . + moreover have "Ifloat (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \ cos (Ifloat x)" + proof (cases "0 < get_even n") + case True show ?thesis using lb[OF True get_even] . + next + case False + hence "get_even n = 0" by auto + have "- (pi / 2) \ Ifloat x" by (rule order_trans[OF _ `0 < Ifloat x`[THEN less_imp_le]], auto) + with `Ifloat x \ pi / 2` + show ?thesis unfolding `get_even n = 0` lb_sin_cos_aux.simps Ifloat_minus Ifloat_0 using cos_ge_zero by auto + qed + ultimately show ?thesis by auto +next + case True + show ?thesis + proof (cases "n = 0") + case True + thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `Ifloat x = 0` lapprox_rat[where x="-1" and y=1] by auto + next + case False with not0_implies_Suc obtain m where "n = Suc m" by blast + thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `Ifloat x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto) + qed +qed + +lemma sin_aux: assumes "0 \ Ifloat x" + shows "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \ (\ i=0.. i=0.. Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub") +proof - + have "0 \ Ifloat (x * x)" unfolding Ifloat_mult by auto + let "?f n" = "fact (2 * n + 1)" + + { fix n + have F: "\m. ((\i. i + 2) ^ n) m = m + 2 * n" by (induct n arbitrary: m, auto) + have "?f (Suc n) = ?f n * ((\i. i + 2) ^ n) 2 * (((\i. i + 2) ^ n) 2 + 1)" + unfolding F by auto } note f_eq = this + + from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0, + OF `0 \ Ifloat (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps] + show "?lb" and "?ub" using `0 \ Ifloat x` unfolding Ifloat_mult + unfolding power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric] + unfolding real_mult_commute + by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "Ifloat x"]) +qed + +lemma sin_boundaries: assumes "0 \ Ifloat x" and "Ifloat x \ pi / 2" + shows "sin (Ifloat x) \ {Ifloat (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. Ifloat (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}" +proof (cases "Ifloat x = 0") + case False hence "Ifloat x \ 0" by auto + hence "0 < x" and "0 < Ifloat x" using `0 \ Ifloat x` unfolding less_float_def by auto + have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0 + using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto + + { fix x n have "(\ j = 0 ..< n. -1 ^ (((2 * j + 1) - Suc 0) div 2) / (real (fact (2 * j + 1))) * x ^(2 * j + 1)) + = (\ i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * x ^ i)" (is "?SUM = _") + proof - + have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" by auto + have "?SUM = (\ j = 0 ..< n. 0) + ?SUM" by auto + also have "\ = (\ i = 0 ..< 2 * n. if even i then 0 else -1 ^ ((i - Suc 0) div 2) / (real (fact i)) * x ^ i)" + unfolding sum_split_even_odd .. + also have "\ = (\ i = 0 ..< 2 * n. (if even i then 0 else -1 ^ ((i - Suc 0) div 2) / (real (fact i))) * x ^ i)" + by (rule setsum_cong2) auto + finally show ?thesis by assumption + qed } note setsum_morph = this + + { fix n :: nat assume "0 < n" + hence "0 < 2 * n + 1" by auto + obtain t where "0 < t" and "t < Ifloat x" and + sin_eq: "sin (Ifloat x) = (\ i = 0 ..< 2 * n + 1. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i) + + (sin (t + 1/2 * real (2 * n + 1) * pi) / real (fact (2*n + 1))) * (Ifloat x)^(2*n + 1)" + (is "_ = ?SUM + ?rest / ?fact * ?pow") + using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < Ifloat x`] by auto + + have "?rest = cos t * -1^n" unfolding sin_add cos_add real_of_nat_add left_distrib right_distrib by auto + moreover + have "t \ pi / 2" using `t < Ifloat x` and `Ifloat x \ pi / 2` by auto + hence "0 \ cos t" using `0 < t` and cos_ge_zero by auto + ultimately have even: "even n \ 0 \ ?rest" and odd: "odd n \ 0 \ - ?rest " by auto + + have "0 < ?fact" by (rule real_of_nat_fact_gt_zero) + have "0 < ?pow" using `0 < Ifloat x` by (rule zero_less_power) + + { + assume "even n" + have "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \ + (\ i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i)" + using sin_aux[OF `0 \ Ifloat x`] unfolding setsum_morph[symmetric] by auto + also have "\ \ ?SUM" by auto + also have "\ \ sin (Ifloat x)" + proof - + from even[OF `even n`] `0 < ?fact` `0 < ?pow` + have "0 \ (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le) + thus ?thesis unfolding sin_eq by auto + qed + finally have "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \ sin (Ifloat x)" . + } note lb = this + + { + assume "odd n" + have "sin (Ifloat x) \ ?SUM" + proof - + from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`] + have "0 \ (- ?rest) / ?fact * ?pow" + by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le) + thus ?thesis unfolding sin_eq by auto + qed + also have "\ \ (\ i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i)" + by auto + also have "\ \ Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" + using sin_aux[OF `0 \ Ifloat x`] unfolding setsum_morph[symmetric] by auto + finally have "sin (Ifloat x) \ Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" . + } note ub = this and lb + } note ub = this(1) and lb = this(2) + + have "sin (Ifloat x) \ Ifloat (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] . + moreover have "Ifloat (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \ sin (Ifloat x)" + proof (cases "0 < get_even n") + case True show ?thesis using lb[OF True get_even] . + next + case False + hence "get_even n = 0" by auto + with `Ifloat x \ pi / 2` `0 \ Ifloat x` + show ?thesis unfolding `get_even n = 0` ub_sin_cos_aux.simps Ifloat_minus Ifloat_0 using sin_ge_zero by auto + qed + ultimately show ?thesis by auto +next + case True + show ?thesis + proof (cases "n = 0") + case True + thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `Ifloat x = 0` lapprox_rat[where x="-1" and y=1] by auto + next + case False with not0_implies_Suc obtain m where "n = Suc m" by blast + thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `Ifloat x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto) + qed +qed + +subsection "Compute the cosinus in the entire domain" + +definition lb_cos :: "nat \ float \ float" where +"lb_cos prec x = (let + horner = \ x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ; + half = \ x. if x < 0 then - 1 else Float 1 1 * x * x - 1 + in if x < Float 1 -1 then horner x +else if x < 1 then half (horner (x * Float 1 -1)) + else half (half (horner (x * Float 1 -2))))" + +definition ub_cos :: "nat \ float \ float" where +"ub_cos prec x = (let + horner = \ x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ; + half = \ x. Float 1 1 * x * x - 1 + in if x < Float 1 -1 then horner x +else if x < 1 then half (horner (x * Float 1 -1)) + else half (half (horner (x * Float 1 -2))))" + +definition bnds_cos :: "nat \ float \ float \ float * float" where +"bnds_cos prec lx ux = (let lpi = lb_pi prec + in if lx < -lpi \ ux > lpi then (Float -1 0, Float 1 0) + else if ux \ 0 then (lb_cos prec (-lx), ub_cos prec (-ux)) + else if 0 \ lx then (lb_cos prec ux, ub_cos prec lx) + else (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0))" + +lemma lb_cos: assumes "0 \ Ifloat x" and "Ifloat x \ pi" + shows "cos (Ifloat x) \ {Ifloat (lb_cos prec x) .. Ifloat (ub_cos prec x)}" (is "?cos x \ { Ifloat (?lb x) .. Ifloat (?ub x) }") +proof - + { fix x :: real + have "cos x = cos (x / 2 + x / 2)" by auto + also have "\ = cos (x / 2) * cos (x / 2) + sin (x / 2) * sin (x / 2) - sin (x / 2) * sin (x / 2) + cos (x / 2) * cos (x / 2) - 1" + unfolding cos_add by auto + also have "\ = 2 * cos (x / 2) * cos (x / 2) - 1" by algebra + finally have "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" . + } note x_half = this[symmetric] + + have "\ x < 0" using `0 \ Ifloat x` unfolding less_float_def by auto + let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)" + let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)" + let "?ub_half x" = "Float 1 1 * x * x - 1" + let "?lb_half x" = "if x < 0 then - 1 else Float 1 1 * x * x - 1" + + show ?thesis + proof (cases "x < Float 1 -1") + case True hence "Ifloat x \ pi / 2" unfolding less_float_def using pi_ge_two by auto + show ?thesis unfolding lb_cos_def[where x=x] ub_cos_def[where x=x] if_not_P[OF `\ x < 0`] if_P[OF `x < Float 1 -1`] Let_def + using cos_boundaries[OF `0 \ Ifloat x` `Ifloat x \ pi / 2`] . + next + case False + + { fix y x :: float let ?x2 = "Ifloat (x * Float 1 -1)" + assume "Ifloat y \ cos ?x2" and "-pi \ Ifloat x" and "Ifloat x \ pi" + hence "- (pi / 2) \ ?x2" and "?x2 \ pi / 2" using pi_ge_two unfolding Ifloat_mult Float_num by auto + hence "0 \ cos ?x2" by (rule cos_ge_zero) + + have "Ifloat (?lb_half y) \ cos (Ifloat x)" + proof (cases "y < 0") + case True show ?thesis using cos_ge_minus_one unfolding if_P[OF True] by auto + next + case False + hence "0 \ Ifloat y" unfolding less_float_def by auto + from mult_mono[OF `Ifloat y \ cos ?x2` `Ifloat y \ cos ?x2` `0 \ cos ?x2` this] + have "Ifloat y * Ifloat y \ cos ?x2 * cos ?x2" . + hence "2 * Ifloat y * Ifloat y \ 2 * cos ?x2 * cos ?x2" by auto + hence "2 * Ifloat y * Ifloat y - 1 \ 2 * cos (Ifloat x / 2) * cos (Ifloat x / 2) - 1" unfolding Float_num Ifloat_mult by auto + thus ?thesis unfolding if_not_P[OF False] x_half Float_num Ifloat_mult Ifloat_sub by auto + qed + } note lb_half = this + + { fix y x :: float let ?x2 = "Ifloat (x * Float 1 -1)" + assume ub: "cos ?x2 \ Ifloat y" and "- pi \ Ifloat x" and "Ifloat x \ pi" + hence "- (pi / 2) \ ?x2" and "?x2 \ pi / 2" using pi_ge_two unfolding Ifloat_mult Float_num by auto + hence "0 \ cos ?x2" by (rule cos_ge_zero) + + have "cos (Ifloat x) \ Ifloat (?ub_half y)" + proof - + have "0 \ Ifloat y" using `0 \ cos ?x2` ub by (rule order_trans) + from mult_mono[OF ub ub this `0 \ cos ?x2`] + have "cos ?x2 * cos ?x2 \ Ifloat y * Ifloat y" . + hence "2 * cos ?x2 * cos ?x2 \ 2 * Ifloat y * Ifloat y" by auto + hence "2 * cos (Ifloat x / 2) * cos (Ifloat x / 2) - 1 \ 2 * Ifloat y * Ifloat y - 1" unfolding Float_num Ifloat_mult by auto + thus ?thesis unfolding x_half Ifloat_mult Float_num Ifloat_sub by auto + qed + } note ub_half = this + + let ?x2 = "x * Float 1 -1" + let ?x4 = "x * Float 1 -1 * Float 1 -1" + + have "-pi \ Ifloat x" using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] `0 \ Ifloat x` by (rule order_trans) + + show ?thesis + proof (cases "x < 1") + case True hence "Ifloat x \ 1" unfolding less_float_def by auto + have "0 \ Ifloat ?x2" and "Ifloat ?x2 \ pi / 2" using pi_ge_two `0 \ Ifloat x` unfolding Ifloat_mult Float_num using assms by auto + from cos_boundaries[OF this] + have lb: "Ifloat (?lb_horner ?x2) \ ?cos ?x2" and ub: "?cos ?x2 \ Ifloat (?ub_horner ?x2)" by auto + + have "Ifloat (?lb x) \ ?cos x" + proof - + from lb_half[OF lb `-pi \ Ifloat x` `Ifloat x \ pi`] + show ?thesis unfolding lb_cos_def[where x=x] Let_def using `\ x < 0` `\ x < Float 1 -1` `x < 1` by auto + qed + moreover have "?cos x \ Ifloat (?ub x)" + proof - + from ub_half[OF ub `-pi \ Ifloat x` `Ifloat x \ pi`] + show ?thesis unfolding ub_cos_def[where x=x] Let_def using `\ x < 0` `\ x < Float 1 -1` `x < 1` by auto + qed + ultimately show ?thesis by auto + next + case False + have "0 \ Ifloat ?x4" and "Ifloat ?x4 \ pi / 2" using pi_ge_two `0 \ Ifloat x` `Ifloat x \ pi` unfolding Ifloat_mult Float_num by auto + from cos_boundaries[OF this] + have lb: "Ifloat (?lb_horner ?x4) \ ?cos ?x4" and ub: "?cos ?x4 \ Ifloat (?ub_horner ?x4)" by auto + + have eq_4: "?x2 * Float 1 -1 = x * Float 1 -2" by (cases x, auto simp add: times_float.simps) + + have "Ifloat (?lb x) \ ?cos x" + proof - + have "-pi \ Ifloat ?x2" and "Ifloat ?x2 \ pi" unfolding Ifloat_mult Float_num using pi_ge_two `0 \ Ifloat x` `Ifloat x \ pi` by auto + from lb_half[OF lb_half[OF lb this] `-pi \ Ifloat x` `Ifloat x \ pi`, unfolded eq_4] + show ?thesis unfolding lb_cos_def[where x=x] if_not_P[OF `\ x < 0`] if_not_P[OF `\ x < Float 1 -1`] if_not_P[OF `\ x < 1`] Let_def . + qed + moreover have "?cos x \ Ifloat (?ub x)" + proof - + have "-pi \ Ifloat ?x2" and "Ifloat ?x2 \ pi" unfolding Ifloat_mult Float_num using pi_ge_two `0 \ Ifloat x` `Ifloat x \ pi` by auto + from ub_half[OF ub_half[OF ub this] `-pi \ Ifloat x` `Ifloat x \ pi`, unfolded eq_4] + show ?thesis unfolding ub_cos_def[where x=x] if_not_P[OF `\ x < 0`] if_not_P[OF `\ x < Float 1 -1`] if_not_P[OF `\ x < 1`] Let_def . + qed + ultimately show ?thesis by auto + qed + qed +qed + +lemma lb_cos_minus: assumes "-pi \ Ifloat x" and "Ifloat x \ 0" + shows "cos (Ifloat (-x)) \ {Ifloat (lb_cos prec (-x)) .. Ifloat (ub_cos prec (-x))}" +proof - + have "0 \ Ifloat (-x)" and "Ifloat (-x) \ pi" using `-pi \ Ifloat x` `Ifloat x \ 0` by auto + from lb_cos[OF this] show ?thesis . +qed + +lemma bnds_cos: "\ x lx ux. (l, u) = bnds_cos prec lx ux \ x \ {Ifloat lx .. Ifloat ux} \ Ifloat l \ cos x \ cos x \ Ifloat u" +proof (rule allI, rule allI, rule allI, rule impI) + fix x lx ux + assume "(l, u) = bnds_cos prec lx ux \ x \ {Ifloat lx .. Ifloat ux}" + hence bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \ {Ifloat lx .. Ifloat ux}" by auto + + let ?lpi = "lb_pi prec" + have [intro!]: "Ifloat lx \ Ifloat ux" using x by auto + hence "lx \ ux" unfolding le_float_def . + + show "Ifloat l \ cos x \ cos x \ Ifloat u" + proof (cases "lx < -?lpi \ ux > ?lpi") + case True + show ?thesis using bnds unfolding bnds_cos_def if_P[OF True] Let_def using cos_le_one cos_ge_minus_one by auto + next + case False note not_out = this + hence lpi_lx: "- Ifloat ?lpi \ Ifloat lx" and lpi_ux: "Ifloat ux \ Ifloat ?lpi" unfolding le_float_def less_float_def by auto + + from pi_boundaries[unfolded atLeastAtMost_iff, THEN conjunct1, THEN le_imp_neg_le] lpi_lx + have "- pi \ Ifloat lx" by (rule order_trans) + hence "- pi \ x" and "- pi \ Ifloat ux" and "x \ Ifloat ux" using x by auto + + from lpi_ux pi_boundaries[unfolded atLeastAtMost_iff, THEN conjunct1] + have "Ifloat ux \ pi" by (rule order_trans) + hence "x \ pi" and "Ifloat lx \ pi" and "Ifloat lx \ x" using x by auto + + note lb_cos_minus_bottom = lb_cos_minus[unfolded atLeastAtMost_iff, THEN conjunct1] + note lb_cos_minus_top = lb_cos_minus[unfolded atLeastAtMost_iff, THEN conjunct2] + note lb_cos_bottom = lb_cos[unfolded atLeastAtMost_iff, THEN conjunct1] + note lb_cos_top = lb_cos[unfolded atLeastAtMost_iff, THEN conjunct2] + + show ?thesis + proof (cases "ux \ 0") + case True hence "Ifloat ux \ 0" unfolding le_float_def by auto + hence "x \ 0" and "Ifloat lx \ 0" using x by auto + + { have "Ifloat (lb_cos prec (-lx)) \ cos (Ifloat (-lx))" using lb_cos_minus_bottom[OF `-pi \ Ifloat lx` `Ifloat lx \ 0`] . + also have "\ \ cos x" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \ Ifloat lx` `Ifloat lx \ x` `x \ 0`] . + finally have "Ifloat (lb_cos prec (-lx)) \ cos x" . } + moreover + { have "cos x \ cos (Ifloat (-ux))" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \ x` `x \ Ifloat ux` `Ifloat ux \ 0`] . + also have "\ \ Ifloat (ub_cos prec (-ux))" using lb_cos_minus_top[OF `-pi \ Ifloat ux` `Ifloat ux \ 0`] . + finally have "cos x \ Ifloat (ub_cos prec (-ux))" . } + ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_P[OF True] by auto + next + case False note not_ux = this + + show ?thesis + proof (cases "0 \ lx") + case True hence "0 \ Ifloat lx" unfolding le_float_def by auto + hence "0 \ x" and "0 \ Ifloat ux" using x by auto + + { have "Ifloat (lb_cos prec ux) \ cos (Ifloat ux)" using lb_cos_bottom[OF `0 \ Ifloat ux` `Ifloat ux \ pi`] . + also have "\ \ cos x" using cos_monotone_0_pi'[OF `0 \ x` `x \ Ifloat ux` `Ifloat ux \ pi`] . + finally have "Ifloat (lb_cos prec ux) \ cos x" . } + moreover + { have "cos x \ cos (Ifloat lx)" using cos_monotone_0_pi'[OF `0 \ Ifloat lx` `Ifloat lx \ x` `x \ pi`] . + also have "\ \ Ifloat (ub_cos prec lx)" using lb_cos_top[OF `0 \ Ifloat lx` `Ifloat lx \ pi`] . + finally have "cos x \ Ifloat (ub_cos prec lx)" . } + ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_not_P[OF not_ux] if_P[OF True] by auto + next + case False with not_ux + have "Ifloat lx \ 0" and "0 \ Ifloat ux" unfolding le_float_def by auto + + have "Ifloat (min (lb_cos prec (-lx)) (lb_cos prec ux)) \ cos x" + proof (cases "x \ 0") + case True + have "Ifloat (lb_cos prec (-lx)) \ cos (Ifloat (-lx))" using lb_cos_minus_bottom[OF `-pi \ Ifloat lx` `Ifloat lx \ 0`] . + also have "\ \ cos x" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \ Ifloat lx` `Ifloat lx \ x` `x \ 0`] . + finally show ?thesis unfolding Ifloat_min by auto + next + case False hence "0 \ x" by auto + have "Ifloat (lb_cos prec ux) \ cos (Ifloat ux)" using lb_cos_bottom[OF `0 \ Ifloat ux` `Ifloat ux \ pi`] . + also have "\ \ cos x" using cos_monotone_0_pi'[OF `0 \ x` `x \ Ifloat ux` `Ifloat ux \ pi`] . + finally show ?thesis unfolding Ifloat_min by auto + qed + moreover have "cos x \ Ifloat (Float 1 0)" by auto + ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_not_P[OF not_ux] if_not_P[OF False] by auto + qed + qed + qed +qed + +subsection "Compute the sinus in the entire domain" + +function lb_sin :: "nat \ float \ float" and ub_sin :: "nat \ float \ float" where +"lb_sin prec x = (let sqr_diff = \ x. if x > 1 then 0 else 1 - x * x + in if x < 0 then - ub_sin prec (- x) +else if x \ Float 1 -1 then x * lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 2 1 (x * x) + else the (lb_sqrt prec (sqr_diff (ub_cos prec x))))" | + +"ub_sin prec x = (let sqr_diff = \ x. if x < 0 then 1 else 1 - x * x + in if x < 0 then - lb_sin prec (- x) +else if x \ Float 1 -1 then x * ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 2 1 (x * x) + else the (ub_sqrt prec (sqr_diff (lb_cos prec x))))" +by pat_completeness auto +termination by (relation "measure (\ v. let (prec, x) = sum_case id id v in (if x < 0 then 1 else 0))", auto simp add: less_float_def) + +definition bnds_sin :: "nat \ float \ float \ float * float" where +"bnds_sin prec lx ux = (let + lpi = lb_pi prec ; + half_pi = lpi * Float 1 -1 + in if lx \ - half_pi \ half_pi \ ux then (Float -1 0, Float 1 0) + else (lb_sin prec lx, ub_sin prec ux))" + +lemma lb_sin: assumes "- (pi / 2) \ Ifloat x" and "Ifloat x \ pi / 2" + shows "sin (Ifloat x) \ { Ifloat (lb_sin prec x) .. Ifloat (ub_sin prec x) }" (is "?sin x \ { ?lb x .. ?ub x}") +proof - + { fix x :: float assume "0 \ Ifloat x" and "Ifloat x \ pi / 2" + hence "\ (x < 0)" and "- (pi / 2) \ Ifloat x" unfolding less_float_def using pi_ge_two by auto + + have "Ifloat x \ pi" using `Ifloat x \ pi / 2` using pi_ge_two by auto + + have "?sin x \ { ?lb x .. ?ub x}" + proof (cases "x \ Float 1 -1") + case True from sin_boundaries[OF `0 \ Ifloat x` `Ifloat x \ pi / 2`] + show ?thesis unfolding lb_sin.simps[of prec x] ub_sin.simps[of prec x] if_not_P[OF `\ (x < 0)`] if_P[OF True] Let_def . + next + case False + have "0 \ cos (Ifloat x)" using cos_ge_zero[OF _ `Ifloat x \ pi /2`] `0 \ Ifloat x` pi_ge_two by auto + have "0 \ sin (Ifloat x)" using `0 \ Ifloat x` and `Ifloat x \ pi / 2` using sin_ge_zero by auto + + have "?sin x \ ?ub x" + proof (cases "lb_cos prec x < 0") + case True + have "?sin x \ 1" using sin_le_one . + also have "\ \ Ifloat (the (ub_sqrt prec 1))" using ub_sqrt_lower_bound[where prec=prec and x=1] unfolding Ifloat_1 by auto + finally show ?thesis unfolding ub_sin.simps if_not_P[OF `\ (x < 0)`] if_not_P[OF `\ x \ Float 1 -1`] if_P[OF True] Let_def . + next + case False hence "0 \ Ifloat (lb_cos prec x)" unfolding less_float_def by auto + + have "sin (Ifloat x) = sqrt (1 - cos (Ifloat x) ^ 2)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \ sin (Ifloat x)` by auto + also have "\ \ sqrt (Ifloat (1 - lb_cos prec x * lb_cos prec x))" + proof (rule real_sqrt_le_mono) + have "Ifloat (lb_cos prec x * lb_cos prec x) \ cos (Ifloat x) ^ 2" unfolding numeral_2_eq_2 power_Suc2 realpow_0 Ifloat_mult + using `0 \ Ifloat (lb_cos prec x)` lb_cos[OF `0 \ Ifloat x` `Ifloat x \ pi`] `0 \ cos (Ifloat x)` by(auto intro!: mult_mono) + thus "1 - cos (Ifloat x) ^ 2 \ Ifloat (1 - lb_cos prec x * lb_cos prec x)" unfolding Ifloat_sub Ifloat_1 by auto + qed + also have "\ \ Ifloat (the (ub_sqrt prec (1 - lb_cos prec x * lb_cos prec x)))" + proof (rule ub_sqrt_lower_bound) + have "Ifloat (lb_cos prec x) \ cos (Ifloat x)" using lb_cos[OF `0 \ Ifloat x` `Ifloat x \ pi`] by auto + from mult_mono[OF order_trans[OF this cos_le_one] order_trans[OF this cos_le_one]] + have "Ifloat (lb_cos prec x) * Ifloat (lb_cos prec x) \ 1" using `0 \ Ifloat (lb_cos prec x)` by auto + thus "0 \ Ifloat (1 - lb_cos prec x * lb_cos prec x)" by auto + qed + finally show ?thesis unfolding ub_sin.simps if_not_P[OF `\ (x < 0)`] if_not_P[OF `\ x \ Float 1 -1`] if_not_P[OF False] Let_def . + qed + moreover + have "?lb x \ ?sin x" + proof (cases "1 < ub_cos prec x") + case True + show ?thesis unfolding lb_sin.simps if_not_P[OF `\ (x < 0)`] if_not_P[OF `\ x \ Float 1 -1`] if_P[OF True] Let_def + by (rule order_trans[OF _ sin_ge_zero[OF `0 \ Ifloat x` `Ifloat x \ pi`]]) + (auto simp add: lb_sqrt_upper_bound[where prec=prec and x=0, unfolded Ifloat_0 real_sqrt_zero]) + next + case False hence "Ifloat (ub_cos prec x) \ 1" unfolding less_float_def by auto + have "0 \ Ifloat (ub_cos prec x)" using order_trans[OF `0 \ cos (Ifloat x)`] lb_cos `0 \ Ifloat x` `Ifloat x \ pi` by auto + + have "Ifloat (the (lb_sqrt prec (1 - ub_cos prec x * ub_cos prec x))) \ sqrt (Ifloat (1 - ub_cos prec x * ub_cos prec x))" + proof (rule lb_sqrt_upper_bound) + from mult_mono[OF `Ifloat (ub_cos prec x) \ 1` `Ifloat (ub_cos prec x) \ 1`] `0 \ Ifloat (ub_cos prec x)` + have "Ifloat (ub_cos prec x) * Ifloat (ub_cos prec x) \ 1" by auto + thus "0 \ Ifloat (1 - ub_cos prec x * ub_cos prec x)" by auto + qed + also have "\ \ sqrt (1 - cos (Ifloat x) ^ 2)" + proof (rule real_sqrt_le_mono) + have "cos (Ifloat x) ^ 2 \ Ifloat (ub_cos prec x * ub_cos prec x)" unfolding numeral_2_eq_2 power_Suc2 realpow_0 Ifloat_mult + using `0 \ Ifloat (ub_cos prec x)` lb_cos[OF `0 \ Ifloat x` `Ifloat x \ pi`] `0 \ cos (Ifloat x)` by(auto intro!: mult_mono) + thus "Ifloat (1 - ub_cos prec x * ub_cos prec x) \ 1 - cos (Ifloat x) ^ 2" unfolding Ifloat_sub Ifloat_1 by auto + qed + also have "\ = sin (Ifloat x)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \ sin (Ifloat x)` by auto + finally show ?thesis unfolding lb_sin.simps if_not_P[OF `\ (x < 0)`] if_not_P[OF `\ x \ Float 1 -1`] if_not_P[OF False] Let_def . + qed + ultimately show ?thesis by auto + qed + } note for_pos = this + + show ?thesis + proof (cases "x < 0") + case True + hence "0 \ Ifloat (-x)" and "Ifloat (- x) \ pi / 2" using `-(pi/2) \ Ifloat x` unfolding less_float_def by auto + from for_pos[OF this] + show ?thesis unfolding Ifloat_minus sin_minus lb_sin.simps[of prec x] ub_sin.simps[of prec x] if_P[OF True] Let_def atLeastAtMost_iff by auto + next + case False hence "0 \ Ifloat x" unfolding less_float_def by auto + from for_pos[OF this `Ifloat x \ pi /2`] + show ?thesis . + qed +qed + +lemma bnds_sin: "\ x lx ux. (l, u) = bnds_sin prec lx ux \ x \ {Ifloat lx .. Ifloat ux} \ Ifloat l \ sin x \ sin x \ Ifloat u" +proof (rule allI, rule allI, rule allI, rule impI) + fix x lx ux + assume "(l, u) = bnds_sin prec lx ux \ x \ {Ifloat lx .. Ifloat ux}" + hence bnds: "(l, u) = bnds_sin prec lx ux" and x: "x \ {Ifloat lx .. Ifloat ux}" by auto + show "Ifloat l \ sin x \ sin x \ Ifloat u" + proof (cases "lx \ - (lb_pi prec * Float 1 -1) \ lb_pi prec * Float 1 -1 \ ux") + case True show ?thesis using bnds unfolding bnds_sin_def if_P[OF True] Let_def by auto + next + case False + hence "- lb_pi prec * Float 1 -1 \ lx" and "ux \ lb_pi prec * Float 1 -1" unfolding le_float_def by auto + moreover have "Ifloat (lb_pi prec * Float 1 -1) \ pi / 2" unfolding Ifloat_mult using pi_boundaries by auto + ultimately have "- (pi / 2) \ Ifloat lx" and "Ifloat ux \ pi / 2" and "Ifloat lx \ Ifloat ux" unfolding le_float_def using x by auto + hence "- (pi / 2) \ Ifloat ux" and "Ifloat lx \ pi / 2" by auto + + have "- (pi / 2) \ x""x \ pi / 2" using `Ifloat ux \ pi / 2` `- (pi /2) \ Ifloat lx` x by auto + + { have "Ifloat (lb_sin prec lx) \ sin (Ifloat lx)" using lb_sin[OF `- (pi / 2) \ Ifloat lx` `Ifloat lx \ pi / 2`] unfolding atLeastAtMost_iff by auto + also have "\ \ sin x" using sin_monotone_2pi' `- (pi / 2) \ Ifloat lx` x `x \ pi / 2` by auto + finally have "Ifloat (lb_sin prec lx) \ sin x" . } + moreover + { have "sin x \ sin (Ifloat ux)" using sin_monotone_2pi' `- (pi / 2) \ x` x `Ifloat ux \ pi / 2` by auto + also have "\ \ Ifloat (ub_sin prec ux)" using lb_sin[OF `- (pi / 2) \ Ifloat ux` `Ifloat ux \ pi / 2`] unfolding atLeastAtMost_iff by auto + finally have "sin x \ Ifloat (ub_sin prec ux)" . } + ultimately + show ?thesis using bnds unfolding bnds_sin_def if_not_P[OF False] Let_def by auto + qed +qed + +section "Exponential function" + +subsection "Compute the series of the exponential function" + +fun ub_exp_horner :: "nat \ nat \ nat \ nat \ float \ float" and lb_exp_horner :: "nat \ nat \ nat \ nat \ float \ float" where +"ub_exp_horner prec 0 i k x = 0" | +"ub_exp_horner prec (Suc n) i k x = rapprox_rat prec 1 (int k) + x * lb_exp_horner prec n (i + 1) (k * i) x" | +"lb_exp_horner prec 0 i k x = 0" | +"lb_exp_horner prec (Suc n) i k x = lapprox_rat prec 1 (int k) + x * ub_exp_horner prec n (i + 1) (k * i) x" + +lemma bnds_exp_horner: assumes "Ifloat x \ 0" + shows "exp (Ifloat x) \ { Ifloat (lb_exp_horner prec (get_even n) 1 1 x) .. Ifloat (ub_exp_horner prec (get_odd n) 1 1 x) }" +proof - + { fix n + have F: "\ m. ((\i. i + 1) ^ n) m = n + m" by (induct n, auto) + have "fact (Suc n) = fact n * ((\i. i + 1) ^ n) 1" unfolding F by auto } note f_eq = this + + note bounds = horner_bounds_nonpos[where f="fact" and lb="lb_exp_horner prec" and ub="ub_exp_horner prec" and j'=0 and s=1, + OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps] + + { have "Ifloat (lb_exp_horner prec (get_even n) 1 1 x) \ (\j = 0.. \ exp (Ifloat x)" + proof - + obtain t where "\t\ \ \Ifloat x\" and "exp (Ifloat x) = (\m = 0.. exp t / real (fact (get_even n)) * (Ifloat x) ^ (get_even n)" + by (auto intro!: mult_nonneg_nonneg divide_nonneg_pos simp add: get_even zero_le_even_power exp_gt_zero) + ultimately show ?thesis + using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg) + qed + finally have "Ifloat (lb_exp_horner prec (get_even n) 1 1 x) \ exp (Ifloat x)" . + } moreover + { + have x_less_zero: "Ifloat x ^ get_odd n \ 0" + proof (cases "Ifloat x = 0") + case True + have "(get_odd n) \ 0" using get_odd[THEN odd_pos] by auto + thus ?thesis unfolding True power_0_left by auto + next + case False hence "Ifloat x < 0" using `Ifloat x \ 0` by auto + show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq get_odd `Ifloat x < 0`) + qed + + obtain t where "\t\ \ \Ifloat x\" and "exp (Ifloat x) = (\m = 0.. 0" + by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero exp_gt_zero) + ultimately have "exp (Ifloat x) \ (\j = 0.. \ Ifloat (ub_exp_horner prec (get_odd n) 1 1 x)" + using bounds(2) by auto + finally have "exp (Ifloat x) \ Ifloat (ub_exp_horner prec (get_odd n) 1 1 x)" . + } ultimately show ?thesis by auto +qed + +subsection "Compute the exponential function on the entire domain" + +function ub_exp :: "nat \ float \ float" and lb_exp :: "nat \ float \ float" where +"lb_exp prec x = (if 0 < x then float_divl prec 1 (ub_exp prec (-x)) + else let + horner = (\ x. let y = lb_exp_horner prec (get_even (prec + 2)) 1 1 x in if y \ 0 then Float 1 -2 else y) + in if x < - 1 then (case floor_fl x of (Float m e) \ (horner (float_divl prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e)) + else horner x)" | +"ub_exp prec x = (if 0 < x then float_divr prec 1 (lb_exp prec (-x)) + else if x < - 1 then (case floor_fl x of (Float m e) \ + (ub_exp_horner prec (get_odd (prec + 2)) 1 1 (float_divr prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e)) + else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)" +by pat_completeness auto +termination by (relation "measure (\ v. let (prec, x) = sum_case id id v in (if 0 < x then 1 else 0))", auto simp add: less_float_def) + +lemma exp_m1_ge_quarter: "(1 / 4 :: real) \ exp (- 1)" +proof - + have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto + + have "1 / 4 = Ifloat (Float 1 -2)" unfolding Float_num by auto + also have "\ \ Ifloat (lb_exp_horner 1 (get_even 4) 1 1 (- 1))" + unfolding get_even_def eq4 + by (auto simp add: lapprox_posrat_def rapprox_posrat_def normfloat.simps) + also have "\ \ exp (Ifloat (- 1))" using bnds_exp_horner[where x="- 1"] by auto + finally show ?thesis unfolding Ifloat_minus Ifloat_1 . +qed + +lemma lb_exp_pos: assumes "\ 0 < x" shows "0 < lb_exp prec x" +proof - + let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x" + let "?horner x" = "let y = ?lb_horner x in if y \ 0 then Float 1 -2 else y" + have pos_horner: "\ x. 0 < ?horner x" unfolding Let_def by (cases "?lb_horner x \ 0", auto simp add: le_float_def less_float_def) + moreover { fix x :: float fix num :: nat + have "0 < Ifloat (?horner x) ^ num" using `0 < ?horner x`[unfolded less_float_def Ifloat_0] by (rule zero_less_power) + also have "\ = Ifloat ((?horner x) ^ num)" using float_power by auto + finally have "0 < Ifloat ((?horner x) ^ num)" . + } + ultimately show ?thesis + unfolding lb_exp.simps if_not_P[OF `\ 0 < x`] Let_def by (cases "floor_fl x", cases "x < - 1", auto simp add: le_float_def less_float_def normfloat) +qed + +lemma exp_boundaries': assumes "x \ 0" + shows "exp (Ifloat x) \ { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}" +proof - + let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x" + let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x" + + have "Ifloat x \ 0" and "\ x > 0" using `x \ 0` unfolding le_float_def less_float_def by auto + show ?thesis + proof (cases "x < - 1") + case False hence "- 1 \ Ifloat x" unfolding less_float_def by auto + show ?thesis + proof (cases "?lb_exp_horner x \ 0") + from `\ x < - 1` have "- 1 \ Ifloat x" unfolding less_float_def by auto + hence "exp (- 1) \ exp (Ifloat x)" unfolding exp_le_cancel_iff . + from order_trans[OF exp_m1_ge_quarter this] + have "Ifloat (Float 1 -2) \ exp (Ifloat x)" unfolding Float_num . + moreover case True + ultimately show ?thesis using bnds_exp_horner `Ifloat x \ 0` `\ x > 0` `\ x < - 1` by auto + next + case False thus ?thesis using bnds_exp_horner `Ifloat x \ 0` `\ x > 0` `\ x < - 1` by (auto simp add: Let_def) + qed + next + case True + + obtain m e where Float_floor: "floor_fl x = Float m e" by (cases "floor_fl x", auto) + let ?num = "nat (- m) * 2 ^ nat e" + + have "Ifloat (floor_fl x) < - 1" using floor_fl `x < - 1` unfolding le_float_def less_float_def Ifloat_minus Ifloat_1 by (rule order_le_less_trans) + hence "Ifloat (floor_fl x) < 0" unfolding Float_floor Ifloat.simps using zero_less_pow2[of xe] by auto + hence "m < 0" + unfolding less_float_def Ifloat_0 Float_floor Ifloat.simps + unfolding pos_prod_lt[OF zero_less_pow2[of e], unfolded real_mult_commute] by auto + hence "1 \ - m" by auto + hence "0 < nat (- m)" by auto + moreover + have "0 \ e" using floor_pos_exp Float_floor[symmetric] by auto + hence "(0::nat) < 2 ^ nat e" by auto + ultimately have "0 < ?num" by auto + hence "real ?num \ 0" by auto + have e_nat: "int (nat e) = e" using `0 \ e` by auto + have num_eq: "real ?num = Ifloat (- floor_fl x)" using `0 < nat (- m)` + unfolding Float_floor Ifloat_minus Ifloat.simps real_of_nat_mult pow2_int[of "nat e", unfolded e_nat] realpow_real_of_nat[symmetric] by auto + have "0 < - floor_fl x" using `0 < ?num`[unfolded real_of_nat_less_iff[symmetric]] unfolding less_float_def num_eq[symmetric] Ifloat_0 real_of_nat_zero . + hence "Ifloat (floor_fl x) < 0" unfolding less_float_def by auto + + have "exp (Ifloat x) \ Ifloat (ub_exp prec x)" + proof - + have div_less_zero: "Ifloat (float_divr prec x (- floor_fl x)) \ 0" + using float_divr_nonpos_pos_upper_bound[OF `x \ 0` `0 < - floor_fl x`] unfolding le_float_def Ifloat_0 . + + have "exp (Ifloat x) = exp (real ?num * (Ifloat x / real ?num))" using `real ?num \ 0` by auto + also have "\ = exp (Ifloat x / real ?num) ^ ?num" unfolding exp_real_of_nat_mult .. + also have "\ \ exp (Ifloat (float_divr prec x (- floor_fl x))) ^ ?num" unfolding num_eq + by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto + also have "\ \ Ifloat ((?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num)" unfolding float_power + by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto) + finally show ?thesis unfolding ub_exp.simps if_not_P[OF `\ 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def . + qed + moreover + have "Ifloat (lb_exp prec x) \ exp (Ifloat x)" + proof - + let ?divl = "float_divl prec x (- Float m e)" + let ?horner = "?lb_exp_horner ?divl" + + show ?thesis + proof (cases "?horner \ 0") + case False hence "0 \ Ifloat ?horner" unfolding le_float_def by auto + + have div_less_zero: "Ifloat (float_divl prec x (- floor_fl x)) \ 0" + using `Ifloat (floor_fl x) < 0` `Ifloat x \ 0` by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg) + + have "Ifloat ((?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num) \ + exp (Ifloat (float_divl prec x (- floor_fl x))) ^ ?num" unfolding float_power + using `0 \ Ifloat ?horner`[unfolded Float_floor[symmetric]] bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1] by (auto intro!: power_mono) + also have "\ \ exp (Ifloat x / real ?num) ^ ?num" unfolding num_eq + using float_divl by (auto intro!: power_mono simp del: Ifloat_minus) + also have "\ = exp (real ?num * (Ifloat x / real ?num))" unfolding exp_real_of_nat_mult .. + also have "\ = exp (Ifloat x)" using `real ?num \ 0` by auto + finally show ?thesis + unfolding lb_exp.simps if_not_P[OF `\ 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def if_not_P[OF False] by auto + next + case True + have "Ifloat (floor_fl x) \ 0" and "Ifloat (floor_fl x) \ 0" using `Ifloat (floor_fl x) < 0` by auto + from divide_right_mono_neg[OF floor_fl[of x] `Ifloat (floor_fl x) \ 0`, unfolded divide_self[OF `Ifloat (floor_fl x) \ 0`]] + have "- 1 \ Ifloat x / Ifloat (- floor_fl x)" unfolding Ifloat_minus by auto + from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]] + have "Ifloat (Float 1 -2) \ exp (Ifloat x / Ifloat (- floor_fl x))" unfolding Float_num . + hence "Ifloat (Float 1 -2) ^ ?num \ exp (Ifloat x / Ifloat (- floor_fl x)) ^ ?num" + by (auto intro!: power_mono simp add: Float_num) + also have "\ = exp (Ifloat x)" unfolding num_eq exp_real_of_nat_mult[symmetric] using `Ifloat (floor_fl x) \ 0` by auto + finally show ?thesis + unfolding lb_exp.simps if_not_P[OF `\ 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def if_P[OF True] float_power . + qed + qed + ultimately show ?thesis by auto + qed +qed + +lemma exp_boundaries: "exp (Ifloat x) \ { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}" +proof - + show ?thesis + proof (cases "0 < x") + case False hence "x \ 0" unfolding less_float_def le_float_def by auto + from exp_boundaries'[OF this] show ?thesis . + next + case True hence "-x \ 0" unfolding less_float_def le_float_def by auto + + have "Ifloat (lb_exp prec x) \ exp (Ifloat x)" + proof - + from exp_boundaries'[OF `-x \ 0`] + have ub_exp: "exp (- Ifloat x) \ Ifloat (ub_exp prec (-x))" unfolding atLeastAtMost_iff Ifloat_minus by auto + + have "Ifloat (float_divl prec 1 (ub_exp prec (-x))) \ Ifloat 1 / Ifloat (ub_exp prec (-x))" using float_divl . + also have "Ifloat 1 / Ifloat (ub_exp prec (-x)) \ exp (Ifloat x)" + using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp] exp_gt_zero, symmetric]] + unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide by auto + finally show ?thesis unfolding lb_exp.simps if_P[OF True] . + qed + moreover + have "exp (Ifloat x) \ Ifloat (ub_exp prec x)" + proof - + have "\ 0 < -x" using `0 < x` unfolding less_float_def by auto + + from exp_boundaries'[OF `-x \ 0`] + have lb_exp: "Ifloat (lb_exp prec (-x)) \ exp (- Ifloat x)" unfolding atLeastAtMost_iff Ifloat_minus by auto + + have "exp (Ifloat x) \ Ifloat 1 / Ifloat (lb_exp prec (-x))" + using lb_exp[unfolded inverse_le_iff_le[OF exp_gt_zero lb_exp_pos[OF `\ 0 < -x`, unfolded less_float_def Ifloat_0], symmetric]] + unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide Ifloat_1 by auto + also have "\ \ Ifloat (float_divr prec 1 (lb_exp prec (-x)))" using float_divr . + finally show ?thesis unfolding ub_exp.simps if_P[OF True] . + qed + ultimately show ?thesis by auto + qed +qed + +lemma bnds_exp: "\ x lx ux. (l, u) = (lb_exp prec lx, ub_exp prec ux) \ x \ {Ifloat lx .. Ifloat ux} \ Ifloat l \ exp x \ exp x \ Ifloat u" +proof (rule allI, rule allI, rule allI, rule impI) + fix x lx ux + assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \ x \ {Ifloat lx .. Ifloat ux}" + hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \ {Ifloat lx .. Ifloat ux}" by auto + + { from exp_boundaries[of lx prec, unfolded l] + have "Ifloat l \ exp (Ifloat lx)" by (auto simp del: lb_exp.simps) + also have "\ \ exp x" using x by auto + finally have "Ifloat l \ exp x" . + } moreover + { have "exp x \ exp (Ifloat ux)" using x by auto + also have "\ \ Ifloat u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps) + finally have "exp x \ Ifloat u" . + } ultimately show "Ifloat l \ exp x \ exp x \ Ifloat u" .. +qed + +section "Logarithm" + +subsection "Compute the logarithm series" + +fun ub_ln_horner :: "nat \ nat \ nat \ float \ float" +and lb_ln_horner :: "nat \ nat \ nat \ float \ float" where +"ub_ln_horner prec 0 i x = 0" | +"ub_ln_horner prec (Suc n) i x = rapprox_rat prec 1 (int i) - x * lb_ln_horner prec n (Suc i) x" | +"lb_ln_horner prec 0 i x = 0" | +"lb_ln_horner prec (Suc n) i x = lapprox_rat prec 1 (int i) - x * ub_ln_horner prec n (Suc i) x" + +lemma ln_bounds: + assumes "0 \ x" and "x < 1" + shows "(\i=0..<2*n. -1^i * (1 / real (i + 1)) * x^(Suc i)) \ ln (x + 1)" (is "?lb") + and "ln (x + 1) \ (\i=0..<2*n + 1. -1^i * (1 / real (i + 1)) * x^(Suc i))" (is "?ub") +proof - + let "?a n" = "(1/real (n +1)) * x^(Suc n)" + + have ln_eq: "(\ i. -1^i * ?a i) = ln (x + 1)" + using ln_series[of "x + 1"] `0 \ x` `x < 1` by auto + + have "norm x < 1" using assms by auto + have "?a ----> 0" unfolding Suc_plus1[symmetric] inverse_eq_divide[symmetric] + using LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF `norm x < 1`]]] by auto + { fix n have "0 \ ?a n" by (rule mult_nonneg_nonneg, auto intro!: mult_nonneg_nonneg simp add: `0 \ x`) } + { fix n have "?a (Suc n) \ ?a n" unfolding inverse_eq_divide[symmetric] + proof (rule mult_mono) + show "0 \ x ^ Suc (Suc n)" by (auto intro!: mult_nonneg_nonneg simp add: `0 \ x`) + have "x ^ Suc (Suc n) \ x ^ Suc n * 1" unfolding power_Suc2 real_mult_assoc[symmetric] + by (rule mult_left_mono, fact less_imp_le[OF `x < 1`], auto intro!: mult_nonneg_nonneg simp add: `0 \ x`) + thus "x ^ Suc (Suc n) \ x ^ Suc n" by auto + qed auto } + from summable_Leibniz'(2,4)[OF `?a ----> 0` `\n. 0 \ ?a n`, OF `\n. ?a (Suc n) \ ?a n`, unfolded ln_eq] + show "?lb" and "?ub" by auto +qed + +lemma ln_float_bounds: + assumes "0 \ Ifloat x" and "Ifloat x < 1" + shows "Ifloat (x * lb_ln_horner prec (get_even n) 1 x) \ ln (Ifloat x + 1)" (is "?lb \ ?ln") + and "ln (Ifloat x + 1) \ Ifloat (x * ub_ln_horner prec (get_odd n) 1 x)" (is "?ln \ ?ub") +proof - + obtain ev where ev: "get_even n = 2 * ev" using get_even_double .. + obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double .. + + let "?s n" = "-1^n * (1 / real (1 + n)) * (Ifloat x)^(Suc n)" + + have "?lb \ setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 real_mult_assoc[symmetric] Ifloat_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "Ifloat x"] ev + using horner_bounds(1)[where G="\ i k. Suc k" and F="\x. x" and f="\x. x" and lb="\n i k x. lb_ln_horner prec n k x" and ub="\n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*ev", + OF `0 \ Ifloat x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \ Ifloat x` + by (rule mult_right_mono) + also have "\ \ ?ln" using ln_bounds(1)[OF `0 \ Ifloat x` `Ifloat x < 1`] by auto + finally show "?lb \ ?ln" . + + have "?ln \ setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \ Ifloat x` `Ifloat x < 1`] by auto + also have "\ \ ?ub" unfolding power_Suc2 real_mult_assoc[symmetric] Ifloat_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "Ifloat x"] od + using horner_bounds(2)[where G="\ i k. Suc k" and F="\x. x" and f="\x. x" and lb="\n i k x. lb_ln_horner prec n k x" and ub="\n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*od+1", + OF `0 \ Ifloat x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \ Ifloat x` + by (rule mult_right_mono) + finally show "?ln \ ?ub" . +qed + +lemma ln_add: assumes "0 < x" and "0 < y" shows "ln (x + y) = ln x + ln (1 + y / x)" +proof - + have "x \ 0" using assms by auto + have "x + y = x * (1 + y / x)" unfolding right_distrib times_divide_eq_right nonzero_mult_divide_cancel_left[OF `x \ 0`] by auto + moreover + have "0 < y / x" using assms divide_pos_pos by auto + hence "0 < 1 + y / x" by auto + ultimately show ?thesis using ln_mult assms by auto +qed + +subsection "Compute the logarithm of 2" + +definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3 + in (Float 1 -1 * ub_ln_horner prec (get_odd prec) 1 (Float 1 -1)) + + (third * ub_ln_horner prec (get_odd prec) 1 third))" +definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3 + in (Float 1 -1 * lb_ln_horner prec (get_even prec) 1 (Float 1 -1)) + + (third * lb_ln_horner prec (get_even prec) 1 third))" + +lemma ub_ln2: "ln 2 \ Ifloat (ub_ln2 prec)" (is "?ub_ln2") + and lb_ln2: "Ifloat (lb_ln2 prec) \ ln 2" (is "?lb_ln2") +proof - + let ?uthird = "rapprox_rat (max prec 1) 1 3" + let ?lthird = "lapprox_rat prec 1 3" + + have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1)" + using ln_add[of "3 / 2" "1 / 2"] by auto + have lb3: "Ifloat ?lthird \ 1 / 3" using lapprox_rat[of prec 1 3] by auto + hence lb3_ub: "Ifloat ?lthird < 1" by auto + have lb3_lb: "0 \ Ifloat ?lthird" using lapprox_rat_bottom[of 1 3] by auto + have ub3: "1 / 3 \ Ifloat ?uthird" using rapprox_rat[of 1 3] by auto + hence ub3_lb: "0 \ Ifloat ?uthird" by auto + + have lb2: "0 \ Ifloat (Float 1 -1)" and ub2: "Ifloat (Float 1 -1) < 1" unfolding Float_num by auto + + have "0 \ (1::int)" and "0 < (3::int)" by auto + have ub3_ub: "Ifloat ?uthird < 1" unfolding rapprox_rat.simps(2)[OF `0 \ 1` `0 < 3`] + by (rule rapprox_posrat_less1, auto) + + have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto + have uthird_gt0: "0 < Ifloat ?uthird + 1" using ub3_lb by auto + have lthird_gt0: "0 < Ifloat ?lthird + 1" using lb3_lb by auto + + show ?ub_ln2 unfolding ub_ln2_def Let_def Ifloat_add ln2_sum Float_num(4)[symmetric] + proof (rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2]) + have "ln (1 / 3 + 1) \ ln (Ifloat ?uthird + 1)" unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto + also have "\ \ Ifloat (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" + using ln_float_bounds(2)[OF ub3_lb ub3_ub] . + finally show "ln (1 / 3 + 1) \ Ifloat (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" . + qed + show ?lb_ln2 unfolding lb_ln2_def Let_def Ifloat_add ln2_sum Float_num(4)[symmetric] + proof (rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2]) + have "Ifloat (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \ ln (Ifloat ?lthird + 1)" + using ln_float_bounds(1)[OF lb3_lb lb3_ub] . + also have "\ \ ln (1 / 3 + 1)" unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] using lb3 by auto + finally show "Ifloat (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \ ln (1 / 3 + 1)" . + qed +qed + +subsection "Compute the logarithm in the entire domain" + +function ub_ln :: "nat \ float \ float option" and lb_ln :: "nat \ float \ float option" where +"ub_ln prec x = (if x \ 0 then None + else if x < 1 then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x))) + else let horner = \x. (x - 1) * ub_ln_horner prec (get_odd prec) 1 (x - 1) in + if x < Float 1 1 then Some (horner x) + else let l = bitlen (mantissa x) - 1 in + Some (ub_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l))))" | +"lb_ln prec x = (if x \ 0 then None + else if x < 1 then Some (- the (ub_ln prec (float_divr prec 1 x))) + else let horner = \x. (x - 1) * lb_ln_horner prec (get_even prec) 1 (x - 1) in + if x < Float 1 1 then Some (horner x) + else let l = bitlen (mantissa x) - 1 in + Some (lb_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l))))" +by pat_completeness auto + +termination proof (relation "measure (\ v. let (prec, x) = sum_case id id v in (if x < 1 then 1 else 0))", auto) + fix prec x assume "\ x \ 0" and "x < 1" and "float_divl (max prec (Suc 0)) 1 x < 1" + hence "0 < x" and "0 < max prec (Suc 0)" unfolding less_float_def le_float_def by auto + from float_divl_pos_less1_bound[OF `0 < x` `x < 1` `0 < max prec (Suc 0)`] + show False using `float_divl (max prec (Suc 0)) 1 x < 1` unfolding less_float_def le_float_def by auto +next + fix prec x assume "\ x \ 0" and "x < 1" and "float_divr prec 1 x < 1" + hence "0 < x" unfolding less_float_def le_float_def by auto + from float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`, of prec] + show False using `float_divr prec 1 x < 1` unfolding less_float_def le_float_def by auto +qed + +lemma ln_shifted_float: assumes "0 < m" shows "ln (Ifloat (Float m e)) = ln 2 * real (e + (bitlen m - 1)) + ln (Ifloat (Float m (- (bitlen m - 1))))" +proof - + let ?B = "2^nat (bitlen m - 1)" + have "0 < real m" and "\X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \ 0" using assms by auto + hence "0 \ bitlen m - 1" using bitlen_ge1[OF `m \ 0`] by auto + show ?thesis + proof (cases "0 \ e") + case True + show ?thesis unfolding normalized_float[OF `m \ 0`] + unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`] + unfolding Ifloat_ge0_exp[OF True] ln_mult[OF `0 < real m` `0 < 2^nat e`] + ln_realpow[OF `0 < 2`] algebra_simps using `0 \ bitlen m - 1` True by auto + next + case False hence "0 < -e" by auto + hence pow_gt0: "(0::real) < 2^nat (-e)" by auto + hence inv_gt0: "(0::real) < inverse (2^nat (-e))" by auto + show ?thesis unfolding normalized_float[OF `m \ 0`] + unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`] + unfolding Ifloat_nge0_exp[OF False] ln_mult[OF `0 < real m` inv_gt0] ln_inverse[OF pow_gt0] + ln_realpow[OF `0 < 2`] algebra_simps using `0 \ bitlen m - 1` False by auto + qed +qed + +lemma ub_ln_lb_ln_bounds': assumes "1 \ x" + shows "Ifloat (the (lb_ln prec x)) \ ln (Ifloat x) \ ln (Ifloat x) \ Ifloat (the (ub_ln prec x))" + (is "?lb \ ?ln \ ?ln \ ?ub") +proof (cases "x < Float 1 1") + case True hence "Ifloat (x - 1) < 1" unfolding less_float_def Float_num by auto + have "\ x \ 0" and "\ x < 1" using `1 \ x` unfolding less_float_def le_float_def by auto + hence "0 \ Ifloat (x - 1)" using `1 \ x` unfolding less_float_def Float_num by auto + show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def + using ln_float_bounds[OF `0 \ Ifloat (x - 1)` `Ifloat (x - 1) < 1`] `\ x \ 0` `\ x < 1` True by auto +next + case False + have "\ x \ 0" and "\ x < 1" "0 < x" using `1 \ x` unfolding less_float_def le_float_def by auto + show ?thesis + proof (cases x) + case (Float m e) + let ?s = "Float (e + (bitlen m - 1)) 0" + let ?x = "Float m (- (bitlen m - 1))" + + have "0 < m" and "m \ 0" using float_pos_m_pos `0 < x` Float by auto + + { + have "Ifloat (lb_ln2 prec * ?s) \ ln 2 * real (e + (bitlen m - 1))" (is "?lb2 \ _") + unfolding Ifloat_mult Ifloat_ge0_exp[OF order_refl] nat_0 realpow_0 mult_1_right + using lb_ln2[of prec] + proof (rule mult_right_mono) + have "1 \ Float m e" using `1 \ x` Float unfolding le_float_def by auto + from float_gt1_scale[OF this] + show "0 \ real (e + (bitlen m - 1))" by auto + qed + moreover + from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \ 0`, symmetric]] + have "0 \ Ifloat (?x - 1)" and "Ifloat (?x - 1) < 1" by auto + from ln_float_bounds(1)[OF this] + have "Ifloat ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \ ln (Ifloat ?x)" (is "?lb_horner \ _") by auto + ultimately have "?lb2 + ?lb_horner \ ln (Ifloat x)" + unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto + } + moreover + { + from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \ 0`, symmetric]] + have "0 \ Ifloat (?x - 1)" and "Ifloat (?x - 1) < 1" by auto + from ln_float_bounds(2)[OF this] + have "ln (Ifloat ?x) \ Ifloat ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))" (is "_ \ ?ub_horner") by auto + moreover + have "ln 2 * real (e + (bitlen m - 1)) \ Ifloat (ub_ln2 prec * ?s)" (is "_ \ ?ub2") + unfolding Ifloat_mult Ifloat_ge0_exp[OF order_refl] nat_0 realpow_0 mult_1_right + using ub_ln2[of prec] + proof (rule mult_right_mono) + have "1 \ Float m e" using `1 \ x` Float unfolding le_float_def by auto + from float_gt1_scale[OF this] + show "0 \ real (e + (bitlen m - 1))" by auto + qed + ultimately have "ln (Ifloat x) \ ?ub2 + ?ub_horner" + unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto + } + ultimately show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps + unfolding if_not_P[OF `\ x \ 0`] if_not_P[OF `\ x < 1`] if_not_P[OF False] Let_def + unfolding scale.simps[of m e, unfolded Float[symmetric]] mantissa.simps[of m e, unfolded Float[symmetric]] Ifloat_add by auto + qed +qed + +lemma ub_ln_lb_ln_bounds: assumes "0 < x" + shows "Ifloat (the (lb_ln prec x)) \ ln (Ifloat x) \ ln (Ifloat x) \ Ifloat (the (ub_ln prec x))" + (is "?lb \ ?ln \ ?ln \ ?ub") +proof (cases "x < 1") + case False hence "1 \ x" unfolding less_float_def le_float_def by auto + show ?thesis using ub_ln_lb_ln_bounds'[OF `1 \ x`] . +next + case True have "\ x \ 0" using `0 < x` unfolding less_float_def le_float_def by auto + + have "0 < Ifloat x" and "Ifloat x \ 0" using `0 < x` unfolding less_float_def by auto + hence A: "0 < 1 / Ifloat x" by auto + + { + let ?divl = "float_divl (max prec 1) 1 x" + have A': "1 \ ?divl" using float_divl_pos_less1_bound[OF `0 < x` `x < 1`] unfolding le_float_def less_float_def by auto + hence B: "0 < Ifloat ?divl" unfolding le_float_def by auto + + have "ln (Ifloat ?divl) \ ln (1 / Ifloat x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto + hence "ln (Ifloat x) \ - ln (Ifloat ?divl)" unfolding nonzero_inverse_eq_divide[OF `Ifloat x \ 0`, symmetric] ln_inverse[OF `0 < Ifloat x`] by auto + from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le] + have "?ln \ Ifloat (- the (lb_ln prec ?divl))" unfolding Ifloat_minus by (rule order_trans) + } moreover + { + let ?divr = "float_divr prec 1 x" + have A': "1 \ ?divr" using float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`] unfolding le_float_def less_float_def by auto + hence B: "0 < Ifloat ?divr" unfolding le_float_def by auto + + have "ln (1 / Ifloat x) \ ln (Ifloat ?divr)" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto + hence "- ln (Ifloat ?divr) \ ln (Ifloat x)" unfolding nonzero_inverse_eq_divide[OF `Ifloat x \ 0`, symmetric] ln_inverse[OF `0 < Ifloat x`] by auto + from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this + have "Ifloat (- the (ub_ln prec ?divr)) \ ?ln" unfolding Ifloat_minus by (rule order_trans) + } + ultimately show ?thesis unfolding lb_ln.simps[where x=x] ub_ln.simps[where x=x] + unfolding if_not_P[OF `\ x \ 0`] if_P[OF True] by auto +qed + +lemma lb_ln: assumes "Some y = lb_ln prec x" + shows "Ifloat y \ ln (Ifloat x)" and "0 < Ifloat x" +proof - + have "0 < x" + proof (rule ccontr) + assume "\ 0 < x" hence "x \ 0" unfolding le_float_def less_float_def by auto + thus False using assms by auto + qed + thus "0 < Ifloat x" unfolding less_float_def by auto + have "Ifloat (the (lb_ln prec x)) \ ln (Ifloat x)" using ub_ln_lb_ln_bounds[OF `0 < x`] .. + thus "Ifloat y \ ln (Ifloat x)" unfolding assms[symmetric] by auto +qed + +lemma ub_ln: assumes "Some y = ub_ln prec x" + shows "ln (Ifloat x) \ Ifloat y" and "0 < Ifloat x" +proof - + have "0 < x" + proof (rule ccontr) + assume "\ 0 < x" hence "x \ 0" unfolding le_float_def less_float_def by auto + thus False using assms by auto + qed + thus "0 < Ifloat x" unfolding less_float_def by auto + have "ln (Ifloat x) \ Ifloat (the (ub_ln prec x))" using ub_ln_lb_ln_bounds[OF `0 < x`] .. + thus "ln (Ifloat x) \ Ifloat y" unfolding assms[symmetric] by auto +qed + +lemma bnds_ln: "\ x lx ux. (Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \ x \ {Ifloat lx .. Ifloat ux} \ Ifloat l \ ln x \ ln x \ Ifloat u" +proof (rule allI, rule allI, rule allI, rule impI) + fix x lx ux + assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \ x \ {Ifloat lx .. Ifloat ux}" + hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \ {Ifloat lx .. Ifloat ux}" by auto + + have "ln (Ifloat ux) \ Ifloat u" and "0 < Ifloat ux" using ub_ln u by auto + have "Ifloat l \ ln (Ifloat lx)" and "0 < Ifloat lx" and "0 < x" using lb_ln[OF l] x by auto + + from ln_le_cancel_iff[OF `0 < Ifloat lx` `0 < x`] `Ifloat l \ ln (Ifloat lx)` + have "Ifloat l \ ln x" using x unfolding atLeastAtMost_iff by auto + moreover + from ln_le_cancel_iff[OF `0 < x` `0 < Ifloat ux`] `ln (Ifloat ux) \ Ifloat u` + have "ln x \ Ifloat u" using x unfolding atLeastAtMost_iff by auto + ultimately show "Ifloat l \ ln x \ ln x \ Ifloat u" .. +qed + + +section "Implement floatarith" + +subsection "Define syntax and semantics" + +datatype floatarith + = Add floatarith floatarith + | Minus floatarith + | Mult floatarith floatarith + | Inverse floatarith + | Sin floatarith + | Cos floatarith + | Arctan floatarith + | Abs floatarith + | Max floatarith floatarith + | Min floatarith floatarith + | Pi + | Sqrt floatarith + | Exp floatarith + | Ln floatarith + | Power floatarith nat + | Atom nat + | Num float + +fun Ifloatarith :: "floatarith \ real list \ real" +where +"Ifloatarith (Add a b) vs = (Ifloatarith a vs) + (Ifloatarith b vs)" | +"Ifloatarith (Minus a) vs = - (Ifloatarith a vs)" | +"Ifloatarith (Mult a b) vs = (Ifloatarith a vs) * (Ifloatarith b vs)" | +"Ifloatarith (Inverse a) vs = inverse (Ifloatarith a vs)" | +"Ifloatarith (Sin a) vs = sin (Ifloatarith a vs)" | +"Ifloatarith (Cos a) vs = cos (Ifloatarith a vs)" | +"Ifloatarith (Arctan a) vs = arctan (Ifloatarith a vs)" | +"Ifloatarith (Min a b) vs = min (Ifloatarith a vs) (Ifloatarith b vs)" | +"Ifloatarith (Max a b) vs = max (Ifloatarith a vs) (Ifloatarith b vs)" | +"Ifloatarith (Abs a) vs = abs (Ifloatarith a vs)" | +"Ifloatarith Pi vs = pi" | +"Ifloatarith (Sqrt a) vs = sqrt (Ifloatarith a vs)" | +"Ifloatarith (Exp a) vs = exp (Ifloatarith a vs)" | +"Ifloatarith (Ln a) vs = ln (Ifloatarith a vs)" | +"Ifloatarith (Power a n) vs = (Ifloatarith a vs)^n" | +"Ifloatarith (Num f) vs = Ifloat f" | +"Ifloatarith (Atom n) vs = vs ! n" + +subsection "Implement approximation function" + +fun lift_bin :: "(float * float) option \ (float * float) option \ (float \ float \ float \ float \ (float option * float option)) \ (float * float) option" where +"lift_bin (Some (l1, u1)) (Some (l2, u2)) f = (case (f l1 u1 l2 u2) of (Some l, Some u) \ Some (l, u) + | t \ None)" | +"lift_bin a b f = None" + +fun lift_bin' :: "(float * float) option \ (float * float) option \ (float \ float \ float \ float \ (float * float)) \ (float * float) option" where +"lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" | +"lift_bin' a b f = None" + +fun lift_un :: "(float * float) option \ (float \ float \ ((float option) * (float option))) \ (float * float) option" where +"lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \ Some (l, u) + | t \ None)" | +"lift_un b f = None" + +fun lift_un' :: "(float * float) option \ (float \ float \ (float * float)) \ (float * float) option" where +"lift_un' (Some (l1, u1)) f = Some (f l1 u1)" | +"lift_un' b f = None" + +fun bounded_by :: "real list \ (float * float) list \ bool " where +bounded_by_Cons: "bounded_by (v#vs) ((l, u)#bs) = ((Ifloat l \ v \ v \ Ifloat u) \ bounded_by vs bs)" | +bounded_by_Nil: "bounded_by [] [] = True" | +"bounded_by _ _ = False" + +lemma bounded_by: assumes "bounded_by vs bs" and "i < length bs" + shows "Ifloat (fst (bs ! i)) \ vs ! i \ vs ! i \ Ifloat (snd (bs ! i))" + using `bounded_by vs bs` and `i < length bs` +proof (induct arbitrary: i rule: bounded_by.induct) + fix v :: real and vs :: "real list" and l u :: float and bs :: "(float * float) list" and i :: nat + assume hyp: "\i. \bounded_by vs bs; i < length bs\ \ Ifloat (fst (bs ! i)) \ vs ! i \ vs ! i \ Ifloat (snd (bs ! i))" + assume bounded: "bounded_by (v # vs) ((l, u) # bs)" and length: "i < length ((l, u) # bs)" + show "Ifloat (fst (((l, u) # bs) ! i)) \ (v # vs) ! i \ (v # vs) ! i \ Ifloat (snd (((l, u) # bs) ! i))" + proof (cases i) + case 0 + show ?thesis using bounded unfolding 0 nth_Cons_0 fst_conv snd_conv bounded_by.simps .. + next + case (Suc i) with length have "i < length bs" by auto + show ?thesis unfolding Suc nth_Cons_Suc bounded_by.simps + using hyp[OF bounded[unfolded bounded_by.simps, THEN conjunct2] `i < length bs`] . + qed +qed auto + +fun approx approx' :: "nat \ floatarith \ (float * float) list \ (float * float) option" where +"approx' prec a bs = (case (approx prec a bs) of Some (l, u) \ Some (round_down prec l, round_up prec u) | None \ None)" | +"approx prec (Add a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (\ l1 u1 l2 u2. (l1 + l2, u1 + u2))" | +"approx prec (Minus a) bs = lift_un' (approx' prec a bs) (\ l u. (-u, -l))" | +"approx prec (Mult a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) + (\ a1 a2 b1 b2. (float_nprt a1 * float_pprt b2 + float_nprt a2 * float_nprt b2 + float_pprt a1 * float_pprt b1 + float_pprt a2 * float_nprt b1, + float_pprt a2 * float_pprt b2 + float_pprt a1 * float_nprt b2 + float_nprt a2 * float_pprt b1 + float_nprt a1 * float_nprt b1))" | +"approx prec (Inverse a) bs = lift_un (approx' prec a bs) (\ l u. if (0 < l \ u < 0) then (Some (float_divl prec 1 u), Some (float_divr prec 1 l)) else (None, None))" | +"approx prec (Sin a) bs = lift_un' (approx' prec a bs) (bnds_sin prec)" | +"approx prec (Cos a) bs = lift_un' (approx' prec a bs) (bnds_cos prec)" | +"approx prec Pi bs = Some (lb_pi prec, ub_pi prec)" | +"approx prec (Min a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (\ l1 u1 l2 u2. (min l1 l2, min u1 u2))" | +"approx prec (Max a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (\ l1 u1 l2 u2. (max l1 l2, max u1 u2))" | +"approx prec (Abs a) bs = lift_un' (approx' prec a bs) (\l u. (if l < 0 \ 0 < u then 0 else min \l\ \u\, max \l\ \u\))" | +"approx prec (Arctan a) bs = lift_un' (approx' prec a bs) (\ l u. (lb_arctan prec l, ub_arctan prec u))" | +"approx prec (Sqrt a) bs = lift_un (approx' prec a bs) (\ l u. (lb_sqrt prec l, ub_sqrt prec u))" | +"approx prec (Exp a) bs = lift_un' (approx' prec a bs) (\ l u. (lb_exp prec l, ub_exp prec u))" | +"approx prec (Ln a) bs = lift_un (approx' prec a bs) (\ l u. (lb_ln prec l, ub_ln prec u))" | +"approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds n)" | +"approx prec (Num f) bs = Some (f, f)" | +"approx prec (Atom i) bs = (if i < length bs then Some (bs ! i) else None)" + +lemma lift_bin'_ex: + assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f" + shows "\ l1 u1 l2 u2. Some (l1, u1) = a \ Some (l2, u2) = b" +proof (cases a) + case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps .. + thus ?thesis using lift_bin'_Some by auto +next + case (Some a') + show ?thesis + proof (cases b) + case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps .. + thus ?thesis using lift_bin'_Some by auto + next + case (Some b') + obtain la ua where a': "a' = (la, ua)" by (cases a', auto) + obtain lb ub where b': "b' = (lb, ub)" by (cases b', auto) + thus ?thesis unfolding `a = Some a'` `b = Some b'` a' b' by auto + qed +qed + +lemma lift_bin'_f: + assumes lift_bin'_Some: "Some (l, u) = lift_bin' (g a) (g b) f" + and Pa: "\l u. Some (l, u) = g a \ P l u a" and Pb: "\l u. Some (l, u) = g b \ P l u b" + shows "\ l1 u1 l2 u2. P l1 u1 a \ P l2 u2 b \ l = fst (f l1 u1 l2 u2) \ u = snd (f l1 u1 l2 u2)" +proof - + obtain l1 u1 l2 u2 + where Sa: "Some (l1, u1) = g a" and Sb: "Some (l2, u2) = g b" using lift_bin'_ex[OF assms(1)] by auto + have lu: "(l, u) = f l1 u1 l2 u2" using lift_bin'_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin'.simps] by auto + have "l = fst (f l1 u1 l2 u2)" and "u = snd (f l1 u1 l2 u2)" unfolding lu[symmetric] by auto + thus ?thesis using Pa[OF Sa] Pb[OF Sb] by auto +qed + +lemma approx_approx': + assumes Pa: "\l u. Some (l, u) = approx prec a vs \ Ifloat l \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u" + and approx': "Some (l, u) = approx' prec a vs" + shows "Ifloat l \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u" +proof - + obtain l' u' where S: "Some (l', u') = approx prec a vs" + using approx' unfolding approx'.simps by (cases "approx prec a vs", auto) + have l': "l = round_down prec l'" and u': "u = round_up prec u'" + using approx' unfolding approx'.simps S[symmetric] by auto + show ?thesis unfolding l' u' + using order_trans[OF Pa[OF S, THEN conjunct2] round_up[of u']] + using order_trans[OF round_down[of _ l'] Pa[OF S, THEN conjunct1]] by auto +qed + +lemma lift_bin': + assumes lift_bin'_Some: "Some (l, u) = lift_bin' (approx' prec a bs) (approx' prec b bs) f" + and Pa: "\l u. Some (l, u) = approx prec a bs \ Ifloat l \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u" (is "\l u. _ = ?g a \ ?P l u a") + and Pb: "\l u. Some (l, u) = approx prec b bs \ Ifloat l \ Ifloatarith b xs \ Ifloatarith b xs \ Ifloat u" + shows "\ l1 u1 l2 u2. (Ifloat l1 \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u1) \ + (Ifloat l2 \ Ifloatarith b xs \ Ifloatarith b xs \ Ifloat u2) \ + l = fst (f l1 u1 l2 u2) \ u = snd (f l1 u1 l2 u2)" +proof - + { fix l u assume "Some (l, u) = approx' prec a bs" + with approx_approx'[of prec a bs, OF _ this] Pa + have "Ifloat l \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u" by auto } note Pa = this + { fix l u assume "Some (l, u) = approx' prec b bs" + with approx_approx'[of prec b bs, OF _ this] Pb + have "Ifloat l \ Ifloatarith b xs \ Ifloatarith b xs \ Ifloat u" by auto } note Pb = this + + from lift_bin'_f[where g="\a. approx' prec a bs" and P = ?P, OF lift_bin'_Some, OF Pa Pb] + show ?thesis by auto +qed + +lemma lift_un'_ex: + assumes lift_un'_Some: "Some (l, u) = lift_un' a f" + shows "\ l u. Some (l, u) = a" +proof (cases a) + case None hence "None = lift_un' a f" unfolding None lift_un'.simps .. + thus ?thesis using lift_un'_Some by auto +next + case (Some a') + obtain la ua where a': "a' = (la, ua)" by (cases a', auto) + thus ?thesis unfolding `a = Some a'` a' by auto +qed + +lemma lift_un'_f: + assumes lift_un'_Some: "Some (l, u) = lift_un' (g a) f" + and Pa: "\l u. Some (l, u) = g a \ P l u a" + shows "\ l1 u1. P l1 u1 a \ l = fst (f l1 u1) \ u = snd (f l1 u1)" +proof - + obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un'_ex[OF assms(1)] by auto + have lu: "(l, u) = f l1 u1" using lift_un'_Some[unfolded Sa[symmetric] lift_un'.simps] by auto + have "l = fst (f l1 u1)" and "u = snd (f l1 u1)" unfolding lu[symmetric] by auto + thus ?thesis using Pa[OF Sa] by auto +qed + +lemma lift_un': + assumes lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f" + and Pa: "\l u. Some (l, u) = approx prec a bs \ Ifloat l \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u" (is "\l u. _ = ?g a \ ?P l u a") + shows "\ l1 u1. (Ifloat l1 \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u1) \ + l = fst (f l1 u1) \ u = snd (f l1 u1)" +proof - + { fix l u assume "Some (l, u) = approx' prec a bs" + with approx_approx'[of prec a bs, OF _ this] Pa + have "Ifloat l \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u" by auto } note Pa = this + from lift_un'_f[where g="\a. approx' prec a bs" and P = ?P, OF lift_un'_Some, OF Pa] + show ?thesis by auto +qed + +lemma lift_un'_bnds: + assumes bnds: "\ x lx ux. (l, u) = f lx ux \ x \ { Ifloat lx .. Ifloat ux } \ Ifloat l \ f' x \ f' x \ Ifloat u" + and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f" + and Pa: "\l u. Some (l, u) = approx prec a bs \ Ifloat l \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u" + shows "Ifloat l \ f' (Ifloatarith a xs) \ f' (Ifloatarith a xs) \ Ifloat u" +proof - + from lift_un'[OF lift_un'_Some Pa] + obtain l1 u1 where "Ifloat l1 \ Ifloatarith a xs" and "Ifloatarith a xs \ Ifloat u1" and "l = fst (f l1 u1)" and "u = snd (f l1 u1)" by blast + hence "(l, u) = f l1 u1" and "Ifloatarith a xs \ {Ifloat l1 .. Ifloat u1}" by auto + thus ?thesis using bnds by auto +qed + +lemma lift_un_ex: + assumes lift_un_Some: "Some (l, u) = lift_un a f" + shows "\ l u. Some (l, u) = a" +proof (cases a) + case None hence "None = lift_un a f" unfolding None lift_un.simps .. + thus ?thesis using lift_un_Some by auto +next + case (Some a') + obtain la ua where a': "a' = (la, ua)" by (cases a', auto) + thus ?thesis unfolding `a = Some a'` a' by auto +qed + +lemma lift_un_f: + assumes lift_un_Some: "Some (l, u) = lift_un (g a) f" + and Pa: "\l u. Some (l, u) = g a \ P l u a" + shows "\ l1 u1. P l1 u1 a \ Some l = fst (f l1 u1) \ Some u = snd (f l1 u1)" +proof - + obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un_ex[OF assms(1)] by auto + have "fst (f l1 u1) \ None \ snd (f l1 u1) \ None" + proof (rule ccontr) + assume "\ (fst (f l1 u1) \ None \ snd (f l1 u1) \ None)" + hence or: "fst (f l1 u1) = None \ snd (f l1 u1) = None" by auto + hence "lift_un (g a) f = None" + proof (cases "fst (f l1 u1) = None") + case True + then obtain b where b: "f l1 u1 = (None, b)" by (cases "f l1 u1", auto) + thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto + next + case False hence "snd (f l1 u1) = None" using or by auto + with False obtain b where b: "f l1 u1 = (Some b, None)" by (cases "f l1 u1", auto) + thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto + qed + thus False using lift_un_Some by auto + qed + then obtain a' b' where f: "f l1 u1 = (Some a', Some b')" by (cases "f l1 u1", auto) + from lift_un_Some[unfolded Sa[symmetric] lift_un.simps f] + have "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" unfolding f by auto + thus ?thesis unfolding Sa[symmetric] lift_un.simps using Pa[OF Sa] by auto +qed + +lemma lift_un: + assumes lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f" + and Pa: "\l u. Some (l, u) = approx prec a bs \ Ifloat l \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u" (is "\l u. _ = ?g a \ ?P l u a") + shows "\ l1 u1. (Ifloat l1 \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u1) \ + Some l = fst (f l1 u1) \ Some u = snd (f l1 u1)" +proof - + { fix l u assume "Some (l, u) = approx' prec a bs" + with approx_approx'[of prec a bs, OF _ this] Pa + have "Ifloat l \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u" by auto } note Pa = this + from lift_un_f[where g="\a. approx' prec a bs" and P = ?P, OF lift_un_Some, OF Pa] + show ?thesis by auto +qed + +lemma lift_un_bnds: + assumes bnds: "\ x lx ux. (Some l, Some u) = f lx ux \ x \ { Ifloat lx .. Ifloat ux } \ Ifloat l \ f' x \ f' x \ Ifloat u" + and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f" + and Pa: "\l u. Some (l, u) = approx prec a bs \ Ifloat l \ Ifloatarith a xs \ Ifloatarith a xs \ Ifloat u" + shows "Ifloat l \ f' (Ifloatarith a xs) \ f' (Ifloatarith a xs) \ Ifloat u" +proof - + from lift_un[OF lift_un_Some Pa] + obtain l1 u1 where "Ifloat l1 \ Ifloatarith a xs" and "Ifloatarith a xs \ Ifloat u1" and "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" by blast + hence "(Some l, Some u) = f l1 u1" and "Ifloatarith a xs \ {Ifloat l1 .. Ifloat u1}" by auto + thus ?thesis using bnds by auto +qed + +lemma approx: + assumes "bounded_by xs vs" + and "Some (l, u) = approx prec arith vs" (is "_ = ?g arith") + shows "Ifloat l \ Ifloatarith arith xs \ Ifloatarith arith xs \ Ifloat u" (is "?P l u arith") + using `Some (l, u) = approx prec arith vs` +proof (induct arith arbitrary: l u x) + case (Add a b) + from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps + obtain l1 u1 l2 u2 where "l = l1 + l2" and "u = u1 + u2" + "Ifloat l1 \ Ifloatarith a xs" and "Ifloatarith a xs \ Ifloat u1" + "Ifloat l2 \ Ifloatarith b xs" and "Ifloatarith b xs \ Ifloat u2" unfolding fst_conv snd_conv by blast + thus ?case unfolding Ifloatarith.simps by auto +next + case (Minus a) + from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps + obtain l1 u1 where "l = -u1" and "u = -l1" + "Ifloat l1 \ Ifloatarith a xs" and "Ifloatarith a xs \ Ifloat u1" unfolding fst_conv snd_conv by blast + thus ?case unfolding Ifloatarith.simps using Ifloat_minus by auto +next + case (Mult a b) + from lift_bin'[OF Mult.prems[unfolded approx.simps]] Mult.hyps + obtain l1 u1 l2 u2 + where l: "l = float_nprt l1 * float_pprt u2 + float_nprt u1 * float_nprt u2 + float_pprt l1 * float_pprt l2 + float_pprt u1 * float_nprt l2" + and u: "u = float_pprt u1 * float_pprt u2 + float_pprt l1 * float_nprt u2 + float_nprt u1 * float_pprt l2 + float_nprt l1 * float_nprt l2" + and "Ifloat l1 \ Ifloatarith a xs" and "Ifloatarith a xs \ Ifloat u1" + and "Ifloat l2 \ Ifloatarith b xs" and "Ifloatarith b xs \ Ifloat u2" unfolding fst_conv snd_conv by blast + thus ?case unfolding Ifloatarith.simps l u Ifloat_add Ifloat_mult Ifloat_nprt Ifloat_pprt + using mult_le_prts mult_ge_prts by auto +next + case (Inverse a) + from lift_un[OF Inverse.prems[unfolded approx.simps], unfolded if_distrib[of fst] if_distrib[of snd] fst_conv snd_conv] Inverse.hyps + obtain l1 u1 where l': "Some l = (if 0 < l1 \ u1 < 0 then Some (float_divl prec 1 u1) else None)" + and u': "Some u = (if 0 < l1 \ u1 < 0 then Some (float_divr prec 1 l1) else None)" + and l1: "Ifloat l1 \ Ifloatarith a xs" and u1: "Ifloatarith a xs \ Ifloat u1" by blast + have either: "0 < l1 \ u1 < 0" proof (rule ccontr) assume P: "\ (0 < l1 \ u1 < 0)" show False using l' unfolding if_not_P[OF P] by auto qed + moreover have l1_le_u1: "Ifloat l1 \ Ifloat u1" using l1 u1 by auto + ultimately have "Ifloat l1 \ 0" and "Ifloat u1 \ 0" unfolding less_float_def by auto + + have inv: "inverse (Ifloat u1) \ inverse (Ifloatarith a xs) + \ inverse (Ifloatarith a xs) \ inverse (Ifloat l1)" + proof (cases "0 < l1") + case True hence "0 < Ifloat u1" and "0 < Ifloat l1" "0 < Ifloatarith a xs" + unfolding less_float_def using l1_le_u1 l1 by auto + show ?thesis + unfolding inverse_le_iff_le[OF `0 < Ifloat u1` `0 < Ifloatarith a xs`] + inverse_le_iff_le[OF `0 < Ifloatarith a xs` `0 < Ifloat l1`] + using l1 u1 by auto + next + case False hence "u1 < 0" using either by blast + hence "Ifloat u1 < 0" and "Ifloat l1 < 0" "Ifloatarith a xs < 0" + unfolding less_float_def using l1_le_u1 u1 by auto + show ?thesis + unfolding inverse_le_iff_le_neg[OF `Ifloat u1 < 0` `Ifloatarith a xs < 0`] + inverse_le_iff_le_neg[OF `Ifloatarith a xs < 0` `Ifloat l1 < 0`] + using l1 u1 by auto + qed + + from l' have "l = float_divl prec 1 u1" by (cases "0 < l1 \ u1 < 0", auto) + hence "Ifloat l \ inverse (Ifloat u1)" unfolding nonzero_inverse_eq_divide[OF `Ifloat u1 \ 0`] using float_divl[of prec 1 u1] by auto + also have "\ \ inverse (Ifloatarith a xs)" using inv by auto + finally have "Ifloat l \ inverse (Ifloatarith a xs)" . + moreover + from u' have "u = float_divr prec 1 l1" by (cases "0 < l1 \ u1 < 0", auto) + hence "inverse (Ifloat l1) \ Ifloat u" unfolding nonzero_inverse_eq_divide[OF `Ifloat l1 \ 0`] using float_divr[of 1 l1 prec] by auto + hence "inverse (Ifloatarith a xs) \ Ifloat u" by (rule order_trans[OF inv[THEN conjunct2]]) + ultimately show ?case unfolding Ifloatarith.simps using l1 u1 by auto +next + case (Abs x) + from lift_un'[OF Abs.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Abs.hyps + obtain l1 u1 where l': "l = (if l1 < 0 \ 0 < u1 then 0 else min \l1\ \u1\)" and u': "u = max \l1\ \u1\" + and l1: "Ifloat l1 \ Ifloatarith x xs" and u1: "Ifloatarith x xs \ Ifloat u1" by blast + thus ?case unfolding l' u' by (cases "l1 < 0 \ 0 < u1", auto simp add: Ifloat_min Ifloat_max Ifloat_abs less_float_def) +next + case (Min a b) + from lift_bin'[OF Min.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Min.hyps + obtain l1 u1 l2 u2 where l': "l = min l1 l2" and u': "u = min u1 u2" + and l1: "Ifloat l1 \ Ifloatarith a xs" and u1: "Ifloatarith a xs \ Ifloat u1" + and l1: "Ifloat l2 \ Ifloatarith b xs" and u1: "Ifloatarith b xs \ Ifloat u2" by blast + thus ?case unfolding l' u' by (auto simp add: Ifloat_min) +next + case (Max a b) + from lift_bin'[OF Max.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Max.hyps + obtain l1 u1 l2 u2 where l': "l = max l1 l2" and u': "u = max u1 u2" + and l1: "Ifloat l1 \ Ifloatarith a xs" and u1: "Ifloatarith a xs \ Ifloat u1" + and l1: "Ifloat l2 \ Ifloatarith b xs" and u1: "Ifloatarith b xs \ Ifloat u2" by blast + thus ?case unfolding l' u' by (auto simp add: Ifloat_max) +next case (Sin a) with lift_un'_bnds[OF bnds_sin] show ?case by auto +next case (Cos a) with lift_un'_bnds[OF bnds_cos] show ?case by auto +next case (Arctan a) with lift_un'_bnds[OF bnds_arctan] show ?case by auto +next case Pi with pi_boundaries show ?case by auto +next case (Sqrt a) with lift_un_bnds[OF bnds_sqrt] show ?case by auto +next case (Exp a) with lift_un'_bnds[OF bnds_exp] show ?case by auto +next case (Ln a) with lift_un_bnds[OF bnds_ln] show ?case by auto +next case (Power a n) with lift_un'_bnds[OF bnds_power] show ?case by auto +next case (Num f) thus ?case by auto +next + case (Atom n) + show ?case + proof (cases "n < length vs") + case True + with Atom have "vs ! n = (l, u)" by auto + thus ?thesis using bounded_by[OF assms(1) True] by auto + next + case False thus ?thesis using Atom by auto + qed +qed + +datatype ApproxEq = Less floatarith floatarith + | LessEqual floatarith floatarith + +fun uneq :: "ApproxEq \ real list \ bool" where +"uneq (Less a b) vs = (Ifloatarith a vs < Ifloatarith b vs)" | +"uneq (LessEqual a b) vs = (Ifloatarith a vs \ Ifloatarith b vs)" + +fun uneq' :: "nat \ ApproxEq \ (float * float) list \ bool" where +"uneq' prec (Less a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \ u < l' | _ \ False)" | +"uneq' prec (LessEqual a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \ u \ l' | _ \ False)" + +lemma uneq_approx: fixes m :: nat assumes "bounded_by vs bs" and "uneq' prec eq bs" + shows "uneq eq vs" +proof (cases eq) + case (Less a b) + show ?thesis + proof (cases "\ u l u' l'. approx prec a bs = Some (l, u) \ + approx prec b bs = Some (l', u')") + case True + then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)" + and b_approx: "approx prec b bs = Some (l', u') " by auto + with `uneq' prec eq bs` have "Ifloat u < Ifloat l'" + unfolding Less uneq'.simps less_float_def by auto + moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs` + have "Ifloatarith a vs \ Ifloat u" and "Ifloat l' \ Ifloatarith b vs" + using approx by auto + ultimately show ?thesis unfolding uneq.simps Less by auto + next + case False + hence "approx prec a bs = None \ approx prec b bs = None" + unfolding not_Some_eq[symmetric] by auto + hence "\ uneq' prec eq bs" unfolding Less uneq'.simps + by (cases "approx prec a bs = None", auto) + thus ?thesis using assms by auto + qed +next + case (LessEqual a b) + show ?thesis + proof (cases "\ u l u' l'. approx prec a bs = Some (l, u) \ + approx prec b bs = Some (l', u')") + case True + then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)" + and b_approx: "approx prec b bs = Some (l', u') " by auto + with `uneq' prec eq bs` have "Ifloat u \ Ifloat l'" + unfolding LessEqual uneq'.simps le_float_def by auto + moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs` + have "Ifloatarith a vs \ Ifloat u" and "Ifloat l' \ Ifloatarith b vs" + using approx by auto + ultimately show ?thesis unfolding uneq.simps LessEqual by auto + next + case False + hence "approx prec a bs = None \ approx prec b bs = None" + unfolding not_Some_eq[symmetric] by auto + hence "\ uneq' prec eq bs" unfolding LessEqual uneq'.simps + by (cases "approx prec a bs = None", auto) + thus ?thesis using assms by auto + qed +qed + +lemma Ifloatarith_divide: "Ifloatarith (Mult a (Inverse b)) vs = (Ifloatarith a vs) / (Ifloatarith b vs)" + unfolding real_divide_def Ifloatarith.simps .. + +lemma Ifloatarith_diff: "Ifloatarith (Add a (Minus b)) vs = (Ifloatarith a vs) - (Ifloatarith b vs)" + unfolding real_diff_def Ifloatarith.simps .. + +lemma Ifloatarith_tan: "Ifloatarith (Mult (Sin a) (Inverse (Cos a))) vs = tan (Ifloatarith a vs)" + unfolding tan_def Ifloatarith.simps real_divide_def .. + +lemma Ifloatarith_powr: "Ifloatarith (Exp (Mult b (Ln a))) vs = (Ifloatarith a vs) powr (Ifloatarith b vs)" + unfolding powr_def Ifloatarith.simps .. + +lemma Ifloatarith_log: "Ifloatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = log (Ifloatarith b vs) (Ifloatarith x vs)" + unfolding log_def Ifloatarith.simps real_divide_def .. + +lemma Ifloatarith_num: shows "Ifloatarith (Num (Float 0 0)) vs = 0" and "Ifloatarith (Num (Float 1 0)) vs = 1" and "Ifloatarith (Num (Float (number_of a) 0)) vs = number_of a" by auto + +subsection {* Implement proof method \texttt{approximation} *} + +lemma bounded_divl: assumes "Ifloat a / Ifloat b \ x" shows "Ifloat (float_divl p a b) \ x" by (rule order_trans[OF _ assms], rule float_divl) +lemma bounded_divr: assumes "x \ Ifloat a / Ifloat b" shows "x \ Ifloat (float_divr p a b)" by (rule order_trans[OF assms _], rule float_divr) +lemma bounded_num: shows "Ifloat (Float 5 1) = 10" and "Ifloat (Float 0 0) = 0" and "Ifloat (Float 1 0) = 1" and "Ifloat (Float (number_of n) 0) = (number_of n)" + and "0 * pow2 e = Ifloat (Float 0 e)" and "1 * pow2 e = Ifloat (Float 1 e)" and "number_of m * pow2 e = Ifloat (Float (number_of m) e)" + by (auto simp add: Ifloat.simps pow2_def) + +lemmas bounded_by_equations = bounded_by_Cons bounded_by_Nil float_power bounded_divl bounded_divr bounded_num HOL.simp_thms +lemmas uneq_equations = uneq.simps Ifloatarith.simps Ifloatarith_num Ifloatarith_divide Ifloatarith_diff Ifloatarith_tan Ifloatarith_powr Ifloatarith_log + +lemma "x div (0::int) = 0" by auto -- "What happens in the zero case for div" +lemma "x mod (0::int) = x" by auto -- "What happens in the zero case for mod" + +text {* The following equations must hold for div & mod + -- see "The Definition of Standard ML" by R. Milner, M. Tofte and R. Harper (pg. 79) *} +lemma "d * (i div d) + i mod d = (i::int)" by auto +lemma "0 < (d :: int) \ 0 \ i mod d \ i mod d < d" by auto +lemma "(d :: int) < 0 \ d < i mod d \ i mod d \ 0" by auto + +code_const "op div :: int \ int \ int" (SML "(fn i => fn d => if d = 0 then 0 else i div d)") +code_const "op mod :: int \ int \ int" (SML "(fn i => fn d => if d = 0 then i else i mod d)") +code_const "divmod :: int \ int \ (int * int)" (SML "(fn i => fn d => if d = 0 then (0, i) else IntInf.divMod (i, d))") + +ML {* + val uneq_equations = PureThy.get_thms @{theory} "uneq_equations"; + val bounded_by_equations = PureThy.get_thms @{theory} "bounded_by_equations"; + val bounded_by_simpset = (HOL_basic_ss addsimps bounded_by_equations) + + fun reify_uneq ctxt i = (fn st => + let + val to = HOLogic.dest_Trueprop (Logic.strip_imp_concl (List.nth (prems_of st, i - 1))) + in (Reflection.genreify_tac ctxt uneq_equations (SOME to) i) st + end) + + fun rule_uneq ctxt prec i thm = let + fun conv_num typ = HOLogic.dest_number #> snd #> HOLogic.mk_number typ + val to_natc = conv_num @{typ "nat"} #> Thm.cterm_of (ProofContext.theory_of ctxt) + val to_nat = conv_num @{typ "nat"} + val to_int = conv_num @{typ "int"} + + val prec' = to_nat prec + + fun bot_float (Const (@{const_name "times"}, _) $ mantisse $ (Const (@{const_name "pow2"}, _) $ exp)) + = @{term "Float"} $ to_int mantisse $ to_int exp + | bot_float (Const (@{const_name "divide"}, _) $ mantisse $ (Const (@{const_name "power"}, _) $ ten $ exp)) + = @{term "float_divl"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ (@{term "power_float (Float 5 1)"} $ to_nat exp) + | bot_float (Const (@{const_name "divide"}, _) $ mantisse $ ten) + = @{term "float_divl"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ @{term "Float 5 1"} + | bot_float mantisse = @{term "Float"} $ to_int mantisse $ @{term "0 :: int"} + + fun top_float (Const (@{const_name "times"}, _) $ mantisse $ (Const (@{const_name "pow2"}, _) $ exp)) + = @{term "Float"} $ to_int mantisse $ to_int exp + | top_float (Const (@{const_name "divide"}, _) $ mantisse $ (Const (@{const_name "power"}, _) $ ten $ exp)) + = @{term "float_divr"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ (@{term "power_float (Float 5 1)"} $ to_nat exp) + | top_float (Const (@{const_name "divide"}, _) $ mantisse $ ten) + = @{term "float_divr"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ @{term "Float 5 1"} + | top_float mantisse = @{term "Float"} $ to_int mantisse $ @{term "0 :: int"} + + val goal' : term = List.nth (prems_of thm, i - 1) + + fun lift_bnd (t as (Const (@{const_name "op &"}, _) $ + (Const (@{const_name "less_eq"}, _) $ + bottom $ (Free (name, _))) $ + (Const (@{const_name "less_eq"}, _) $ _ $ top))) + = ((name, HOLogic.mk_prod (bot_float bottom, top_float top)) + handle TERM (txt, ts) => raise TERM ("Premisse needs format ' <= & <= ', but found " ^ + (Syntax.string_of_term ctxt t), [t])) + | lift_bnd t = raise TERM ("Premisse needs format ' <= & <= ', but found " ^ + (Syntax.string_of_term ctxt t), [t]) + val bound_eqs = map (HOLogic.dest_Trueprop #> lift_bnd) (Logic.strip_imp_prems goal') + + fun lift_var (Free (varname, _)) = (case AList.lookup (op =) bound_eqs varname of + SOME bound => bound + | NONE => raise TERM ("No bound equations found for " ^ varname, [])) + | lift_var t = raise TERM ("Can not convert expression " ^ + (Syntax.string_of_term ctxt t), [t]) + + val _ $ vs = HOLogic.dest_Trueprop (Logic.strip_imp_concl goal') + + val bs = (HOLogic.dest_list #> map lift_var #> HOLogic.mk_list @{typ "float * float"}) vs + val map = [(@{cpat "?prec::nat"}, to_natc prec), + (@{cpat "?bs::(float * float) list"}, Thm.cterm_of (ProofContext.theory_of ctxt) bs)] + in rtac (Thm.instantiate ([], map) @{thm "uneq_approx"}) i thm end + + val eval_tac = CSUBGOAL (fn (ct, i) => rtac (eval_oracle ct) i) + + fun gen_eval_tac conv ctxt = CONVERSION (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt) + THEN' rtac TrueI + +*} + +method_setup approximation = {* fn src => + Method.syntax Args.term src #> + (fn (prec, ctxt) => let + in Method.SIMPLE_METHOD' (fn i => + (DETERM (reify_uneq ctxt i) + THEN rule_uneq ctxt prec i + THEN Simplifier.asm_full_simp_tac bounded_by_simpset i + THEN (TRY (filter_prems_tac (fn t => false) i)) + THEN (gen_eval_tac eval_oracle ctxt) i)) + end) +*} "real number approximation" + +end diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Decision_Procs/Cooper.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Decision_Procs/Cooper.thy Fri Feb 06 15:15:46 2009 +0100 @@ -0,0 +1,2174 @@ +(* Title: HOL/Reflection/Cooper.thy + Author: Amine Chaieb +*) + +theory Cooper +imports Complex_Main Efficient_Nat +uses ("cooper_tac.ML") +begin + +function iupt :: "int \ int \ int list" where + "iupt i j = (if j < i then [] else i # iupt (i+1) j)" +by pat_completeness auto +termination by (relation "measure (\ (i, j). nat (j-i+1))") auto + +lemma iupt_set: "set (iupt i j) = {i..j}" + by (induct rule: iupt.induct) (simp add: simp_from_to) + +(* Periodicity of dvd *) + + (*********************************************************************************) + (**** SHADOW SYNTAX AND SEMANTICS ****) + (*********************************************************************************) + +datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num + | Mul int num + + (* A size for num to make inductive proofs simpler*) +primrec num_size :: "num \ nat" where + "num_size (C c) = 1" +| "num_size (Bound n) = 1" +| "num_size (Neg a) = 1 + num_size a" +| "num_size (Add a b) = 1 + num_size a + num_size b" +| "num_size (Sub a b) = 3 + num_size a + num_size b" +| "num_size (CN n c a) = 4 + num_size a" +| "num_size (Mul c a) = 1 + num_size a" + +primrec Inum :: "int list \ num \ int" where + "Inum bs (C c) = c" +| "Inum bs (Bound n) = bs!n" +| "Inum bs (CN n c a) = c * (bs!n) + (Inum bs a)" +| "Inum bs (Neg a) = -(Inum bs a)" +| "Inum bs (Add a b) = Inum bs a + Inum bs b" +| "Inum bs (Sub a b) = Inum bs a - Inum bs b" +| "Inum bs (Mul c a) = c* Inum bs a" + +datatype fm = + T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| Dvd int num| NDvd int num| + NOT fm| And fm fm| Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm + | Closed nat | NClosed nat + + + (* A size for fm *) +consts fmsize :: "fm \ nat" +recdef fmsize "measure size" + "fmsize (NOT p) = 1 + fmsize p" + "fmsize (And p q) = 1 + fmsize p + fmsize q" + "fmsize (Or p q) = 1 + fmsize p + fmsize q" + "fmsize (Imp p q) = 3 + fmsize p + fmsize q" + "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)" + "fmsize (E p) = 1 + fmsize p" + "fmsize (A p) = 4+ fmsize p" + "fmsize (Dvd i t) = 2" + "fmsize (NDvd i t) = 2" + "fmsize p = 1" + (* several lemmas about fmsize *) +lemma fmsize_pos: "fmsize p > 0" +by (induct p rule: fmsize.induct) simp_all + + (* Semantics of formulae (fm) *) +consts Ifm ::"bool list \ int list \ fm \ bool" +primrec + "Ifm bbs bs T = True" + "Ifm bbs bs F = False" + "Ifm bbs bs (Lt a) = (Inum bs a < 0)" + "Ifm bbs bs (Gt a) = (Inum bs a > 0)" + "Ifm bbs bs (Le a) = (Inum bs a \ 0)" + "Ifm bbs bs (Ge a) = (Inum bs a \ 0)" + "Ifm bbs bs (Eq a) = (Inum bs a = 0)" + "Ifm bbs bs (NEq a) = (Inum bs a \ 0)" + "Ifm bbs bs (Dvd i b) = (i dvd Inum bs b)" + "Ifm bbs bs (NDvd i b) = (\(i dvd Inum bs b))" + "Ifm bbs bs (NOT p) = (\ (Ifm bbs bs p))" + "Ifm bbs bs (And p q) = (Ifm bbs bs p \ Ifm bbs bs q)" + "Ifm bbs bs (Or p q) = (Ifm bbs bs p \ Ifm bbs bs q)" + "Ifm bbs bs (Imp p q) = ((Ifm bbs bs p) \ (Ifm bbs bs q))" + "Ifm bbs bs (Iff p q) = (Ifm bbs bs p = Ifm bbs bs q)" + "Ifm bbs bs (E p) = (\ x. Ifm bbs (x#bs) p)" + "Ifm bbs bs (A p) = (\ x. Ifm bbs (x#bs) p)" + "Ifm bbs bs (Closed n) = bbs!n" + "Ifm bbs bs (NClosed n) = (\ bbs!n)" + +consts prep :: "fm \ fm" +recdef prep "measure fmsize" + "prep (E T) = T" + "prep (E F) = F" + "prep (E (Or p q)) = Or (prep (E p)) (prep (E q))" + "prep (E (Imp p q)) = Or (prep (E (NOT p))) (prep (E q))" + "prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" + "prep (E (NOT (And p q))) = Or (prep (E (NOT p))) (prep (E(NOT q)))" + "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))" + "prep (E (NOT (Iff p q))) = Or (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))" + "prep (E p) = E (prep p)" + "prep (A (And p q)) = And (prep (A p)) (prep (A q))" + "prep (A p) = prep (NOT (E (NOT p)))" + "prep (NOT (NOT p)) = prep p" + "prep (NOT (And p q)) = Or (prep (NOT p)) (prep (NOT q))" + "prep (NOT (A p)) = prep (E (NOT p))" + "prep (NOT (Or p q)) = And (prep (NOT p)) (prep (NOT q))" + "prep (NOT (Imp p q)) = And (prep p) (prep (NOT q))" + "prep (NOT (Iff p q)) = Or (prep (And p (NOT q))) (prep (And (NOT p) q))" + "prep (NOT p) = NOT (prep p)" + "prep (Or p q) = Or (prep p) (prep q)" + "prep (And p q) = And (prep p) (prep q)" + "prep (Imp p q) = prep (Or (NOT p) q)" + "prep (Iff p q) = Or (prep (And p q)) (prep (And (NOT p) (NOT q)))" + "prep p = p" +(hints simp add: fmsize_pos) +lemma prep: "Ifm bbs bs (prep p) = Ifm bbs bs p" +by (induct p arbitrary: bs rule: prep.induct, auto) + + + (* Quantifier freeness *) +consts qfree:: "fm \ bool" +recdef qfree "measure size" + "qfree (E p) = False" + "qfree (A p) = False" + "qfree (NOT p) = qfree p" + "qfree (And p q) = (qfree p \ qfree q)" + "qfree (Or p q) = (qfree p \ qfree q)" + "qfree (Imp p q) = (qfree p \ qfree q)" + "qfree (Iff p q) = (qfree p \ qfree q)" + "qfree p = True" + + (* Boundedness and substitution *) +consts + numbound0:: "num \ bool" (* a num is INDEPENDENT of Bound 0 *) + bound0:: "fm \ bool" (* A Formula is independent of Bound 0 *) + subst0:: "num \ fm \ fm" (* substitue a num into a formula for Bound 0 *) +primrec + "numbound0 (C c) = True" + "numbound0 (Bound n) = (n>0)" + "numbound0 (CN n i a) = (n >0 \ numbound0 a)" + "numbound0 (Neg a) = numbound0 a" + "numbound0 (Add a b) = (numbound0 a \ numbound0 b)" + "numbound0 (Sub a b) = (numbound0 a \ numbound0 b)" + "numbound0 (Mul i a) = numbound0 a" + +lemma numbound0_I: + assumes nb: "numbound0 a" + shows "Inum (b#bs) a = Inum (b'#bs) a" +using nb +by (induct a rule: numbound0.induct) (auto simp add: gr0_conv_Suc) + +primrec + "bound0 T = True" + "bound0 F = True" + "bound0 (Lt a) = numbound0 a" + "bound0 (Le a) = numbound0 a" + "bound0 (Gt a) = numbound0 a" + "bound0 (Ge a) = numbound0 a" + "bound0 (Eq a) = numbound0 a" + "bound0 (NEq a) = numbound0 a" + "bound0 (Dvd i a) = numbound0 a" + "bound0 (NDvd i a) = numbound0 a" + "bound0 (NOT p) = bound0 p" + "bound0 (And p q) = (bound0 p \ bound0 q)" + "bound0 (Or p q) = (bound0 p \ bound0 q)" + "bound0 (Imp p q) = ((bound0 p) \ (bound0 q))" + "bound0 (Iff p q) = (bound0 p \ bound0 q)" + "bound0 (E p) = False" + "bound0 (A p) = False" + "bound0 (Closed P) = True" + "bound0 (NClosed P) = True" +lemma bound0_I: + assumes bp: "bound0 p" + shows "Ifm bbs (b#bs) p = Ifm bbs (b'#bs) p" +using bp numbound0_I[where b="b" and bs="bs" and b'="b'"] +by (induct p rule: bound0.induct) (auto simp add: gr0_conv_Suc) + +fun numsubst0:: "num \ num \ num" where + "numsubst0 t (C c) = (C c)" +| "numsubst0 t (Bound n) = (if n=0 then t else Bound n)" +| "numsubst0 t (CN 0 i a) = Add (Mul i t) (numsubst0 t a)" +| "numsubst0 t (CN n i a) = CN n i (numsubst0 t a)" +| "numsubst0 t (Neg a) = Neg (numsubst0 t a)" +| "numsubst0 t (Add a b) = Add (numsubst0 t a) (numsubst0 t b)" +| "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)" +| "numsubst0 t (Mul i a) = Mul i (numsubst0 t a)" + +lemma numsubst0_I: + "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t" +by (induct t rule: numsubst0.induct,auto simp:nth_Cons') + +lemma numsubst0_I': + "numbound0 a \ Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b'#bs) a)#bs) t" +by (induct t rule: numsubst0.induct, auto simp: nth_Cons' numbound0_I[where b="b" and b'="b'"]) + +primrec + "subst0 t T = T" + "subst0 t F = F" + "subst0 t (Lt a) = Lt (numsubst0 t a)" + "subst0 t (Le a) = Le (numsubst0 t a)" + "subst0 t (Gt a) = Gt (numsubst0 t a)" + "subst0 t (Ge a) = Ge (numsubst0 t a)" + "subst0 t (Eq a) = Eq (numsubst0 t a)" + "subst0 t (NEq a) = NEq (numsubst0 t a)" + "subst0 t (Dvd i a) = Dvd i (numsubst0 t a)" + "subst0 t (NDvd i a) = NDvd i (numsubst0 t a)" + "subst0 t (NOT p) = NOT (subst0 t p)" + "subst0 t (And p q) = And (subst0 t p) (subst0 t q)" + "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)" + "subst0 t (Imp p q) = Imp (subst0 t p) (subst0 t q)" + "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)" + "subst0 t (Closed P) = (Closed P)" + "subst0 t (NClosed P) = (NClosed P)" + +lemma subst0_I: assumes qfp: "qfree p" + shows "Ifm bbs (b#bs) (subst0 a p) = Ifm bbs ((Inum (b#bs) a)#bs) p" + using qfp numsubst0_I[where b="b" and bs="bs" and a="a"] + by (induct p) (simp_all add: gr0_conv_Suc) + + +consts + decrnum:: "num \ num" + decr :: "fm \ fm" + +recdef decrnum "measure size" + "decrnum (Bound n) = Bound (n - 1)" + "decrnum (Neg a) = Neg (decrnum a)" + "decrnum (Add a b) = Add (decrnum a) (decrnum b)" + "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)" + "decrnum (Mul c a) = Mul c (decrnum a)" + "decrnum (CN n i a) = (CN (n - 1) i (decrnum a))" + "decrnum a = a" + +recdef decr "measure size" + "decr (Lt a) = Lt (decrnum a)" + "decr (Le a) = Le (decrnum a)" + "decr (Gt a) = Gt (decrnum a)" + "decr (Ge a) = Ge (decrnum a)" + "decr (Eq a) = Eq (decrnum a)" + "decr (NEq a) = NEq (decrnum a)" + "decr (Dvd i a) = Dvd i (decrnum a)" + "decr (NDvd i a) = NDvd i (decrnum a)" + "decr (NOT p) = NOT (decr p)" + "decr (And p q) = And (decr p) (decr q)" + "decr (Or p q) = Or (decr p) (decr q)" + "decr (Imp p q) = Imp (decr p) (decr q)" + "decr (Iff p q) = Iff (decr p) (decr q)" + "decr p = p" + +lemma decrnum: assumes nb: "numbound0 t" + shows "Inum (x#bs) t = Inum bs (decrnum t)" + using nb by (induct t rule: decrnum.induct, auto simp add: gr0_conv_Suc) + +lemma decr: assumes nb: "bound0 p" + shows "Ifm bbs (x#bs) p = Ifm bbs bs (decr p)" + using nb + by (induct p rule: decr.induct, simp_all add: gr0_conv_Suc decrnum) + +lemma decr_qf: "bound0 p \ qfree (decr p)" +by (induct p, simp_all) + +consts + isatom :: "fm \ bool" (* test for atomicity *) +recdef isatom "measure size" + "isatom T = True" + "isatom F = True" + "isatom (Lt a) = True" + "isatom (Le a) = True" + "isatom (Gt a) = True" + "isatom (Ge a) = True" + "isatom (Eq a) = True" + "isatom (NEq a) = True" + "isatom (Dvd i b) = True" + "isatom (NDvd i b) = True" + "isatom (Closed P) = True" + "isatom (NClosed P) = True" + "isatom p = False" + +lemma numsubst0_numbound0: assumes nb: "numbound0 t" + shows "numbound0 (numsubst0 t a)" +using nb apply (induct a rule: numbound0.induct) +apply simp_all +apply (case_tac n, simp_all) +done + +lemma subst0_bound0: assumes qf: "qfree p" and nb: "numbound0 t" + shows "bound0 (subst0 t p)" +using qf numsubst0_numbound0[OF nb] by (induct p rule: subst0.induct, auto) + +lemma bound0_qf: "bound0 p \ qfree p" +by (induct p, simp_all) + + +constdefs djf:: "('a \ fm) \ 'a \ fm \ fm" + "djf f p q \ (if q=T then T else if q=F then f p else + (let fp = f p in case fp of T \ T | F \ q | _ \ Or (f p) q))" +constdefs evaldjf:: "('a \ fm) \ 'a list \ fm" + "evaldjf f ps \ foldr (djf f) ps F" + +lemma djf_Or: "Ifm bbs bs (djf f p q) = Ifm bbs bs (Or (f p) q)" +by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) +(cases "f p", simp_all add: Let_def djf_def) + +lemma evaldjf_ex: "Ifm bbs bs (evaldjf f ps) = (\ p \ set ps. Ifm bbs bs (f p))" + by(induct ps, simp_all add: evaldjf_def djf_Or) + +lemma evaldjf_bound0: + assumes nb: "\ x\ set xs. bound0 (f x)" + shows "bound0 (evaldjf f xs)" + using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) + +lemma evaldjf_qf: + assumes nb: "\ x\ set xs. qfree (f x)" + shows "qfree (evaldjf f xs)" + using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) + +consts disjuncts :: "fm \ fm list" +recdef disjuncts "measure size" + "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)" + "disjuncts F = []" + "disjuncts p = [p]" + +lemma disjuncts: "(\ q\ set (disjuncts p). Ifm bbs bs q) = Ifm bbs bs p" +by(induct p rule: disjuncts.induct, auto) + +lemma disjuncts_nb: "bound0 p \ \ q\ set (disjuncts p). bound0 q" +proof- + assume nb: "bound0 p" + hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto) + thus ?thesis by (simp only: list_all_iff) +qed + +lemma disjuncts_qf: "qfree p \ \ q\ set (disjuncts p). qfree q" +proof- + assume qf: "qfree p" + hence "list_all qfree (disjuncts p)" + by (induct p rule: disjuncts.induct, auto) + thus ?thesis by (simp only: list_all_iff) +qed + +constdefs DJ :: "(fm \ fm) \ fm \ fm" + "DJ f p \ evaldjf f (disjuncts p)" + +lemma DJ: assumes fdj: "\ p q. f (Or p q) = Or (f p) (f q)" + and fF: "f F = F" + shows "Ifm bbs bs (DJ f p) = Ifm bbs bs (f p)" +proof- + have "Ifm bbs bs (DJ f p) = (\ q \ set (disjuncts p). Ifm bbs bs (f q))" + by (simp add: DJ_def evaldjf_ex) + also have "\ = Ifm bbs bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto) + finally show ?thesis . +qed + +lemma DJ_qf: assumes + fqf: "\ p. qfree p \ qfree (f p)" + shows "\p. qfree p \ qfree (DJ f p) " +proof(clarify) + fix p assume qf: "qfree p" + have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def) + from disjuncts_qf[OF qf] have "\ q\ set (disjuncts p). qfree q" . + with fqf have th':"\ q\ set (disjuncts p). qfree (f q)" by blast + + from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp +qed + +lemma DJ_qe: assumes qe: "\ bs p. qfree p \ qfree (qe p) \ (Ifm bbs bs (qe p) = Ifm bbs bs (E p))" + shows "\ bs p. qfree p \ qfree (DJ qe p) \ (Ifm bbs bs ((DJ qe p)) = Ifm bbs bs (E p))" +proof(clarify) + fix p::fm and bs + assume qf: "qfree p" + from qe have qth: "\ p. qfree p \ qfree (qe p)" by blast + from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto + have "Ifm bbs bs (DJ qe p) = (\ q\ set (disjuncts p). Ifm bbs bs (qe q))" + by (simp add: DJ_def evaldjf_ex) + also have "\ = (\ q \ set(disjuncts p). Ifm bbs bs (E q))" using qe disjuncts_qf[OF qf] by auto + also have "\ = Ifm bbs bs (E p)" by (induct p rule: disjuncts.induct, auto) + finally show "qfree (DJ qe p) \ Ifm bbs bs (DJ qe p) = Ifm bbs bs (E p)" using qfth by blast +qed + (* Simplification *) + + (* Algebraic simplifications for nums *) +consts bnds:: "num \ nat list" + lex_ns:: "nat list \ nat list \ bool" +recdef bnds "measure size" + "bnds (Bound n) = [n]" + "bnds (CN n c a) = n#(bnds a)" + "bnds (Neg a) = bnds a" + "bnds (Add a b) = (bnds a)@(bnds b)" + "bnds (Sub a b) = (bnds a)@(bnds b)" + "bnds (Mul i a) = bnds a" + "bnds a = []" +recdef lex_ns "measure (\ (xs,ys). length xs + length ys)" + "lex_ns ([], ms) = True" + "lex_ns (ns, []) = False" + "lex_ns (n#ns, m#ms) = (n ((n = m) \ lex_ns (ns,ms))) " +constdefs lex_bnd :: "num \ num \ bool" + "lex_bnd t s \ lex_ns (bnds t, bnds s)" + +consts + numadd:: "num \ num \ num" +recdef numadd "measure (\ (t,s). num_size t + num_size s)" + "numadd (CN n1 c1 r1 ,CN n2 c2 r2) = + (if n1=n2 then + (let c = c1 + c2 + in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2)))) + else if n1 \ n2 then CN n1 c1 (numadd (r1,Add (Mul c2 (Bound n2)) r2)) + else CN n2 c2 (numadd (Add (Mul c1 (Bound n1)) r1,r2)))" + "numadd (CN n1 c1 r1, t) = CN n1 c1 (numadd (r1, t))" + "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" + "numadd (C b1, C b2) = C (b1+b2)" + "numadd (a,b) = Add a b" + +(*function (sequential) + numadd :: "num \ num \ num" +where + "numadd (Add (Mul c1 (Bound n1)) r1) (Add (Mul c2 (Bound n2)) r2) = + (if n1 = n2 then (let c = c1 + c2 + in (if c = 0 then numadd r1 r2 else + Add (Mul c (Bound n1)) (numadd r1 r2))) + else if n1 \ n2 then + Add (Mul c1 (Bound n1)) (numadd r1 (Add (Mul c2 (Bound n2)) r2)) + else + Add (Mul c2 (Bound n2)) (numadd (Add (Mul c1 (Bound n1)) r1) r2))" + | "numadd (Add (Mul c1 (Bound n1)) r1) t = + Add (Mul c1 (Bound n1)) (numadd r1 t)" + | "numadd t (Add (Mul c2 (Bound n2)) r2) = + Add (Mul c2 (Bound n2)) (numadd t r2)" + | "numadd (C b1) (C b2) = C (b1 + b2)" + | "numadd a b = Add a b" +apply pat_completeness apply auto*) + +lemma numadd: "Inum bs (numadd (t,s)) = Inum bs (Add t s)" +apply (induct t s rule: numadd.induct, simp_all add: Let_def) +apply (case_tac "c1+c2 = 0",case_tac "n1 \ n2", simp_all) + apply (case_tac "n1 = n2") + apply(simp_all add: algebra_simps) +apply(simp add: left_distrib[symmetric]) +done + +lemma numadd_nb: "\ numbound0 t ; numbound0 s\ \ numbound0 (numadd (t,s))" +by (induct t s rule: numadd.induct, auto simp add: Let_def) + +fun + nummul :: "int \ num \ num" +where + "nummul i (C j) = C (i * j)" + | "nummul i (CN n c t) = CN n (c*i) (nummul i t)" + | "nummul i t = Mul i t" + +lemma nummul: "\ i. Inum bs (nummul i t) = Inum bs (Mul i t)" +by (induct t rule: nummul.induct, auto simp add: algebra_simps numadd) + +lemma nummul_nb: "\ i. numbound0 t \ numbound0 (nummul i t)" +by (induct t rule: nummul.induct, auto simp add: numadd_nb) + +constdefs numneg :: "num \ num" + "numneg t \ nummul (- 1) t" + +constdefs numsub :: "num \ num \ num" + "numsub s t \ (if s = t then C 0 else numadd (s, numneg t))" + +lemma numneg: "Inum bs (numneg t) = Inum bs (Neg t)" +using numneg_def nummul by simp + +lemma numneg_nb: "numbound0 t \ numbound0 (numneg t)" +using numneg_def nummul_nb by simp + +lemma numsub: "Inum bs (numsub a b) = Inum bs (Sub a b)" +using numneg numadd numsub_def by simp + +lemma numsub_nb: "\ numbound0 t ; numbound0 s\ \ numbound0 (numsub t s)" +using numsub_def numadd_nb numneg_nb by simp + +fun + simpnum :: "num \ num" +where + "simpnum (C j) = C j" + | "simpnum (Bound n) = CN n 1 (C 0)" + | "simpnum (Neg t) = numneg (simpnum t)" + | "simpnum (Add t s) = numadd (simpnum t, simpnum s)" + | "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)" + | "simpnum (Mul i t) = (if i = 0 then C 0 else nummul i (simpnum t))" + | "simpnum t = t" + +lemma simpnum_ci: "Inum bs (simpnum t) = Inum bs t" +by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul) + +lemma simpnum_numbound0: + "numbound0 t \ numbound0 (simpnum t)" +by (induct t rule: simpnum.induct, auto simp add: numadd_nb numsub_nb nummul_nb numneg_nb) + +fun + not :: "fm \ fm" +where + "not (NOT p) = p" + | "not T = F" + | "not F = T" + | "not p = NOT p" +lemma not: "Ifm bbs bs (not p) = Ifm bbs bs (NOT p)" +by (cases p) auto +lemma not_qf: "qfree p \ qfree (not p)" +by (cases p, auto) +lemma not_bn: "bound0 p \ bound0 (not p)" +by (cases p, auto) + +constdefs conj :: "fm \ fm \ fm" + "conj p q \ (if (p = F \ q=F) then F else if p=T then q else if q=T then p else And p q)" +lemma conj: "Ifm bbs bs (conj p q) = Ifm bbs bs (And p q)" +by (cases "p=F \ q=F",simp_all add: conj_def) (cases p,simp_all) + +lemma conj_qf: "\qfree p ; qfree q\ \ qfree (conj p q)" +using conj_def by auto +lemma conj_nb: "\bound0 p ; bound0 q\ \ bound0 (conj p q)" +using conj_def by auto + +constdefs disj :: "fm \ fm \ fm" + "disj p q \ (if (p = T \ q=T) then T else if p=F then q else if q=F then p else Or p q)" + +lemma disj: "Ifm bbs bs (disj p q) = Ifm bbs bs (Or p q)" +by (cases "p=T \ q=T",simp_all add: disj_def) (cases p,simp_all) +lemma disj_qf: "\qfree p ; qfree q\ \ qfree (disj p q)" +using disj_def by auto +lemma disj_nb: "\bound0 p ; bound0 q\ \ bound0 (disj p q)" +using disj_def by auto + +constdefs imp :: "fm \ fm \ fm" + "imp p q \ (if (p = F \ q=T) then T else if p=T then q else if q=F then not p else Imp p q)" +lemma imp: "Ifm bbs bs (imp p q) = Ifm bbs bs (Imp p q)" +by (cases "p=F \ q=T",simp_all add: imp_def,cases p) (simp_all add: not) +lemma imp_qf: "\qfree p ; qfree q\ \ qfree (imp p q)" +using imp_def by (cases "p=F \ q=T",simp_all add: imp_def,cases p) (simp_all add: not_qf) +lemma imp_nb: "\bound0 p ; bound0 q\ \ bound0 (imp p q)" +using imp_def by (cases "p=F \ q=T",simp_all add: imp_def,cases p) simp_all + +constdefs iff :: "fm \ fm \ fm" + "iff p q \ (if (p = q) then T else if (p = not q \ not p = q) then F else + if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else + Iff p q)" +lemma iff: "Ifm bbs bs (iff p q) = Ifm bbs bs (Iff p q)" + by (unfold iff_def,cases "p=q", simp,cases "p=not q", simp add:not) +(cases "not p= q", auto simp add:not) +lemma iff_qf: "\qfree p ; qfree q\ \ qfree (iff p q)" + by (unfold iff_def,cases "p=q", auto simp add: not_qf) +lemma iff_nb: "\bound0 p ; bound0 q\ \ bound0 (iff p q)" +using iff_def by (unfold iff_def,cases "p=q", auto simp add: not_bn) + +function (sequential) + simpfm :: "fm \ fm" +where + "simpfm (And p q) = conj (simpfm p) (simpfm q)" + | "simpfm (Or p q) = disj (simpfm p) (simpfm q)" + | "simpfm (Imp p q) = imp (simpfm p) (simpfm q)" + | "simpfm (Iff p q) = iff (simpfm p) (simpfm q)" + | "simpfm (NOT p) = not (simpfm p)" + | "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \ if (v < 0) then T else F + | _ \ Lt a')" + | "simpfm (Le a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ Le a')" + | "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \ if (v > 0) then T else F | _ \ Gt a')" + | "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ Ge a')" + | "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \ if (v = 0) then T else F | _ \ Eq a')" + | "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ NEq a')" + | "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a) + else if (abs i = 1) then T + else let a' = simpnum a in case a' of C v \ if (i dvd v) then T else F | _ \ Dvd i a')" + | "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) + else if (abs i = 1) then F + else let a' = simpnum a in case a' of C v \ if (\(i dvd v)) then T else F | _ \ NDvd i a')" + | "simpfm p = p" +by pat_completeness auto +termination by (relation "measure fmsize") auto + +lemma simpfm: "Ifm bbs bs (simpfm p) = Ifm bbs bs p" +proof(induct p rule: simpfm.induct) + case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\ (\ v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (7 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\ (\ v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (8 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\ (\ v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (9 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\ (\ v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (10 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\ (\ v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (11 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\ (\ v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (12 i a) let ?sa = "simpnum a" from simpnum_ci + have sa: "Inum bs ?sa = Inum bs a" by simp + have "i=0 \ abs i = 1 \ (i\0 \ (abs i \ 1))" by auto + {assume "i=0" hence ?case using "12.hyps" by (simp add: dvd_def Let_def)} + moreover + {assume i1: "abs i = 1" + from zdvd_1_left[where m = "Inum bs a"] uminus_dvd_conv[where d="1" and t="Inum bs a"] + have ?case using i1 apply (cases "i=0", simp_all add: Let_def) + by (cases "i > 0", simp_all)} + moreover + {assume inz: "i\0" and cond: "abs i \ 1" + {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond + by (cases "abs i = 1", auto) } + moreover {assume "\ (\ v. ?sa = C v)" + hence "simpfm (Dvd i a) = Dvd i ?sa" using inz cond + by (cases ?sa, auto simp add: Let_def) + hence ?case using sa by simp} + ultimately have ?case by blast} + ultimately show ?case by blast +next + case (13 i a) let ?sa = "simpnum a" from simpnum_ci + have sa: "Inum bs ?sa = Inum bs a" by simp + have "i=0 \ abs i = 1 \ (i\0 \ (abs i \ 1))" by auto + {assume "i=0" hence ?case using "13.hyps" by (simp add: dvd_def Let_def)} + moreover + {assume i1: "abs i = 1" + from zdvd_1_left[where m = "Inum bs a"] uminus_dvd_conv[where d="1" and t="Inum bs a"] + have ?case using i1 apply (cases "i=0", simp_all add: Let_def) + apply (cases "i > 0", simp_all) done} + moreover + {assume inz: "i\0" and cond: "abs i \ 1" + {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond + by (cases "abs i = 1", auto) } + moreover {assume "\ (\ v. ?sa = C v)" + hence "simpfm (NDvd i a) = NDvd i ?sa" using inz cond + by (cases ?sa, auto simp add: Let_def) + hence ?case using sa by simp} + ultimately have ?case by blast} + ultimately show ?case by blast +qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not) + +lemma simpfm_bound0: "bound0 p \ bound0 (simpfm p)" +proof(induct p rule: simpfm.induct) + case (6 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (7 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (8 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (9 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (10 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (11 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (12 i a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (13 i a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +qed(auto simp add: disj_def imp_def iff_def conj_def not_bn) + +lemma simpfm_qf: "qfree p \ qfree (simpfm p)" +by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def) + (case_tac "simpnum a",auto)+ + + (* Generic quantifier elimination *) +consts qelim :: "fm \ (fm \ fm) \ fm" +recdef qelim "measure fmsize" + "qelim (E p) = (\ qe. DJ qe (qelim p qe))" + "qelim (A p) = (\ qe. not (qe ((qelim (NOT p) qe))))" + "qelim (NOT p) = (\ qe. not (qelim p qe))" + "qelim (And p q) = (\ qe. conj (qelim p qe) (qelim q qe))" + "qelim (Or p q) = (\ qe. disj (qelim p qe) (qelim q qe))" + "qelim (Imp p q) = (\ qe. imp (qelim p qe) (qelim q qe))" + "qelim (Iff p q) = (\ qe. iff (qelim p qe) (qelim q qe))" + "qelim p = (\ y. simpfm p)" + +(*function (sequential) + qelim :: "(fm \ fm) \ fm \ fm" +where + "qelim qe (E p) = DJ qe (qelim qe p)" + | "qelim qe (A p) = not (qe ((qelim qe (NOT p))))" + | "qelim qe (NOT p) = not (qelim qe p)" + | "qelim qe (And p q) = conj (qelim qe p) (qelim qe q)" + | "qelim qe (Or p q) = disj (qelim qe p) (qelim qe q)" + | "qelim qe (Imp p q) = imp (qelim qe p) (qelim qe q)" + | "qelim qe (Iff p q) = iff (qelim qe p) (qelim qe q)" + | "qelim qe p = simpfm p" +by pat_completeness auto +termination by (relation "measure (fmsize o snd)") auto*) + +lemma qelim_ci: + assumes qe_inv: "\ bs p. qfree p \ qfree (qe p) \ (Ifm bbs bs (qe p) = Ifm bbs bs (E p))" + shows "\ bs. qfree (qelim p qe) \ (Ifm bbs bs (qelim p qe) = Ifm bbs bs p)" +using qe_inv DJ_qe[OF qe_inv] +by(induct p rule: qelim.induct) +(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf + simpfm simpfm_qf simp del: simpfm.simps) + (* Linearity for fm where Bound 0 ranges over \ *) + +fun + zsplit0 :: "num \ int \ num" (* splits the bounded from the unbounded part*) +where + "zsplit0 (C c) = (0,C c)" + | "zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))" + | "zsplit0 (CN n i a) = + (let (i',a') = zsplit0 a + in if n=0 then (i+i', a') else (i',CN n i a'))" + | "zsplit0 (Neg a) = (let (i',a') = zsplit0 a in (-i', Neg a'))" + | "zsplit0 (Add a b) = (let (ia,a') = zsplit0 a ; + (ib,b') = zsplit0 b + in (ia+ib, Add a' b'))" + | "zsplit0 (Sub a b) = (let (ia,a') = zsplit0 a ; + (ib,b') = zsplit0 b + in (ia-ib, Sub a' b'))" + | "zsplit0 (Mul i a) = (let (i',a') = zsplit0 a in (i*i', Mul i a'))" + +lemma zsplit0_I: + shows "\ n a. zsplit0 t = (n,a) \ (Inum ((x::int) #bs) (CN 0 n a) = Inum (x #bs) t) \ numbound0 a" + (is "\ n a. ?S t = (n,a) \ (?I x (CN 0 n a) = ?I x t) \ ?N a") +proof(induct t rule: zsplit0.induct) + case (1 c n a) thus ?case by auto +next + case (2 m n a) thus ?case by (cases "m=0") auto +next + case (3 m i a n a') + let ?j = "fst (zsplit0 a)" + let ?b = "snd (zsplit0 a)" + have abj: "zsplit0 a = (?j,?b)" by simp + {assume "m\0" + with prems(1)[OF abj] prems(2) have ?case by (auto simp add: Let_def split_def)} + moreover + {assume m0: "m =0" + from abj have th: "a'=?b \ n=i+?j" using prems + by (simp add: Let_def split_def) + from abj prems have th2: "(?I x (CN 0 ?j ?b) = ?I x a) \ ?N ?b" by blast + from th have "?I x (CN 0 n a') = ?I x (CN 0 (i+?j) ?b)" by simp + also from th2 have "\ = ?I x (CN 0 i (CN 0 ?j ?b))" by (simp add: left_distrib) + finally have "?I x (CN 0 n a') = ?I x (CN 0 i a)" using th2 by simp + with th2 th have ?case using m0 by blast} +ultimately show ?case by blast +next + case (4 t n a) + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Neg ?at \ n=-?nt" using prems + by (simp add: Let_def split_def) + from abj prems have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \ ?N ?at" by blast + from th2[simplified] th[simplified] show ?case by simp +next + case (5 s t n a) + let ?ns = "fst (zsplit0 s)" + let ?as = "snd (zsplit0 s)" + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abjs: "zsplit0 s = (?ns,?as)" by simp + moreover have abjt: "zsplit0 t = (?nt,?at)" by simp + ultimately have th: "a=Add ?as ?at \ n=?ns + ?nt" using prems + by (simp add: Let_def split_def) + from abjs[symmetric] have bluddy: "\ x y. (x,y) = zsplit0 s" by blast + from prems have "(\ x y. (x,y) = zsplit0 s) \ (\xa xb. zsplit0 t = (xa, xb) \ Inum (x # bs) (CN 0 xa xb) = Inum (x # bs) t \ numbound0 xb)" by auto + with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \ ?N ?at" by blast + from abjs prems have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \ ?N ?as" by blast + from th3[simplified] th2[simplified] th[simplified] show ?case + by (simp add: left_distrib) +next + case (6 s t n a) + let ?ns = "fst (zsplit0 s)" + let ?as = "snd (zsplit0 s)" + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abjs: "zsplit0 s = (?ns,?as)" by simp + moreover have abjt: "zsplit0 t = (?nt,?at)" by simp + ultimately have th: "a=Sub ?as ?at \ n=?ns - ?nt" using prems + by (simp add: Let_def split_def) + from abjs[symmetric] have bluddy: "\ x y. (x,y) = zsplit0 s" by blast + from prems have "(\ x y. (x,y) = zsplit0 s) \ (\xa xb. zsplit0 t = (xa, xb) \ Inum (x # bs) (CN 0 xa xb) = Inum (x # bs) t \ numbound0 xb)" by auto + with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \ ?N ?at" by blast + from abjs prems have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \ ?N ?as" by blast + from th3[simplified] th2[simplified] th[simplified] show ?case + by (simp add: left_diff_distrib) +next + case (7 i t n a) + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Mul i ?at \ n=i*?nt" using prems + by (simp add: Let_def split_def) + from abj prems have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \ ?N ?at" by blast + hence " ?I x (Mul i t) = i * ?I x (CN 0 ?nt ?at)" by simp + also have "\ = ?I x (CN 0 (i*?nt) (Mul i ?at))" by (simp add: right_distrib) + finally show ?case using th th2 by simp +qed + +consts + iszlfm :: "fm \ bool" (* Linearity test for fm *) +recdef iszlfm "measure size" + "iszlfm (And p q) = (iszlfm p \ iszlfm q)" + "iszlfm (Or p q) = (iszlfm p \ iszlfm q)" + "iszlfm (Eq (CN 0 c e)) = (c>0 \ numbound0 e)" + "iszlfm (NEq (CN 0 c e)) = (c>0 \ numbound0 e)" + "iszlfm (Lt (CN 0 c e)) = (c>0 \ numbound0 e)" + "iszlfm (Le (CN 0 c e)) = (c>0 \ numbound0 e)" + "iszlfm (Gt (CN 0 c e)) = (c>0 \ numbound0 e)" + "iszlfm (Ge (CN 0 c e)) = ( c>0 \ numbound0 e)" + "iszlfm (Dvd i (CN 0 c e)) = + (c>0 \ i>0 \ numbound0 e)" + "iszlfm (NDvd i (CN 0 c e))= + (c>0 \ i>0 \ numbound0 e)" + "iszlfm p = (isatom p \ (bound0 p))" + +lemma zlin_qfree: "iszlfm p \ qfree p" + by (induct p rule: iszlfm.induct) auto + +consts + zlfm :: "fm \ fm" (* Linearity transformation for fm *) +recdef zlfm "measure fmsize" + "zlfm (And p q) = And (zlfm p) (zlfm q)" + "zlfm (Or p q) = Or (zlfm p) (zlfm q)" + "zlfm (Imp p q) = Or (zlfm (NOT p)) (zlfm q)" + "zlfm (Iff p q) = Or (And (zlfm p) (zlfm q)) (And (zlfm (NOT p)) (zlfm (NOT q)))" + "zlfm (Lt a) = (let (c,r) = zsplit0 a in + if c=0 then Lt r else + if c>0 then (Lt (CN 0 c r)) else (Gt (CN 0 (- c) (Neg r))))" + "zlfm (Le a) = (let (c,r) = zsplit0 a in + if c=0 then Le r else + if c>0 then (Le (CN 0 c r)) else (Ge (CN 0 (- c) (Neg r))))" + "zlfm (Gt a) = (let (c,r) = zsplit0 a in + if c=0 then Gt r else + if c>0 then (Gt (CN 0 c r)) else (Lt (CN 0 (- c) (Neg r))))" + "zlfm (Ge a) = (let (c,r) = zsplit0 a in + if c=0 then Ge r else + if c>0 then (Ge (CN 0 c r)) else (Le (CN 0 (- c) (Neg r))))" + "zlfm (Eq a) = (let (c,r) = zsplit0 a in + if c=0 then Eq r else + if c>0 then (Eq (CN 0 c r)) else (Eq (CN 0 (- c) (Neg r))))" + "zlfm (NEq a) = (let (c,r) = zsplit0 a in + if c=0 then NEq r else + if c>0 then (NEq (CN 0 c r)) else (NEq (CN 0 (- c) (Neg r))))" + "zlfm (Dvd i a) = (if i=0 then zlfm (Eq a) + else (let (c,r) = zsplit0 a in + if c=0 then (Dvd (abs i) r) else + if c>0 then (Dvd (abs i) (CN 0 c r)) + else (Dvd (abs i) (CN 0 (- c) (Neg r)))))" + "zlfm (NDvd i a) = (if i=0 then zlfm (NEq a) + else (let (c,r) = zsplit0 a in + if c=0 then (NDvd (abs i) r) else + if c>0 then (NDvd (abs i) (CN 0 c r)) + else (NDvd (abs i) (CN 0 (- c) (Neg r)))))" + "zlfm (NOT (And p q)) = Or (zlfm (NOT p)) (zlfm (NOT q))" + "zlfm (NOT (Or p q)) = And (zlfm (NOT p)) (zlfm (NOT q))" + "zlfm (NOT (Imp p q)) = And (zlfm p) (zlfm (NOT q))" + "zlfm (NOT (Iff p q)) = Or (And(zlfm p) (zlfm(NOT q))) (And (zlfm(NOT p)) (zlfm q))" + "zlfm (NOT (NOT p)) = zlfm p" + "zlfm (NOT T) = F" + "zlfm (NOT F) = T" + "zlfm (NOT (Lt a)) = zlfm (Ge a)" + "zlfm (NOT (Le a)) = zlfm (Gt a)" + "zlfm (NOT (Gt a)) = zlfm (Le a)" + "zlfm (NOT (Ge a)) = zlfm (Lt a)" + "zlfm (NOT (Eq a)) = zlfm (NEq a)" + "zlfm (NOT (NEq a)) = zlfm (Eq a)" + "zlfm (NOT (Dvd i a)) = zlfm (NDvd i a)" + "zlfm (NOT (NDvd i a)) = zlfm (Dvd i a)" + "zlfm (NOT (Closed P)) = NClosed P" + "zlfm (NOT (NClosed P)) = Closed P" + "zlfm p = p" (hints simp add: fmsize_pos) + +lemma zlfm_I: + assumes qfp: "qfree p" + shows "(Ifm bbs (i#bs) (zlfm p) = Ifm bbs (i# bs) p) \ iszlfm (zlfm p)" + (is "(?I (?l p) = ?I p) \ ?L (?l p)") +using qfp +proof(induct p rule: zlfm.induct) + case (5 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (i#bs) t" + from prems Ia nb show ?case + apply (auto simp add: Let_def split_def algebra_simps) + apply (cases "?r",auto) + apply (case_tac nat, auto) + done +next + case (6 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (i#bs) t" + from prems Ia nb show ?case + apply (auto simp add: Let_def split_def algebra_simps) + apply (cases "?r",auto) + apply (case_tac nat, auto) + done +next + case (7 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (i#bs) t" + from prems Ia nb show ?case + apply (auto simp add: Let_def split_def algebra_simps) + apply (cases "?r",auto) + apply (case_tac nat, auto) + done +next + case (8 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (i#bs) t" + from prems Ia nb show ?case + apply (auto simp add: Let_def split_def algebra_simps) + apply (cases "?r",auto) + apply (case_tac nat, auto) + done +next + case (9 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (i#bs) t" + from prems Ia nb show ?case + apply (auto simp add: Let_def split_def algebra_simps) + apply (cases "?r",auto) + apply (case_tac nat, auto) + done +next + case (10 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (i#bs) t" + from prems Ia nb show ?case + apply (auto simp add: Let_def split_def algebra_simps) + apply (cases "?r",auto) + apply (case_tac nat, auto) + done +next + case (11 j a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (i#bs) t" + have "j=0 \ (j\0 \ ?c = 0) \ (j\0 \ ?c >0) \ (j\ 0 \ ?c<0)" by arith + moreover + {assume "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def) + hence ?case using prems by (simp del: zlfm.simps add: zdvd_0_left)} + moreover + {assume "?c=0" and "j\0" hence ?case + using zsplit0_I[OF spl, where x="i" and bs="bs"] + apply (auto simp add: Let_def split_def algebra_simps) + apply (cases "?r",auto) + apply (case_tac nat, auto) + done} + moreover + {assume cp: "?c > 0" and jnz: "j\0" hence l: "?L (?l (Dvd j a))" + by (simp add: nb Let_def split_def) + hence ?case using Ia cp jnz by (simp add: Let_def split_def)} + moreover + {assume cn: "?c < 0" and jnz: "j\0" hence l: "?L (?l (Dvd j a))" + by (simp add: nb Let_def split_def) + hence ?case using Ia cn jnz zdvd_zminus_iff[where m="abs j" and n="?c*i + ?N ?r" ] + by (simp add: Let_def split_def) } + ultimately show ?case by blast +next + case (12 j a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (i#bs) t" + have "j=0 \ (j\0 \ ?c = 0) \ (j\0 \ ?c >0) \ (j\ 0 \ ?c<0)" by arith + moreover + {assume "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def) + hence ?case using prems by (simp del: zlfm.simps add: zdvd_0_left)} + moreover + {assume "?c=0" and "j\0" hence ?case + using zsplit0_I[OF spl, where x="i" and bs="bs"] + apply (auto simp add: Let_def split_def algebra_simps) + apply (cases "?r",auto) + apply (case_tac nat, auto) + done} + moreover + {assume cp: "?c > 0" and jnz: "j\0" hence l: "?L (?l (Dvd j a))" + by (simp add: nb Let_def split_def) + hence ?case using Ia cp jnz by (simp add: Let_def split_def) } + moreover + {assume cn: "?c < 0" and jnz: "j\0" hence l: "?L (?l (Dvd j a))" + by (simp add: nb Let_def split_def) + hence ?case using Ia cn jnz zdvd_zminus_iff[where m="abs j" and n="?c*i + ?N ?r" ] + by (simp add: Let_def split_def)} + ultimately show ?case by blast +qed auto + +consts + plusinf:: "fm \ fm" (* Virtual substitution of +\*) + minusinf:: "fm \ fm" (* Virtual substitution of -\*) + \ :: "fm \ int" (* Compute lcm {d| N\<^isup>?\<^isup> Dvd c*x+t \ p}*) + d\ :: "fm \ int \ bool" (* checks if a given l divides all the ds above*) + +recdef minusinf "measure size" + "minusinf (And p q) = And (minusinf p) (minusinf q)" + "minusinf (Or p q) = Or (minusinf p) (minusinf q)" + "minusinf (Eq (CN 0 c e)) = F" + "minusinf (NEq (CN 0 c e)) = T" + "minusinf (Lt (CN 0 c e)) = T" + "minusinf (Le (CN 0 c e)) = T" + "minusinf (Gt (CN 0 c e)) = F" + "minusinf (Ge (CN 0 c e)) = F" + "minusinf p = p" + +lemma minusinf_qfree: "qfree p \ qfree (minusinf p)" + by (induct p rule: minusinf.induct, auto) + +recdef plusinf "measure size" + "plusinf (And p q) = And (plusinf p) (plusinf q)" + "plusinf (Or p q) = Or (plusinf p) (plusinf q)" + "plusinf (Eq (CN 0 c e)) = F" + "plusinf (NEq (CN 0 c e)) = T" + "plusinf (Lt (CN 0 c e)) = F" + "plusinf (Le (CN 0 c e)) = F" + "plusinf (Gt (CN 0 c e)) = T" + "plusinf (Ge (CN 0 c e)) = T" + "plusinf p = p" + +recdef \ "measure size" + "\ (And p q) = zlcm (\ p) (\ q)" + "\ (Or p q) = zlcm (\ p) (\ q)" + "\ (Dvd i (CN 0 c e)) = i" + "\ (NDvd i (CN 0 c e)) = i" + "\ p = 1" + +recdef d\ "measure size" + "d\ (And p q) = (\ d. d\ p d \ d\ q d)" + "d\ (Or p q) = (\ d. d\ p d \ d\ q d)" + "d\ (Dvd i (CN 0 c e)) = (\ d. i dvd d)" + "d\ (NDvd i (CN 0 c e)) = (\ d. i dvd d)" + "d\ p = (\ d. True)" + +lemma delta_mono: + assumes lin: "iszlfm p" + and d: "d dvd d'" + and ad: "d\ p d" + shows "d\ p d'" + using lin ad d +proof(induct p rule: iszlfm.induct) + case (9 i c e) thus ?case using d + by (simp add: zdvd_trans[where m="i" and n="d" and k="d'"]) +next + case (10 i c e) thus ?case using d + by (simp add: zdvd_trans[where m="i" and n="d" and k="d'"]) +qed simp_all + +lemma \ : assumes lin:"iszlfm p" + shows "d\ p (\ p) \ \ p >0" +using lin +proof (induct p rule: iszlfm.induct) + case (1 p q) + let ?d = "\ (And p q)" + from prems zlcm_pos have dp: "?d >0" by simp + have d1: "\ p dvd \ (And p q)" using prems by simp + hence th: "d\ p ?d" using delta_mono prems(3-4) by(simp del:dvd_zlcm_self1) + have "\ q dvd \ (And p q)" using prems by simp + hence th': "d\ q ?d" using delta_mono prems by(simp del:dvd_zlcm_self2) + from th th' dp show ?case by simp +next + case (2 p q) + let ?d = "\ (And p q)" + from prems zlcm_pos have dp: "?d >0" by simp + have "\ p dvd \ (And p q)" using prems by simp + hence th: "d\ p ?d" using delta_mono prems by(simp del:dvd_zlcm_self1) + have "\ q dvd \ (And p q)" using prems by simp + hence th': "d\ q ?d" using delta_mono prems by(simp del:dvd_zlcm_self2) + from th th' dp show ?case by simp +qed simp_all + + +consts + a\ :: "fm \ int \ fm" (* adjusts the coeffitients of a formula *) + d\ :: "fm \ int \ bool" (* tests if all coeffs c of c divide a given l*) + \ :: "fm \ int" (* computes the lcm of all coefficients of x*) + \ :: "fm \ num list" + \ :: "fm \ num list" + +recdef a\ "measure size" + "a\ (And p q) = (\ k. And (a\ p k) (a\ q k))" + "a\ (Or p q) = (\ k. Or (a\ p k) (a\ q k))" + "a\ (Eq (CN 0 c e)) = (\ k. Eq (CN 0 1 (Mul (k div c) e)))" + "a\ (NEq (CN 0 c e)) = (\ k. NEq (CN 0 1 (Mul (k div c) e)))" + "a\ (Lt (CN 0 c e)) = (\ k. Lt (CN 0 1 (Mul (k div c) e)))" + "a\ (Le (CN 0 c e)) = (\ k. Le (CN 0 1 (Mul (k div c) e)))" + "a\ (Gt (CN 0 c e)) = (\ k. Gt (CN 0 1 (Mul (k div c) e)))" + "a\ (Ge (CN 0 c e)) = (\ k. Ge (CN 0 1 (Mul (k div c) e)))" + "a\ (Dvd i (CN 0 c e)) =(\ k. Dvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))" + "a\ (NDvd i (CN 0 c e))=(\ k. NDvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))" + "a\ p = (\ k. p)" + +recdef d\ "measure size" + "d\ (And p q) = (\ k. (d\ p k) \ (d\ q k))" + "d\ (Or p q) = (\ k. (d\ p k) \ (d\ q k))" + "d\ (Eq (CN 0 c e)) = (\ k. c dvd k)" + "d\ (NEq (CN 0 c e)) = (\ k. c dvd k)" + "d\ (Lt (CN 0 c e)) = (\ k. c dvd k)" + "d\ (Le (CN 0 c e)) = (\ k. c dvd k)" + "d\ (Gt (CN 0 c e)) = (\ k. c dvd k)" + "d\ (Ge (CN 0 c e)) = (\ k. c dvd k)" + "d\ (Dvd i (CN 0 c e)) =(\ k. c dvd k)" + "d\ (NDvd i (CN 0 c e))=(\ k. c dvd k)" + "d\ p = (\ k. True)" + +recdef \ "measure size" + "\ (And p q) = zlcm (\ p) (\ q)" + "\ (Or p q) = zlcm (\ p) (\ q)" + "\ (Eq (CN 0 c e)) = c" + "\ (NEq (CN 0 c e)) = c" + "\ (Lt (CN 0 c e)) = c" + "\ (Le (CN 0 c e)) = c" + "\ (Gt (CN 0 c e)) = c" + "\ (Ge (CN 0 c e)) = c" + "\ (Dvd i (CN 0 c e)) = c" + "\ (NDvd i (CN 0 c e))= c" + "\ p = 1" + +recdef \ "measure size" + "\ (And p q) = (\ p @ \ q)" + "\ (Or p q) = (\ p @ \ q)" + "\ (Eq (CN 0 c e)) = [Sub (C -1) e]" + "\ (NEq (CN 0 c e)) = [Neg e]" + "\ (Lt (CN 0 c e)) = []" + "\ (Le (CN 0 c e)) = []" + "\ (Gt (CN 0 c e)) = [Neg e]" + "\ (Ge (CN 0 c e)) = [Sub (C -1) e]" + "\ p = []" + +recdef \ "measure size" + "\ (And p q) = (\ p @ \ q)" + "\ (Or p q) = (\ p @ \ q)" + "\ (Eq (CN 0 c e)) = [Add (C -1) e]" + "\ (NEq (CN 0 c e)) = [e]" + "\ (Lt (CN 0 c e)) = [e]" + "\ (Le (CN 0 c e)) = [Add (C -1) e]" + "\ (Gt (CN 0 c e)) = []" + "\ (Ge (CN 0 c e)) = []" + "\ p = []" +consts mirror :: "fm \ fm" +recdef mirror "measure size" + "mirror (And p q) = And (mirror p) (mirror q)" + "mirror (Or p q) = Or (mirror p) (mirror q)" + "mirror (Eq (CN 0 c e)) = Eq (CN 0 c (Neg e))" + "mirror (NEq (CN 0 c e)) = NEq (CN 0 c (Neg e))" + "mirror (Lt (CN 0 c e)) = Gt (CN 0 c (Neg e))" + "mirror (Le (CN 0 c e)) = Ge (CN 0 c (Neg e))" + "mirror (Gt (CN 0 c e)) = Lt (CN 0 c (Neg e))" + "mirror (Ge (CN 0 c e)) = Le (CN 0 c (Neg e))" + "mirror (Dvd i (CN 0 c e)) = Dvd i (CN 0 c (Neg e))" + "mirror (NDvd i (CN 0 c e)) = NDvd i (CN 0 c (Neg e))" + "mirror p = p" + (* Lemmas for the correctness of \\ *) +lemma dvd1_eq1: "x >0 \ (x::int) dvd 1 = (x = 1)" +by simp + +lemma minusinf_inf: + assumes linp: "iszlfm p" + and u: "d\ p 1" + shows "\ (z::int). \ x < z. Ifm bbs (x#bs) (minusinf p) = Ifm bbs (x#bs) p" + (is "?P p" is "\ (z::int). \ x < z. ?I x (?M p) = ?I x p") +using linp u +proof (induct p rule: minusinf.induct) + case (1 p q) thus ?case + by auto (rule_tac x="min z za" in exI,simp) +next + case (2 p q) thus ?case + by auto (rule_tac x="min z za" in exI,simp) +next + case (3 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+ + fix a + from 3 have "\ x<(- Inum (a#bs) e). c*x + Inum (x#bs) e \ 0" + proof(clarsimp) + fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e = 0" + with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"] + show "False" by simp + qed + thus ?case by auto +next + case (4 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+ + fix a + from 4 have "\ x<(- Inum (a#bs) e). c*x + Inum (x#bs) e \ 0" + proof(clarsimp) + fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e = 0" + with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"] + show "False" by simp + qed + thus ?case by auto +next + case (5 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+ + fix a + from 5 have "\ x<(- Inum (a#bs) e). c*x + Inum (x#bs) e < 0" + proof(clarsimp) + fix x assume "x < (- Inum (a#bs) e)" + with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"] + show "x + Inum (x#bs) e < 0" by simp + qed + thus ?case by auto +next + case (6 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+ + fix a + from 6 have "\ x<(- Inum (a#bs) e). c*x + Inum (x#bs) e \ 0" + proof(clarsimp) + fix x assume "x < (- Inum (a#bs) e)" + with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"] + show "x + Inum (x#bs) e \ 0" by simp + qed + thus ?case by auto +next + case (7 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+ + fix a + from 7 have "\ x<(- Inum (a#bs) e). \ (c*x + Inum (x#bs) e > 0)" + proof(clarsimp) + fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e > 0" + with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"] + show "False" by simp + qed + thus ?case by auto +next + case (8 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+ + fix a + from 8 have "\ x<(- Inum (a#bs) e). \ (c*x + Inum (x#bs) e \ 0)" + proof(clarsimp) + fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e \ 0" + with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"] + show "False" by simp + qed + thus ?case by auto +qed auto + +lemma minusinf_repeats: + assumes d: "d\ p d" and linp: "iszlfm p" + shows "Ifm bbs ((x - k*d)#bs) (minusinf p) = Ifm bbs (x #bs) (minusinf p)" +using linp d +proof(induct p rule: iszlfm.induct) + case (9 i c e) hence nbe: "numbound0 e" and id: "i dvd d" by simp+ + hence "\ k. d=i*k" by (simp add: dvd_def) + then obtain "di" where di_def: "d=i*di" by blast + show ?case + proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="x - k * d" and b'="x"] right_diff_distrib, rule iffI) + assume + "i dvd c * x - c*(k*d) + Inum (x # bs) e" + (is "?ri dvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri dvd ?rt") + hence "\ (l::int). ?rt = i * l" by (simp add: dvd_def) + hence "\ (l::int). c*x+ ?I x e = i*l+c*(k * i*di)" + by (simp add: algebra_simps di_def) + hence "\ (l::int). c*x+ ?I x e = i*(l + c*k*di)" + by (simp add: algebra_simps) + hence "\ (l::int). c*x+ ?I x e = i*l" by blast + thus "i dvd c*x + Inum (x # bs) e" by (simp add: dvd_def) + next + assume + "i dvd c*x + Inum (x # bs) e" (is "?ri dvd ?rc*?rx+?e") + hence "\ (l::int). c*x+?e = i*l" by (simp add: dvd_def) + hence "\ (l::int). c*x - c*(k*d) +?e = i*l - c*(k*d)" by simp + hence "\ (l::int). c*x - c*(k*d) +?e = i*l - c*(k*i*di)" by (simp add: di_def) + hence "\ (l::int). c*x - c*(k*d) +?e = i*((l - c*k*di))" by (simp add: algebra_simps) + hence "\ (l::int). c*x - c * (k*d) +?e = i*l" + by blast + thus "i dvd c*x - c*(k*d) + Inum (x # bs) e" by (simp add: dvd_def) + qed +next + case (10 i c e) hence nbe: "numbound0 e" and id: "i dvd d" by simp+ + hence "\ k. d=i*k" by (simp add: dvd_def) + then obtain "di" where di_def: "d=i*di" by blast + show ?case + proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="x - k * d" and b'="x"] right_diff_distrib, rule iffI) + assume + "i dvd c * x - c*(k*d) + Inum (x # bs) e" + (is "?ri dvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri dvd ?rt") + hence "\ (l::int). ?rt = i * l" by (simp add: dvd_def) + hence "\ (l::int). c*x+ ?I x e = i*l+c*(k * i*di)" + by (simp add: algebra_simps di_def) + hence "\ (l::int). c*x+ ?I x e = i*(l + c*k*di)" + by (simp add: algebra_simps) + hence "\ (l::int). c*x+ ?I x e = i*l" by blast + thus "i dvd c*x + Inum (x # bs) e" by (simp add: dvd_def) + next + assume + "i dvd c*x + Inum (x # bs) e" (is "?ri dvd ?rc*?rx+?e") + hence "\ (l::int). c*x+?e = i*l" by (simp add: dvd_def) + hence "\ (l::int). c*x - c*(k*d) +?e = i*l - c*(k*d)" by simp + hence "\ (l::int). c*x - c*(k*d) +?e = i*l - c*(k*i*di)" by (simp add: di_def) + hence "\ (l::int). c*x - c*(k*d) +?e = i*((l - c*k*di))" by (simp add: algebra_simps) + hence "\ (l::int). c*x - c * (k*d) +?e = i*l" + by blast + thus "i dvd c*x - c*(k*d) + Inum (x # bs) e" by (simp add: dvd_def) + qed +qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="x - k*d" and b'="x"]) + +lemma mirror\\: + assumes lp: "iszlfm p" + shows "(Inum (i#bs)) ` set (\ p) = (Inum (i#bs)) ` set (\ (mirror p))" +using lp +by (induct p rule: mirror.induct, auto) + +lemma mirror: + assumes lp: "iszlfm p" + shows "Ifm bbs (x#bs) (mirror p) = Ifm bbs ((- x)#bs) p" +using lp +proof(induct p rule: iszlfm.induct) + case (9 j c e) hence nb: "numbound0 e" by simp + have "Ifm bbs (x#bs) (mirror (Dvd j (CN 0 c e))) = (j dvd c*x - Inum (x#bs) e)" (is "_ = (j dvd c*x - ?e)") by simp + also have "\ = (j dvd (- (c*x - ?e)))" + by (simp only: zdvd_zminus_iff) + also have "\ = (j dvd (c* (- x)) + ?e)" + apply (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] diff_def zadd_ac zminus_zadd_distrib) + by (simp add: algebra_simps) + also have "\ = Ifm bbs ((- x)#bs) (Dvd j (CN 0 c e))" + using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"] + by simp + finally show ?case . +next + case (10 j c e) hence nb: "numbound0 e" by simp + have "Ifm bbs (x#bs) (mirror (Dvd j (CN 0 c e))) = (j dvd c*x - Inum (x#bs) e)" (is "_ = (j dvd c*x - ?e)") by simp + also have "\ = (j dvd (- (c*x - ?e)))" + by (simp only: zdvd_zminus_iff) + also have "\ = (j dvd (c* (- x)) + ?e)" + apply (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] diff_def zadd_ac zminus_zadd_distrib) + by (simp add: algebra_simps) + also have "\ = Ifm bbs ((- x)#bs) (Dvd j (CN 0 c e))" + using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"] + by simp + finally show ?case by simp +qed (auto simp add: numbound0_I[where bs="bs" and b="x" and b'="- x"] gr0_conv_Suc) + +lemma mirror_l: "iszlfm p \ d\ p 1 + \ iszlfm (mirror p) \ d\ (mirror p) 1" +by (induct p rule: mirror.induct, auto) + +lemma mirror_\: "iszlfm p \ \ (mirror p) = \ p" +by (induct p rule: mirror.induct,auto) + +lemma \_numbound0: assumes lp: "iszlfm p" + shows "\ b\ set (\ p). numbound0 b" + using lp by (induct p rule: \.induct,auto) + +lemma d\_mono: + assumes linp: "iszlfm p" + and dr: "d\ p l" + and d: "l dvd l'" + shows "d\ p l'" +using dr linp zdvd_trans[where n="l" and k="l'", simplified d] +by (induct p rule: iszlfm.induct) simp_all + +lemma \_l: assumes lp: "iszlfm p" + shows "\ b\ set (\ p). numbound0 b" +using lp +by(induct p rule: \.induct, auto) + +lemma \: + assumes linp: "iszlfm p" + shows "\ p > 0 \ d\ p (\ p)" +using linp +proof(induct p rule: iszlfm.induct) + case (1 p q) + from prems have dl1: "\ p dvd zlcm (\ p) (\ q)" by simp + from prems have dl2: "\ q dvd zlcm (\ p) (\ q)" by simp + from prems d\_mono[where p = "p" and l="\ p" and l'="zlcm (\ p) (\ q)"] + d\_mono[where p = "q" and l="\ q" and l'="zlcm (\ p) (\ q)"] + dl1 dl2 show ?case by (auto simp add: zlcm_pos) +next + case (2 p q) + from prems have dl1: "\ p dvd zlcm (\ p) (\ q)" by simp + from prems have dl2: "\ q dvd zlcm (\ p) (\ q)" by simp + from prems d\_mono[where p = "p" and l="\ p" and l'="zlcm (\ p) (\ q)"] + d\_mono[where p = "q" and l="\ q" and l'="zlcm (\ p) (\ q)"] + dl1 dl2 show ?case by (auto simp add: zlcm_pos) +qed (auto simp add: zlcm_pos) + +lemma a\: assumes linp: "iszlfm p" and d: "d\ p l" and lp: "l > 0" + shows "iszlfm (a\ p l) \ d\ (a\ p l) 1 \ (Ifm bbs (l*x #bs) (a\ p l) = Ifm bbs (x#bs) p)" +using linp d +proof (induct p rule: iszlfm.induct) + case (5 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(l*x + (l div c) * Inum (x # bs) e < 0) = + ((c * (l div c)) * x + (l div c) * Inum (x # bs) e < 0)" + by simp + also have "\ = ((l div c) * (c*x + Inum (x # bs) e) < (l div c) * 0)" by (simp add: algebra_simps) + also have "\ = (c*x + Inum (x # bs) e < 0)" + using mult_less_0_iff [where a="(l div c)" and b="c*x + Inum (x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] be by simp +next + case (6 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(l*x + (l div c) * Inum (x# bs) e \ 0) = + ((c * (l div c)) * x + (l div c) * Inum (x # bs) e \ 0)" + by simp + also have "\ = ((l div c) * (c * x + Inum (x # bs) e) \ ((l div c)) * 0)" by (simp add: algebra_simps) + also have "\ = (c*x + Inum (x # bs) e \ 0)" + using mult_le_0_iff [where a="(l div c)" and b="c*x + Inum (x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] be by simp +next + case (7 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(l*x + (l div c)* Inum (x # bs) e > 0) = + ((c * (l div c)) * x + (l div c) * Inum (x # bs) e > 0)" + by simp + also have "\ = ((l div c) * (c * x + Inum (x # bs) e) > ((l div c)) * 0)" by (simp add: algebra_simps) + also have "\ = (c * x + Inum (x # bs) e > 0)" + using zero_less_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be by simp +next + case (8 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(l*x + (l div c)* Inum (x # bs) e \ 0) = + ((c*(l div c))*x + (l div c)* Inum (x # bs) e \ 0)" + by simp + also have "\ = ((l div c)*(c*x + Inum (x # bs) e) \ ((l div c)) * 0)" + by (simp add: algebra_simps) + also have "\ = (c*x + Inum (x # bs) e \ 0)" using ldcp + zero_le_mult_iff [where a="l div c" and b="c*x + Inum (x # bs) e"] by simp + finally show ?case using be numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] + by simp +next + case (3 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(l * x + (l div c) * Inum (x # bs) e = 0) = + ((c * (l div c)) * x + (l div c) * Inum (x # bs) e = 0)" + by simp + also have "\ = ((l div c) * (c * x + Inum (x # bs) e) = ((l div c)) * 0)" by (simp add: algebra_simps) + also have "\ = (c * x + Inum (x # bs) e = 0)" + using mult_eq_0_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be by simp +next + case (4 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(l * x + (l div c) * Inum (x # bs) e \ 0) = + ((c * (l div c)) * x + (l div c) * Inum (x # bs) e \ 0)" + by simp + also have "\ = ((l div c) * (c * x + Inum (x # bs) e) \ ((l div c)) * 0)" by (simp add: algebra_simps) + also have "\ = (c * x + Inum (x # bs) e \ 0)" + using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be by simp +next + case (9 j c e) hence cp: "c>0" and be: "numbound0 e" and jp: "j > 0" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(\ (k::int). l * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k) = (\ (k::int). (c * (l div c)) * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k)" by simp + also have "\ = (\ (k::int). (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c)*0)" by (simp add: algebra_simps) + also fix k have "\ = (\ (k::int). c * x + Inum (x # bs) e - j * k = 0)" + using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp by simp + also have "\ = (\ (k::int). c * x + Inum (x # bs) e = j * k)" by simp + finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be mult_strict_mono[OF ldcp jp ldcp ] by (simp add: dvd_def) +next + case (10 j c e) hence cp: "c>0" and be: "numbound0 e" and jp: "j > 0" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(\ (k::int). l * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k) = (\ (k::int). (c * (l div c)) * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k)" by simp + also have "\ = (\ (k::int). (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c)*0)" by (simp add: algebra_simps) + also fix k have "\ = (\ (k::int). c * x + Inum (x # bs) e - j * k = 0)" + using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp by simp + also have "\ = (\ (k::int). c * x + Inum (x # bs) e = j * k)" by simp + finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be mult_strict_mono[OF ldcp jp ldcp ] by (simp add: dvd_def) +qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="(l * x)" and b'="x"]) + +lemma a\_ex: assumes linp: "iszlfm p" and d: "d\ p l" and lp: "l>0" + shows "(\ x. l dvd x \ Ifm bbs (x #bs) (a\ p l)) = (\ (x::int). Ifm bbs (x#bs) p)" + (is "(\ x. l dvd x \ ?P x) = (\ x. ?P' x)") +proof- + have "(\ x. l dvd x \ ?P x) = (\ (x::int). ?P (l*x))" + using unity_coeff_ex[where l="l" and P="?P", simplified] by simp + also have "\ = (\ (x::int). ?P' x)" using a\[OF linp d lp] by simp + finally show ?thesis . +qed + +lemma \: + assumes lp: "iszlfm p" + and u: "d\ p 1" + and d: "d\ p d" + and dp: "d > 0" + and nob: "\(\(j::int) \ {1 .. d}. \ b\ (Inum (a#bs)) ` set(\ p). x = b + j)" + and p: "Ifm bbs (x#bs) p" (is "?P x") + shows "?P (x - d)" +using lp u d dp nob p +proof(induct p rule: iszlfm.induct) + case (5 c e) hence c1: "c=1" and bn:"numbound0 e" by simp+ + with dp p c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] prems + show ?case by simp +next + case (6 c e) hence c1: "c=1" and bn:"numbound0 e" by simp+ + with dp p c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] prems + show ?case by simp +next + case (7 c e) hence p: "Ifm bbs (x #bs) (Gt (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" by simp+ + let ?e = "Inum (x # bs) e" + {assume "(x-d) +?e > 0" hence ?case using c1 + numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] by simp} + moreover + {assume H: "\ (x-d) + ?e > 0" + let ?v="Neg e" + have vb: "?v \ set (\ (Gt (CN 0 c e)))" by simp + from prems(11)[simplified simp_thms Inum.simps \.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="x" and bs="bs"]] + have nob: "\ (\ j\ {1 ..d}. x = - ?e + j)" by auto + from H p have "x + ?e > 0 \ x + ?e \ d" by (simp add: c1) + hence "x + ?e \ 1 \ x + ?e \ d" by simp + hence "\ (j::int) \ {1 .. d}. j = x + ?e" by simp + hence "\ (j::int) \ {1 .. d}. x = (- ?e + j)" + by (simp add: algebra_simps) + with nob have ?case by auto} + ultimately show ?case by blast +next + case (8 c e) hence p: "Ifm bbs (x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" + by simp+ + let ?e = "Inum (x # bs) e" + {assume "(x-d) +?e \ 0" hence ?case using c1 + numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] + by simp} + moreover + {assume H: "\ (x-d) + ?e \ 0" + let ?v="Sub (C -1) e" + have vb: "?v \ set (\ (Ge (CN 0 c e)))" by simp + from prems(11)[simplified simp_thms Inum.simps \.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="x" and bs="bs"]] + have nob: "\ (\ j\ {1 ..d}. x = - ?e - 1 + j)" by auto + from H p have "x + ?e \ 0 \ x + ?e < d" by (simp add: c1) + hence "x + ?e +1 \ 1 \ x + ?e + 1 \ d" by simp + hence "\ (j::int) \ {1 .. d}. j = x + ?e + 1" by simp + hence "\ (j::int) \ {1 .. d}. x= - ?e - 1 + j" by (simp add: algebra_simps) + with nob have ?case by simp } + ultimately show ?case by blast +next + case (3 c e) hence p: "Ifm bbs (x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp+ + let ?e = "Inum (x # bs) e" + let ?v="(Sub (C -1) e)" + have vb: "?v \ set (\ (Eq (CN 0 c e)))" by simp + from p have "x= - ?e" by (simp add: c1) with prems(11) show ?case using dp + by simp (erule ballE[where x="1"], + simp_all add:algebra_simps numbound0_I[OF bn,where b="x"and b'="a"and bs="bs"]) +next + case (4 c e)hence p: "Ifm bbs (x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp+ + let ?e = "Inum (x # bs) e" + let ?v="Neg e" + have vb: "?v \ set (\ (NEq (CN 0 c e)))" by simp + {assume "x - d + Inum (((x -d)) # bs) e \ 0" + hence ?case by (simp add: c1)} + moreover + {assume H: "x - d + Inum (((x -d)) # bs) e = 0" + hence "x = - Inum (((x -d)) # bs) e + d" by simp + hence "x = - Inum (a # bs) e + d" + by (simp add: numbound0_I[OF bn,where b="x - d"and b'="a"and bs="bs"]) + with prems(11) have ?case using dp by simp} + ultimately show ?case by blast +next + case (9 j c e) hence p: "Ifm bbs (x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp+ + let ?e = "Inum (x # bs) e" + from prems have id: "j dvd d" by simp + from c1 have "?p x = (j dvd (x+ ?e))" by simp + also have "\ = (j dvd x - d + ?e)" + using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp + finally show ?case + using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p by simp +next + case (10 j c e) hence p: "Ifm bbs (x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp+ + let ?e = "Inum (x # bs) e" + from prems have id: "j dvd d" by simp + from c1 have "?p x = (\ j dvd (x+ ?e))" by simp + also have "\ = (\ j dvd x - d + ?e)" + using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp + finally show ?case using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p by simp +qed (auto simp add: numbound0_I[where bs="bs" and b="(x - d)" and b'="x"] gr0_conv_Suc) + +lemma \': + assumes lp: "iszlfm p" + and u: "d\ p 1" + and d: "d\ p d" + and dp: "d > 0" + shows "\ x. \(\(j::int) \ {1 .. d}. \ b\ set(\ p). Ifm bbs ((Inum (a#bs) b + j) #bs) p) \ Ifm bbs (x#bs) p \ Ifm bbs ((x - d)#bs) p" (is "\ x. ?b \ ?P x \ ?P (x - d)") +proof(clarify) + fix x + assume nb:"?b" and px: "?P x" + hence nb2: "\(\(j::int) \ {1 .. d}. \ b\ (Inum (a#bs)) ` set(\ p). x = b + j)" + by auto + from \[OF lp u d dp nb2 px] show "?P (x -d )" . +qed +lemma cpmi_eq: "0 < D \ (EX z::int. ALL x. x < z --> (P x = P1 x)) +==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D) +==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D)))) +==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))" +apply(rule iffI) +prefer 2 +apply(drule minusinfinity) +apply assumption+ +apply(fastsimp) +apply clarsimp +apply(subgoal_tac "!!k. 0<=k \ !x. P x \ P (x - k*D)") +apply(frule_tac x = x and z=z in decr_lemma) +apply(subgoal_tac "P1(x - (\x - z\ + 1) * D)") +prefer 2 +apply(subgoal_tac "0 <= (\x - z\ + 1)") +prefer 2 apply arith + apply fastsimp +apply(drule (1) periodic_finite_ex) +apply blast +apply(blast dest:decr_mult_lemma) +done + +theorem cp_thm: + assumes lp: "iszlfm p" + and u: "d\ p 1" + and d: "d\ p d" + and dp: "d > 0" + shows "(\ (x::int). Ifm bbs (x #bs) p) = (\ j\ {1.. d}. Ifm bbs (j #bs) (minusinf p) \ (\ b \ set (\ p). Ifm bbs ((Inum (i#bs) b + j) #bs) p))" + (is "(\ (x::int). ?P (x)) = (\ j\ ?D. ?M j \ (\ b\ ?B. ?P (?I b + j)))") +proof- + from minusinf_inf[OF lp u] + have th: "\(z::int). \xj\?D. \b\ ?B. ?P (?I b +j)) = (\ j \ ?D. \ b \ ?B'. ?P (b + j))" by auto + hence th2: "\ x. \ (\ j \ ?D. \ b \ ?B'. ?P ((b + j))) \ ?P (x) \ ?P ((x - d))" + using \'[OF lp u d dp, where a="i" and bbs = "bbs"] by blast + from minusinf_repeats[OF d lp] + have th3: "\ x k. ?M x = ?M (x-k*d)" by simp + from cpmi_eq[OF dp th th2 th3] BB' show ?thesis by blast +qed + + (* Implement the right hand sides of Cooper's theorem and Ferrante and Rackoff. *) +lemma mirror_ex: + assumes lp: "iszlfm p" + shows "(\ x. Ifm bbs (x#bs) (mirror p)) = (\ x. Ifm bbs (x#bs) p)" + (is "(\ x. ?I x ?mp) = (\ x. ?I x p)") +proof(auto) + fix x assume "?I x ?mp" hence "?I (- x) p" using mirror[OF lp] by blast + thus "\ x. ?I x p" by blast +next + fix x assume "?I x p" hence "?I (- x) ?mp" + using mirror[OF lp, where x="- x", symmetric] by auto + thus "\ x. ?I x ?mp" by blast +qed + + +lemma cp_thm': + assumes lp: "iszlfm p" + and up: "d\ p 1" and dd: "d\ p d" and dp: "d > 0" + shows "(\ x. Ifm bbs (x#bs) p) = ((\ j\ {1 .. d}. Ifm bbs (j#bs) (minusinf p)) \ (\ j\ {1.. d}. \ b\ (Inum (i#bs)) ` set (\ p). Ifm bbs ((b+j)#bs) p))" + using cp_thm[OF lp up dd dp,where i="i"] by auto + +constdefs unit:: "fm \ fm \ num list \ int" + "unit p \ (let p' = zlfm p ; l = \ p' ; q = And (Dvd l (CN 0 1 (C 0))) (a\ p' l); d = \ q; + B = remdups (map simpnum (\ q)) ; a = remdups (map simpnum (\ q)) + in if length B \ length a then (q,B,d) else (mirror q, a,d))" + +lemma unit: assumes qf: "qfree p" + shows "\ q B d. unit p = (q,B,d) \ ((\ x. Ifm bbs (x#bs) p) = (\ x. Ifm bbs (x#bs) q)) \ (Inum (i#bs)) ` set B = (Inum (i#bs)) ` set (\ q) \ d\ q 1 \ d\ q d \ d >0 \ iszlfm q \ (\ b\ set B. numbound0 b)" +proof- + fix q B d + assume qBd: "unit p = (q,B,d)" + let ?thes = "((\ x. Ifm bbs (x#bs) p) = (\ x. Ifm bbs (x#bs) q)) \ + Inum (i#bs) ` set B = Inum (i#bs) ` set (\ q) \ + d\ q 1 \ d\ q d \ 0 < d \ iszlfm q \ (\ b\ set B. numbound0 b)" + let ?I = "\ x p. Ifm bbs (x#bs) p" + let ?p' = "zlfm p" + let ?l = "\ ?p'" + let ?q = "And (Dvd ?l (CN 0 1 (C 0))) (a\ ?p' ?l)" + let ?d = "\ ?q" + let ?B = "set (\ ?q)" + let ?B'= "remdups (map simpnum (\ ?q))" + let ?A = "set (\ ?q)" + let ?A'= "remdups (map simpnum (\ ?q))" + from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] + have pp': "\ i. ?I i ?p' = ?I i p" by auto + from conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]] + have lp': "iszlfm ?p'" . + from lp' \[where p="?p'"] have lp: "?l >0" and dl: "d\ ?p' ?l" by auto + from a\_ex[where p="?p'" and l="?l" and bs="bs", OF lp' dl lp] pp' + have pq_ex:"(\ (x::int). ?I x p) = (\ x. ?I x ?q)" by simp + from lp' lp a\[OF lp' dl lp] have lq:"iszlfm ?q" and uq: "d\ ?q 1" by auto + from \[OF lq] have dp:"?d >0" and dd: "d\ ?q ?d" by blast+ + let ?N = "\ t. Inum (i#bs) t" + have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by auto + also have "\ = ?N ` ?B" using simpnum_ci[where bs="i#bs"] by auto + finally have BB': "?N ` set ?B' = ?N ` ?B" . + have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by auto + also have "\ = ?N ` ?A" using simpnum_ci[where bs="i#bs"] by auto + finally have AA': "?N ` set ?A' = ?N ` ?A" . + from \_numbound0[OF lq] have B_nb:"\ b\ set ?B'. numbound0 b" + by (simp add: simpnum_numbound0) + from \_l[OF lq] have A_nb: "\ b\ set ?A'. numbound0 b" + by (simp add: simpnum_numbound0) + {assume "length ?B' \ length ?A'" + hence q:"q=?q" and "B = ?B'" and d:"d = ?d" + using qBd by (auto simp add: Let_def unit_def) + with BB' B_nb have b: "?N ` (set B) = ?N ` set (\ q)" + and bn: "\b\ set B. numbound0 b" by simp+ + with pq_ex dp uq dd lq q d have ?thes by simp} + moreover + {assume "\ (length ?B' \ length ?A')" + hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d" + using qBd by (auto simp add: Let_def unit_def) + with AA' mirror\\[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\ q)" + and bn: "\b\ set B. numbound0 b" by simp+ + from mirror_ex[OF lq] pq_ex q + have pqm_eq:"(\ (x::int). ?I x p) = (\ (x::int). ?I x q)" by simp + from lq uq q mirror_l[where p="?q"] + have lq': "iszlfm q" and uq: "d\ q 1" by auto + from \[OF lq'] mirror_\[OF lq] q d have dq:"d\ q d " by auto + from pqm_eq b bn uq lq' dp dq q dp d have ?thes by simp + } + ultimately show ?thes by blast +qed + (* Cooper's Algorithm *) + +constdefs cooper :: "fm \ fm" + "cooper p \ + (let (q,B,d) = unit p; js = iupt 1 d; + mq = simpfm (minusinf q); + md = evaldjf (\ j. simpfm (subst0 (C j) mq)) js + in if md = T then T else + (let qd = evaldjf (\ (b,j). simpfm (subst0 (Add b (C j)) q)) + [(b,j). b\B,j\js] + in decr (disj md qd)))" +lemma cooper: assumes qf: "qfree p" + shows "((\ x. Ifm bbs (x#bs) p) = (Ifm bbs bs (cooper p))) \ qfree (cooper p)" + (is "(?lhs = ?rhs) \ _") +proof- + let ?I = "\ x p. Ifm bbs (x#bs) p" + let ?q = "fst (unit p)" + let ?B = "fst (snd(unit p))" + let ?d = "snd (snd (unit p))" + let ?js = "iupt 1 ?d" + let ?mq = "minusinf ?q" + let ?smq = "simpfm ?mq" + let ?md = "evaldjf (\ j. simpfm (subst0 (C j) ?smq)) ?js" + fix i + let ?N = "\ t. Inum (i#bs) t" + let ?Bjs = "[(b,j). b\?B,j\?js]" + let ?qd = "evaldjf (\ (b,j). simpfm (subst0 (Add b (C j)) ?q)) ?Bjs" + have qbf:"unit p = (?q,?B,?d)" by simp + from unit[OF qf qbf] have pq_ex: "(\(x::int). ?I x p) = (\ (x::int). ?I x ?q)" and + B:"?N ` set ?B = ?N ` set (\ ?q)" and + uq:"d\ ?q 1" and dd: "d\ ?q ?d" and dp: "?d > 0" and + lq: "iszlfm ?q" and + Bn: "\ b\ set ?B. numbound0 b" by auto + from zlin_qfree[OF lq] have qfq: "qfree ?q" . + from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq". + have jsnb: "\ j \ set ?js. numbound0 (C j)" by simp + hence "\ j\ set ?js. bound0 (subst0 (C j) ?smq)" + by (auto simp only: subst0_bound0[OF qfmq]) + hence th: "\ j\ set ?js. bound0 (simpfm (subst0 (C j) ?smq))" + by (auto simp add: simpfm_bound0) + from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp + from Bn jsnb have "\ (b,j) \ set ?Bjs. numbound0 (Add b (C j))" + by simp + hence "\ (b,j) \ set ?Bjs. bound0 (subst0 (Add b (C j)) ?q)" + using subst0_bound0[OF qfq] by blast + hence "\ (b,j) \ set ?Bjs. bound0 (simpfm (subst0 (Add b (C j)) ?q))" + using simpfm_bound0 by blast + hence th': "\ x \ set ?Bjs. bound0 ((\ (b,j). simpfm (subst0 (Add b (C j)) ?q)) x)" + by auto + from evaldjf_bound0 [OF th'] have qdb: "bound0 ?qd" by simp + from mdb qdb + have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \ ?qd=T", simp_all) + from trans [OF pq_ex cp_thm'[OF lq uq dd dp,where i="i"]] B + have "?lhs = (\ j\ {1.. ?d}. ?I j ?mq \ (\ b\ ?N ` set ?B. Ifm bbs ((b+ j)#bs) ?q))" by auto + also have "\ = (\ j\ {1.. ?d}. ?I j ?mq \ (\ b\ set ?B. Ifm bbs ((?N b+ j)#bs) ?q))" by simp + also have "\ = ((\ j\ {1.. ?d}. ?I j ?mq ) \ (\ j\ {1.. ?d}. \ b\ set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q))" by (simp only: Inum.simps) blast + also have "\ = ((\ j\ {1.. ?d}. ?I j ?smq ) \ (\ j\ {1.. ?d}. \ b\ set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q))" by (simp add: simpfm) + also have "\ = ((\ j\ set ?js. (\ j. ?I i (simpfm (subst0 (C j) ?smq))) j) \ (\ j\ set ?js. \ b\ set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q))" + by (simp only: simpfm subst0_I[OF qfmq] iupt_set) auto + also have "\ = (?I i (evaldjf (\ j. simpfm (subst0 (C j) ?smq)) ?js) \ (\ j\ set ?js. \ b\ set ?B. ?I i (subst0 (Add b (C j)) ?q)))" + by (simp only: evaldjf_ex subst0_I[OF qfq]) + also have "\= (?I i ?md \ (\ (b,j) \ set ?Bjs. (\ (b,j). ?I i (simpfm (subst0 (Add b (C j)) ?q))) (b,j)))" + by (simp only: simpfm set_concat set_map concat_map_singleton UN_simps) blast + also have "\ = (?I i ?md \ (?I i (evaldjf (\ (b,j). simpfm (subst0 (Add b (C j)) ?q)) ?Bjs)))" + by (simp only: evaldjf_ex[where bs="i#bs" and f="\ (b,j). simpfm (subst0 (Add b (C j)) ?q)" and ps="?Bjs"]) (auto simp add: split_def) + finally have mdqd: "?lhs = (?I i ?md \ ?I i ?qd)" by simp + also have "\ = (?I i (disj ?md ?qd))" by (simp add: disj) + also have "\ = (Ifm bbs bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb]) + finally have mdqd2: "?lhs = (Ifm bbs bs (decr (disj ?md ?qd)))" . + {assume mdT: "?md = T" + hence cT:"cooper p = T" + by (simp only: cooper_def unit_def split_def Let_def if_True) simp + from mdT have lhs:"?lhs" using mdqd by simp + from mdT have "?rhs" by (simp add: cooper_def unit_def split_def) + with lhs cT have ?thesis by simp } + moreover + {assume mdT: "?md \ T" hence "cooper p = decr (disj ?md ?qd)" + by (simp only: cooper_def unit_def split_def Let_def if_False) + with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp } + ultimately show ?thesis by blast +qed + +definition pa :: "fm \ fm" where + "pa p = qelim (prep p) cooper" + +theorem mirqe: "(Ifm bbs bs (pa p) = Ifm bbs bs p) \ qfree (pa p)" + using qelim_ci cooper prep by (auto simp add: pa_def) + +definition + cooper_test :: "unit \ fm" +where + "cooper_test u = pa (E (A (Imp (Ge (Sub (Bound 0) (Bound 1))) + (E (E (Eq (Sub (Add (Mul 3 (Bound 1)) (Mul 5 (Bound 0))) + (Bound 2))))))))" + +ML {* @{code cooper_test} () *} + +(* +code_reserved SML oo +export_code pa in SML module_name GeneratedCooper file "~~/src/HOL/Tools/Qelim/raw_generated_cooper.ML" +*) + +oracle linzqe_oracle = {* +let + +fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t + of NONE => error "Variable not found in the list!" + | SOME n => @{code Bound} n) + | num_of_term vs @{term "0::int"} = @{code C} 0 + | num_of_term vs @{term "1::int"} = @{code C} 1 + | num_of_term vs (@{term "number_of :: int \ int"} $ t) = @{code C} (HOLogic.dest_numeral t) + | num_of_term vs (Bound i) = @{code Bound} i + | num_of_term vs (@{term "uminus :: int \ int"} $ t') = @{code Neg} (num_of_term vs t') + | num_of_term vs (@{term "op + :: int \ int \ int"} $ t1 $ t2) = + @{code Add} (num_of_term vs t1, num_of_term vs t2) + | num_of_term vs (@{term "op - :: int \ int \ int"} $ t1 $ t2) = + @{code Sub} (num_of_term vs t1, num_of_term vs t2) + | num_of_term vs (@{term "op * :: int \ int \ int"} $ t1 $ t2) = + (case try HOLogic.dest_number t1 + of SOME (_, i) => @{code Mul} (i, num_of_term vs t2) + | NONE => (case try HOLogic.dest_number t2 + of SOME (_, i) => @{code Mul} (i, num_of_term vs t1) + | NONE => error "num_of_term: unsupported multiplication")) + | num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t); + +fun fm_of_term ps vs @{term True} = @{code T} + | fm_of_term ps vs @{term False} = @{code F} + | fm_of_term ps vs (@{term "op < :: int \ int \ bool"} $ t1 $ t2) = + @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) + | fm_of_term ps vs (@{term "op \ :: int \ int \ bool"} $ t1 $ t2) = + @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) + | fm_of_term ps vs (@{term "op = :: int \ int \ bool"} $ t1 $ t2) = + @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) + | fm_of_term ps vs (@{term "op dvd :: int \ int \ bool"} $ t1 $ t2) = + (case try HOLogic.dest_number t1 + of SOME (_, i) => @{code Dvd} (i, num_of_term vs t2) + | NONE => error "num_of_term: unsupported dvd") + | fm_of_term ps vs (@{term "op = :: bool \ bool \ bool"} $ t1 $ t2) = + @{code Iff} (fm_of_term ps vs t1, fm_of_term ps vs t2) + | fm_of_term ps vs (@{term "op &"} $ t1 $ t2) = + @{code And} (fm_of_term ps vs t1, fm_of_term ps vs t2) + | fm_of_term ps vs (@{term "op |"} $ t1 $ t2) = + @{code Or} (fm_of_term ps vs t1, fm_of_term ps vs t2) + | fm_of_term ps vs (@{term "op -->"} $ t1 $ t2) = + @{code Imp} (fm_of_term ps vs t1, fm_of_term ps vs t2) + | fm_of_term ps vs (@{term "Not"} $ t') = + @{code NOT} (fm_of_term ps vs t') + | fm_of_term ps vs (Const ("Ex", _) $ Abs (xn, xT, p)) = + let + val (xn', p') = variant_abs (xn, xT, p); + val vs' = (Free (xn', xT), 0) :: map (fn (v, n) => (v, n + 1)) vs; + in @{code E} (fm_of_term ps vs' p) end + | fm_of_term ps vs (Const ("All", _) $ Abs (xn, xT, p)) = + let + val (xn', p') = variant_abs (xn, xT, p); + val vs' = (Free (xn', xT), 0) :: map (fn (v, n) => (v, n + 1)) vs; + in @{code A} (fm_of_term ps vs' p) end + | fm_of_term ps vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t); + +fun term_of_num vs (@{code C} i) = HOLogic.mk_number HOLogic.intT i + | term_of_num vs (@{code Bound} n) = fst (the (find_first (fn (_, m) => n = m) vs)) + | term_of_num vs (@{code Neg} t') = @{term "uminus :: int \ int"} $ term_of_num vs t' + | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: int \ int \ int"} $ + term_of_num vs t1 $ term_of_num vs t2 + | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: int \ int \ int"} $ + term_of_num vs t1 $ term_of_num vs t2 + | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: int \ int \ int"} $ + term_of_num vs (@{code C} i) $ term_of_num vs t2 + | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t)); + +fun term_of_fm ps vs @{code T} = HOLogic.true_const + | term_of_fm ps vs @{code F} = HOLogic.false_const + | term_of_fm ps vs (@{code Lt} t) = + @{term "op < :: int \ int \ bool"} $ term_of_num vs t $ @{term "0::int"} + | term_of_fm ps vs (@{code Le} t) = + @{term "op \ :: int \ int \ bool"} $ term_of_num vs t $ @{term "0::int"} + | term_of_fm ps vs (@{code Gt} t) = + @{term "op < :: int \ int \ bool"} $ @{term "0::int"} $ term_of_num vs t + | term_of_fm ps vs (@{code Ge} t) = + @{term "op \ :: int \ int \ bool"} $ @{term "0::int"} $ term_of_num vs t + | term_of_fm ps vs (@{code Eq} t) = + @{term "op = :: int \ int \ bool"} $ term_of_num vs t $ @{term "0::int"} + | term_of_fm ps vs (@{code NEq} t) = + term_of_fm ps vs (@{code NOT} (@{code Eq} t)) + | term_of_fm ps vs (@{code Dvd} (i, t)) = + @{term "op dvd :: int \ int \ bool"} $ term_of_num vs (@{code C} i) $ term_of_num vs t + | term_of_fm ps vs (@{code NDvd} (i, t)) = + term_of_fm ps vs (@{code NOT} (@{code Dvd} (i, t))) + | term_of_fm ps vs (@{code NOT} t') = + HOLogic.Not $ term_of_fm ps vs t' + | term_of_fm ps vs (@{code And} (t1, t2)) = + HOLogic.conj $ term_of_fm ps vs t1 $ term_of_fm ps vs t2 + | term_of_fm ps vs (@{code Or} (t1, t2)) = + HOLogic.disj $ term_of_fm ps vs t1 $ term_of_fm ps vs t2 + | term_of_fm ps vs (@{code Imp} (t1, t2)) = + HOLogic.imp $ term_of_fm ps vs t1 $ term_of_fm ps vs t2 + | term_of_fm ps vs (@{code Iff} (t1, t2)) = + @{term "op = :: bool \ bool \ bool"} $ term_of_fm ps vs t1 $ term_of_fm ps vs t2 + | term_of_fm ps vs (@{code Closed} n) = (fst o the) (find_first (fn (_, m) => m = n) ps) + | term_of_fm ps vs (@{code NClosed} n) = term_of_fm ps vs (@{code NOT} (@{code Closed} n)); + +fun term_bools acc t = + let + val is_op = member (op =) [@{term "op &"}, @{term "op |"}, @{term "op -->"}, @{term "op = :: bool => _"}, + @{term "op = :: int => _"}, @{term "op < :: int => _"}, + @{term "op <= :: int => _"}, @{term "Not"}, @{term "All :: (int => _) => _"}, + @{term "Ex :: (int => _) => _"}, @{term "True"}, @{term "False"}] + fun is_ty t = not (fastype_of t = HOLogic.boolT) + in case t + of (l as f $ a) $ b => if is_ty t orelse is_op t then term_bools (term_bools acc l)b + else insert (op aconv) t acc + | f $ a => if is_ty t orelse is_op t then term_bools (term_bools acc f) a + else insert (op aconv) t acc + | Abs p => term_bools acc (snd (variant_abs p)) + | _ => if is_ty t orelse is_op t then acc else insert (op aconv) t acc + end; + +in fn ct => + let + val thy = Thm.theory_of_cterm ct; + val t = Thm.term_of ct; + val fs = OldTerm.term_frees t; + val bs = term_bools [] t; + val vs = fs ~~ (0 upto (length fs - 1)) + val ps = bs ~~ (0 upto (length bs - 1)) + val t' = (term_of_fm ps vs o @{code pa} o fm_of_term ps vs) t; + in (Thm.cterm_of thy o HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, t') end +end; +*} + +use "cooper_tac.ML" +setup "Cooper_Tac.setup" + +text {* Tests *} + +lemma "\ (j::int). \ x\j. (\ a b. x = 3*a+5*b)" + by cooper + +lemma "ALL (x::int) >=8. EX i j. 5*i + 3*j = x" + by cooper + +theorem "(\(y::int). 3 dvd y) ==> \(x::int). b < x --> a \ x" + by cooper + +theorem "!! (y::int) (z::int) (n::int). 3 dvd z ==> 2 dvd (y::int) ==> + (\(x::int). 2*x = y) & (\(k::int). 3*k = z)" + by cooper + +theorem "!! (y::int) (z::int) n. Suc(n::nat) < 6 ==> 3 dvd z ==> + 2 dvd (y::int) ==> (\(x::int). 2*x = y) & (\(k::int). 3*k = z)" + by cooper + +theorem "\(x::nat). \(y::nat). (0::nat) \ 5 --> y = 5 + x " + by cooper + +lemma "ALL (x::int) >=8. EX i j. 5*i + 3*j = x" + by cooper + +lemma "ALL (y::int) (z::int) (n::int). 3 dvd z --> 2 dvd (y::int) --> (EX (x::int). 2*x = y) & (EX (k::int). 3*k = z)" + by cooper + +lemma "ALL(x::int) y. x < y --> 2 * x + 1 < 2 * y" + by cooper + +lemma "ALL(x::int) y. 2 * x + 1 ~= 2 * y" + by cooper + +lemma "EX(x::int) y. 0 < x & 0 <= y & 3 * x - 5 * y = 1" + by cooper + +lemma "~ (EX(x::int) (y::int) (z::int). 4*x + (-6::int)*y = 1)" + by cooper + +lemma "ALL(x::int). (2 dvd x) --> (EX(y::int). x = 2*y)" + by cooper + +lemma "ALL(x::int). (2 dvd x) = (EX(y::int). x = 2*y)" + by cooper + +lemma "ALL(x::int). ((2 dvd x) = (ALL(y::int). x ~= 2*y + 1))" + by cooper + +lemma "~ (ALL(x::int). ((2 dvd x) = (ALL(y::int). x ~= 2*y+1) | (EX(q::int) (u::int) i. 3*i + 2*q - u < 17) --> 0 < x | ((~ 3 dvd x) &(x + 8 = 0))))" + by cooper + +lemma "~ (ALL(i::int). 4 <= i --> (EX x y. 0 <= x & 0 <= y & 3 * x + 5 * y = i))" + by cooper + +lemma "EX j. ALL (x::int) >= j. EX i j. 5*i + 3*j = x" + by cooper + +theorem "(\(y::int). 3 dvd y) ==> \(x::int). b < x --> a \ x" + by cooper + +theorem "!! (y::int) (z::int) (n::int). 3 dvd z ==> 2 dvd (y::int) ==> + (\(x::int). 2*x = y) & (\(k::int). 3*k = z)" + by cooper + +theorem "!! (y::int) (z::int) n. Suc(n::nat) < 6 ==> 3 dvd z ==> + 2 dvd (y::int) ==> (\(x::int). 2*x = y) & (\(k::int). 3*k = z)" + by cooper + +theorem "\(x::nat). \(y::nat). (0::nat) \ 5 --> y = 5 + x " + by cooper + +theorem "\(x::nat). \(y::nat). y = 5 + x | x div 6 + 1= 2" + by cooper + +theorem "\(x::int). 0 < x" + by cooper + +theorem "\(x::int) y. x < y --> 2 * x + 1 < 2 * y" + by cooper + +theorem "\(x::int) y. 2 * x + 1 \ 2 * y" + by cooper + +theorem "\(x::int) y. 0 < x & 0 \ y & 3 * x - 5 * y = 1" + by cooper + +theorem "~ (\(x::int) (y::int) (z::int). 4*x + (-6::int)*y = 1)" + by cooper + +theorem "~ (\(x::int). False)" + by cooper + +theorem "\(x::int). (2 dvd x) --> (\(y::int). x = 2*y)" + by cooper + +theorem "\(x::int). (2 dvd x) --> (\(y::int). x = 2*y)" + by cooper + +theorem "\(x::int). (2 dvd x) = (\(y::int). x = 2*y)" + by cooper + +theorem "\(x::int). ((2 dvd x) = (\(y::int). x \ 2*y + 1))" + by cooper + +theorem "~ (\(x::int). + ((2 dvd x) = (\(y::int). x \ 2*y+1) | + (\(q::int) (u::int) i. 3*i + 2*q - u < 17) + --> 0 < x | ((~ 3 dvd x) &(x + 8 = 0))))" + by cooper + +theorem "~ (\(i::int). 4 \ i --> (\x y. 0 \ x & 0 \ y & 3 * x + 5 * y = i))" + by cooper + +theorem "\(i::int). 8 \ i --> (\x y. 0 \ x & 0 \ y & 3 * x + 5 * y = i)" + by cooper + +theorem "\(j::int). \i. j \ i --> (\x y. 0 \ x & 0 \ y & 3 * x + 5 * y = i)" + by cooper + +theorem "~ (\j (i::int). j \ i --> (\x y. 0 \ x & 0 \ y & 3 * x + 5 * y = i))" + by cooper + +theorem "(\m::nat. n = 2 * m) --> (n + 1) div 2 = n div 2" + by cooper + +end diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Decision_Procs/Dense_Linear_Order.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Decision_Procs/Dense_Linear_Order.thy Fri Feb 06 15:15:46 2009 +0100 @@ -0,0 +1,879 @@ +(* Title : HOL/Dense_Linear_Order.thy + Author : Amine Chaieb, TU Muenchen +*) + +header {* Dense linear order without endpoints + and a quantifier elimination procedure in Ferrante and Rackoff style *} + +theory Dense_Linear_Order +imports Plain Groebner_Basis Main +uses + "~~/src/HOL/Tools/Qelim/langford_data.ML" + "~~/src/HOL/Tools/Qelim/ferrante_rackoff_data.ML" + ("~~/src/HOL/Tools/Qelim/langford.ML") + ("~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML") +begin + +setup {* Langford_Data.setup #> Ferrante_Rackoff_Data.setup *} + +context linorder +begin + +lemma less_not_permute[noatp]: "\ (x < y \ y < x)" by (simp add: not_less linear) + +lemma gather_simps[noatp]: + shows + "(\x. (\y \ L. y < x) \ (\y \ U. x < y) \ x < u \ P x) \ (\x. (\y \ L. y < x) \ (\y \ (insert u U). x < y) \ P x)" + and "(\x. (\y \ L. y < x) \ (\y \ U. x < y) \ l < x \ P x) \ (\x. (\y \ (insert l L). y < x) \ (\y \ U. x < y) \ P x)" + "(\x. (\y \ L. y < x) \ (\y \ U. x < y) \ x < u) \ (\x. (\y \ L. y < x) \ (\y \ (insert u U). x < y))" + and "(\x. (\y \ L. y < x) \ (\y \ U. x < y) \ l < x) \ (\x. (\y \ (insert l L). y < x) \ (\y \ U. x < y))" by auto + +lemma + gather_start[noatp]: "(\x. P x) \ (\x. (\y \ {}. y < x) \ (\y\ {}. x < y) \ P x)" + by simp + +text{* Theorems for @{text "\z. \x. x < z \ (P x \ P\<^bsub>-\\<^esub>)"}*} +lemma minf_lt[noatp]: "\z . \x. x < z \ (x < t \ True)" by auto +lemma minf_gt[noatp]: "\z . \x. x < z \ (t < x \ False)" + by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le) + +lemma minf_le[noatp]: "\z. \x. x < z \ (x \ t \ True)" by (auto simp add: less_le) +lemma minf_ge[noatp]: "\z. \x. x < z \ (t \ x \ False)" + by (auto simp add: less_le not_less not_le) +lemma minf_eq[noatp]: "\z. \x. x < z \ (x = t \ False)" by auto +lemma minf_neq[noatp]: "\z. \x. x < z \ (x \ t \ True)" by auto +lemma minf_P[noatp]: "\z. \x. x < z \ (P \ P)" by blast + +text{* Theorems for @{text "\z. \x. x < z \ (P x \ P\<^bsub>+\\<^esub>)"}*} +lemma pinf_gt[noatp]: "\z . \x. z < x \ (t < x \ True)" by auto +lemma pinf_lt[noatp]: "\z . \x. z < x \ (x < t \ False)" + by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le) + +lemma pinf_ge[noatp]: "\z. \x. z < x \ (t \ x \ True)" by (auto simp add: less_le) +lemma pinf_le[noatp]: "\z. \x. z < x \ (x \ t \ False)" + by (auto simp add: less_le not_less not_le) +lemma pinf_eq[noatp]: "\z. \x. z < x \ (x = t \ False)" by auto +lemma pinf_neq[noatp]: "\z. \x. z < x \ (x \ t \ True)" by auto +lemma pinf_P[noatp]: "\z. \x. z < x \ (P \ P)" by blast + +lemma nmi_lt[noatp]: "t \ U \ \x. \True \ x < t \ (\ u\ U. u \ x)" by auto +lemma nmi_gt[noatp]: "t \ U \ \x. \False \ t < x \ (\ u\ U. u \ x)" + by (auto simp add: le_less) +lemma nmi_le[noatp]: "t \ U \ \x. \True \ x\ t \ (\ u\ U. u \ x)" by auto +lemma nmi_ge[noatp]: "t \ U \ \x. \False \ t\ x \ (\ u\ U. u \ x)" by auto +lemma nmi_eq[noatp]: "t \ U \ \x. \False \ x = t \ (\ u\ U. u \ x)" by auto +lemma nmi_neq[noatp]: "t \ U \\x. \True \ x \ t \ (\ u\ U. u \ x)" by auto +lemma nmi_P[noatp]: "\ x. ~P \ P \ (\ u\ U. u \ x)" by auto +lemma nmi_conj[noatp]: "\\x. \P1' \ P1 x \ (\ u\ U. u \ x) ; + \x. \P2' \ P2 x \ (\ u\ U. u \ x)\ \ + \x. \(P1' \ P2') \ (P1 x \ P2 x) \ (\ u\ U. u \ x)" by auto +lemma nmi_disj[noatp]: "\\x. \P1' \ P1 x \ (\ u\ U. u \ x) ; + \x. \P2' \ P2 x \ (\ u\ U. u \ x)\ \ + \x. \(P1' \ P2') \ (P1 x \ P2 x) \ (\ u\ U. u \ x)" by auto + +lemma npi_lt[noatp]: "t \ U \ \x. \False \ x < t \ (\ u\ U. x \ u)" by (auto simp add: le_less) +lemma npi_gt[noatp]: "t \ U \ \x. \True \ t < x \ (\ u\ U. x \ u)" by auto +lemma npi_le[noatp]: "t \ U \ \x. \False \ x \ t \ (\ u\ U. x \ u)" by auto +lemma npi_ge[noatp]: "t \ U \ \x. \True \ t \ x \ (\ u\ U. x \ u)" by auto +lemma npi_eq[noatp]: "t \ U \ \x. \False \ x = t \ (\ u\ U. x \ u)" by auto +lemma npi_neq[noatp]: "t \ U \ \x. \True \ x \ t \ (\ u\ U. x \ u )" by auto +lemma npi_P[noatp]: "\ x. ~P \ P \ (\ u\ U. x \ u)" by auto +lemma npi_conj[noatp]: "\\x. \P1' \ P1 x \ (\ u\ U. x \ u) ; \x. \P2' \ P2 x \ (\ u\ U. x \ u)\ + \ \x. \(P1' \ P2') \ (P1 x \ P2 x) \ (\ u\ U. x \ u)" by auto +lemma npi_disj[noatp]: "\\x. \P1' \ P1 x \ (\ u\ U. x \ u) ; \x. \P2' \ P2 x \ (\ u\ U. x \ u)\ + \ \x. \(P1' \ P2') \ (P1 x \ P2 x) \ (\ u\ U. x \ u)" by auto + +lemma lin_dense_lt[noatp]: "t \ U \ \x l u. (\ t. l < t \ t < u \ t \ U) \ l< x \ x < u \ x < t \ (\ y. l < y \ y < u \ y < t)" +proof(clarsimp) + fix x l u y assume tU: "t \ U" and noU: "\t. l < t \ t < u \ t \ U" and lx: "l < x" + and xu: "xy" by auto + {assume H: "t < y" + from less_trans[OF lx px] less_trans[OF H yu] + have "l < t \ t < u" by simp + with tU noU have "False" by auto} + hence "\ t < y" by auto hence "y \ t" by (simp add: not_less) + thus "y < t" using tny by (simp add: less_le) +qed + +lemma lin_dense_gt[noatp]: "t \ U \ \x l u. (\ t. l < t \ t< u \ t \ U) \ l < x \ x < u \ t < x \ (\ y. l < y \ y < u \ t < y)" +proof(clarsimp) + fix x l u y + assume tU: "t \ U" and noU: "\t. l < t \ t < u \ t \ U" and lx: "l < x" and xu: "xy" by auto + {assume H: "y< t" + from less_trans[OF ly H] less_trans[OF px xu] have "l < t \ t < u" by simp + with tU noU have "False" by auto} + hence "\ y y" by (auto simp add: not_less) + thus "t < y" using tny by (simp add:less_le) +qed + +lemma lin_dense_le[noatp]: "t \ U \ \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ x \ t \ (\ y. l < y \ y < u \ y\ t)" +proof(clarsimp) + fix x l u y + assume tU: "t \ U" and noU: "\t. l < t \ t < u \ t \ U" and lx: "l < x" and xu: "x t" and ly: "ly" by auto + {assume H: "t < y" + from less_le_trans[OF lx px] less_trans[OF H yu] + have "l < t \ t < u" by simp + with tU noU have "False" by auto} + hence "\ t < y" by auto thus "y \ t" by (simp add: not_less) +qed + +lemma lin_dense_ge[noatp]: "t \ U \ \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ t \ x \ (\ y. l < y \ y < u \ t \ y)" +proof(clarsimp) + fix x l u y + assume tU: "t \ U" and noU: "\t. l < t \ t < u \ t \ U" and lx: "l < x" and xu: "x x" and ly: "ly" by auto + {assume H: "y< t" + from less_trans[OF ly H] le_less_trans[OF px xu] + have "l < t \ t < u" by simp + with tU noU have "False" by auto} + hence "\ y y" by (simp add: not_less) +qed +lemma lin_dense_eq[noatp]: "t \ U \ \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ x = t \ (\ y. l < y \ y < u \ y= t)" by auto +lemma lin_dense_neq[noatp]: "t \ U \ \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ x \ t \ (\ y. l < y \ y < u \ y\ t)" by auto +lemma lin_dense_P[noatp]: "\x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ P \ (\ y. l < y \ y < u \ P)" by auto + +lemma lin_dense_conj[noatp]: + "\\x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ P1 x + \ (\ y. l < y \ y < u \ P1 y) ; + \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ P2 x + \ (\ y. l < y \ y < u \ P2 y)\ \ + \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ (P1 x \ P2 x) + \ (\ y. l < y \ y < u \ (P1 y \ P2 y))" + by blast +lemma lin_dense_disj[noatp]: + "\\x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ P1 x + \ (\ y. l < y \ y < u \ P1 y) ; + \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ P2 x + \ (\ y. l < y \ y < u \ P2 y)\ \ + \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ (P1 x \ P2 x) + \ (\ y. l < y \ y < u \ (P1 y \ P2 y))" + by blast + +lemma npmibnd[noatp]: "\\x. \ MP \ P x \ (\ u\ U. u \ x); \x. \PP \ P x \ (\ u\ U. x \ u)\ + \ \x. \ MP \ \PP \ P x \ (\ u\ U. \ u' \ U. u \ x \ x \ u')" +by auto + +lemma finite_set_intervals[noatp]: + assumes px: "P x" and lx: "l \ x" and xu: "x \ u" and linS: "l\ S" + and uinS: "u \ S" and fS:"finite S" and lS: "\ x\ S. l \ x" and Su: "\ x\ S. x \ u" + shows "\ a \ S. \ b \ S. (\ y. a < y \ y < b \ y \ S) \ a \ x \ x \ b \ P x" +proof- + let ?Mx = "{y. y\ S \ y \ x}" + let ?xM = "{y. y\ S \ x \ y}" + let ?a = "Max ?Mx" + let ?b = "Min ?xM" + have MxS: "?Mx \ S" by blast + hence fMx: "finite ?Mx" using fS finite_subset by auto + from lx linS have linMx: "l \ ?Mx" by blast + hence Mxne: "?Mx \ {}" by blast + have xMS: "?xM \ S" by blast + hence fxM: "finite ?xM" using fS finite_subset by auto + from xu uinS have linxM: "u \ ?xM" by blast + hence xMne: "?xM \ {}" by blast + have ax:"?a \ x" using Mxne fMx by auto + have xb:"x \ ?b" using xMne fxM by auto + have "?a \ ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \ S" using MxS by blast + have "?b \ ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \ S" using xMS by blast + have noy:"\ y. ?a < y \ y < ?b \ y \ S" + proof(clarsimp) + fix y assume ay: "?a < y" and yb: "y < ?b" and yS: "y \ S" + from yS have "y\ ?Mx \ y\ ?xM" by (auto simp add: linear) + moreover {assume "y \ ?Mx" hence "y \ ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])} + moreover {assume "y \ ?xM" hence "?b \ y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])} + ultimately show "False" by blast + qed + from ainS binS noy ax xb px show ?thesis by blast +qed + +lemma finite_set_intervals2[noatp]: + assumes px: "P x" and lx: "l \ x" and xu: "x \ u" and linS: "l\ S" + and uinS: "u \ S" and fS:"finite S" and lS: "\ x\ S. l \ x" and Su: "\ x\ S. x \ u" + shows "(\ s\ S. P s) \ (\ a \ S. \ b \ S. (\ y. a < y \ y < b \ y \ S) \ a < x \ x < b \ P x)" +proof- + from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su] + obtain a and b where + as: "a\ S" and bs: "b\ S" and noS:"\y. a < y \ y < b \ y \ S" + and axb: "a \ x \ x \ b \ P x" by auto + from axb have "x= a \ x= b \ (a < x \ x < b)" by (auto simp add: le_less) + thus ?thesis using px as bs noS by blast +qed + +end + +section {* The classical QE after Langford for dense linear orders *} + +context dense_linear_order +begin + +lemma interval_empty_iff: + "{y. x < y \ y < z} = {} \ \ x < z" + by (auto dest: dense) + +lemma dlo_qe_bnds[noatp]: + assumes ne: "L \ {}" and neU: "U \ {}" and fL: "finite L" and fU: "finite U" + shows "(\x. (\y \ L. y < x) \ (\y \ U. x < y)) \ (\ l \ L. \u \ U. l < u)" +proof (simp only: atomize_eq, rule iffI) + assume H: "\x. (\y\L. y < x) \ (\y\U. x < y)" + then obtain x where xL: "\y\L. y < x" and xU: "\y\U. x < y" by blast + {fix l u assume l: "l \ L" and u: "u \ U" + have "l < x" using xL l by blast + also have "x < u" using xU u by blast + finally (less_trans) have "l < u" .} + thus "\l\L. \u\U. l < u" by blast +next + assume H: "\l\L. \u\U. l < u" + let ?ML = "Max L" + let ?MU = "Min U" + from fL ne have th1: "?ML \ L" and th1': "\l\L. l \ ?ML" by auto + from fU neU have th2: "?MU \ U" and th2': "\u\U. ?MU \ u" by auto + from th1 th2 H have "?ML < ?MU" by auto + with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" by blast + from th3 th1' have "\l \ L. l < w" by auto + moreover from th4 th2' have "\u \ U. w < u" by auto + ultimately show "\x. (\y\L. y < x) \ (\y\U. x < y)" by auto +qed + +lemma dlo_qe_noub[noatp]: + assumes ne: "L \ {}" and fL: "finite L" + shows "(\x. (\y \ L. y < x) \ (\y \ {}. x < y)) \ True" +proof(simp add: atomize_eq) + from gt_ex[of "Max L"] obtain M where M: "Max L < M" by blast + from ne fL have "\x \ L. x \ Max L" by simp + with M have "\x\L. x < M" by (auto intro: le_less_trans) + thus "\x. \y\L. y < x" by blast +qed + +lemma dlo_qe_nolb[noatp]: + assumes ne: "U \ {}" and fU: "finite U" + shows "(\x. (\y \ {}. y < x) \ (\y \ U. x < y)) \ True" +proof(simp add: atomize_eq) + from lt_ex[of "Min U"] obtain M where M: "M < Min U" by blast + from ne fU have "\x \ U. Min U \ x" by simp + with M have "\x\U. M < x" by (auto intro: less_le_trans) + thus "\x. \y\U. x < y" by blast +qed + +lemma exists_neq[noatp]: "\(x::'a). x \ t" "\(x::'a). t \ x" + using gt_ex[of t] by auto + +lemmas dlo_simps[noatp] = order_refl less_irrefl not_less not_le exists_neq + le_less neq_iff linear less_not_permute + +lemma axiom[noatp]: "dense_linear_order (op \) (op <)" by (rule dense_linear_order_axioms) +lemma atoms[noatp]: + shows "TERM (less :: 'a \ _)" + and "TERM (less_eq :: 'a \ _)" + and "TERM (op = :: 'a \ _)" . + +declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms] +declare dlo_simps[langfordsimp] + +end + +(* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *) +lemma dnf[noatp]: + "(P & (Q | R)) = ((P&Q) | (P&R))" + "((Q | R) & P) = ((Q&P) | (R&P))" + by blast+ + +lemmas weak_dnf_simps[noatp] = simp_thms dnf + +lemma nnf_simps[noatp]: + "(\(P \ Q)) = (\P \ \Q)" "(\(P \ Q)) = (\P \ \Q)" "(P \ Q) = (\P \ Q)" + "(P = Q) = ((P \ Q) \ (\P \ \ Q))" "(\ \(P)) = P" + by blast+ + +lemma ex_distrib[noatp]: "(\x. P x \ Q x) \ ((\x. P x) \ (\x. Q x))" by blast + +lemmas dnf_simps[noatp] = weak_dnf_simps nnf_simps ex_distrib + +use "~~/src/HOL/Tools/Qelim/langford.ML" +method_setup dlo = {* + Method.ctxt_args (Method.SIMPLE_METHOD' o LangfordQE.dlo_tac) +*} "Langford's algorithm for quantifier elimination in dense linear orders" + + +section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields -- see @{text "Arith_Tools.thy"} *} + +text {* Linear order without upper bounds *} + +locale linorder_stupid_syntax = linorder +begin +notation + less_eq ("op \") and + less_eq ("(_/ \ _)" [51, 51] 50) and + less ("op \") and + less ("(_/ \ _)" [51, 51] 50) + +end + +locale linorder_no_ub = linorder_stupid_syntax + + assumes gt_ex: "\y. less x y" +begin +lemma ge_ex[noatp]: "\y. x \ y" using gt_ex by auto + +text {* Theorems for @{text "\z. \x. z \ x \ (P x \ P\<^bsub>+\\<^esub>)"} *} +lemma pinf_conj[noatp]: + assumes ex1: "\z1. \x. z1 \ x \ (P1 x \ P1')" + and ex2: "\z2. \x. z2 \ x \ (P2 x \ P2')" + shows "\z. \x. z \ x \ ((P1 x \ P2 x) \ (P1' \ P2'))" +proof- + from ex1 ex2 obtain z1 and z2 where z1: "\x. z1 \ x \ (P1 x \ P1')" + and z2: "\x. z2 \ x \ (P2 x \ P2')" by blast + from gt_ex obtain z where z:"ord.max less_eq z1 z2 \ z" by blast + from z have zz1: "z1 \ z" and zz2: "z2 \ z" by simp_all + {fix x assume H: "z \ x" + from less_trans[OF zz1 H] less_trans[OF zz2 H] + have "(P1 x \ P2 x) \ (P1' \ P2')" using z1 zz1 z2 zz2 by auto + } + thus ?thesis by blast +qed + +lemma pinf_disj[noatp]: + assumes ex1: "\z1. \x. z1 \ x \ (P1 x \ P1')" + and ex2: "\z2. \x. z2 \ x \ (P2 x \ P2')" + shows "\z. \x. z \ x \ ((P1 x \ P2 x) \ (P1' \ P2'))" +proof- + from ex1 ex2 obtain z1 and z2 where z1: "\x. z1 \ x \ (P1 x \ P1')" + and z2: "\x. z2 \ x \ (P2 x \ P2')" by blast + from gt_ex obtain z where z:"ord.max less_eq z1 z2 \ z" by blast + from z have zz1: "z1 \ z" and zz2: "z2 \ z" by simp_all + {fix x assume H: "z \ x" + from less_trans[OF zz1 H] less_trans[OF zz2 H] + have "(P1 x \ P2 x) \ (P1' \ P2')" using z1 zz1 z2 zz2 by auto + } + thus ?thesis by blast +qed + +lemma pinf_ex[noatp]: assumes ex:"\z. \x. z \ x \ (P x \ P1)" and p1: P1 shows "\ x. P x" +proof- + from ex obtain z where z: "\x. z \ x \ (P x \ P1)" by blast + from gt_ex obtain x where x: "z \ x" by blast + from z x p1 show ?thesis by blast +qed + +end + +text {* Linear order without upper bounds *} + +locale linorder_no_lb = linorder_stupid_syntax + + assumes lt_ex: "\y. less y x" +begin +lemma le_ex[noatp]: "\y. y \ x" using lt_ex by auto + + +text {* Theorems for @{text "\z. \x. x \ z \ (P x \ P\<^bsub>-\\<^esub>)"} *} +lemma minf_conj[noatp]: + assumes ex1: "\z1. \x. x \ z1 \ (P1 x \ P1')" + and ex2: "\z2. \x. x \ z2 \ (P2 x \ P2')" + shows "\z. \x. x \ z \ ((P1 x \ P2 x) \ (P1' \ P2'))" +proof- + from ex1 ex2 obtain z1 and z2 where z1: "\x. x \ z1 \ (P1 x \ P1')"and z2: "\x. x \ z2 \ (P2 x \ P2')" by blast + from lt_ex obtain z where z:"z \ ord.min less_eq z1 z2" by blast + from z have zz1: "z \ z1" and zz2: "z \ z2" by simp_all + {fix x assume H: "x \ z" + from less_trans[OF H zz1] less_trans[OF H zz2] + have "(P1 x \ P2 x) \ (P1' \ P2')" using z1 zz1 z2 zz2 by auto + } + thus ?thesis by blast +qed + +lemma minf_disj[noatp]: + assumes ex1: "\z1. \x. x \ z1 \ (P1 x \ P1')" + and ex2: "\z2. \x. x \ z2 \ (P2 x \ P2')" + shows "\z. \x. x \ z \ ((P1 x \ P2 x) \ (P1' \ P2'))" +proof- + from ex1 ex2 obtain z1 and z2 where z1: "\x. x \ z1 \ (P1 x \ P1')"and z2: "\x. x \ z2 \ (P2 x \ P2')" by blast + from lt_ex obtain z where z:"z \ ord.min less_eq z1 z2" by blast + from z have zz1: "z \ z1" and zz2: "z \ z2" by simp_all + {fix x assume H: "x \ z" + from less_trans[OF H zz1] less_trans[OF H zz2] + have "(P1 x \ P2 x) \ (P1' \ P2')" using z1 zz1 z2 zz2 by auto + } + thus ?thesis by blast +qed + +lemma minf_ex[noatp]: assumes ex:"\z. \x. x \ z \ (P x \ P1)" and p1: P1 shows "\ x. P x" +proof- + from ex obtain z where z: "\x. x \ z \ (P x \ P1)" by blast + from lt_ex obtain x where x: "x \ z" by blast + from z x p1 show ?thesis by blast +qed + +end + + +locale constr_dense_linear_order = linorder_no_lb + linorder_no_ub + + fixes between + assumes between_less: "less x y \ less x (between x y) \ less (between x y) y" + and between_same: "between x x = x" + +sublocale constr_dense_linear_order < dense_linear_order + apply unfold_locales + using gt_ex lt_ex between_less + by (auto, rule_tac x="between x y" in exI, simp) + +context constr_dense_linear_order +begin + +lemma rinf_U[noatp]: + assumes fU: "finite U" + and lin_dense: "\x l u. (\ t. l \ t \ t\ u \ t \ U) \ l\ x \ x \ u \ P x + \ (\ y. l \ y \ y \ u \ P y )" + and nmpiU: "\x. \ MP \ \PP \ P x \ (\ u\ U. \ u' \ U. u \ x \ x \ u')" + and nmi: "\ MP" and npi: "\ PP" and ex: "\ x. P x" + shows "\ u\ U. \ u' \ U. P (between u u')" +proof- + from ex obtain x where px: "P x" by blast + from px nmi npi nmpiU have "\ u\ U. \ u' \ U. u \ x \ x \ u'" by auto + then obtain u and u' where uU:"u\ U" and uU': "u' \ U" and ux:"u \ x" and xu':"x \ u'" by auto + from uU have Une: "U \ {}" by auto + term "linorder.Min less_eq" + let ?l = "linorder.Min less_eq U" + let ?u = "linorder.Max less_eq U" + have linM: "?l \ U" using fU Une by simp + have uinM: "?u \ U" using fU Une by simp + have lM: "\ t\ U. ?l \ t" using Une fU by auto + have Mu: "\ t\ U. t \ ?u" using Une fU by auto + have th:"?l \ u" using uU Une lM by auto + from order_trans[OF th ux] have lx: "?l \ x" . + have th: "u' \ ?u" using uU' Une Mu by simp + from order_trans[OF xu' th] have xu: "x \ ?u" . + from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu] + have "(\ s\ U. P s) \ + (\ t1\ U. \ t2 \ U. (\ y. t1 \ y \ y \ t2 \ y \ U) \ t1 \ x \ x \ t2 \ P x)" . + moreover { fix u assume um: "u\U" and pu: "P u" + have "between u u = u" by (simp add: between_same) + with um pu have "P (between u u)" by simp + with um have ?thesis by blast} + moreover{ + assume "\ t1\ U. \ t2 \ U. (\ y. t1 \ y \ y \ t2 \ y \ U) \ t1 \ x \ x \ t2 \ P x" + then obtain t1 and t2 where t1M: "t1 \ U" and t2M: "t2\ U" + and noM: "\ y. t1 \ y \ y \ t2 \ y \ U" and t1x: "t1 \ x" and xt2: "x \ t2" and px: "P x" + by blast + from less_trans[OF t1x xt2] have t1t2: "t1 \ t2" . + let ?u = "between t1 t2" + from between_less t1t2 have t1lu: "t1 \ ?u" and ut2: "?u \ t2" by auto + from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast + with t1M t2M have ?thesis by blast} + ultimately show ?thesis by blast + qed + +theorem fr_eq[noatp]: + assumes fU: "finite U" + and lin_dense: "\x l u. (\ t. l \ t \ t\ u \ t \ U) \ l\ x \ x \ u \ P x + \ (\ y. l \ y \ y \ u \ P y )" + and nmibnd: "\x. \ MP \ P x \ (\ u\ U. u \ x)" + and npibnd: "\x. \PP \ P x \ (\ u\ U. x \ u)" + and mi: "\z. \x. x \ z \ (P x = MP)" and pi: "\z. \x. z \ x \ (P x = PP)" + shows "(\ x. P x) \ (MP \ PP \ (\ u \ U. \ u'\ U. P (between u u')))" + (is "_ \ (_ \ _ \ ?F)" is "?E \ ?D") +proof- + { + assume px: "\ x. P x" + have "MP \ PP \ (\ MP \ \ PP)" by blast + moreover {assume "MP \ PP" hence "?D" by blast} + moreover {assume nmi: "\ MP" and npi: "\ PP" + from npmibnd[OF nmibnd npibnd] + have nmpiU: "\x. \ MP \ \PP \ P x \ (\ u\ U. \ u' \ U. u \ x \ x \ u')" . + from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast} + ultimately have "?D" by blast} + moreover + { assume "?D" + moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .} + moreover {assume p: "PP" from pinf_ex[OF pi p] have "?E" . } + moreover {assume f:"?F" hence "?E" by blast} + ultimately have "?E" by blast} + ultimately have "?E = ?D" by blast thus "?E \ ?D" by simp +qed + +lemmas minf_thms[noatp] = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P +lemmas pinf_thms[noatp] = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P + +lemmas nmi_thms[noatp] = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P +lemmas npi_thms[noatp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P +lemmas lin_dense_thms[noatp] = lin_dense_conj lin_dense_disj lin_dense_eq lin_dense_neq lin_dense_lt lin_dense_le lin_dense_gt lin_dense_ge lin_dense_P + +lemma ferrack_axiom[noatp]: "constr_dense_linear_order less_eq less between" + by (rule constr_dense_linear_order_axioms) +lemma atoms[noatp]: + shows "TERM (less :: 'a \ _)" + and "TERM (less_eq :: 'a \ _)" + and "TERM (op = :: 'a \ _)" . + +declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms + nmi: nmi_thms npi: npi_thms lindense: + lin_dense_thms qe: fr_eq atoms: atoms] + +declaration {* +let +fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}] +fun generic_whatis phi = + let + val [lt, le] = map (Morphism.term phi) [@{term "op \"}, @{term "op \"}] + fun h x t = + case term_of t of + Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq + else Ferrante_Rackoff_Data.Nox + | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq + else Ferrante_Rackoff_Data.Nox + | b$y$z => if Term.could_unify (b, lt) then + if term_of x aconv y then Ferrante_Rackoff_Data.Lt + else if term_of x aconv z then Ferrante_Rackoff_Data.Gt + else Ferrante_Rackoff_Data.Nox + else if Term.could_unify (b, le) then + if term_of x aconv y then Ferrante_Rackoff_Data.Le + else if term_of x aconv z then Ferrante_Rackoff_Data.Ge + else Ferrante_Rackoff_Data.Nox + else Ferrante_Rackoff_Data.Nox + | _ => Ferrante_Rackoff_Data.Nox + in h end + fun ss phi = HOL_ss addsimps (simps phi) +in + Ferrante_Rackoff_Data.funs @{thm "ferrack_axiom"} + {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss} +end +*} + +end + +use "~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML" + +method_setup ferrack = {* + Method.ctxt_args (Method.SIMPLE_METHOD' o FerranteRackoff.dlo_tac) +*} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders" + +subsection {* Ferrante and Rackoff algorithm over ordered fields *} + +lemma neg_prod_lt:"(c\'a\ordered_field) < 0 \ ((c*x < 0) == (x > 0))" +proof- + assume H: "c < 0" + have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps) + also have "\ = (0 < x)" by simp + finally show "(c*x < 0) == (x > 0)" by simp +qed + +lemma pos_prod_lt:"(c\'a\ordered_field) > 0 \ ((c*x < 0) == (x < 0))" +proof- + assume H: "c > 0" + hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps) + also have "\ = (0 > x)" by simp + finally show "(c*x < 0) == (x < 0)" by simp +qed + +lemma neg_prod_sum_lt: "(c\'a\ordered_field) < 0 \ ((c*x + t< 0) == (x > (- 1/c)*t))" +proof- + assume H: "c < 0" + have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp) + also have "\ = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps) + also have "\ = ((- 1/c)*t < x)" by simp + finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simp +qed + +lemma pos_prod_sum_lt:"(c\'a\ordered_field) > 0 \ ((c*x + t < 0) == (x < (- 1/c)*t))" +proof- + assume H: "c > 0" + have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp) + also have "\ = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps) + also have "\ = ((- 1/c)*t > x)" by simp + finally show "(c*x + t < 0) == (x < (- 1/c)*t)" by simp +qed + +lemma sum_lt:"((x::'a::pordered_ab_group_add) + t < 0) == (x < - t)" + using less_diff_eq[where a= x and b=t and c=0] by simp + +lemma neg_prod_le:"(c\'a\ordered_field) < 0 \ ((c*x <= 0) == (x >= 0))" +proof- + assume H: "c < 0" + have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps) + also have "\ = (0 <= x)" by simp + finally show "(c*x <= 0) == (x >= 0)" by simp +qed + +lemma pos_prod_le:"(c\'a\ordered_field) > 0 \ ((c*x <= 0) == (x <= 0))" +proof- + assume H: "c > 0" + hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps) + also have "\ = (0 >= x)" by simp + finally show "(c*x <= 0) == (x <= 0)" by simp +qed + +lemma neg_prod_sum_le: "(c\'a\ordered_field) < 0 \ ((c*x + t <= 0) == (x >= (- 1/c)*t))" +proof- + assume H: "c < 0" + have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp) + also have "\ = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps) + also have "\ = ((- 1/c)*t <= x)" by simp + finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp +qed + +lemma pos_prod_sum_le:"(c\'a\ordered_field) > 0 \ ((c*x + t <= 0) == (x <= (- 1/c)*t))" +proof- + assume H: "c > 0" + have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp) + also have "\ = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps) + also have "\ = ((- 1/c)*t >= x)" by simp + finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp +qed + +lemma sum_le:"((x::'a::pordered_ab_group_add) + t <= 0) == (x <= - t)" + using le_diff_eq[where a= x and b=t and c=0] by simp + +lemma nz_prod_eq:"(c\'a\ordered_field) \ 0 \ ((c*x = 0) == (x = 0))" by simp +lemma nz_prod_sum_eq: "(c\'a\ordered_field) \ 0 \ ((c*x + t = 0) == (x = (- 1/c)*t))" +proof- + assume H: "c \ 0" + have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp) + also have "\ = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] algebra_simps) + finally show "(c*x + t = 0) == (x = (- 1/c)*t)" by simp +qed +lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)" + using eq_diff_eq[where a= x and b=t and c=0] by simp + + +interpretation class_ordered_field_dense_linear_order!: constr_dense_linear_order + "op <=" "op <" + "\ x y. 1/2 * ((x::'a::{ordered_field,recpower,number_ring}) + y)" +proof (unfold_locales, dlo, dlo, auto) + fix x y::'a assume lt: "x < y" + from less_half_sum[OF lt] show "x < (x + y) /2" by simp +next + fix x y::'a assume lt: "x < y" + from gt_half_sum[OF lt] show "(x + y) /2 < y" by simp +qed + +declaration{* +let +fun earlier [] x y = false + | earlier (h::t) x y = + if h aconvc y then false else if h aconvc x then true else earlier t x y; + +fun dest_frac ct = case term_of ct of + Const (@{const_name "HOL.divide"},_) $ a $ b=> + Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b)) + | t => Rat.rat_of_int (snd (HOLogic.dest_number t)) + +fun mk_frac phi cT x = + let val (a, b) = Rat.quotient_of_rat x + in if b = 1 then Numeral.mk_cnumber cT a + else Thm.capply + (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"}) + (Numeral.mk_cnumber cT a)) + (Numeral.mk_cnumber cT b) + end + +fun whatis x ct = case term_of ct of + Const(@{const_name "HOL.plus"}, _)$(Const(@{const_name "HOL.times"},_)$_$y)$_ => + if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct]) + else ("Nox",[]) +| Const(@{const_name "HOL.plus"}, _)$y$_ => + if y aconv term_of x then ("x+t",[Thm.dest_arg ct]) + else ("Nox",[]) +| Const(@{const_name "HOL.times"}, _)$_$y => + if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct]) + else ("Nox",[]) +| t => if t aconv term_of x then ("x",[]) else ("Nox",[]); + +fun xnormalize_conv ctxt [] ct = reflexive ct +| xnormalize_conv ctxt (vs as (x::_)) ct = + case term_of ct of + Const(@{const_name HOL.less},_)$_$Const(@{const_name "HOL.zero"},_) => + (case whatis x (Thm.dest_arg1 ct) of + ("c*x+t",[c,t]) => + let + val cr = dest_frac c + val clt = Thm.dest_fun2 ct + val cz = Thm.dest_arg ct + val neg = cr + let + val T = ctyp_of_term x + val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"} + val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv + (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th + in rth end + | ("c*x",[c]) => + let + val cr = dest_frac c + val clt = Thm.dest_fun2 ct + val cz = Thm.dest_arg ct + val neg = cr reflexive ct) + + +| Const(@{const_name HOL.less_eq},_)$_$Const(@{const_name "HOL.zero"},_) => + (case whatis x (Thm.dest_arg1 ct) of + ("c*x+t",[c,t]) => + let + val T = ctyp_of_term x + val cr = dest_frac c + val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"} + val cz = Thm.dest_arg ct + val neg = cr + let + val T = ctyp_of_term x + val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"} + val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv + (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th + in rth end + | ("c*x",[c]) => + let + val T = ctyp_of_term x + val cr = dest_frac c + val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"} + val cz = Thm.dest_arg ct + val neg = cr reflexive ct) + +| Const("op =",_)$_$Const(@{const_name "HOL.zero"},_) => + (case whatis x (Thm.dest_arg1 ct) of + ("c*x+t",[c,t]) => + let + val T = ctyp_of_term x + val cr = dest_frac c + val ceq = Thm.dest_fun2 ct + val cz = Thm.dest_arg ct + val cthp = Simplifier.rewrite (local_simpset_of ctxt) + (Thm.capply @{cterm "Trueprop"} + (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz))) + val cth = equal_elim (symmetric cthp) TrueI + val th = implies_elim + (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth + val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv + (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th + in rth end + | ("x+t",[t]) => + let + val T = ctyp_of_term x + val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"} + val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv + (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th + in rth end + | ("c*x",[c]) => + let + val T = ctyp_of_term x + val cr = dest_frac c + val ceq = Thm.dest_fun2 ct + val cz = Thm.dest_arg ct + val cthp = Simplifier.rewrite (local_simpset_of ctxt) + (Thm.capply @{cterm "Trueprop"} + (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz))) + val cth = equal_elim (symmetric cthp) TrueI + val rth = implies_elim + (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth + in rth end + | _ => reflexive ct); + +local + val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"} + val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"} + val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"} +in +fun field_isolate_conv phi ctxt vs ct = case term_of ct of + Const(@{const_name HOL.less},_)$a$b => + let val (ca,cb) = Thm.dest_binop ct + val T = ctyp_of_term ca + val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0 + val nth = Conv.fconv_rule + (Conv.arg_conv (Conv.arg1_conv + (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th + val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) + in rth end +| Const(@{const_name HOL.less_eq},_)$a$b => + let val (ca,cb) = Thm.dest_binop ct + val T = ctyp_of_term ca + val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0 + val nth = Conv.fconv_rule + (Conv.arg_conv (Conv.arg1_conv + (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th + val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) + in rth end + +| Const("op =",_)$a$b => + let val (ca,cb) = Thm.dest_binop ct + val T = ctyp_of_term ca + val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0 + val nth = Conv.fconv_rule + (Conv.arg_conv (Conv.arg1_conv + (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th + val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) + in rth end +| @{term "Not"} $(Const("op =",_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct +| _ => reflexive ct +end; + +fun classfield_whatis phi = + let + fun h x t = + case term_of t of + Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq + else Ferrante_Rackoff_Data.Nox + | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq + else Ferrante_Rackoff_Data.Nox + | Const(@{const_name HOL.less},_)$y$z => + if term_of x aconv y then Ferrante_Rackoff_Data.Lt + else if term_of x aconv z then Ferrante_Rackoff_Data.Gt + else Ferrante_Rackoff_Data.Nox + | Const (@{const_name HOL.less_eq},_)$y$z => + if term_of x aconv y then Ferrante_Rackoff_Data.Le + else if term_of x aconv z then Ferrante_Rackoff_Data.Ge + else Ferrante_Rackoff_Data.Nox + | _ => Ferrante_Rackoff_Data.Nox + in h end; +fun class_field_ss phi = + HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}]) + addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}] + +in +Ferrante_Rackoff_Data.funs @{thm "class_ordered_field_dense_linear_order.ferrack_axiom"} + {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss} +end +*} + + +end diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Decision_Procs/Ferrack.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Decision_Procs/Ferrack.thy Fri Feb 06 15:15:46 2009 +0100 @@ -0,0 +1,2101 @@ +(* Title: HOL/Reflection/Ferrack.thy + Author: Amine Chaieb +*) + +theory Ferrack +imports Complex_Main Dense_Linear_Order Efficient_Nat +uses ("ferrack_tac.ML") +begin + +section {* Quantifier elimination for @{text "\ (0, 1, +, <)"} *} + + (*********************************************************************************) + (* SOME GENERAL STUFF< HAS TO BE MOVED IN SOME LIB *) + (*********************************************************************************) + +consts alluopairs:: "'a list \ ('a \ 'a) list" +primrec + "alluopairs [] = []" + "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)" + +lemma alluopairs_set1: "set (alluopairs xs) \ {(x,y). x\ set xs \ y\ set xs}" +by (induct xs, auto) + +lemma alluopairs_set: + "\x\ set xs ; y \ set xs\ \ (x,y) \ set (alluopairs xs) \ (y,x) \ set (alluopairs xs) " +by (induct xs, auto) + +lemma alluopairs_ex: + assumes Pc: "\ x y. P x y = P y x" + shows "(\ x \ set xs. \ y \ set xs. P x y) = (\ (x,y) \ set (alluopairs xs). P x y)" +proof + assume "\x\set xs. \y\set xs. P x y" + then obtain x y where x: "x \ set xs" and y:"y \ set xs" and P: "P x y" by blast + from alluopairs_set[OF x y] P Pc show"\(x, y)\set (alluopairs xs). P x y" + by auto +next + assume "\(x, y)\set (alluopairs xs). P x y" + then obtain "x" and "y" where xy:"(x,y) \ set (alluopairs xs)" and P: "P x y" by blast+ + from xy have "x \ set xs \ y\ set xs" using alluopairs_set1 by blast + with P show "\x\set xs. \y\set xs. P x y" by blast +qed + +lemma nth_pos2: "0 < n \ (x#xs) ! n = xs ! (n - 1)" +using Nat.gr0_conv_Suc +by clarsimp + +lemma filter_length: "length (List.filter P xs) < Suc (length xs)" + apply (induct xs, auto) done + +consts remdps:: "'a list \ 'a list" + +recdef remdps "measure size" + "remdps [] = []" + "remdps (x#xs) = (x#(remdps (List.filter (\ y. y \ x) xs)))" +(hints simp add: filter_length[rule_format]) + +lemma remdps_set[simp]: "set (remdps xs) = set xs" + by (induct xs rule: remdps.induct, auto) + + + + (*********************************************************************************) + (**** SHADOW SYNTAX AND SEMANTICS ****) + (*********************************************************************************) + +datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num + | Mul int num + + (* A size for num to make inductive proofs simpler*) +consts num_size :: "num \ nat" +primrec + "num_size (C c) = 1" + "num_size (Bound n) = 1" + "num_size (Neg a) = 1 + num_size a" + "num_size (Add a b) = 1 + num_size a + num_size b" + "num_size (Sub a b) = 3 + num_size a + num_size b" + "num_size (Mul c a) = 1 + num_size a" + "num_size (CN n c a) = 3 + num_size a " + + (* Semantics of numeral terms (num) *) +consts Inum :: "real list \ num \ real" +primrec + "Inum bs (C c) = (real c)" + "Inum bs (Bound n) = bs!n" + "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)" + "Inum bs (Neg a) = -(Inum bs a)" + "Inum bs (Add a b) = Inum bs a + Inum bs b" + "Inum bs (Sub a b) = Inum bs a - Inum bs b" + "Inum bs (Mul c a) = (real c) * Inum bs a" + (* FORMULAE *) +datatype fm = + T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| + NOT fm| And fm fm| Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm + + + (* A size for fm *) +consts fmsize :: "fm \ nat" +recdef fmsize "measure size" + "fmsize (NOT p) = 1 + fmsize p" + "fmsize (And p q) = 1 + fmsize p + fmsize q" + "fmsize (Or p q) = 1 + fmsize p + fmsize q" + "fmsize (Imp p q) = 3 + fmsize p + fmsize q" + "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)" + "fmsize (E p) = 1 + fmsize p" + "fmsize (A p) = 4+ fmsize p" + "fmsize p = 1" + (* several lemmas about fmsize *) +lemma fmsize_pos: "fmsize p > 0" +by (induct p rule: fmsize.induct) simp_all + + (* Semantics of formulae (fm) *) +consts Ifm ::"real list \ fm \ bool" +primrec + "Ifm bs T = True" + "Ifm bs F = False" + "Ifm bs (Lt a) = (Inum bs a < 0)" + "Ifm bs (Gt a) = (Inum bs a > 0)" + "Ifm bs (Le a) = (Inum bs a \ 0)" + "Ifm bs (Ge a) = (Inum bs a \ 0)" + "Ifm bs (Eq a) = (Inum bs a = 0)" + "Ifm bs (NEq a) = (Inum bs a \ 0)" + "Ifm bs (NOT p) = (\ (Ifm bs p))" + "Ifm bs (And p q) = (Ifm bs p \ Ifm bs q)" + "Ifm bs (Or p q) = (Ifm bs p \ Ifm bs q)" + "Ifm bs (Imp p q) = ((Ifm bs p) \ (Ifm bs q))" + "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)" + "Ifm bs (E p) = (\ x. Ifm (x#bs) p)" + "Ifm bs (A p) = (\ x. Ifm (x#bs) p)" + +lemma IfmLeSub: "\ Inum bs s = s' ; Inum bs t = t' \ \ Ifm bs (Le (Sub s t)) = (s' \ t')" +apply simp +done + +lemma IfmLtSub: "\ Inum bs s = s' ; Inum bs t = t' \ \ Ifm bs (Lt (Sub s t)) = (s' < t')" +apply simp +done +lemma IfmEqSub: "\ Inum bs s = s' ; Inum bs t = t' \ \ Ifm bs (Eq (Sub s t)) = (s' = t')" +apply simp +done +lemma IfmNOT: " (Ifm bs p = P) \ (Ifm bs (NOT p) = (\P))" +apply simp +done +lemma IfmAnd: " \ Ifm bs p = P ; Ifm bs q = Q\ \ (Ifm bs (And p q) = (P \ Q))" +apply simp +done +lemma IfmOr: " \ Ifm bs p = P ; Ifm bs q = Q\ \ (Ifm bs (Or p q) = (P \ Q))" +apply simp +done +lemma IfmImp: " \ Ifm bs p = P ; Ifm bs q = Q\ \ (Ifm bs (Imp p q) = (P \ Q))" +apply simp +done +lemma IfmIff: " \ Ifm bs p = P ; Ifm bs q = Q\ \ (Ifm bs (Iff p q) = (P = Q))" +apply simp +done + +lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \ (Ifm bs (E p) = (\x. P x))" +apply simp +done +lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \ (Ifm bs (A p) = (\x. P x))" +apply simp +done + +consts not:: "fm \ fm" +recdef not "measure size" + "not (NOT p) = p" + "not T = F" + "not F = T" + "not p = NOT p" +lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)" +by (cases p) auto + +constdefs conj :: "fm \ fm \ fm" + "conj p q \ (if (p = F \ q=F) then F else if p=T then q else if q=T then p else + if p = q then p else And p q)" +lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)" +by (cases "p=F \ q=F",simp_all add: conj_def) (cases p,simp_all) + +constdefs disj :: "fm \ fm \ fm" + "disj p q \ (if (p = T \ q=T) then T else if p=F then q else if q=F then p + else if p=q then p else Or p q)" + +lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)" +by (cases "p=T \ q=T",simp_all add: disj_def) (cases p,simp_all) + +constdefs imp :: "fm \ fm \ fm" + "imp p q \ (if (p = F \ q=T \ p=q) then T else if p=T then q else if q=F then not p + else Imp p q)" +lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)" +by (cases "p=F \ q=T",simp_all add: imp_def) + +constdefs iff :: "fm \ fm \ fm" + "iff p q \ (if (p = q) then T else if (p = NOT q \ NOT p = q) then F else + if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else + Iff p q)" +lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)" + by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto) + +lemma conj_simps: + "conj F Q = F" + "conj P F = F" + "conj T Q = Q" + "conj P T = P" + "conj P P = P" + "P \ T \ P \ F \ Q \ T \ Q \ F \ P \ Q \ conj P Q = And P Q" + by (simp_all add: conj_def) + +lemma disj_simps: + "disj T Q = T" + "disj P T = T" + "disj F Q = Q" + "disj P F = P" + "disj P P = P" + "P \ T \ P \ F \ Q \ T \ Q \ F \ P \ Q \ disj P Q = Or P Q" + by (simp_all add: disj_def) +lemma imp_simps: + "imp F Q = T" + "imp P T = T" + "imp T Q = Q" + "imp P F = not P" + "imp P P = T" + "P \ T \ P \ F \ P \ Q \ Q \ T \ Q \ F \ imp P Q = Imp P Q" + by (simp_all add: imp_def) +lemma trivNOT: "p \ NOT p" "NOT p \ p" +apply (induct p, auto) +done + +lemma iff_simps: + "iff p p = T" + "iff p (NOT p) = F" + "iff (NOT p) p = F" + "iff p F = not p" + "iff F p = not p" + "p \ NOT T \ iff T p = p" + "p\ NOT T \ iff p T = p" + "p\q \ p\ NOT q \ q\ NOT p \ p\ F \ q\ F \ p \ T \ q \ T \ iff p q = Iff p q" + using trivNOT + by (simp_all add: iff_def, cases p, auto) + (* Quantifier freeness *) +consts qfree:: "fm \ bool" +recdef qfree "measure size" + "qfree (E p) = False" + "qfree (A p) = False" + "qfree (NOT p) = qfree p" + "qfree (And p q) = (qfree p \ qfree q)" + "qfree (Or p q) = (qfree p \ qfree q)" + "qfree (Imp p q) = (qfree p \ qfree q)" + "qfree (Iff p q) = (qfree p \ qfree q)" + "qfree p = True" + + (* Boundedness and substitution *) +consts + numbound0:: "num \ bool" (* a num is INDEPENDENT of Bound 0 *) + bound0:: "fm \ bool" (* A Formula is independent of Bound 0 *) +primrec + "numbound0 (C c) = True" + "numbound0 (Bound n) = (n>0)" + "numbound0 (CN n c a) = (n\0 \ numbound0 a)" + "numbound0 (Neg a) = numbound0 a" + "numbound0 (Add a b) = (numbound0 a \ numbound0 b)" + "numbound0 (Sub a b) = (numbound0 a \ numbound0 b)" + "numbound0 (Mul i a) = numbound0 a" +lemma numbound0_I: + assumes nb: "numbound0 a" + shows "Inum (b#bs) a = Inum (b'#bs) a" +using nb +by (induct a rule: numbound0.induct,auto simp add: nth_pos2) + +primrec + "bound0 T = True" + "bound0 F = True" + "bound0 (Lt a) = numbound0 a" + "bound0 (Le a) = numbound0 a" + "bound0 (Gt a) = numbound0 a" + "bound0 (Ge a) = numbound0 a" + "bound0 (Eq a) = numbound0 a" + "bound0 (NEq a) = numbound0 a" + "bound0 (NOT p) = bound0 p" + "bound0 (And p q) = (bound0 p \ bound0 q)" + "bound0 (Or p q) = (bound0 p \ bound0 q)" + "bound0 (Imp p q) = ((bound0 p) \ (bound0 q))" + "bound0 (Iff p q) = (bound0 p \ bound0 q)" + "bound0 (E p) = False" + "bound0 (A p) = False" + +lemma bound0_I: + assumes bp: "bound0 p" + shows "Ifm (b#bs) p = Ifm (b'#bs) p" +using bp numbound0_I[where b="b" and bs="bs" and b'="b'"] +by (induct p rule: bound0.induct) (auto simp add: nth_pos2) + +lemma not_qf[simp]: "qfree p \ qfree (not p)" +by (cases p, auto) +lemma not_bn[simp]: "bound0 p \ bound0 (not p)" +by (cases p, auto) + + +lemma conj_qf[simp]: "\qfree p ; qfree q\ \ qfree (conj p q)" +using conj_def by auto +lemma conj_nb[simp]: "\bound0 p ; bound0 q\ \ bound0 (conj p q)" +using conj_def by auto + +lemma disj_qf[simp]: "\qfree p ; qfree q\ \ qfree (disj p q)" +using disj_def by auto +lemma disj_nb[simp]: "\bound0 p ; bound0 q\ \ bound0 (disj p q)" +using disj_def by auto + +lemma imp_qf[simp]: "\qfree p ; qfree q\ \ qfree (imp p q)" +using imp_def by (cases "p=F \ q=T",simp_all add: imp_def) +lemma imp_nb[simp]: "\bound0 p ; bound0 q\ \ bound0 (imp p q)" +using imp_def by (cases "p=F \ q=T \ p=q",simp_all add: imp_def) + +lemma iff_qf[simp]: "\qfree p ; qfree q\ \ qfree (iff p q)" + by (unfold iff_def,cases "p=q", auto) +lemma iff_nb[simp]: "\bound0 p ; bound0 q\ \ bound0 (iff p q)" +using iff_def by (unfold iff_def,cases "p=q", auto) + +consts + decrnum:: "num \ num" + decr :: "fm \ fm" + +recdef decrnum "measure size" + "decrnum (Bound n) = Bound (n - 1)" + "decrnum (Neg a) = Neg (decrnum a)" + "decrnum (Add a b) = Add (decrnum a) (decrnum b)" + "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)" + "decrnum (Mul c a) = Mul c (decrnum a)" + "decrnum (CN n c a) = CN (n - 1) c (decrnum a)" + "decrnum a = a" + +recdef decr "measure size" + "decr (Lt a) = Lt (decrnum a)" + "decr (Le a) = Le (decrnum a)" + "decr (Gt a) = Gt (decrnum a)" + "decr (Ge a) = Ge (decrnum a)" + "decr (Eq a) = Eq (decrnum a)" + "decr (NEq a) = NEq (decrnum a)" + "decr (NOT p) = NOT (decr p)" + "decr (And p q) = conj (decr p) (decr q)" + "decr (Or p q) = disj (decr p) (decr q)" + "decr (Imp p q) = imp (decr p) (decr q)" + "decr (Iff p q) = iff (decr p) (decr q)" + "decr p = p" + +lemma decrnum: assumes nb: "numbound0 t" + shows "Inum (x#bs) t = Inum bs (decrnum t)" + using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2) + +lemma decr: assumes nb: "bound0 p" + shows "Ifm (x#bs) p = Ifm bs (decr p)" + using nb + by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum) + +lemma decr_qf: "bound0 p \ qfree (decr p)" +by (induct p, simp_all) + +consts + isatom :: "fm \ bool" (* test for atomicity *) +recdef isatom "measure size" + "isatom T = True" + "isatom F = True" + "isatom (Lt a) = True" + "isatom (Le a) = True" + "isatom (Gt a) = True" + "isatom (Ge a) = True" + "isatom (Eq a) = True" + "isatom (NEq a) = True" + "isatom p = False" + +lemma bound0_qf: "bound0 p \ qfree p" +by (induct p, simp_all) + +constdefs djf:: "('a \ fm) \ 'a \ fm \ fm" + "djf f p q \ (if q=T then T else if q=F then f p else + (let fp = f p in case fp of T \ T | F \ q | _ \ Or (f p) q))" +constdefs evaldjf:: "('a \ fm) \ 'a list \ fm" + "evaldjf f ps \ foldr (djf f) ps F" + +lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)" +by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) +(cases "f p", simp_all add: Let_def djf_def) + + +lemma djf_simps: + "djf f p T = T" + "djf f p F = f p" + "q\T \ q\F \ djf f p q = (let fp = f p in case fp of T \ T | F \ q | _ \ Or (f p) q)" + by (simp_all add: djf_def) + +lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\ p \ set ps. Ifm bs (f p))" + by(induct ps, simp_all add: evaldjf_def djf_Or) + +lemma evaldjf_bound0: + assumes nb: "\ x\ set xs. bound0 (f x)" + shows "bound0 (evaldjf f xs)" + using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) + +lemma evaldjf_qf: + assumes nb: "\ x\ set xs. qfree (f x)" + shows "qfree (evaldjf f xs)" + using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) + +consts disjuncts :: "fm \ fm list" +recdef disjuncts "measure size" + "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)" + "disjuncts F = []" + "disjuncts p = [p]" + +lemma disjuncts: "(\ q\ set (disjuncts p). Ifm bs q) = Ifm bs p" +by(induct p rule: disjuncts.induct, auto) + +lemma disjuncts_nb: "bound0 p \ \ q\ set (disjuncts p). bound0 q" +proof- + assume nb: "bound0 p" + hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto) + thus ?thesis by (simp only: list_all_iff) +qed + +lemma disjuncts_qf: "qfree p \ \ q\ set (disjuncts p). qfree q" +proof- + assume qf: "qfree p" + hence "list_all qfree (disjuncts p)" + by (induct p rule: disjuncts.induct, auto) + thus ?thesis by (simp only: list_all_iff) +qed + +constdefs DJ :: "(fm \ fm) \ fm \ fm" + "DJ f p \ evaldjf f (disjuncts p)" + +lemma DJ: assumes fdj: "\ p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))" + and fF: "f F = F" + shows "Ifm bs (DJ f p) = Ifm bs (f p)" +proof- + have "Ifm bs (DJ f p) = (\ q \ set (disjuncts p). Ifm bs (f q))" + by (simp add: DJ_def evaldjf_ex) + also have "\ = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto) + finally show ?thesis . +qed + +lemma DJ_qf: assumes + fqf: "\ p. qfree p \ qfree (f p)" + shows "\p. qfree p \ qfree (DJ f p) " +proof(clarify) + fix p assume qf: "qfree p" + have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def) + from disjuncts_qf[OF qf] have "\ q\ set (disjuncts p). qfree q" . + with fqf have th':"\ q\ set (disjuncts p). qfree (f q)" by blast + + from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp +qed + +lemma DJ_qe: assumes qe: "\ bs p. qfree p \ qfree (qe p) \ (Ifm bs (qe p) = Ifm bs (E p))" + shows "\ bs p. qfree p \ qfree (DJ qe p) \ (Ifm bs ((DJ qe p)) = Ifm bs (E p))" +proof(clarify) + fix p::fm and bs + assume qf: "qfree p" + from qe have qth: "\ p. qfree p \ qfree (qe p)" by blast + from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto + have "Ifm bs (DJ qe p) = (\ q\ set (disjuncts p). Ifm bs (qe q))" + by (simp add: DJ_def evaldjf_ex) + also have "\ = (\ q \ set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto + also have "\ = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto) + finally show "qfree (DJ qe p) \ Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast +qed + (* Simplification *) +consts + numgcd :: "num \ int" + numgcdh:: "num \ int \ int" + reducecoeffh:: "num \ int \ num" + reducecoeff :: "num \ num" + dvdnumcoeff:: "num \ int \ bool" +consts maxcoeff:: "num \ int" +recdef maxcoeff "measure size" + "maxcoeff (C i) = abs i" + "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)" + "maxcoeff t = 1" + +lemma maxcoeff_pos: "maxcoeff t \ 0" + by (induct t rule: maxcoeff.induct, auto) + +recdef numgcdh "measure size" + "numgcdh (C i) = (\g. zgcd i g)" + "numgcdh (CN n c t) = (\g. zgcd c (numgcdh t g))" + "numgcdh t = (\g. 1)" +defs numgcd_def [code]: "numgcd t \ numgcdh t (maxcoeff t)" + +recdef reducecoeffh "measure size" + "reducecoeffh (C i) = (\ g. C (i div g))" + "reducecoeffh (CN n c t) = (\ g. CN n (c div g) (reducecoeffh t g))" + "reducecoeffh t = (\g. t)" + +defs reducecoeff_def: "reducecoeff t \ + (let g = numgcd t in + if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)" + +recdef dvdnumcoeff "measure size" + "dvdnumcoeff (C i) = (\ g. g dvd i)" + "dvdnumcoeff (CN n c t) = (\ g. g dvd c \ (dvdnumcoeff t g))" + "dvdnumcoeff t = (\g. False)" + +lemma dvdnumcoeff_trans: + assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'" + shows "dvdnumcoeff t g" + using dgt' gdg + by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg zdvd_trans[OF gdg]) + +declare zdvd_trans [trans add] + +lemma natabs0: "(nat (abs x) = 0) = (x = 0)" +by arith + +lemma numgcd0: + assumes g0: "numgcd t = 0" + shows "Inum bs t = 0" + using g0[simplified numgcd_def] + by (induct t rule: numgcdh.induct, auto simp add: zgcd_def gcd_zero natabs0 max_def maxcoeff_pos) + +lemma numgcdh_pos: assumes gp: "g \ 0" shows "numgcdh t g \ 0" + using gp + by (induct t rule: numgcdh.induct, auto simp add: zgcd_def) + +lemma numgcd_pos: "numgcd t \0" + by (simp add: numgcd_def numgcdh_pos maxcoeff_pos) + +lemma reducecoeffh: + assumes gt: "dvdnumcoeff t g" and gp: "g > 0" + shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t" + using gt +proof(induct t rule: reducecoeffh.induct) + case (1 i) hence gd: "g dvd i" by simp + from gp have gnz: "g \ 0" by simp + from prems show ?case by (simp add: real_of_int_div[OF gnz gd]) +next + case (2 n c t) hence gd: "g dvd c" by simp + from gp have gnz: "g \ 0" by simp + from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps) +qed (auto simp add: numgcd_def gp) +consts ismaxcoeff:: "num \ int \ bool" +recdef ismaxcoeff "measure size" + "ismaxcoeff (C i) = (\ x. abs i \ x)" + "ismaxcoeff (CN n c t) = (\x. abs c \ x \ (ismaxcoeff t x))" + "ismaxcoeff t = (\x. True)" + +lemma ismaxcoeff_mono: "ismaxcoeff t c \ c \ c' \ ismaxcoeff t c'" +by (induct t rule: ismaxcoeff.induct, auto) + +lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)" +proof (induct t rule: maxcoeff.induct) + case (2 n c t) + hence H:"ismaxcoeff t (maxcoeff t)" . + have thh: "maxcoeff t \ max (abs c) (maxcoeff t)" by (simp add: le_maxI2) + from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1) +qed simp_all + +lemma zgcd_gt1: "zgcd i j > 1 \ ((abs i > 1 \ abs j > 1) \ (abs i = 0 \ abs j > 1) \ (abs i > 1 \ abs j = 0))" + apply (cases "abs i = 0", simp_all add: zgcd_def) + apply (cases "abs j = 0", simp_all) + apply (cases "abs i = 1", simp_all) + apply (cases "abs j = 1", simp_all) + apply auto + done +lemma numgcdh0:"numgcdh t m = 0 \ m =0" + by (induct t rule: numgcdh.induct, auto simp add:zgcd0) + +lemma dvdnumcoeff_aux: + assumes "ismaxcoeff t m" and mp:"m \ 0" and "numgcdh t m > 1" + shows "dvdnumcoeff t (numgcdh t m)" +using prems +proof(induct t rule: numgcdh.induct) + case (2 n c t) + let ?g = "numgcdh t m" + from prems have th:"zgcd c ?g > 1" by simp + from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"] + have "(abs c > 1 \ ?g > 1) \ (abs c = 0 \ ?g > 1) \ (abs c > 1 \ ?g = 0)" by simp + moreover {assume "abs c > 1" and gp: "?g > 1" with prems + have th: "dvdnumcoeff t ?g" by simp + have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2) + from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)} + moreover {assume "abs c = 0 \ ?g > 1" + with prems have th: "dvdnumcoeff t ?g" by simp + have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2) + from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1) + hence ?case by simp } + moreover {assume "abs c > 1" and g0:"?g = 0" + from numgcdh0[OF g0] have "m=0". with prems have ?case by simp } + ultimately show ?case by blast +qed(auto simp add: zgcd_zdvd1) + +lemma dvdnumcoeff_aux2: + assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \ numgcd t > 0" + using prems +proof (simp add: numgcd_def) + let ?mc = "maxcoeff t" + let ?g = "numgcdh t ?mc" + have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff) + have th2: "?mc \ 0" by (rule maxcoeff_pos) + assume H: "numgcdh t ?mc > 1" + from dvdnumcoeff_aux[OF th1 th2 H] show "dvdnumcoeff t ?g" . +qed + +lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t" +proof- + let ?g = "numgcd t" + have "?g \ 0" by (simp add: numgcd_pos) + hence "?g = 0 \ ?g = 1 \ ?g > 1" by auto + moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} + moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} + moreover { assume g1:"?g > 1" + from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+ + from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis + by (simp add: reducecoeff_def Let_def)} + ultimately show ?thesis by blast +qed + +lemma reducecoeffh_numbound0: "numbound0 t \ numbound0 (reducecoeffh t g)" +by (induct t rule: reducecoeffh.induct, auto) + +lemma reducecoeff_numbound0: "numbound0 t \ numbound0 (reducecoeff t)" +using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def) + +consts + simpnum:: "num \ num" + numadd:: "num \ num \ num" + nummul:: "num \ int \ num" +recdef numadd "measure (\ (t,s). size t + size s)" + "numadd (CN n1 c1 r1,CN n2 c2 r2) = + (if n1=n2 then + (let c = c1 + c2 + in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2)))) + else if n1 \ n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2))) + else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))" + "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))" + "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" + "numadd (C b1, C b2) = C (b1+b2)" + "numadd (a,b) = Add a b" + +lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)" +apply (induct t s rule: numadd.induct, simp_all add: Let_def) +apply (case_tac "c1+c2 = 0",case_tac "n1 \ n2", simp_all) +apply (case_tac "n1 = n2", simp_all add: algebra_simps) +by (simp only: left_distrib[symmetric],simp) + +lemma numadd_nb[simp]: "\ numbound0 t ; numbound0 s\ \ numbound0 (numadd (t,s))" +by (induct t s rule: numadd.induct, auto simp add: Let_def) + +recdef nummul "measure size" + "nummul (C j) = (\ i. C (i*j))" + "nummul (CN n c a) = (\ i. CN n (i*c) (nummul a i))" + "nummul t = (\ i. Mul i t)" + +lemma nummul[simp]: "\ i. Inum bs (nummul t i) = Inum bs (Mul i t)" +by (induct t rule: nummul.induct, auto simp add: algebra_simps) + +lemma nummul_nb[simp]: "\ i. numbound0 t \ numbound0 (nummul t i)" +by (induct t rule: nummul.induct, auto ) + +constdefs numneg :: "num \ num" + "numneg t \ nummul t (- 1)" + +constdefs numsub :: "num \ num \ num" + "numsub s t \ (if s = t then C 0 else numadd (s,numneg t))" + +lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)" +using numneg_def by simp + +lemma numneg_nb[simp]: "numbound0 t \ numbound0 (numneg t)" +using numneg_def by simp + +lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)" +using numsub_def by simp + +lemma numsub_nb[simp]: "\ numbound0 t ; numbound0 s\ \ numbound0 (numsub t s)" +using numsub_def by simp + +recdef simpnum "measure size" + "simpnum (C j) = C j" + "simpnum (Bound n) = CN n 1 (C 0)" + "simpnum (Neg t) = numneg (simpnum t)" + "simpnum (Add t s) = numadd (simpnum t,simpnum s)" + "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)" + "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)" + "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))" + +lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t" +by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul) + +lemma simpnum_numbound0[simp]: + "numbound0 t \ numbound0 (simpnum t)" +by (induct t rule: simpnum.induct, auto) + +consts nozerocoeff:: "num \ bool" +recdef nozerocoeff "measure size" + "nozerocoeff (C c) = True" + "nozerocoeff (CN n c t) = (c\0 \ nozerocoeff t)" + "nozerocoeff t = True" + +lemma numadd_nz : "nozerocoeff a \ nozerocoeff b \ nozerocoeff (numadd (a,b))" +by (induct a b rule: numadd.induct,auto simp add: Let_def) + +lemma nummul_nz : "\ i. i\0 \ nozerocoeff a \ nozerocoeff (nummul a i)" +by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz) + +lemma numneg_nz : "nozerocoeff a \ nozerocoeff (numneg a)" +by (simp add: numneg_def nummul_nz) + +lemma numsub_nz: "nozerocoeff a \ nozerocoeff b \ nozerocoeff (numsub a b)" +by (simp add: numsub_def numneg_nz numadd_nz) + +lemma simpnum_nz: "nozerocoeff (simpnum t)" +by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz) + +lemma maxcoeff_nz: "nozerocoeff t \ maxcoeff t = 0 \ t = C 0" +proof (induct t rule: maxcoeff.induct) + case (2 n c t) + hence cnz: "c \0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+ + have "max (abs c) (maxcoeff t) \ abs c" by (simp add: le_maxI1) + with cnz have "max (abs c) (maxcoeff t) > 0" by arith + with prems show ?case by simp +qed auto + +lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0" +proof- + from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def) + from numgcdh0[OF th] have th:"maxcoeff t = 0" . + from maxcoeff_nz[OF nz th] show ?thesis . +qed + +constdefs simp_num_pair:: "(num \ int) \ num \ int" + "simp_num_pair \ (\ (t,n). (if n = 0 then (C 0, 0) else + (let t' = simpnum t ; g = numgcd t' in + if g > 1 then (let g' = zgcd n g in + if g' = 1 then (t',n) + else (reducecoeffh t' g', n div g')) + else (t',n))))" + +lemma simp_num_pair_ci: + shows "((\ (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\ (t,n). Inum bs t / real n) (t,n))" + (is "?lhs = ?rhs") +proof- + let ?t' = "simpnum t" + let ?g = "numgcd ?t'" + let ?g' = "zgcd n ?g" + {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)} + moreover + { assume nnz: "n \ 0" + {assume "\ ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)} + moreover + {assume g1:"?g>1" hence g0: "?g > 0" by simp + from zgcd0 g1 nnz have gp0: "?g' \ 0" by simp + hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith + hence "?g'= 1 \ ?g' > 1" by arith + moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)} + moreover {assume g'1:"?g'>1" + from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" .. + let ?tt = "reducecoeffh ?t' ?g'" + let ?t = "Inum bs ?tt" + have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2) + have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1) + have gpdgp: "?g' dvd ?g'" by simp + from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] + have th2:"real ?g' * ?t = Inum bs ?t'" by simp + from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def) + also have "\ = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp + also have "\ = (Inum bs ?t' / real n)" + using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp + finally have "?lhs = Inum bs t / real n" by (simp add: simpnum_ci) + then have ?thesis using prems by (simp add: simp_num_pair_def)} + ultimately have ?thesis by blast} + ultimately have ?thesis by blast} + ultimately show ?thesis by blast +qed + +lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')" + shows "numbound0 t' \ n' >0" +proof- + let ?t' = "simpnum t" + let ?g = "numgcd ?t'" + let ?g' = "zgcd n ?g" + {assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)} + moreover + { assume nnz: "n \ 0" + {assume "\ ?g > 1" hence ?thesis using prems by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)} + moreover + {assume g1:"?g>1" hence g0: "?g > 0" by simp + from zgcd0 g1 nnz have gp0: "?g' \ 0" by simp + hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith + hence "?g'= 1 \ ?g' > 1" by arith + moreover {assume "?g'=1" hence ?thesis using prems + by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)} + moreover {assume g'1:"?g'>1" + have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2) + have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1) + have gpdgp: "?g' dvd ?g'" by simp + from zdvd_imp_le[OF gpdd np] have g'n: "?g' \ n" . + from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]] + have "n div ?g' >0" by simp + hence ?thesis using prems + by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0)} + ultimately have ?thesis by blast} + ultimately have ?thesis by blast} + ultimately show ?thesis by blast +qed + +consts simpfm :: "fm \ fm" +recdef simpfm "measure fmsize" + "simpfm (And p q) = conj (simpfm p) (simpfm q)" + "simpfm (Or p q) = disj (simpfm p) (simpfm q)" + "simpfm (Imp p q) = imp (simpfm p) (simpfm q)" + "simpfm (Iff p q) = iff (simpfm p) (simpfm q)" + "simpfm (NOT p) = not (simpfm p)" + "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \ if (v < 0) then T else F + | _ \ Lt a')" + "simpfm (Le a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ Le a')" + "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \ if (v > 0) then T else F | _ \ Gt a')" + "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ Ge a')" + "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \ if (v = 0) then T else F | _ \ Eq a')" + "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ NEq a')" + "simpfm p = p" +lemma simpfm: "Ifm bs (simpfm p) = Ifm bs p" +proof(induct p rule: simpfm.induct) + case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\ (\ v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (7 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\ (\ v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (8 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\ (\ v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (9 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\ (\ v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (10 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\ (\ v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (11 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\ (\ v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not) + + +lemma simpfm_bound0: "bound0 p \ bound0 (simpfm p)" +proof(induct p rule: simpfm.induct) + case (6 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (7 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (8 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (9 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (10 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (11 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +qed(auto simp add: disj_def imp_def iff_def conj_def not_bn) + +lemma simpfm_qf: "qfree p \ qfree (simpfm p)" +by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def) + (case_tac "simpnum a",auto)+ + +consts prep :: "fm \ fm" +recdef prep "measure fmsize" + "prep (E T) = T" + "prep (E F) = F" + "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))" + "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))" + "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" + "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))" + "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))" + "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))" + "prep (E p) = E (prep p)" + "prep (A (And p q)) = conj (prep (A p)) (prep (A q))" + "prep (A p) = prep (NOT (E (NOT p)))" + "prep (NOT (NOT p)) = prep p" + "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))" + "prep (NOT (A p)) = prep (E (NOT p))" + "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))" + "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))" + "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))" + "prep (NOT p) = not (prep p)" + "prep (Or p q) = disj (prep p) (prep q)" + "prep (And p q) = conj (prep p) (prep q)" + "prep (Imp p q) = prep (Or (NOT p) q)" + "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))" + "prep p = p" +(hints simp add: fmsize_pos) +lemma prep: "\ bs. Ifm bs (prep p) = Ifm bs p" +by (induct p rule: prep.induct, auto) + + (* Generic quantifier elimination *) +consts qelim :: "fm \ (fm \ fm) \ fm" +recdef qelim "measure fmsize" + "qelim (E p) = (\ qe. DJ qe (qelim p qe))" + "qelim (A p) = (\ qe. not (qe ((qelim (NOT p) qe))))" + "qelim (NOT p) = (\ qe. not (qelim p qe))" + "qelim (And p q) = (\ qe. conj (qelim p qe) (qelim q qe))" + "qelim (Or p q) = (\ qe. disj (qelim p qe) (qelim q qe))" + "qelim (Imp p q) = (\ qe. imp (qelim p qe) (qelim q qe))" + "qelim (Iff p q) = (\ qe. iff (qelim p qe) (qelim q qe))" + "qelim p = (\ y. simpfm p)" + +lemma qelim_ci: + assumes qe_inv: "\ bs p. qfree p \ qfree (qe p) \ (Ifm bs (qe p) = Ifm bs (E p))" + shows "\ bs. qfree (qelim p qe) \ (Ifm bs (qelim p qe) = Ifm bs p)" +using qe_inv DJ_qe[OF qe_inv] +by(induct p rule: qelim.induct) +(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf + simpfm simpfm_qf simp del: simpfm.simps) + +consts + plusinf:: "fm \ fm" (* Virtual substitution of +\*) + minusinf:: "fm \ fm" (* Virtual substitution of -\*) +recdef minusinf "measure size" + "minusinf (And p q) = conj (minusinf p) (minusinf q)" + "minusinf (Or p q) = disj (minusinf p) (minusinf q)" + "minusinf (Eq (CN 0 c e)) = F" + "minusinf (NEq (CN 0 c e)) = T" + "minusinf (Lt (CN 0 c e)) = T" + "minusinf (Le (CN 0 c e)) = T" + "minusinf (Gt (CN 0 c e)) = F" + "minusinf (Ge (CN 0 c e)) = F" + "minusinf p = p" + +recdef plusinf "measure size" + "plusinf (And p q) = conj (plusinf p) (plusinf q)" + "plusinf (Or p q) = disj (plusinf p) (plusinf q)" + "plusinf (Eq (CN 0 c e)) = F" + "plusinf (NEq (CN 0 c e)) = T" + "plusinf (Lt (CN 0 c e)) = F" + "plusinf (Le (CN 0 c e)) = F" + "plusinf (Gt (CN 0 c e)) = T" + "plusinf (Ge (CN 0 c e)) = T" + "plusinf p = p" + +consts + isrlfm :: "fm \ bool" (* Linearity test for fm *) +recdef isrlfm "measure size" + "isrlfm (And p q) = (isrlfm p \ isrlfm q)" + "isrlfm (Or p q) = (isrlfm p \ isrlfm q)" + "isrlfm (Eq (CN 0 c e)) = (c>0 \ numbound0 e)" + "isrlfm (NEq (CN 0 c e)) = (c>0 \ numbound0 e)" + "isrlfm (Lt (CN 0 c e)) = (c>0 \ numbound0 e)" + "isrlfm (Le (CN 0 c e)) = (c>0 \ numbound0 e)" + "isrlfm (Gt (CN 0 c e)) = (c>0 \ numbound0 e)" + "isrlfm (Ge (CN 0 c e)) = (c>0 \ numbound0 e)" + "isrlfm p = (isatom p \ (bound0 p))" + + (* splits the bounded from the unbounded part*) +consts rsplit0 :: "num \ int \ num" +recdef rsplit0 "measure num_size" + "rsplit0 (Bound 0) = (1,C 0)" + "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b + in (ca+cb, Add ta tb))" + "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))" + "rsplit0 (Neg a) = (let (c,t) = rsplit0 a in (-c,Neg t))" + "rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c*ca,Mul c ta))" + "rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c+ca,ta))" + "rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca,CN n c ta))" + "rsplit0 t = (0,t)" +lemma rsplit0: + shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \ numbound0 (snd (rsplit0 t))" +proof (induct t rule: rsplit0.induct) + case (2 a b) + let ?sa = "rsplit0 a" let ?sb = "rsplit0 b" + let ?ca = "fst ?sa" let ?cb = "fst ?sb" + let ?ta = "snd ?sa" let ?tb = "snd ?sb" + from prems have nb: "numbound0 (snd(rsplit0 (Add a b)))" + by(cases "rsplit0 a",auto simp add: Let_def split_def) + have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) = + Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)" + by (simp add: Let_def split_def algebra_simps) + also have "\ = Inum bs a + Inum bs b" using prems by (cases "rsplit0 a", simp_all) + finally show ?case using nb by simp +qed(auto simp add: Let_def split_def algebra_simps , simp add: right_distrib[symmetric]) + + (* Linearize a formula*) +definition + lt :: "int \ num \ fm" +where + "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) + else (Gt (CN 0 (-c) (Neg t))))" + +definition + le :: "int \ num \ fm" +where + "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) + else (Ge (CN 0 (-c) (Neg t))))" + +definition + gt :: "int \ num \ fm" +where + "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) + else (Lt (CN 0 (-c) (Neg t))))" + +definition + ge :: "int \ num \ fm" +where + "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) + else (Le (CN 0 (-c) (Neg t))))" + +definition + eq :: "int \ num \ fm" +where + "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) + else (Eq (CN 0 (-c) (Neg t))))" + +definition + neq :: "int \ num \ fm" +where + "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) + else (NEq (CN 0 (-c) (Neg t))))" + +lemma lt: "numnoabs t \ Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \ isrlfm (split lt (rsplit0 t))" +using rsplit0[where bs = "bs" and t="t"] +by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto) + +lemma le: "numnoabs t \ Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \ isrlfm (split le (rsplit0 t))" +using rsplit0[where bs = "bs" and t="t"] +by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) + +lemma gt: "numnoabs t \ Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \ isrlfm (split gt (rsplit0 t))" +using rsplit0[where bs = "bs" and t="t"] +by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) + +lemma ge: "numnoabs t \ Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \ isrlfm (split ge (rsplit0 t))" +using rsplit0[where bs = "bs" and t="t"] +by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) + +lemma eq: "numnoabs t \ Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \ isrlfm (split eq (rsplit0 t))" +using rsplit0[where bs = "bs" and t="t"] +by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) + +lemma neq: "numnoabs t \ Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \ isrlfm (split neq (rsplit0 t))" +using rsplit0[where bs = "bs" and t="t"] +by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) + +lemma conj_lin: "isrlfm p \ isrlfm q \ isrlfm (conj p q)" +by (auto simp add: conj_def) +lemma disj_lin: "isrlfm p \ isrlfm q \ isrlfm (disj p q)" +by (auto simp add: disj_def) + +consts rlfm :: "fm \ fm" +recdef rlfm "measure fmsize" + "rlfm (And p q) = conj (rlfm p) (rlfm q)" + "rlfm (Or p q) = disj (rlfm p) (rlfm q)" + "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)" + "rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))" + "rlfm (Lt a) = split lt (rsplit0 a)" + "rlfm (Le a) = split le (rsplit0 a)" + "rlfm (Gt a) = split gt (rsplit0 a)" + "rlfm (Ge a) = split ge (rsplit0 a)" + "rlfm (Eq a) = split eq (rsplit0 a)" + "rlfm (NEq a) = split neq (rsplit0 a)" + "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))" + "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))" + "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))" + "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))" + "rlfm (NOT (NOT p)) = rlfm p" + "rlfm (NOT T) = F" + "rlfm (NOT F) = T" + "rlfm (NOT (Lt a)) = rlfm (Ge a)" + "rlfm (NOT (Le a)) = rlfm (Gt a)" + "rlfm (NOT (Gt a)) = rlfm (Le a)" + "rlfm (NOT (Ge a)) = rlfm (Lt a)" + "rlfm (NOT (Eq a)) = rlfm (NEq a)" + "rlfm (NOT (NEq a)) = rlfm (Eq a)" + "rlfm p = p" (hints simp add: fmsize_pos) + +lemma rlfm_I: + assumes qfp: "qfree p" + shows "(Ifm bs (rlfm p) = Ifm bs p) \ isrlfm (rlfm p)" + using qfp +by (induct p rule: rlfm.induct, auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin) + + (* Operations needed for Ferrante and Rackoff *) +lemma rminusinf_inf: + assumes lp: "isrlfm p" + shows "\ z. \ x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\ z. \ x. ?P z x p") +using lp +proof (induct p rule: minusinf.induct) + case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto +next + case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto +next + case (3 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + hence "real c * x + ?e \ 0" by simp + with xz have "?P ?z x (Eq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp + thus ?case by blast +next + case (4 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + hence "real c * x + ?e \ 0" by simp + with xz have "?P ?z x (NEq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp + thus ?case by blast +next + case (5 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Lt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp + thus ?case by blast +next + case (6 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Le (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x < ?z. ?P ?z x (Le (CN 0 c e))" by simp + thus ?case by blast +next + case (7 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Gt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp + thus ?case by blast +next + case (8 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Ge (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp + thus ?case by blast +qed simp_all + +lemma rplusinf_inf: + assumes lp: "isrlfm p" + shows "\ z. \ x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\ z. \ x. ?P z x p") +using lp +proof (induct p rule: isrlfm.induct) + case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto +next + case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto +next + case (3 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + hence "real c * x + ?e \ 0" by simp + with xz have "?P ?z x (Eq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp + thus ?case by blast +next + case (4 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + hence "real c * x + ?e \ 0" by simp + with xz have "?P ?z x (NEq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp + thus ?case by blast +next + case (5 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Lt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp + thus ?case by blast +next + case (6 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Le (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x > ?z. ?P ?z x (Le (CN 0 c e))" by simp + thus ?case by blast +next + case (7 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Gt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp + thus ?case by blast +next + case (8 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Ge (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp + thus ?case by blast +qed simp_all + +lemma rminusinf_bound0: + assumes lp: "isrlfm p" + shows "bound0 (minusinf p)" + using lp + by (induct p rule: minusinf.induct) simp_all + +lemma rplusinf_bound0: + assumes lp: "isrlfm p" + shows "bound0 (plusinf p)" + using lp + by (induct p rule: plusinf.induct) simp_all + +lemma rminusinf_ex: + assumes lp: "isrlfm p" + and ex: "Ifm (a#bs) (minusinf p)" + shows "\ x. Ifm (x#bs) p" +proof- + from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex + have th: "\ x. Ifm (x#bs) (minusinf p)" by auto + from rminusinf_inf[OF lp, where bs="bs"] + obtain z where z_def: "\x x. Ifm (x#bs) p" +proof- + from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex + have th: "\ x. Ifm (x#bs) (plusinf p)" by auto + from rplusinf_inf[OF lp, where bs="bs"] + obtain z where z_def: "\x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast + from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp + moreover have "z + 1 > z" by simp + ultimately show ?thesis using z_def by auto +qed + +consts + uset:: "fm \ (num \ int) list" + usubst :: "fm \ (num \ int) \ fm " +recdef uset "measure size" + "uset (And p q) = (uset p @ uset q)" + "uset (Or p q) = (uset p @ uset q)" + "uset (Eq (CN 0 c e)) = [(Neg e,c)]" + "uset (NEq (CN 0 c e)) = [(Neg e,c)]" + "uset (Lt (CN 0 c e)) = [(Neg e,c)]" + "uset (Le (CN 0 c e)) = [(Neg e,c)]" + "uset (Gt (CN 0 c e)) = [(Neg e,c)]" + "uset (Ge (CN 0 c e)) = [(Neg e,c)]" + "uset p = []" +recdef usubst "measure size" + "usubst (And p q) = (\ (t,n). And (usubst p (t,n)) (usubst q (t,n)))" + "usubst (Or p q) = (\ (t,n). Or (usubst p (t,n)) (usubst q (t,n)))" + "usubst (Eq (CN 0 c e)) = (\ (t,n). Eq (Add (Mul c t) (Mul n e)))" + "usubst (NEq (CN 0 c e)) = (\ (t,n). NEq (Add (Mul c t) (Mul n e)))" + "usubst (Lt (CN 0 c e)) = (\ (t,n). Lt (Add (Mul c t) (Mul n e)))" + "usubst (Le (CN 0 c e)) = (\ (t,n). Le (Add (Mul c t) (Mul n e)))" + "usubst (Gt (CN 0 c e)) = (\ (t,n). Gt (Add (Mul c t) (Mul n e)))" + "usubst (Ge (CN 0 c e)) = (\ (t,n). Ge (Add (Mul c t) (Mul n e)))" + "usubst p = (\ (t,n). p)" + +lemma usubst_I: assumes lp: "isrlfm p" + and np: "real n > 0" and nbt: "numbound0 t" + shows "(Ifm (x#bs) (usubst p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \ bound0 (usubst p (t,n))" (is "(?I x (usubst p (t,n)) = ?I ?u p) \ ?B p" is "(_ = ?I (?t/?n) p) \ _" is "(_ = ?I (?N x t /_) p) \ _") + using lp +proof(induct p rule: usubst.induct) + case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\ = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)" + by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: algebra_simps) + also have "\ = (real c *?t + ?n* (?N x e) < 0)" + using np by simp + finally show ?case using nbt nb by (simp add: algebra_simps) +next + case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \ 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\ = (?n*(real c *(?t/?n)) + ?n*(?N x e) \ 0)" + by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: algebra_simps) + also have "\ = (real c *?t + ?n* (?N x e) \ 0)" + using np by simp + finally show ?case using nbt nb by (simp add: algebra_simps) +next + case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\ = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)" + by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: algebra_simps) + also have "\ = (real c *?t + ?n* (?N x e) > 0)" + using np by simp + finally show ?case using nbt nb by (simp add: algebra_simps) +next + case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \ 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\ = (?n*(real c *(?t/?n)) + ?n*(?N x e) \ 0)" + by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: algebra_simps) + also have "\ = (real c *?t + ?n* (?N x e) \ 0)" + using np by simp + finally show ?case using nbt nb by (simp add: algebra_simps) +next + case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + from np have np: "real n \ 0" by simp + have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\ = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)" + by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: algebra_simps) + also have "\ = (real c *?t + ?n* (?N x e) = 0)" + using np by simp + finally show ?case using nbt nb by (simp add: algebra_simps) +next + case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + from np have np: "real n \ 0" by simp + have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \ 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\ = (?n*(real c *(?t/?n)) + ?n*(?N x e) \ 0)" + by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: algebra_simps) + also have "\ = (real c *?t + ?n* (?N x e) \ 0)" + using np by simp + finally show ?case using nbt nb by (simp add: algebra_simps) +qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2) + +lemma uset_l: + assumes lp: "isrlfm p" + shows "\ (t,k) \ set (uset p). numbound0 t \ k >0" +using lp +by(induct p rule: uset.induct,auto) + +lemma rminusinf_uset: + assumes lp: "isrlfm p" + and nmi: "\ (Ifm (a#bs) (minusinf p))" (is "\ (Ifm (a#bs) (?M p))") + and ex: "Ifm (x#bs) p" (is "?I x p") + shows "\ (s,m) \ set (uset p). x \ Inum (a#bs) s / real m" (is "\ (s,m) \ ?U p. x \ ?N a s / real m") +proof- + have "\ (s,m) \ set (uset p). real m * x \ Inum (a#bs) s " (is "\ (s,m) \ ?U p. real m *x \ ?N a s") + using lp nmi ex + by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2) + then obtain s m where smU: "(s,m) \ set (uset p)" and mx: "real m * x \ ?N a s" by blast + from uset_l[OF lp] smU have mp: "real m > 0" by auto + from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \ ?N a s / real m" + by (auto simp add: mult_commute) + thus ?thesis using smU by auto +qed + +lemma rplusinf_uset: + assumes lp: "isrlfm p" + and nmi: "\ (Ifm (a#bs) (plusinf p))" (is "\ (Ifm (a#bs) (?M p))") + and ex: "Ifm (x#bs) p" (is "?I x p") + shows "\ (s,m) \ set (uset p). x \ Inum (a#bs) s / real m" (is "\ (s,m) \ ?U p. x \ ?N a s / real m") +proof- + have "\ (s,m) \ set (uset p). real m * x \ Inum (a#bs) s " (is "\ (s,m) \ ?U p. real m *x \ ?N a s") + using lp nmi ex + by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2) + then obtain s m where smU: "(s,m) \ set (uset p)" and mx: "real m * x \ ?N a s" by blast + from uset_l[OF lp] smU have mp: "real m > 0" by auto + from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \ ?N a s / real m" + by (auto simp add: mult_commute) + thus ?thesis using smU by auto +qed + +lemma lin_dense: + assumes lp: "isrlfm p" + and noS: "\ t. l < t \ t< u \ t \ (\ (t,n). Inum (x#bs) t / real n) ` set (uset p)" + (is "\ t. _ \ _ \ t \ (\ (t,n). ?N x t / real n ) ` (?U p)") + and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p" + and ly: "l < y" and yu: "y < u" + shows "Ifm (y#bs) p" +using lp px noS +proof (induct p rule: isrlfm.induct) + case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from prems have "x * real c + ?N x e < 0" by (simp add: algebra_simps) + hence pxc: "x < (- ?N x e) / real c" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\ t. l < t \ t < u \ t \ (- ?N x e) / real c" by auto + with ly yu have yne: "y \ - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \ y > (-?N x e) / real c" by auto + moreover {assume y: "y < (-?N x e)/ real c" + hence "y * real c < - ?N x e" + by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e < 0" by (simp add: algebra_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y > (- ?N x e) / real c" + with yu have eu: "u > (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \ l" by (cases "(- ?N x e) / real c > l", auto) + with lx pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp + + from prems have "x * real c + ?N x e \ 0" by (simp add: algebra_simps) + hence pxc: "x \ (- ?N x e) / real c" + by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\ t. l < t \ t < u \ t \ (- ?N x e) / real c" by auto + with ly yu have yne: "y \ - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \ y > (-?N x e) / real c" by auto + moreover {assume y: "y < (-?N x e)/ real c" + hence "y * real c < - ?N x e" + by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e < 0" by (simp add: algebra_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y > (- ?N x e) / real c" + with yu have eu: "u > (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \ l" by (cases "(- ?N x e) / real c > l", auto) + with lx pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from prems have "x * real c + ?N x e > 0" by (simp add: algebra_simps) + hence pxc: "x > (- ?N x e) / real c" + by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\ t. l < t \ t < u \ t \ (- ?N x e) / real c" by auto + with ly yu have yne: "y \ - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \ y > (-?N x e) / real c" by auto + moreover {assume y: "y > (-?N x e)/ real c" + hence "y * real c > - ?N x e" + by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e > 0" by (simp add: algebra_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y < (- ?N x e) / real c" + with ly have eu: "l < (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \ u" by (cases "(- ?N x e) / real c > l", auto) + with xu pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from prems have "x * real c + ?N x e \ 0" by (simp add: algebra_simps) + hence pxc: "x \ (- ?N x e) / real c" + by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\ t. l < t \ t < u \ t \ (- ?N x e) / real c" by auto + with ly yu have yne: "y \ - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \ y > (-?N x e) / real c" by auto + moreover {assume y: "y > (-?N x e)/ real c" + hence "y * real c > - ?N x e" + by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e > 0" by (simp add: algebra_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y < (- ?N x e) / real c" + with ly have eu: "l < (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \ u" by (cases "(- ?N x e) / real c > l", auto) + with xu pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from cp have cnz: "real c \ 0" by simp + from prems have "x * real c + ?N x e = 0" by (simp add: algebra_simps) + hence pxc: "x = (- ?N x e) / real c" + by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"]) + from prems have noSc:"\ t. l < t \ t < u \ t \ (- ?N x e) / real c" by auto + with lx xu have yne: "x \ - ?N x e / real c" by auto + with pxc show ?case by simp +next + case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from cp have cnz: "real c \ 0" by simp + from prems have noSc:"\ t. l < t \ t < u \ t \ (- ?N x e) / real c" by auto + with ly yu have yne: "y \ - ?N x e / real c" by auto + hence "y* real c \ -?N x e" + by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp + hence "y* real c + ?N x e \ 0" by (simp add: algebra_simps) + thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] + by (simp add: algebra_simps) +qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"]) + +lemma finite_set_intervals: + assumes px: "P (x::real)" + and lx: "l \ x" and xu: "x \ u" + and linS: "l\ S" and uinS: "u \ S" + and fS:"finite S" and lS: "\ x\ S. l \ x" and Su: "\ x\ S. x \ u" + shows "\ a \ S. \ b \ S. (\ y. a < y \ y < b \ y \ S) \ a \ x \ x \ b \ P x" +proof- + let ?Mx = "{y. y\ S \ y \ x}" + let ?xM = "{y. y\ S \ x \ y}" + let ?a = "Max ?Mx" + let ?b = "Min ?xM" + have MxS: "?Mx \ S" by blast + hence fMx: "finite ?Mx" using fS finite_subset by auto + from lx linS have linMx: "l \ ?Mx" by blast + hence Mxne: "?Mx \ {}" by blast + have xMS: "?xM \ S" by blast + hence fxM: "finite ?xM" using fS finite_subset by auto + from xu uinS have linxM: "u \ ?xM" by blast + hence xMne: "?xM \ {}" by blast + have ax:"?a \ x" using Mxne fMx by auto + have xb:"x \ ?b" using xMne fxM by auto + have "?a \ ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \ S" using MxS by blast + have "?b \ ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \ S" using xMS by blast + have noy:"\ y. ?a < y \ y < ?b \ y \ S" + proof(clarsimp) + fix y + assume ay: "?a < y" and yb: "y < ?b" and yS: "y \ S" + from yS have "y\ ?Mx \ y\ ?xM" by auto + moreover {assume "y \ ?Mx" hence "y \ ?a" using Mxne fMx by auto with ay have "False" by simp} + moreover {assume "y \ ?xM" hence "y \ ?b" using xMne fxM by auto with yb have "False" by simp} + ultimately show "False" by blast + qed + from ainS binS noy ax xb px show ?thesis by blast +qed + +lemma finite_set_intervals2: + assumes px: "P (x::real)" + and lx: "l \ x" and xu: "x \ u" + and linS: "l\ S" and uinS: "u \ S" + and fS:"finite S" and lS: "\ x\ S. l \ x" and Su: "\ x\ S. x \ u" + shows "(\ s\ S. P s) \ (\ a \ S. \ b \ S. (\ y. a < y \ y < b \ y \ S) \ a < x \ x < b \ P x)" +proof- + from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su] + obtain a and b where + as: "a\ S" and bs: "b\ S" and noS:"\y. a < y \ y < b \ y \ S" and axb: "a \ x \ x \ b \ P x" by auto + from axb have "x= a \ x= b \ (a < x \ x < b)" by auto + thus ?thesis using px as bs noS by blast +qed + +lemma rinf_uset: + assumes lp: "isrlfm p" + and nmi: "\ (Ifm (x#bs) (minusinf p))" (is "\ (Ifm (x#bs) (?M p))") + and npi: "\ (Ifm (x#bs) (plusinf p))" (is "\ (Ifm (x#bs) (?P p))") + and ex: "\ x. Ifm (x#bs) p" (is "\ x. ?I x p") + shows "\ (l,n) \ set (uset p). \ (s,m) \ set (uset p). ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p" +proof- + let ?N = "\ x t. Inum (x#bs) t" + let ?U = "set (uset p)" + from ex obtain a where pa: "?I a p" by blast + from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi + have nmi': "\ (?I a (?M p))" by simp + from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi + have npi': "\ (?I a (?P p))" by simp + have "\ (l,n) \ set (uset p). \ (s,m) \ set (uset p). ?I ((?N a l/real n + ?N a s /real m) / 2) p" + proof- + let ?M = "(\ (t,c). ?N a t / real c) ` ?U" + have fM: "finite ?M" by auto + from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa] + have "\ (l,n) \ set (uset p). \ (s,m) \ set (uset p). a \ ?N x l / real n \ a \ ?N x s / real m" by blast + then obtain "t" "n" "s" "m" where + tnU: "(t,n) \ ?U" and smU: "(s,m) \ ?U" + and xs1: "a \ ?N x s / real m" and tx1: "a \ ?N x t / real n" by blast + from uset_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \ ?N a s / real m" and tx: "a \ ?N a t / real n" by auto + from tnU have Mne: "?M \ {}" by auto + hence Une: "?U \ {}" by simp + let ?l = "Min ?M" + let ?u = "Max ?M" + have linM: "?l \ ?M" using fM Mne by simp + have uinM: "?u \ ?M" using fM Mne by simp + have tnM: "?N a t / real n \ ?M" using tnU by auto + have smM: "?N a s / real m \ ?M" using smU by auto + have lM: "\ t\ ?M. ?l \ t" using Mne fM by auto + have Mu: "\ t\ ?M. t \ ?u" using Mne fM by auto + have "?l \ ?N a t / real n" using tnM Mne by simp hence lx: "?l \ a" using tx by simp + have "?N a s / real m \ ?u" using smM Mne by simp hence xu: "a \ ?u" using xs by simp + from finite_set_intervals2[where P="\ x. ?I x p",OF pa lx xu linM uinM fM lM Mu] + have "(\ s\ ?M. ?I s p) \ + (\ t1\ ?M. \ t2 \ ?M. (\ y. t1 < y \ y < t2 \ y \ ?M) \ t1 < a \ a < t2 \ ?I a p)" . + moreover { fix u assume um: "u\ ?M" and pu: "?I u p" + hence "\ (tu,nu) \ ?U. u = ?N a tu / real nu" by auto + then obtain "tu" "nu" where tuU: "(tu,nu) \ ?U" and tuu:"u= ?N a tu / real nu" by blast + have "(u + u) / 2 = u" by auto with pu tuu + have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp + with tuU have ?thesis by blast} + moreover{ + assume "\ t1\ ?M. \ t2 \ ?M. (\ y. t1 < y \ y < t2 \ y \ ?M) \ t1 < a \ a < t2 \ ?I a p" + then obtain t1 and t2 where t1M: "t1 \ ?M" and t2M: "t2\ ?M" + and noM: "\ y. t1 < y \ y < t2 \ y \ ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p" + by blast + from t1M have "\ (t1u,t1n) \ ?U. t1 = ?N a t1u / real t1n" by auto + then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \ ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast + from t2M have "\ (t2u,t2n) \ ?U. t2 = ?N a t2u / real t2n" by auto + then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \ ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast + from t1x xt2 have t1t2: "t1 < t2" by simp + let ?u = "(t1 + t2) / 2" + from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto + from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" . + with t1uU t2uU t1u t2u have ?thesis by blast} + ultimately show ?thesis by blast + qed + then obtain "l" "n" "s" "m" where lnU: "(l,n) \ ?U" and smU:"(s,m) \ ?U" + and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast + from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto + from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] + numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu + have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp + with lnU smU + show ?thesis by auto +qed + (* The Ferrante - Rackoff Theorem *) + +theorem fr_eq: + assumes lp: "isrlfm p" + shows "(\ x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \ (Ifm (x#bs) (plusinf p)) \ (\ (t,n) \ set (uset p). \ (s,m) \ set (uset p). Ifm ((((Inum (x#bs) t)/ real n + (Inum (x#bs) s) / real m) /2)#bs) p))" + (is "(\ x. ?I x p) = (?M \ ?P \ ?F)" is "?E = ?D") +proof + assume px: "\ x. ?I x p" + have "?M \ ?P \ (\ ?M \ \ ?P)" by blast + moreover {assume "?M \ ?P" hence "?D" by blast} + moreover {assume nmi: "\ ?M" and npi: "\ ?P" + from rinf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast} + ultimately show "?D" by blast +next + assume "?D" + moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .} + moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . } + moreover {assume f:"?F" hence "?E" by blast} + ultimately show "?E" by blast +qed + + +lemma fr_equsubst: + assumes lp: "isrlfm p" + shows "(\ x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \ (Ifm (x#bs) (plusinf p)) \ (\ (t,k) \ set (uset p). \ (s,l) \ set (uset p). Ifm (x#bs) (usubst p (Add(Mul l t) (Mul k s) , 2*k*l))))" + (is "(\ x. ?I x p) = (?M \ ?P \ ?F)" is "?E = ?D") +proof + assume px: "\ x. ?I x p" + have "?M \ ?P \ (\ ?M \ \ ?P)" by blast + moreover {assume "?M \ ?P" hence "?D" by blast} + moreover {assume nmi: "\ ?M" and npi: "\ ?P" + let ?f ="\ (t,n). Inum (x#bs) t / real n" + let ?N = "\ t. Inum (x#bs) t" + {fix t n s m assume "(t,n)\ set (uset p)" and "(s,m) \ set (uset p)" + with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0" + by auto + let ?st = "Add (Mul m t) (Mul n s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute) + from tnb snb have st_nb: "numbound0 ?st" by simp + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mnp mp np by (simp add: algebra_simps add_divide_distrib) + from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"] + have "?I x (usubst p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])} + with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast} + ultimately show "?D" by blast +next + assume "?D" + moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .} + moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . } + moreover {fix t k s l assume "(t,k) \ set (uset p)" and "(s,l) \ set (uset p)" + and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))" + with uset_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto + let ?st = "Add (Mul l t) (Mul k s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0" + by (simp add: mult_commute) + from tnb snb have st_nb: "numbound0 ?st" by simp + from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto} + ultimately show "?E" by blast +qed + + + (* Implement the right hand side of Ferrante and Rackoff's Theorem. *) +constdefs ferrack:: "fm \ fm" + "ferrack p \ (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p' + in if (mp = T \ pp = T) then T else + (let U = remdps(map simp_num_pair + (map (\ ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) + (alluopairs (uset p')))) + in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))" + +lemma uset_cong_aux: + assumes Ul: "\ (t,n) \ set U. numbound0 t \ n >0" + shows "((\ (t,n). Inum (x#bs) t /real n) ` (set (map (\ ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\ ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \ set U))" + (is "?lhs = ?rhs") +proof(auto) + fix t n s m + assume "((t,n),(s,m)) \ set (alluopairs U)" + hence th: "((t,n),(s,m)) \ (set U \ set U)" + using alluopairs_set1[where xs="U"] by blast + let ?N = "\ t. Inum (x#bs) t" + let ?st= "Add (Mul m t) (Mul n s)" + from Ul th have mnz: "m \ 0" by auto + from Ul th have nnz: "n \ 0" by auto + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mnz nnz by (simp add: algebra_simps add_divide_distrib) + + thus "(real m * Inum (x # bs) t + real n * Inum (x # bs) s) / + (2 * real n * real m) + \ (\((t, n), s, m). + (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) ` + (set U \ set U)"using mnz nnz th + apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def) + by (rule_tac x="(s,m)" in bexI,simp_all) + (rule_tac x="(t,n)" in bexI,simp_all) +next + fix t n s m + assume tnU: "(t,n) \ set U" and smU:"(s,m) \ set U" + let ?N = "\ t. Inum (x#bs) t" + let ?st= "Add (Mul m t) (Mul n s)" + from Ul smU have mnz: "m \ 0" by auto + from Ul tnU have nnz: "n \ 0" by auto + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mnz nnz by (simp add: algebra_simps add_divide_distrib) + let ?P = "\ (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" + have Pc:"\ a b. ?P a b = ?P b a" + by auto + from Ul alluopairs_set1 have Up:"\ ((t,n),(s,m)) \ set (alluopairs U). n \ 0 \ m \ 0" by blast + from alluopairs_ex[OF Pc, where xs="U"] tnU smU + have th':"\ ((t',n'),(s',m')) \ set (alluopairs U). ?P (t',n') (s',m')" + by blast + then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \ set (alluopairs U)" + and Pts': "?P (t',n') (s',m')" by blast + from ts'_U Up have mnz': "m' \ 0" and nnz': "n'\ 0" by auto + let ?st' = "Add (Mul m' t') (Mul n' s')" + have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')" + using mnz' nnz' by (simp add: algebra_simps add_divide_distrib) + from Pts' have + "(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp + also have "\ = ((\(t, n). Inum (x # bs) t / real n) ((\((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st') + finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2 + \ (\(t, n). Inum (x # bs) t / real n) ` + (\((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ` + set (alluopairs U)" + using ts'_U by blast +qed + +lemma uset_cong: + assumes lp: "isrlfm p" + and UU': "((\ (t,n). Inum (x#bs) t /real n) ` U') = ((\ ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (U \ U))" (is "?f ` U' = ?g ` (U\U)") + and U: "\ (t,n) \ U. numbound0 t \ n > 0" + and U': "\ (t,n) \ U'. numbound0 t \ n > 0" + shows "(\ (t,n) \ U. \ (s,m) \ U. Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))) = (\ (t,n) \ U'. Ifm (x#bs) (usubst p (t,n)))" + (is "?lhs = ?rhs") +proof + assume ?lhs + then obtain t n s m where tnU: "(t,n) \ U" and smU:"(s,m) \ U" and + Pst: "Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))" by blast + let ?N = "\ t. Inum (x#bs) t" + from tnU smU U have tnb: "numbound0 t" and np: "n > 0" + and snb: "numbound0 s" and mp:"m > 0" by auto + let ?st= "Add (Mul m t) (Mul n s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult) + from tnb snb have stnb: "numbound0 ?st" by simp + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mp np by (simp add: algebra_simps add_divide_distrib) + from tnU smU UU' have "?g ((t,n),(s,m)) \ ?f ` U'" by blast + hence "\ (t',n') \ U'. ?g ((t,n),(s,m)) = ?f (t',n')" + by auto (rule_tac x="(a,b)" in bexI, auto) + then obtain t' n' where tnU': "(t',n') \ U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast + from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto + from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst + have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp + from conjunct1[OF usubst_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]] + have "Ifm (x # bs) (usubst p (t', n')) " by (simp only: st) + then show ?rhs using tnU' by auto +next + assume ?rhs + then obtain t' n' where tnU': "(t',n') \ U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))" + by blast + from tnU' UU' have "?f (t',n') \ ?g ` (U\U)" by blast + hence "\ ((t,n),(s,m)) \ (U\U). ?f (t',n') = ?g ((t,n),(s,m))" + by auto (rule_tac x="(a,b)" in bexI, auto) + then obtain t n s m where tnU: "(t,n) \ U" and smU:"(s,m) \ U" and + th: "?f (t',n') = ?g((t,n),(s,m)) "by blast + let ?N = "\ t. Inum (x#bs) t" + from tnU smU U have tnb: "numbound0 t" and np: "n > 0" + and snb: "numbound0 s" and mp:"m > 0" by auto + let ?st= "Add (Mul m t) (Mul n s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult) + from tnb snb have stnb: "numbound0 ?st" by simp + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mp np by (simp add: algebra_simps add_divide_distrib) + from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto + from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt' + have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp + with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast +qed + +lemma ferrack: + assumes qf: "qfree p" + shows "qfree (ferrack p) \ ((Ifm bs (ferrack p)) = (\ x. Ifm (x#bs) p))" + (is "_ \ (?rhs = ?lhs)") +proof- + let ?I = "\ x p. Ifm (x#bs) p" + fix x + let ?N = "\ t. Inum (x#bs) t" + let ?q = "rlfm (simpfm p)" + let ?U = "uset ?q" + let ?Up = "alluopairs ?U" + let ?g = "\ ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)" + let ?S = "map ?g ?Up" + let ?SS = "map simp_num_pair ?S" + let ?Y = "remdps ?SS" + let ?f= "(\ (t,n). ?N t / real n)" + let ?h = "\ ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2" + let ?F = "\ p. \ a \ set (uset p). \ b \ set (uset p). ?I x (usubst p (?g(a,b)))" + let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y" + from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast + from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \ (set ?U \ set ?U)" by simp + from uset_l[OF lq] have U_l: "\ (t,n) \ set ?U. numbound0 t \ n > 0" . + from U_l UpU + have "\ ((t,n),(s,m)) \ set ?Up. numbound0 t \ n> 0 \ numbound0 s \ m > 0" by auto + hence Snb: "\ (t,n) \ set ?S. numbound0 t \ n > 0 " + by (auto simp add: mult_pos_pos) + have Y_l: "\ (t,n) \ set ?Y. numbound0 t \ n > 0" + proof- + { fix t n assume tnY: "(t,n) \ set ?Y" + hence "(t,n) \ set ?SS" by simp + hence "\ (t',n') \ set ?S. simp_num_pair (t',n') = (t,n)" + by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all) + then obtain t' n' where tn'S: "(t',n') \ set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast + from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto + from simp_num_pair_l[OF tnb np tns] + have "numbound0 t \ n > 0" . } + thus ?thesis by blast + qed + + have YU: "(?f ` set ?Y) = (?h ` (set ?U \ set ?U))" + proof- + from simp_num_pair_ci[where bs="x#bs"] have + "\x. (?f o simp_num_pair) x = ?f x" by auto + hence th: "?f o simp_num_pair = ?f" using ext by blast + have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose) + also have "\ = (?f ` set ?S)" by (simp add: th) + also have "\ = ((?f o ?g) ` set ?Up)" + by (simp only: set_map o_def image_compose[symmetric]) + also have "\ = (?h ` (set ?U \ set ?U))" + using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast + finally show ?thesis . + qed + have "\ (t,n) \ set ?Y. bound0 (simpfm (usubst ?q (t,n)))" + proof- + { fix t n assume tnY: "(t,n) \ set ?Y" + with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto + from usubst_I[OF lq np tnb] + have "bound0 (usubst ?q (t,n))" by simp hence "bound0 (simpfm (usubst ?q (t,n)))" + using simpfm_bound0 by simp} + thus ?thesis by blast + qed + hence ep_nb: "bound0 ?ep" using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto + let ?mp = "minusinf ?q" + let ?pp = "plusinf ?q" + let ?M = "?I x ?mp" + let ?P = "?I x ?pp" + let ?res = "disj ?mp (disj ?pp ?ep)" + from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb + have nbth: "bound0 ?res" by auto + + from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm + + have th: "?lhs = (\ x. ?I x ?q)" by auto + from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \ ?P \ ?F ?q)" + by (simp only: split_def fst_conv snd_conv) + also have "\ = (?M \ ?P \ (\ (t,n) \ set ?Y. ?I x (simpfm (usubst ?q (t,n)))))" + using uset_cong[OF lq YU U_l Y_l] by (simp only: split_def fst_conv snd_conv simpfm) + also have "\ = (Ifm (x#bs) ?res)" + using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric] + by (simp add: split_def pair_collapse) + finally have lheq: "?lhs = (Ifm bs (decr ?res))" using decr[OF nbth] by blast + hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def) + by (cases "?mp = T \ ?pp = T", auto) (simp add: disj_def)+ + from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def) + with lr show ?thesis by blast +qed + +definition linrqe:: "fm \ fm" where + "linrqe p = qelim (prep p) ferrack" + +theorem linrqe: "Ifm bs (linrqe p) = Ifm bs p \ qfree (linrqe p)" +using ferrack qelim_ci prep +unfolding linrqe_def by auto + +definition ferrack_test :: "unit \ fm" where + "ferrack_test u = linrqe (A (A (Imp (Lt (Sub (Bound 1) (Bound 0))) + (E (Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))" + +ML {* @{code ferrack_test} () *} + +oracle linr_oracle = {* +let + +fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t + of NONE => error "Variable not found in the list!" + | SOME n => @{code Bound} n) + | num_of_term vs @{term "real (0::int)"} = @{code C} 0 + | num_of_term vs @{term "real (1::int)"} = @{code C} 1 + | num_of_term vs @{term "0::real"} = @{code C} 0 + | num_of_term vs @{term "1::real"} = @{code C} 1 + | num_of_term vs (Bound i) = @{code Bound} i + | num_of_term vs (@{term "uminus :: real \ real"} $ t') = @{code Neg} (num_of_term vs t') + | num_of_term vs (@{term "op + :: real \ real \ real"} $ t1 $ t2) = @{code Add} (num_of_term vs t1, num_of_term vs t2) + | num_of_term vs (@{term "op - :: real \ real \ real"} $ t1 $ t2) = @{code Sub} (num_of_term vs t1, num_of_term vs t2) + | num_of_term vs (@{term "op * :: real \ real \ real"} $ t1 $ t2) = (case (num_of_term vs t1) + of @{code C} i => @{code Mul} (i, num_of_term vs t2) + | _ => error "num_of_term: unsupported Multiplication") + | num_of_term vs (@{term "real :: int \ real"} $ (@{term "number_of :: int \ int"} $ t')) = @{code C} (HOLogic.dest_numeral t') + | num_of_term vs (@{term "number_of :: int \ real"} $ t') = @{code C} (HOLogic.dest_numeral t') + | num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t); + +fun fm_of_term vs @{term True} = @{code T} + | fm_of_term vs @{term False} = @{code F} + | fm_of_term vs (@{term "op < :: real \ real \ bool"} $ t1 $ t2) = @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) + | fm_of_term vs (@{term "op \ :: real \ real \ bool"} $ t1 $ t2) = @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) + | fm_of_term vs (@{term "op = :: real \ real \ bool"} $ t1 $ t2) = @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) + | fm_of_term vs (@{term "op \ :: bool \ bool \ bool"} $ t1 $ t2) = @{code Iff} (fm_of_term vs t1, fm_of_term vs t2) + | fm_of_term vs (@{term "op &"} $ t1 $ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2) + | fm_of_term vs (@{term "op |"} $ t1 $ t2) = @{code Or} (fm_of_term vs t1, fm_of_term vs t2) + | fm_of_term vs (@{term "op -->"} $ t1 $ t2) = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2) + | fm_of_term vs (@{term "Not"} $ t') = @{code NOT} (fm_of_term vs t') + | fm_of_term vs (Const ("Ex", _) $ Abs (xn, xT, p)) = + @{code E} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p) + | fm_of_term vs (Const ("All", _) $ Abs (xn, xT, p)) = + @{code A} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p) + | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t); + +fun term_of_num vs (@{code C} i) = @{term "real :: int \ real"} $ HOLogic.mk_number HOLogic.intT i + | term_of_num vs (@{code Bound} n) = fst (the (find_first (fn (_, m) => n = m) vs)) + | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \ real"} $ term_of_num vs t' + | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \ real \ real"} $ + term_of_num vs t1 $ term_of_num vs t2 + | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \ real \ real"} $ + term_of_num vs t1 $ term_of_num vs t2 + | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \ real \ real"} $ + term_of_num vs (@{code C} i) $ term_of_num vs t2 + | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t)); + +fun term_of_fm vs @{code T} = HOLogic.true_const + | term_of_fm vs @{code F} = HOLogic.false_const + | term_of_fm vs (@{code Lt} t) = @{term "op < :: real \ real \ bool"} $ + term_of_num vs t $ @{term "0::real"} + | term_of_fm vs (@{code Le} t) = @{term "op \ :: real \ real \ bool"} $ + term_of_num vs t $ @{term "0::real"} + | term_of_fm vs (@{code Gt} t) = @{term "op < :: real \ real \ bool"} $ + @{term "0::real"} $ term_of_num vs t + | term_of_fm vs (@{code Ge} t) = @{term "op \ :: real \ real \ bool"} $ + @{term "0::real"} $ term_of_num vs t + | term_of_fm vs (@{code Eq} t) = @{term "op = :: real \ real \ bool"} $ + term_of_num vs t $ @{term "0::real"} + | term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code NOT} (@{code Eq} t)) + | term_of_fm vs (@{code NOT} t') = HOLogic.Not $ term_of_fm vs t' + | term_of_fm vs (@{code And} (t1, t2)) = HOLogic.conj $ term_of_fm vs t1 $ term_of_fm vs t2 + | term_of_fm vs (@{code Or} (t1, t2)) = HOLogic.disj $ term_of_fm vs t1 $ term_of_fm vs t2 + | term_of_fm vs (@{code Imp} (t1, t2)) = HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2 + | term_of_fm vs (@{code Iff} (t1, t2)) = @{term "op \ :: bool \ bool \ bool"} $ + term_of_fm vs t1 $ term_of_fm vs t2 + | term_of_fm vs _ = error "If this is raised, Isabelle/HOL or generate_code is inconsistent."; + +in fn ct => + let + val thy = Thm.theory_of_cterm ct; + val t = Thm.term_of ct; + val fs = OldTerm.term_frees t; + val vs = fs ~~ (0 upto (length fs - 1)); + val res = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, term_of_fm vs (@{code linrqe} (fm_of_term vs t)))); + in Thm.cterm_of thy res end +end; +*} + +use "ferrack_tac.ML" +setup Ferrack_Tac.setup + +lemma + fixes x :: real + shows "2 * x \ 2 * x \ 2 * x \ 2 * x + 1" +apply rferrack +done + +lemma + fixes x :: real + shows "\y \ x. x = y + 1" +apply rferrack +done + +lemma + fixes x :: real + shows "\ (\z. x + z = x + z + 1)" +apply rferrack +done + +end diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Decision_Procs/MIR.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Decision_Procs/MIR.thy Fri Feb 06 15:15:46 2009 +0100 @@ -0,0 +1,5933 @@ +(* Title: HOL/Reflection/MIR.thy + Author: Amine Chaieb +*) + +theory MIR +imports Complex_Main Dense_Linear_Order Efficient_Nat +uses ("mir_tac.ML") +begin + +section {* Quantifier elimination for @{text "\ (0, 1, +, floor, <)"} *} + +declare real_of_int_floor_cancel [simp del] + +primrec alluopairs:: "'a list \ ('a \ 'a) list" where + "alluopairs [] = []" +| "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)" + +lemma alluopairs_set1: "set (alluopairs xs) \ {(x,y). x\ set xs \ y\ set xs}" +by (induct xs, auto) + +lemma alluopairs_set: + "\x\ set xs ; y \ set xs\ \ (x,y) \ set (alluopairs xs) \ (y,x) \ set (alluopairs xs) " +by (induct xs, auto) + +lemma alluopairs_ex: + assumes Pc: "\ x y. P x y = P y x" + shows "(\ x \ set xs. \ y \ set xs. P x y) = (\ (x,y) \ set (alluopairs xs). P x y)" +proof + assume "\x\set xs. \y\set xs. P x y" + then obtain x y where x: "x \ set xs" and y:"y \ set xs" and P: "P x y" by blast + from alluopairs_set[OF x y] P Pc show"\(x, y)\set (alluopairs xs). P x y" + by auto +next + assume "\(x, y)\set (alluopairs xs). P x y" + then obtain "x" and "y" where xy:"(x,y) \ set (alluopairs xs)" and P: "P x y" by blast+ + from xy have "x \ set xs \ y\ set xs" using alluopairs_set1 by blast + with P show "\x\set xs. \y\set xs. P x y" by blast +qed + + (* generate a list from i to j*) +consts iupt :: "int \ int \ int list" +recdef iupt "measure (\ (i,j). nat (j-i +1))" + "iupt (i,j) = (if j (x#xs) ! n = xs ! (n - 1)" +using Nat.gr0_conv_Suc +by clarsimp + + +lemma myl: "\ (a::'a::{pordered_ab_group_add}) (b::'a). (a \ b) = (0 \ b - a)" +proof(clarify) + fix x y ::"'a" + have "(x \ y) = (x - y \ 0)" by (simp only: le_iff_diff_le_0[where a="x" and b="y"]) + also have "\ = (- (y - x) \ 0)" by simp + also have "\ = (0 \ y - x)" by (simp only: neg_le_0_iff_le[where a="y-x"]) + finally show "(x \ y) = (0 \ y - x)" . +qed + +lemma myless: "\ (a::'a::{pordered_ab_group_add}) (b::'a). (a < b) = (0 < b - a)" +proof(clarify) + fix x y ::"'a" + have "(x < y) = (x - y < 0)" by (simp only: less_iff_diff_less_0[where a="x" and b="y"]) + also have "\ = (- (y - x) < 0)" by simp + also have "\ = (0 < y - x)" by (simp only: neg_less_0_iff_less[where a="y-x"]) + finally show "(x < y) = (0 < y - x)" . +qed + +lemma myeq: "\ (a::'a::{pordered_ab_group_add}) (b::'a). (a = b) = (0 = b - a)" + by auto + + (* Maybe should be added to the library \ *) +lemma floor_int_eq: "(real n\ x \ x < real (n+1)) = (floor x = n)" +proof( auto) + assume lb: "real n \ x" + and ub: "x < real n + 1" + have "real (floor x) \ x" by simp + hence "real (floor x) < real (n + 1) " using ub by arith + hence "floor x < n+1" by simp + moreover from lb have "n \ floor x" using floor_mono2[where x="real n" and y="x"] + by simp ultimately show "floor x = n" by simp +qed + +(* Periodicity of dvd *) +lemma dvd_period: + assumes advdd: "(a::int) dvd d" + shows "(a dvd (x + t)) = (a dvd ((x+ c*d) + t))" + using advdd +proof- + {fix x k + from inf_period(3)[OF advdd, rule_format, where x=x and k="-k"] + have " ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))" by simp} + hence "\x.\k. ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))" by simp + then show ?thesis by simp +qed + + (* The Divisibility relation between reals *) +definition + rdvd:: "real \ real \ bool" (infixl "rdvd" 50) +where + rdvd_def: "x rdvd y \ (\k\int. y = x * real k)" + +lemma int_rdvd_real: + shows "real (i::int) rdvd x = (i dvd (floor x) \ real (floor x) = x)" (is "?l = ?r") +proof + assume "?l" + hence th: "\ k. x=real (i*k)" by (simp add: rdvd_def) + hence th': "real (floor x) = x" by (auto simp del: real_of_int_mult) + with th have "\ k. real (floor x) = real (i*k)" by simp + hence "\ k. floor x = i*k" by (simp only: real_of_int_inject) + thus ?r using th' by (simp add: dvd_def) +next + assume "?r" hence "(i\int) dvd \x\real\" .. + hence "\ k. real (floor x) = real (i*k)" + by (simp only: real_of_int_inject) (simp add: dvd_def) + thus ?l using prems by (simp add: rdvd_def) +qed + +lemma int_rdvd_iff: "(real (i::int) rdvd real t) = (i dvd t)" +by (auto simp add: rdvd_def dvd_def) (rule_tac x="k" in exI, simp only :real_of_int_mult[symmetric]) + + +lemma rdvd_abs1: + "(abs (real d) rdvd t) = (real (d ::int) rdvd t)" +proof + assume d: "real d rdvd t" + from d int_rdvd_real have d2: "d dvd (floor t)" and ti: "real (floor t) = t" by auto + + from iffD2[OF zdvd_abs1] d2 have "(abs d) dvd (floor t)" by blast + with ti int_rdvd_real[symmetric] have "real (abs d) rdvd t" by blast + thus "abs (real d) rdvd t" by simp +next + assume "abs (real d) rdvd t" hence "real (abs d) rdvd t" by simp + with int_rdvd_real[where i="abs d" and x="t"] have d2: "abs d dvd floor t" and ti: "real (floor t) =t" by auto + from iffD1[OF zdvd_abs1] d2 have "d dvd floor t" by blast + with ti int_rdvd_real[symmetric] show "real d rdvd t" by blast +qed + +lemma rdvd_minus: "(real (d::int) rdvd t) = (real d rdvd -t)" + apply (auto simp add: rdvd_def) + apply (rule_tac x="-k" in exI, simp) + apply (rule_tac x="-k" in exI, simp) +done + +lemma rdvd_left_0_eq: "(0 rdvd t) = (t=0)" +by (auto simp add: rdvd_def) + +lemma rdvd_mult: + assumes knz: "k\0" + shows "(real (n::int) * real (k::int) rdvd x * real k) = (real n rdvd x)" +using knz by (simp add:rdvd_def) + +lemma rdvd_trans: assumes mn:"m rdvd n" and nk:"n rdvd k" + shows "m rdvd k" +proof- + from rdvd_def mn obtain c where nmc:"n = m * real (c::int)" by auto + from rdvd_def nk obtain c' where nkc:"k = n * real (c'::int)" by auto + hence "k = m * real (c * c')" using nmc by simp + thus ?thesis using rdvd_def by blast +qed + + (*********************************************************************************) + (**** SHADOW SYNTAX AND SEMANTICS ****) + (*********************************************************************************) + +datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num + | Mul int num | Floor num| CF int num num + + (* A size for num to make inductive proofs simpler*) +primrec num_size :: "num \ nat" where + "num_size (C c) = 1" +| "num_size (Bound n) = 1" +| "num_size (Neg a) = 1 + num_size a" +| "num_size (Add a b) = 1 + num_size a + num_size b" +| "num_size (Sub a b) = 3 + num_size a + num_size b" +| "num_size (CN n c a) = 4 + num_size a " +| "num_size (CF c a b) = 4 + num_size a + num_size b" +| "num_size (Mul c a) = 1 + num_size a" +| "num_size (Floor a) = 1 + num_size a" + + (* Semantics of numeral terms (num) *) +primrec Inum :: "real list \ num \ real" where + "Inum bs (C c) = (real c)" +| "Inum bs (Bound n) = bs!n" +| "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)" +| "Inum bs (Neg a) = -(Inum bs a)" +| "Inum bs (Add a b) = Inum bs a + Inum bs b" +| "Inum bs (Sub a b) = Inum bs a - Inum bs b" +| "Inum bs (Mul c a) = (real c) * Inum bs a" +| "Inum bs (Floor a) = real (floor (Inum bs a))" +| "Inum bs (CF c a b) = real c * real (floor (Inum bs a)) + Inum bs b" +definition "isint t bs \ real (floor (Inum bs t)) = Inum bs t" + +lemma isint_iff: "isint n bs = (real (floor (Inum bs n)) = Inum bs n)" +by (simp add: isint_def) + +lemma isint_Floor: "isint (Floor n) bs" + by (simp add: isint_iff) + +lemma isint_Mul: "isint e bs \ isint (Mul c e) bs" +proof- + let ?e = "Inum bs e" + let ?fe = "floor ?e" + assume be: "isint e bs" hence efe:"real ?fe = ?e" by (simp add: isint_iff) + have "real ((floor (Inum bs (Mul c e)))) = real (floor (real (c * ?fe)))" using efe by simp + also have "\ = real (c* ?fe)" by (simp only: floor_real_of_int) + also have "\ = real c * ?e" using efe by simp + finally show ?thesis using isint_iff by simp +qed + +lemma isint_neg: "isint e bs \ isint (Neg e) bs" +proof- + let ?I = "\ t. Inum bs t" + assume ie: "isint e bs" + hence th: "real (floor (?I e)) = ?I e" by (simp add: isint_def) + have "real (floor (?I (Neg e))) = real (floor (- (real (floor (?I e)))))" by (simp add: th) + also have "\ = - real (floor (?I e))" by(simp add: floor_minus_real_of_int) + finally show "isint (Neg e) bs" by (simp add: isint_def th) +qed + +lemma isint_sub: + assumes ie: "isint e bs" shows "isint (Sub (C c) e) bs" +proof- + let ?I = "\ t. Inum bs t" + from ie have th: "real (floor (?I e)) = ?I e" by (simp add: isint_def) + have "real (floor (?I (Sub (C c) e))) = real (floor ((real (c -floor (?I e)))))" by (simp add: th) + also have "\ = real (c- floor (?I e))" by(simp add: floor_minus_real_of_int) + finally show "isint (Sub (C c) e) bs" by (simp add: isint_def th) +qed + +lemma isint_add: assumes + ai:"isint a bs" and bi: "isint b bs" shows "isint (Add a b) bs" +proof- + let ?a = "Inum bs a" + let ?b = "Inum bs b" + from ai bi isint_iff have "real (floor (?a + ?b)) = real (floor (real (floor ?a) + real (floor ?b)))" by simp + also have "\ = real (floor ?a) + real (floor ?b)" by simp + also have "\ = ?a + ?b" using ai bi isint_iff by simp + finally show "isint (Add a b) bs" by (simp add: isint_iff) +qed + +lemma isint_c: "isint (C j) bs" + by (simp add: isint_iff) + + + (* FORMULAE *) +datatype fm = + T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| Dvd int num| NDvd int num| + NOT fm| And fm fm| Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm + + + (* A size for fm *) +fun fmsize :: "fm \ nat" where + "fmsize (NOT p) = 1 + fmsize p" +| "fmsize (And p q) = 1 + fmsize p + fmsize q" +| "fmsize (Or p q) = 1 + fmsize p + fmsize q" +| "fmsize (Imp p q) = 3 + fmsize p + fmsize q" +| "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)" +| "fmsize (E p) = 1 + fmsize p" +| "fmsize (A p) = 4+ fmsize p" +| "fmsize (Dvd i t) = 2" +| "fmsize (NDvd i t) = 2" +| "fmsize p = 1" + (* several lemmas about fmsize *) +lemma fmsize_pos: "fmsize p > 0" +by (induct p rule: fmsize.induct) simp_all + + (* Semantics of formulae (fm) *) +primrec Ifm ::"real list \ fm \ bool" where + "Ifm bs T = True" +| "Ifm bs F = False" +| "Ifm bs (Lt a) = (Inum bs a < 0)" +| "Ifm bs (Gt a) = (Inum bs a > 0)" +| "Ifm bs (Le a) = (Inum bs a \ 0)" +| "Ifm bs (Ge a) = (Inum bs a \ 0)" +| "Ifm bs (Eq a) = (Inum bs a = 0)" +| "Ifm bs (NEq a) = (Inum bs a \ 0)" +| "Ifm bs (Dvd i b) = (real i rdvd Inum bs b)" +| "Ifm bs (NDvd i b) = (\(real i rdvd Inum bs b))" +| "Ifm bs (NOT p) = (\ (Ifm bs p))" +| "Ifm bs (And p q) = (Ifm bs p \ Ifm bs q)" +| "Ifm bs (Or p q) = (Ifm bs p \ Ifm bs q)" +| "Ifm bs (Imp p q) = ((Ifm bs p) \ (Ifm bs q))" +| "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)" +| "Ifm bs (E p) = (\ x. Ifm (x#bs) p)" +| "Ifm bs (A p) = (\ x. Ifm (x#bs) p)" + +consts prep :: "fm \ fm" +recdef prep "measure fmsize" + "prep (E T) = T" + "prep (E F) = F" + "prep (E (Or p q)) = Or (prep (E p)) (prep (E q))" + "prep (E (Imp p q)) = Or (prep (E (NOT p))) (prep (E q))" + "prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" + "prep (E (NOT (And p q))) = Or (prep (E (NOT p))) (prep (E(NOT q)))" + "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))" + "prep (E (NOT (Iff p q))) = Or (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))" + "prep (E p) = E (prep p)" + "prep (A (And p q)) = And (prep (A p)) (prep (A q))" + "prep (A p) = prep (NOT (E (NOT p)))" + "prep (NOT (NOT p)) = prep p" + "prep (NOT (And p q)) = Or (prep (NOT p)) (prep (NOT q))" + "prep (NOT (A p)) = prep (E (NOT p))" + "prep (NOT (Or p q)) = And (prep (NOT p)) (prep (NOT q))" + "prep (NOT (Imp p q)) = And (prep p) (prep (NOT q))" + "prep (NOT (Iff p q)) = Or (prep (And p (NOT q))) (prep (And (NOT p) q))" + "prep (NOT p) = NOT (prep p)" + "prep (Or p q) = Or (prep p) (prep q)" + "prep (And p q) = And (prep p) (prep q)" + "prep (Imp p q) = prep (Or (NOT p) q)" + "prep (Iff p q) = Or (prep (And p q)) (prep (And (NOT p) (NOT q)))" + "prep p = p" +(hints simp add: fmsize_pos) +lemma prep: "\ bs. Ifm bs (prep p) = Ifm bs p" +by (induct p rule: prep.induct, auto) + + + (* Quantifier freeness *) +fun qfree:: "fm \ bool" where + "qfree (E p) = False" + | "qfree (A p) = False" + | "qfree (NOT p) = qfree p" + | "qfree (And p q) = (qfree p \ qfree q)" + | "qfree (Or p q) = (qfree p \ qfree q)" + | "qfree (Imp p q) = (qfree p \ qfree q)" + | "qfree (Iff p q) = (qfree p \ qfree q)" + | "qfree p = True" + + (* Boundedness and substitution *) +primrec numbound0 :: "num \ bool" (* a num is INDEPENDENT of Bound 0 *) where + "numbound0 (C c) = True" + | "numbound0 (Bound n) = (n>0)" + | "numbound0 (CN n i a) = (n > 0 \ numbound0 a)" + | "numbound0 (Neg a) = numbound0 a" + | "numbound0 (Add a b) = (numbound0 a \ numbound0 b)" + | "numbound0 (Sub a b) = (numbound0 a \ numbound0 b)" + | "numbound0 (Mul i a) = numbound0 a" + | "numbound0 (Floor a) = numbound0 a" + | "numbound0 (CF c a b) = (numbound0 a \ numbound0 b)" + +lemma numbound0_I: + assumes nb: "numbound0 a" + shows "Inum (b#bs) a = Inum (b'#bs) a" + using nb by (induct a) (auto simp add: nth_pos2) + +lemma numbound0_gen: + assumes nb: "numbound0 t" and ti: "isint t (x#bs)" + shows "\ y. isint t (y#bs)" +using nb ti +proof(clarify) + fix y + from numbound0_I[OF nb, where bs="bs" and b="y" and b'="x"] ti[simplified isint_def] + show "isint t (y#bs)" + by (simp add: isint_def) +qed + +primrec bound0:: "fm \ bool" (* A Formula is independent of Bound 0 *) where + "bound0 T = True" + | "bound0 F = True" + | "bound0 (Lt a) = numbound0 a" + | "bound0 (Le a) = numbound0 a" + | "bound0 (Gt a) = numbound0 a" + | "bound0 (Ge a) = numbound0 a" + | "bound0 (Eq a) = numbound0 a" + | "bound0 (NEq a) = numbound0 a" + | "bound0 (Dvd i a) = numbound0 a" + | "bound0 (NDvd i a) = numbound0 a" + | "bound0 (NOT p) = bound0 p" + | "bound0 (And p q) = (bound0 p \ bound0 q)" + | "bound0 (Or p q) = (bound0 p \ bound0 q)" + | "bound0 (Imp p q) = ((bound0 p) \ (bound0 q))" + | "bound0 (Iff p q) = (bound0 p \ bound0 q)" + | "bound0 (E p) = False" + | "bound0 (A p) = False" + +lemma bound0_I: + assumes bp: "bound0 p" + shows "Ifm (b#bs) p = Ifm (b'#bs) p" + using bp numbound0_I [where b="b" and bs="bs" and b'="b'"] + by (induct p) (auto simp add: nth_pos2) + +primrec numsubst0:: "num \ num \ num" (* substitute a num into a num for Bound 0 *) where + "numsubst0 t (C c) = (C c)" + | "numsubst0 t (Bound n) = (if n=0 then t else Bound n)" + | "numsubst0 t (CN n i a) = (if n=0 then Add (Mul i t) (numsubst0 t a) else CN n i (numsubst0 t a))" + | "numsubst0 t (CF i a b) = CF i (numsubst0 t a) (numsubst0 t b)" + | "numsubst0 t (Neg a) = Neg (numsubst0 t a)" + | "numsubst0 t (Add a b) = Add (numsubst0 t a) (numsubst0 t b)" + | "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)" + | "numsubst0 t (Mul i a) = Mul i (numsubst0 t a)" + | "numsubst0 t (Floor a) = Floor (numsubst0 t a)" + +lemma numsubst0_I: + shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t" + by (induct t) (simp_all add: nth_pos2) + +lemma numsubst0_I': + assumes nb: "numbound0 a" + shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b'#bs) a)#bs) t" + by (induct t) (simp_all add: nth_pos2 numbound0_I[OF nb, where b="b" and b'="b'"]) + +primrec subst0:: "num \ fm \ fm" (* substitue a num into a formula for Bound 0 *) where + "subst0 t T = T" + | "subst0 t F = F" + | "subst0 t (Lt a) = Lt (numsubst0 t a)" + | "subst0 t (Le a) = Le (numsubst0 t a)" + | "subst0 t (Gt a) = Gt (numsubst0 t a)" + | "subst0 t (Ge a) = Ge (numsubst0 t a)" + | "subst0 t (Eq a) = Eq (numsubst0 t a)" + | "subst0 t (NEq a) = NEq (numsubst0 t a)" + | "subst0 t (Dvd i a) = Dvd i (numsubst0 t a)" + | "subst0 t (NDvd i a) = NDvd i (numsubst0 t a)" + | "subst0 t (NOT p) = NOT (subst0 t p)" + | "subst0 t (And p q) = And (subst0 t p) (subst0 t q)" + | "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)" + | "subst0 t (Imp p q) = Imp (subst0 t p) (subst0 t q)" + | "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)" + +lemma subst0_I: assumes qfp: "qfree p" + shows "Ifm (b#bs) (subst0 a p) = Ifm ((Inum (b#bs) a)#bs) p" + using qfp numsubst0_I[where b="b" and bs="bs" and a="a"] + by (induct p) (simp_all add: nth_pos2 ) + +consts + decrnum:: "num \ num" + decr :: "fm \ fm" + +recdef decrnum "measure size" + "decrnum (Bound n) = Bound (n - 1)" + "decrnum (Neg a) = Neg (decrnum a)" + "decrnum (Add a b) = Add (decrnum a) (decrnum b)" + "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)" + "decrnum (Mul c a) = Mul c (decrnum a)" + "decrnum (Floor a) = Floor (decrnum a)" + "decrnum (CN n c a) = CN (n - 1) c (decrnum a)" + "decrnum (CF c a b) = CF c (decrnum a) (decrnum b)" + "decrnum a = a" + +recdef decr "measure size" + "decr (Lt a) = Lt (decrnum a)" + "decr (Le a) = Le (decrnum a)" + "decr (Gt a) = Gt (decrnum a)" + "decr (Ge a) = Ge (decrnum a)" + "decr (Eq a) = Eq (decrnum a)" + "decr (NEq a) = NEq (decrnum a)" + "decr (Dvd i a) = Dvd i (decrnum a)" + "decr (NDvd i a) = NDvd i (decrnum a)" + "decr (NOT p) = NOT (decr p)" + "decr (And p q) = And (decr p) (decr q)" + "decr (Or p q) = Or (decr p) (decr q)" + "decr (Imp p q) = Imp (decr p) (decr q)" + "decr (Iff p q) = Iff (decr p) (decr q)" + "decr p = p" + +lemma decrnum: assumes nb: "numbound0 t" + shows "Inum (x#bs) t = Inum bs (decrnum t)" + using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2) + +lemma decr: assumes nb: "bound0 p" + shows "Ifm (x#bs) p = Ifm bs (decr p)" + using nb + by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum) + +lemma decr_qf: "bound0 p \ qfree (decr p)" +by (induct p, simp_all) + +consts + isatom :: "fm \ bool" (* test for atomicity *) +recdef isatom "measure size" + "isatom T = True" + "isatom F = True" + "isatom (Lt a) = True" + "isatom (Le a) = True" + "isatom (Gt a) = True" + "isatom (Ge a) = True" + "isatom (Eq a) = True" + "isatom (NEq a) = True" + "isatom (Dvd i b) = True" + "isatom (NDvd i b) = True" + "isatom p = False" + +lemma numsubst0_numbound0: assumes nb: "numbound0 t" + shows "numbound0 (numsubst0 t a)" +using nb by (induct a, auto) + +lemma subst0_bound0: assumes qf: "qfree p" and nb: "numbound0 t" + shows "bound0 (subst0 t p)" +using qf numsubst0_numbound0[OF nb] by (induct p, auto) + +lemma bound0_qf: "bound0 p \ qfree p" +by (induct p, simp_all) + + +definition djf:: "('a \ fm) \ 'a \ fm \ fm" where + "djf f p q = (if q=T then T else if q=F then f p else + (let fp = f p in case fp of T \ T | F \ q | _ \ Or fp q))" + +definition evaldjf:: "('a \ fm) \ 'a list \ fm" where + "evaldjf f ps = foldr (djf f) ps F" + +lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)" +by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) +(cases "f p", simp_all add: Let_def djf_def) + +lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\ p \ set ps. Ifm bs (f p))" + by(induct ps, simp_all add: evaldjf_def djf_Or) + +lemma evaldjf_bound0: + assumes nb: "\ x\ set xs. bound0 (f x)" + shows "bound0 (evaldjf f xs)" + using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) + +lemma evaldjf_qf: + assumes nb: "\ x\ set xs. qfree (f x)" + shows "qfree (evaldjf f xs)" + using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) + +consts + disjuncts :: "fm \ fm list" + conjuncts :: "fm \ fm list" +recdef disjuncts "measure size" + "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)" + "disjuncts F = []" + "disjuncts p = [p]" + +recdef conjuncts "measure size" + "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)" + "conjuncts T = []" + "conjuncts p = [p]" +lemma disjuncts: "(\ q\ set (disjuncts p). Ifm bs q) = Ifm bs p" +by(induct p rule: disjuncts.induct, auto) +lemma conjuncts: "(\ q\ set (conjuncts p). Ifm bs q) = Ifm bs p" +by(induct p rule: conjuncts.induct, auto) + +lemma disjuncts_nb: "bound0 p \ \ q\ set (disjuncts p). bound0 q" +proof- + assume nb: "bound0 p" + hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto) + thus ?thesis by (simp only: list_all_iff) +qed +lemma conjuncts_nb: "bound0 p \ \ q\ set (conjuncts p). bound0 q" +proof- + assume nb: "bound0 p" + hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct,auto) + thus ?thesis by (simp only: list_all_iff) +qed + +lemma disjuncts_qf: "qfree p \ \ q\ set (disjuncts p). qfree q" +proof- + assume qf: "qfree p" + hence "list_all qfree (disjuncts p)" + by (induct p rule: disjuncts.induct, auto) + thus ?thesis by (simp only: list_all_iff) +qed +lemma conjuncts_qf: "qfree p \ \ q\ set (conjuncts p). qfree q" +proof- + assume qf: "qfree p" + hence "list_all qfree (conjuncts p)" + by (induct p rule: conjuncts.induct, auto) + thus ?thesis by (simp only: list_all_iff) +qed + +constdefs DJ :: "(fm \ fm) \ fm \ fm" + "DJ f p \ evaldjf f (disjuncts p)" + +lemma DJ: assumes fdj: "\ p q. f (Or p q) = Or (f p) (f q)" + and fF: "f F = F" + shows "Ifm bs (DJ f p) = Ifm bs (f p)" +proof- + have "Ifm bs (DJ f p) = (\ q \ set (disjuncts p). Ifm bs (f q))" + by (simp add: DJ_def evaldjf_ex) + also have "\ = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto) + finally show ?thesis . +qed + +lemma DJ_qf: assumes + fqf: "\ p. qfree p \ qfree (f p)" + shows "\p. qfree p \ qfree (DJ f p) " +proof(clarify) + fix p assume qf: "qfree p" + have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def) + from disjuncts_qf[OF qf] have "\ q\ set (disjuncts p). qfree q" . + with fqf have th':"\ q\ set (disjuncts p). qfree (f q)" by blast + + from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp +qed + +lemma DJ_qe: assumes qe: "\ bs p. qfree p \ qfree (qe p) \ (Ifm bs (qe p) = Ifm bs (E p))" + shows "\ bs p. qfree p \ qfree (DJ qe p) \ (Ifm bs ((DJ qe p)) = Ifm bs (E p))" +proof(clarify) + fix p::fm and bs + assume qf: "qfree p" + from qe have qth: "\ p. qfree p \ qfree (qe p)" by blast + from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto + have "Ifm bs (DJ qe p) = (\ q\ set (disjuncts p). Ifm bs (qe q))" + by (simp add: DJ_def evaldjf_ex) + also have "\ = (\ q \ set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto + also have "\ = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto) + finally show "qfree (DJ qe p) \ Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast +qed + (* Simplification *) + + (* Algebraic simplifications for nums *) +consts bnds:: "num \ nat list" + lex_ns:: "nat list \ nat list \ bool" +recdef bnds "measure size" + "bnds (Bound n) = [n]" + "bnds (CN n c a) = n#(bnds a)" + "bnds (Neg a) = bnds a" + "bnds (Add a b) = (bnds a)@(bnds b)" + "bnds (Sub a b) = (bnds a)@(bnds b)" + "bnds (Mul i a) = bnds a" + "bnds (Floor a) = bnds a" + "bnds (CF c a b) = (bnds a)@(bnds b)" + "bnds a = []" +recdef lex_ns "measure (\ (xs,ys). length xs + length ys)" + "lex_ns ([], ms) = True" + "lex_ns (ns, []) = False" + "lex_ns (n#ns, m#ms) = (n ((n = m) \ lex_ns (ns,ms))) " +constdefs lex_bnd :: "num \ num \ bool" + "lex_bnd t s \ lex_ns (bnds t, bnds s)" + +consts + numgcdh:: "num \ int \ int" + reducecoeffh:: "num \ int \ num" + dvdnumcoeff:: "num \ int \ bool" +consts maxcoeff:: "num \ int" +recdef maxcoeff "measure size" + "maxcoeff (C i) = abs i" + "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)" + "maxcoeff (CF c t s) = max (abs c) (maxcoeff s)" + "maxcoeff t = 1" + +lemma maxcoeff_pos: "maxcoeff t \ 0" + apply (induct t rule: maxcoeff.induct, auto) + done + +recdef numgcdh "measure size" + "numgcdh (C i) = (\g. zgcd i g)" + "numgcdh (CN n c t) = (\g. zgcd c (numgcdh t g))" + "numgcdh (CF c s t) = (\g. zgcd c (numgcdh t g))" + "numgcdh t = (\g. 1)" + +definition + numgcd :: "num \ int" +where + numgcd_def: "numgcd t = numgcdh t (maxcoeff t)" + +recdef reducecoeffh "measure size" + "reducecoeffh (C i) = (\ g. C (i div g))" + "reducecoeffh (CN n c t) = (\ g. CN n (c div g) (reducecoeffh t g))" + "reducecoeffh (CF c s t) = (\ g. CF (c div g) s (reducecoeffh t g))" + "reducecoeffh t = (\g. t)" + +definition + reducecoeff :: "num \ num" +where + reducecoeff_def: "reducecoeff t = + (let g = numgcd t in + if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)" + +recdef dvdnumcoeff "measure size" + "dvdnumcoeff (C i) = (\ g. g dvd i)" + "dvdnumcoeff (CN n c t) = (\ g. g dvd c \ (dvdnumcoeff t g))" + "dvdnumcoeff (CF c s t) = (\ g. g dvd c \ (dvdnumcoeff t g))" + "dvdnumcoeff t = (\g. False)" + +lemma dvdnumcoeff_trans: + assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'" + shows "dvdnumcoeff t g" + using dgt' gdg + by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg zdvd_trans[OF gdg]) + +declare zdvd_trans [trans add] + +lemma natabs0: "(nat (abs x) = 0) = (x = 0)" +by arith + +lemma numgcd0: + assumes g0: "numgcd t = 0" + shows "Inum bs t = 0" +proof- + have "\x. numgcdh t x= 0 \ Inum bs t = 0" + by (induct t rule: numgcdh.induct, auto simp add: zgcd_def gcd_zero natabs0 max_def maxcoeff_pos) + thus ?thesis using g0[simplified numgcd_def] by blast +qed + +lemma numgcdh_pos: assumes gp: "g \ 0" shows "numgcdh t g \ 0" + using gp + by (induct t rule: numgcdh.induct, auto simp add: zgcd_def) + +lemma numgcd_pos: "numgcd t \0" + by (simp add: numgcd_def numgcdh_pos maxcoeff_pos) + +lemma reducecoeffh: + assumes gt: "dvdnumcoeff t g" and gp: "g > 0" + shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t" + using gt +proof(induct t rule: reducecoeffh.induct) + case (1 i) hence gd: "g dvd i" by simp + from gp have gnz: "g \ 0" by simp + from prems show ?case by (simp add: real_of_int_div[OF gnz gd]) +next + case (2 n c t) hence gd: "g dvd c" by simp + from gp have gnz: "g \ 0" by simp + from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps) +next + case (3 c s t) hence gd: "g dvd c" by simp + from gp have gnz: "g \ 0" by simp + from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps) +qed (auto simp add: numgcd_def gp) +consts ismaxcoeff:: "num \ int \ bool" +recdef ismaxcoeff "measure size" + "ismaxcoeff (C i) = (\ x. abs i \ x)" + "ismaxcoeff (CN n c t) = (\x. abs c \ x \ (ismaxcoeff t x))" + "ismaxcoeff (CF c s t) = (\x. abs c \ x \ (ismaxcoeff t x))" + "ismaxcoeff t = (\x. True)" + +lemma ismaxcoeff_mono: "ismaxcoeff t c \ c \ c' \ ismaxcoeff t c'" +by (induct t rule: ismaxcoeff.induct, auto) + +lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)" +proof (induct t rule: maxcoeff.induct) + case (2 n c t) + hence H:"ismaxcoeff t (maxcoeff t)" . + have thh: "maxcoeff t \ max (abs c) (maxcoeff t)" by (simp add: le_maxI2) + from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1) +next + case (3 c t s) + hence H1:"ismaxcoeff s (maxcoeff s)" by auto + have thh1: "maxcoeff s \ max \c\ (maxcoeff s)" by (simp add: max_def) + from ismaxcoeff_mono[OF H1 thh1] show ?case by (simp add: le_maxI1) +qed simp_all + +lemma zgcd_gt1: "zgcd i j > 1 \ ((abs i > 1 \ abs j > 1) \ (abs i = 0 \ abs j > 1) \ (abs i > 1 \ abs j = 0))" + apply (unfold zgcd_def) + apply (cases "i = 0", simp_all) + apply (cases "j = 0", simp_all) + apply (cases "abs i = 1", simp_all) + apply (cases "abs j = 1", simp_all) + apply auto + done +lemma numgcdh0:"numgcdh t m = 0 \ m =0" + by (induct t rule: numgcdh.induct, auto simp add:zgcd0) + +lemma dvdnumcoeff_aux: + assumes "ismaxcoeff t m" and mp:"m \ 0" and "numgcdh t m > 1" + shows "dvdnumcoeff t (numgcdh t m)" +using prems +proof(induct t rule: numgcdh.induct) + case (2 n c t) + let ?g = "numgcdh t m" + from prems have th:"zgcd c ?g > 1" by simp + from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"] + have "(abs c > 1 \ ?g > 1) \ (abs c = 0 \ ?g > 1) \ (abs c > 1 \ ?g = 0)" by simp + moreover {assume "abs c > 1" and gp: "?g > 1" with prems + have th: "dvdnumcoeff t ?g" by simp + have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2) + from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)} + moreover {assume "abs c = 0 \ ?g > 1" + with prems have th: "dvdnumcoeff t ?g" by simp + have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2) + from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1) + hence ?case by simp } + moreover {assume "abs c > 1" and g0:"?g = 0" + from numgcdh0[OF g0] have "m=0". with prems have ?case by simp } + ultimately show ?case by blast +next + case (3 c s t) + let ?g = "numgcdh t m" + from prems have th:"zgcd c ?g > 1" by simp + from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"] + have "(abs c > 1 \ ?g > 1) \ (abs c = 0 \ ?g > 1) \ (abs c > 1 \ ?g = 0)" by simp + moreover {assume "abs c > 1" and gp: "?g > 1" with prems + have th: "dvdnumcoeff t ?g" by simp + have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2) + from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)} + moreover {assume "abs c = 0 \ ?g > 1" + with prems have th: "dvdnumcoeff t ?g" by simp + have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2) + from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1) + hence ?case by simp } + moreover {assume "abs c > 1" and g0:"?g = 0" + from numgcdh0[OF g0] have "m=0". with prems have ?case by simp } + ultimately show ?case by blast +qed(auto simp add: zgcd_zdvd1) + +lemma dvdnumcoeff_aux2: + assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \ numgcd t > 0" + using prems +proof (simp add: numgcd_def) + let ?mc = "maxcoeff t" + let ?g = "numgcdh t ?mc" + have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff) + have th2: "?mc \ 0" by (rule maxcoeff_pos) + assume H: "numgcdh t ?mc > 1" + from dvdnumcoeff_aux[OF th1 th2 H] show "dvdnumcoeff t ?g" . +qed + +lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t" +proof- + let ?g = "numgcd t" + have "?g \ 0" by (simp add: numgcd_pos) + hence "?g = 0 \ ?g = 1 \ ?g > 1" by auto + moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} + moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} + moreover { assume g1:"?g > 1" + from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+ + from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis + by (simp add: reducecoeff_def Let_def)} + ultimately show ?thesis by blast +qed + +lemma reducecoeffh_numbound0: "numbound0 t \ numbound0 (reducecoeffh t g)" +by (induct t rule: reducecoeffh.induct, auto) + +lemma reducecoeff_numbound0: "numbound0 t \ numbound0 (reducecoeff t)" +using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def) + +consts + simpnum:: "num \ num" + numadd:: "num \ num \ num" + nummul:: "num \ int \ num" + +recdef numadd "measure (\ (t,s). size t + size s)" + "numadd (CN n1 c1 r1,CN n2 c2 r2) = + (if n1=n2 then + (let c = c1 + c2 + in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2)))) + else if n1 \ n2 then CN n1 c1 (numadd (r1,CN n2 c2 r2)) + else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))" + "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))" + "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" + "numadd (CF c1 t1 r1,CF c2 t2 r2) = + (if t1 = t2 then + (let c=c1+c2; s= numadd(r1,r2) in (if c=0 then s else CF c t1 s)) + else if lex_bnd t1 t2 then CF c1 t1 (numadd(r1,CF c2 t2 r2)) + else CF c2 t2 (numadd(CF c1 t1 r1,r2)))" + "numadd (CF c1 t1 r1,C c) = CF c1 t1 (numadd (r1, C c))" + "numadd (C c,CF c1 t1 r1) = CF c1 t1 (numadd (r1, C c))" + "numadd (C b1, C b2) = C (b1+b2)" + "numadd (a,b) = Add a b" + +lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)" +apply (induct t s rule: numadd.induct, simp_all add: Let_def) + apply (case_tac "c1+c2 = 0",case_tac "n1 \ n2", simp_all) + apply (case_tac "n1 = n2", simp_all add: algebra_simps) + apply (simp only: left_distrib[symmetric]) + apply simp +apply (case_tac "lex_bnd t1 t2", simp_all) + apply (case_tac "c1+c2 = 0") + by (case_tac "t1 = t2", simp_all add: algebra_simps left_distrib[symmetric] real_of_int_mult[symmetric] real_of_int_add[symmetric]del: real_of_int_mult real_of_int_add left_distrib) + +lemma numadd_nb[simp]: "\ numbound0 t ; numbound0 s\ \ numbound0 (numadd (t,s))" +by (induct t s rule: numadd.induct, auto simp add: Let_def) + +recdef nummul "measure size" + "nummul (C j) = (\ i. C (i*j))" + "nummul (CN n c t) = (\ i. CN n (c*i) (nummul t i))" + "nummul (CF c t s) = (\ i. CF (c*i) t (nummul s i))" + "nummul (Mul c t) = (\ i. nummul t (i*c))" + "nummul t = (\ i. Mul i t)" + +lemma nummul[simp]: "\ i. Inum bs (nummul t i) = Inum bs (Mul i t)" +by (induct t rule: nummul.induct, auto simp add: algebra_simps) + +lemma nummul_nb[simp]: "\ i. numbound0 t \ numbound0 (nummul t i)" +by (induct t rule: nummul.induct, auto) + +constdefs numneg :: "num \ num" + "numneg t \ nummul t (- 1)" + +constdefs numsub :: "num \ num \ num" + "numsub s t \ (if s = t then C 0 else numadd (s,numneg t))" + +lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)" +using numneg_def nummul by simp + +lemma numneg_nb[simp]: "numbound0 t \ numbound0 (numneg t)" +using numneg_def by simp + +lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)" +using numsub_def by simp + +lemma numsub_nb[simp]: "\ numbound0 t ; numbound0 s\ \ numbound0 (numsub t s)" +using numsub_def by simp + +lemma isint_CF: assumes si: "isint s bs" shows "isint (CF c t s) bs" +proof- + have cti: "isint (Mul c (Floor t)) bs" by (simp add: isint_Mul isint_Floor) + + have "?thesis = isint (Add (Mul c (Floor t)) s) bs" by (simp add: isint_def) + also have "\" by (simp add: isint_add cti si) + finally show ?thesis . +qed + +consts split_int:: "num \ num\num" +recdef split_int "measure num_size" + "split_int (C c) = (C 0, C c)" + "split_int (CN n c b) = + (let (bv,bi) = split_int b + in (CN n c bv, bi))" + "split_int (CF c a b) = + (let (bv,bi) = split_int b + in (bv, CF c a bi))" + "split_int a = (a,C 0)" + +lemma split_int:"\ tv ti. split_int t = (tv,ti) \ (Inum bs (Add tv ti) = Inum bs t) \ isint ti bs" +proof (induct t rule: split_int.induct) + case (2 c n b tv ti) + let ?bv = "fst (split_int b)" + let ?bi = "snd (split_int b)" + have "split_int b = (?bv,?bi)" by simp + with prems(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+ + from prems(2) have tibi: "ti = ?bi" by (simp add: Let_def split_def) + from prems(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def) +next + case (3 c a b tv ti) + let ?bv = "fst (split_int b)" + let ?bi = "snd (split_int b)" + have "split_int b = (?bv,?bi)" by simp + with prems(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+ + from prems(2) have tibi: "ti = CF c a ?bi" by (simp add: Let_def split_def) + from prems(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def isint_Floor isint_add isint_Mul isint_CF) +qed (auto simp add: Let_def isint_iff isint_Floor isint_add isint_Mul split_def algebra_simps) + +lemma split_int_nb: "numbound0 t \ numbound0 (fst (split_int t)) \ numbound0 (snd (split_int t)) " +by (induct t rule: split_int.induct, auto simp add: Let_def split_def) + +definition + numfloor:: "num \ num" +where + numfloor_def: "numfloor t = (let (tv,ti) = split_int t in + (case tv of C i \ numadd (tv,ti) + | _ \ numadd(CF 1 tv (C 0),ti)))" + +lemma numfloor[simp]: "Inum bs (numfloor t) = Inum bs (Floor t)" (is "?n t = ?N (Floor t)") +proof- + let ?tv = "fst (split_int t)" + let ?ti = "snd (split_int t)" + have tvti:"split_int t = (?tv,?ti)" by simp + {assume H: "\ v. ?tv \ C v" + hence th1: "?n t = ?N (Add (Floor ?tv) ?ti)" + by (cases ?tv, auto simp add: numfloor_def Let_def split_def numadd) + from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+ + hence "?N (Floor t) = real (floor (?N (Add ?tv ?ti)))" by simp + also have "\ = real (floor (?N ?tv) + (floor (?N ?ti)))" + by (simp,subst tii[simplified isint_iff, symmetric]) simp + also have "\ = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff]) + finally have ?thesis using th1 by simp} + moreover {fix v assume H:"?tv = C v" + from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+ + hence "?N (Floor t) = real (floor (?N (Add ?tv ?ti)))" by simp + also have "\ = real (floor (?N ?tv) + (floor (?N ?ti)))" + by (simp,subst tii[simplified isint_iff, symmetric]) simp + also have "\ = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff]) + finally have ?thesis by (simp add: H numfloor_def Let_def split_def numadd) } + ultimately show ?thesis by auto +qed + +lemma numfloor_nb[simp]: "numbound0 t \ numbound0 (numfloor t)" + using split_int_nb[where t="t"] + by (cases "fst(split_int t)" , auto simp add: numfloor_def Let_def split_def numadd_nb) + +recdef simpnum "measure num_size" + "simpnum (C j) = C j" + "simpnum (Bound n) = CN n 1 (C 0)" + "simpnum (Neg t) = numneg (simpnum t)" + "simpnum (Add t s) = numadd (simpnum t,simpnum s)" + "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)" + "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)" + "simpnum (Floor t) = numfloor (simpnum t)" + "simpnum (CN n c t) = (if c=0 then simpnum t else CN n c (simpnum t))" + "simpnum (CF c t s) = simpnum(Add (Mul c (Floor t)) s)" + +lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t" +by (induct t rule: simpnum.induct, auto) + +lemma simpnum_numbound0[simp]: + "numbound0 t \ numbound0 (simpnum t)" +by (induct t rule: simpnum.induct, auto) + +consts nozerocoeff:: "num \ bool" +recdef nozerocoeff "measure size" + "nozerocoeff (C c) = True" + "nozerocoeff (CN n c t) = (c\0 \ nozerocoeff t)" + "nozerocoeff (CF c s t) = (c \ 0 \ nozerocoeff t)" + "nozerocoeff (Mul c t) = (c\0 \ nozerocoeff t)" + "nozerocoeff t = True" + +lemma numadd_nz : "nozerocoeff a \ nozerocoeff b \ nozerocoeff (numadd (a,b))" +by (induct a b rule: numadd.induct,auto simp add: Let_def) + +lemma nummul_nz : "\ i. i\0 \ nozerocoeff a \ nozerocoeff (nummul a i)" + by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz) + +lemma numneg_nz : "nozerocoeff a \ nozerocoeff (numneg a)" +by (simp add: numneg_def nummul_nz) + +lemma numsub_nz: "nozerocoeff a \ nozerocoeff b \ nozerocoeff (numsub a b)" +by (simp add: numsub_def numneg_nz numadd_nz) + +lemma split_int_nz: "nozerocoeff t \ nozerocoeff (fst (split_int t)) \ nozerocoeff (snd (split_int t))" +by (induct t rule: split_int.induct,auto simp add: Let_def split_def) + +lemma numfloor_nz: "nozerocoeff t \ nozerocoeff (numfloor t)" +by (simp add: numfloor_def Let_def split_def) +(cases "fst (split_int t)", simp_all add: split_int_nz numadd_nz) + +lemma simpnum_nz: "nozerocoeff (simpnum t)" +by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz numfloor_nz) + +lemma maxcoeff_nz: "nozerocoeff t \ maxcoeff t = 0 \ t = C 0" +proof (induct t rule: maxcoeff.induct) + case (2 n c t) + hence cnz: "c \0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+ + have "max (abs c) (maxcoeff t) \ abs c" by (simp add: le_maxI1) + with cnz have "max (abs c) (maxcoeff t) > 0" by arith + with prems show ?case by simp +next + case (3 c s t) + hence cnz: "c \0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+ + have "max (abs c) (maxcoeff t) \ abs c" by (simp add: le_maxI1) + with cnz have "max (abs c) (maxcoeff t) > 0" by arith + with prems show ?case by simp +qed auto + +lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0" +proof- + from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def) + from numgcdh0[OF th] have th:"maxcoeff t = 0" . + from maxcoeff_nz[OF nz th] show ?thesis . +qed + +constdefs simp_num_pair:: "(num \ int) \ num \ int" + "simp_num_pair \ (\ (t,n). (if n = 0 then (C 0, 0) else + (let t' = simpnum t ; g = numgcd t' in + if g > 1 then (let g' = zgcd n g in + if g' = 1 then (t',n) + else (reducecoeffh t' g', n div g')) + else (t',n))))" + +lemma simp_num_pair_ci: + shows "((\ (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\ (t,n). Inum bs t / real n) (t,n))" + (is "?lhs = ?rhs") +proof- + let ?t' = "simpnum t" + let ?g = "numgcd ?t'" + let ?g' = "zgcd n ?g" + {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)} + moreover + { assume nnz: "n \ 0" + {assume "\ ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def)} + moreover + {assume g1:"?g>1" hence g0: "?g > 0" by simp + from zgcd0 g1 nnz have gp0: "?g' \ 0" by simp + hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith + hence "?g'= 1 \ ?g' > 1" by arith + moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def)} + moreover {assume g'1:"?g'>1" + from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" .. + let ?tt = "reducecoeffh ?t' ?g'" + let ?t = "Inum bs ?tt" + have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2) + have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1) + have gpdgp: "?g' dvd ?g'" by simp + from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] + have th2:"real ?g' * ?t = Inum bs ?t'" by simp + from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def) + also have "\ = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp + also have "\ = (Inum bs ?t' / real n)" + using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp + finally have "?lhs = Inum bs t / real n" by simp + then have ?thesis using prems by (simp add: simp_num_pair_def)} + ultimately have ?thesis by blast} + ultimately have ?thesis by blast} + ultimately show ?thesis by blast +qed + +lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')" + shows "numbound0 t' \ n' >0" +proof- + let ?t' = "simpnum t" + let ?g = "numgcd ?t'" + let ?g' = "zgcd n ?g" + {assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)} + moreover + { assume nnz: "n \ 0" + {assume "\ ?g > 1" hence ?thesis using prems by (auto simp add: Let_def simp_num_pair_def)} + moreover + {assume g1:"?g>1" hence g0: "?g > 0" by simp + from zgcd0 g1 nnz have gp0: "?g' \ 0" by simp + hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith + hence "?g'= 1 \ ?g' > 1" by arith + moreover {assume "?g'=1" hence ?thesis using prems + by (auto simp add: Let_def simp_num_pair_def)} + moreover {assume g'1:"?g'>1" + have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2) + have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1) + have gpdgp: "?g' dvd ?g'" by simp + from zdvd_imp_le[OF gpdd np] have g'n: "?g' \ n" . + from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]] + have "n div ?g' >0" by simp + hence ?thesis using prems + by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0)} + ultimately have ?thesis by blast} + ultimately have ?thesis by blast} + ultimately show ?thesis by blast +qed + +consts not:: "fm \ fm" +recdef not "measure size" + "not (NOT p) = p" + "not T = F" + "not F = T" + "not (Lt t) = Ge t" + "not (Le t) = Gt t" + "not (Gt t) = Le t" + "not (Ge t) = Lt t" + "not (Eq t) = NEq t" + "not (NEq t) = Eq t" + "not (Dvd i t) = NDvd i t" + "not (NDvd i t) = Dvd i t" + "not (And p q) = Or (not p) (not q)" + "not (Or p q) = And (not p) (not q)" + "not p = NOT p" +lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)" +by (induct p) auto +lemma not_qf[simp]: "qfree p \ qfree (not p)" +by (induct p, auto) +lemma not_nb[simp]: "bound0 p \ bound0 (not p)" +by (induct p, auto) + +constdefs conj :: "fm \ fm \ fm" + "conj p q \ (if (p = F \ q=F) then F else if p=T then q else if q=T then p else + if p = q then p else And p q)" +lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)" +by (cases "p=F \ q=F",simp_all add: conj_def) (cases p,simp_all) + +lemma conj_qf[simp]: "\qfree p ; qfree q\ \ qfree (conj p q)" +using conj_def by auto +lemma conj_nb[simp]: "\bound0 p ; bound0 q\ \ bound0 (conj p q)" +using conj_def by auto + +constdefs disj :: "fm \ fm \ fm" + "disj p q \ (if (p = T \ q=T) then T else if p=F then q else if q=F then p + else if p=q then p else Or p q)" + +lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)" +by (cases "p=T \ q=T",simp_all add: disj_def) (cases p,simp_all) +lemma disj_qf[simp]: "\qfree p ; qfree q\ \ qfree (disj p q)" +using disj_def by auto +lemma disj_nb[simp]: "\bound0 p ; bound0 q\ \ bound0 (disj p q)" +using disj_def by auto + +constdefs imp :: "fm \ fm \ fm" + "imp p q \ (if (p = F \ q=T \ p=q) then T else if p=T then q else if q=F then not p + else Imp p q)" +lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)" +by (cases "p=F \ q=T",simp_all add: imp_def) +lemma imp_qf[simp]: "\qfree p ; qfree q\ \ qfree (imp p q)" +using imp_def by (cases "p=F \ q=T",simp_all add: imp_def) +lemma imp_nb[simp]: "\bound0 p ; bound0 q\ \ bound0 (imp p q)" +using imp_def by (cases "p=F \ q=T \ p=q",simp_all add: imp_def) + +constdefs iff :: "fm \ fm \ fm" + "iff p q \ (if (p = q) then T else if (p = not q \ not p = q) then F else + if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else + Iff p q)" +lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)" + by (unfold iff_def,cases "p=q", simp,cases "p=not q", simp add:not) +(cases "not p= q", auto simp add:not) +lemma iff_qf[simp]: "\qfree p ; qfree q\ \ qfree (iff p q)" + by (unfold iff_def,cases "p=q", auto) +lemma iff_nb[simp]: "\bound0 p ; bound0 q\ \ bound0 (iff p q)" +using iff_def by (unfold iff_def,cases "p=q", auto) + +consts check_int:: "num \ bool" +recdef check_int "measure size" + "check_int (C i) = True" + "check_int (Floor t) = True" + "check_int (Mul i t) = check_int t" + "check_int (Add t s) = (check_int t \ check_int s)" + "check_int (Neg t) = check_int t" + "check_int (CF c t s) = check_int s" + "check_int t = False" +lemma check_int: "check_int t \ isint t bs" +by (induct t, auto simp add: isint_add isint_Floor isint_Mul isint_neg isint_c isint_CF) + +lemma rdvd_left1_int: "real \t\ = t \ 1 rdvd t" + by (simp add: rdvd_def,rule_tac x="\t\" in exI) simp + +lemma rdvd_reduce: + assumes gd:"g dvd d" and gc:"g dvd c" and gp: "g > 0" + shows "real (d::int) rdvd real (c::int)*t = (real (d div g) rdvd real (c div g)*t)" +proof + assume d: "real d rdvd real c * t" + from d rdvd_def obtain k where k_def: "real c * t = real d* real (k::int)" by auto + from gd dvd_def obtain kd where kd_def: "d = g * kd" by auto + from gc dvd_def obtain kc where kc_def: "c = g * kc" by auto + from k_def kd_def kc_def have "real g * real kc * t = real g * real kd * real k" by simp + hence "real kc * t = real kd * real k" using gp by simp + hence th:"real kd rdvd real kc * t" using rdvd_def by blast + from kd_def gp have th':"kd = d div g" by simp + from kc_def gp have "kc = c div g" by simp + with th th' show "real (d div g) rdvd real (c div g) * t" by simp +next + assume d: "real (d div g) rdvd real (c div g) * t" + from gp have gnz: "g \ 0" by simp + thus "real d rdvd real c * t" using d rdvd_mult[OF gnz, where n="d div g" and x="real (c div g) * t"] real_of_int_div[OF gnz gd] real_of_int_div[OF gnz gc] by simp +qed + +constdefs simpdvd:: "int \ num \ (int \ num)" + "simpdvd d t \ + (let g = numgcd t in + if g > 1 then (let g' = zgcd d g in + if g' = 1 then (d, t) + else (d div g',reducecoeffh t g')) + else (d, t))" +lemma simpdvd: + assumes tnz: "nozerocoeff t" and dnz: "d \ 0" + shows "Ifm bs (Dvd (fst (simpdvd d t)) (snd (simpdvd d t))) = Ifm bs (Dvd d t)" +proof- + let ?g = "numgcd t" + let ?g' = "zgcd d ?g" + {assume "\ ?g > 1" hence ?thesis by (simp add: Let_def simpdvd_def)} + moreover + {assume g1:"?g>1" hence g0: "?g > 0" by simp + from zgcd0 g1 dnz have gp0: "?g' \ 0" by simp + hence g'p: "?g' > 0" using zgcd_pos[where i="d" and j="numgcd t"] by arith + hence "?g'= 1 \ ?g' > 1" by arith + moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simpdvd_def)} + moreover {assume g'1:"?g'>1" + from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" .. + let ?tt = "reducecoeffh t ?g'" + let ?t = "Inum bs ?tt" + have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2) + have gpdd: "?g' dvd d" by (simp add: zgcd_zdvd1) + have gpdgp: "?g' dvd ?g'" by simp + from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] + have th2:"real ?g' * ?t = Inum bs t" by simp + from prems have "Ifm bs (Dvd (fst (simpdvd d t)) (snd(simpdvd d t))) = Ifm bs (Dvd (d div ?g') ?tt)" + by (simp add: simpdvd_def Let_def) + also have "\ = (real d rdvd (Inum bs t))" + using rdvd_reduce[OF gpdd gpdgp g'p, where t="?t", simplified zdiv_self[OF gp0]] + th2[symmetric] by simp + finally have ?thesis by simp } + ultimately have ?thesis by blast + } + ultimately show ?thesis by blast +qed + +consts simpfm :: "fm \ fm" +recdef simpfm "measure fmsize" + "simpfm (And p q) = conj (simpfm p) (simpfm q)" + "simpfm (Or p q) = disj (simpfm p) (simpfm q)" + "simpfm (Imp p q) = imp (simpfm p) (simpfm q)" + "simpfm (Iff p q) = iff (simpfm p) (simpfm q)" + "simpfm (NOT p) = not (simpfm p)" + "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \ if (v < 0) then T else F + | _ \ Lt (reducecoeff a'))" + "simpfm (Le a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ Le (reducecoeff a'))" + "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \ if (v > 0) then T else F | _ \ Gt (reducecoeff a'))" + "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ Ge (reducecoeff a'))" + "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \ if (v = 0) then T else F | _ \ Eq (reducecoeff a'))" + "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ NEq (reducecoeff a'))" + "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a) + else if (abs i = 1) \ check_int a then T + else let a' = simpnum a in case a' of C v \ if (i dvd v) then T else F | _ \ (let (d,t) = simpdvd i a' in Dvd d t))" + "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) + else if (abs i = 1) \ check_int a then F + else let a' = simpnum a in case a' of C v \ if (\(i dvd v)) then T else F | _ \ (let (d,t) = simpdvd i a' in NDvd d t))" + "simpfm p = p" + +lemma simpfm[simp]: "Ifm bs (simpfm p) = Ifm bs p" +proof(induct p rule: simpfm.induct) + case (6 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume H:"\ (\ v. ?sa = C v)" + let ?g = "numgcd ?sa" + let ?rsa = "reducecoeff ?sa" + let ?r = "Inum bs ?rsa" + have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) + {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} + with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) + hence gp: "real ?g > 0" by simp + have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff) + with sa have "Inum bs a < 0 = (real ?g * ?r < real ?g * 0)" by simp + also have "\ = (?r < 0)" using gp + by (simp only: mult_less_cancel_left) simp + finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} + ultimately show ?case by blast +next + case (7 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume H:"\ (\ v. ?sa = C v)" + let ?g = "numgcd ?sa" + let ?rsa = "reducecoeff ?sa" + let ?r = "Inum bs ?rsa" + have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) + {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} + with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) + hence gp: "real ?g > 0" by simp + have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff) + with sa have "Inum bs a \ 0 = (real ?g * ?r \ real ?g * 0)" by simp + also have "\ = (?r \ 0)" using gp + by (simp only: mult_le_cancel_left) simp + finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} + ultimately show ?case by blast +next + case (8 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume H:"\ (\ v. ?sa = C v)" + let ?g = "numgcd ?sa" + let ?rsa = "reducecoeff ?sa" + let ?r = "Inum bs ?rsa" + have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) + {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} + with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) + hence gp: "real ?g > 0" by simp + have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff) + with sa have "Inum bs a > 0 = (real ?g * ?r > real ?g * 0)" by simp + also have "\ = (?r > 0)" using gp + by (simp only: mult_less_cancel_left) simp + finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} + ultimately show ?case by blast +next + case (9 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume H:"\ (\ v. ?sa = C v)" + let ?g = "numgcd ?sa" + let ?rsa = "reducecoeff ?sa" + let ?r = "Inum bs ?rsa" + have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) + {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} + with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) + hence gp: "real ?g > 0" by simp + have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff) + with sa have "Inum bs a \ 0 = (real ?g * ?r \ real ?g * 0)" by simp + also have "\ = (?r \ 0)" using gp + by (simp only: mult_le_cancel_left) simp + finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} + ultimately show ?case by blast +next + case (10 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume H:"\ (\ v. ?sa = C v)" + let ?g = "numgcd ?sa" + let ?rsa = "reducecoeff ?sa" + let ?r = "Inum bs ?rsa" + have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) + {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} + with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) + hence gp: "real ?g > 0" by simp + have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff) + with sa have "Inum bs a = 0 = (real ?g * ?r = 0)" by simp + also have "\ = (?r = 0)" using gp + by (simp add: mult_eq_0_iff) + finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} + ultimately show ?case by blast +next + case (11 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume H:"\ (\ v. ?sa = C v)" + let ?g = "numgcd ?sa" + let ?rsa = "reducecoeff ?sa" + let ?r = "Inum bs ?rsa" + have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) + {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} + with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) + hence gp: "real ?g > 0" by simp + have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff) + with sa have "Inum bs a \ 0 = (real ?g * ?r \ 0)" by simp + also have "\ = (?r \ 0)" using gp + by (simp add: mult_eq_0_iff) + finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} + ultimately show ?case by blast +next + case (12 i a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + have "i=0 \ (abs i = 1 \ check_int a) \ (i\0 \ ((abs i \ 1) \ (\ check_int a)))" by auto + {assume "i=0" hence ?case using "12.hyps" by (simp add: rdvd_left_0_eq Let_def)} + moreover + {assume ai1: "abs i = 1" and ai: "check_int a" + hence "i=1 \ i= - 1" by arith + moreover {assume i1: "i = 1" + from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] + have ?case using i1 ai by simp } + moreover {assume i1: "i = - 1" + from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] + rdvd_abs1[where d="- 1" and t="Inum bs a"] + have ?case using i1 ai by simp } + ultimately have ?case by blast} + moreover + {assume inz: "i\0" and cond: "(abs i \ 1) \ (\ check_int a)" + {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond + by (cases "abs i = 1", auto simp add: int_rdvd_iff) } + moreover {assume H:"\ (\ v. ?sa = C v)" + hence th: "simpfm (Dvd i a) = Dvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond by (cases ?sa, auto simp add: Let_def split_def) + from simpnum_nz have nz:"nozerocoeff ?sa" by simp + from simpdvd [OF nz inz] th have ?case using sa by simp} + ultimately have ?case by blast} + ultimately show ?case by blast +next + case (13 i a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + have "i=0 \ (abs i = 1 \ check_int a) \ (i\0 \ ((abs i \ 1) \ (\ check_int a)))" by auto + {assume "i=0" hence ?case using "13.hyps" by (simp add: rdvd_left_0_eq Let_def)} + moreover + {assume ai1: "abs i = 1" and ai: "check_int a" + hence "i=1 \ i= - 1" by arith + moreover {assume i1: "i = 1" + from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] + have ?case using i1 ai by simp } + moreover {assume i1: "i = - 1" + from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] + rdvd_abs1[where d="- 1" and t="Inum bs a"] + have ?case using i1 ai by simp } + ultimately have ?case by blast} + moreover + {assume inz: "i\0" and cond: "(abs i \ 1) \ (\ check_int a)" + {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond + by (cases "abs i = 1", auto simp add: int_rdvd_iff) } + moreover {assume H:"\ (\ v. ?sa = C v)" + hence th: "simpfm (NDvd i a) = NDvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond + by (cases ?sa, auto simp add: Let_def split_def) + from simpnum_nz have nz:"nozerocoeff ?sa" by simp + from simpdvd [OF nz inz] th have ?case using sa by simp} + ultimately have ?case by blast} + ultimately show ?case by blast +qed (induct p rule: simpfm.induct, simp_all) + +lemma simpdvd_numbound0: "numbound0 t \ numbound0 (snd (simpdvd d t))" + by (simp add: simpdvd_def Let_def split_def reducecoeffh_numbound0) + +lemma simpfm_bound0[simp]: "bound0 p \ bound0 (simpfm p)" +proof(induct p rule: simpfm.induct) + case (6 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) +next + case (7 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) +next + case (8 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) +next + case (9 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) +next + case (10 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) +next + case (11 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) +next + case (12 i a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0 simpdvd_numbound0 split_def) +next + case (13 i a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0 simpdvd_numbound0 split_def) +qed(auto simp add: disj_def imp_def iff_def conj_def) + +lemma simpfm_qf[simp]: "qfree p \ qfree (simpfm p)" +by (induct p rule: simpfm.induct, auto simp add: Let_def) +(case_tac "simpnum a",auto simp add: split_def Let_def)+ + + + (* Generic quantifier elimination *) + +constdefs list_conj :: "fm list \ fm" + "list_conj ps \ foldr conj ps T" +lemma list_conj: "Ifm bs (list_conj ps) = (\p\ set ps. Ifm bs p)" + by (induct ps, auto simp add: list_conj_def) +lemma list_conj_qf: " \p\ set ps. qfree p \ qfree (list_conj ps)" + by (induct ps, auto simp add: list_conj_def) +lemma list_conj_nb: " \p\ set ps. bound0 p \ bound0 (list_conj ps)" + by (induct ps, auto simp add: list_conj_def) +constdefs CJNB:: "(fm \ fm) \ fm \ fm" + "CJNB f p \ (let cjs = conjuncts p ; (yes,no) = List.partition bound0 cjs + in conj (decr (list_conj yes)) (f (list_conj no)))" + +lemma CJNB_qe: + assumes qe: "\ bs p. qfree p \ qfree (qe p) \ (Ifm bs (qe p) = Ifm bs (E p))" + shows "\ bs p. qfree p \ qfree (CJNB qe p) \ (Ifm bs ((CJNB qe p)) = Ifm bs (E p))" +proof(clarify) + fix bs p + assume qfp: "qfree p" + let ?cjs = "conjuncts p" + let ?yes = "fst (List.partition bound0 ?cjs)" + let ?no = "snd (List.partition bound0 ?cjs)" + let ?cno = "list_conj ?no" + let ?cyes = "list_conj ?yes" + have part: "List.partition bound0 ?cjs = (?yes,?no)" by simp + from partition_P[OF part] have "\ q\ set ?yes. bound0 q" by blast + hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb) + hence yes_qf: "qfree (decr ?cyes )" by (simp add: decr_qf) + from conjuncts_qf[OF qfp] partition_set[OF part] + have " \q\ set ?no. qfree q" by auto + hence no_qf: "qfree ?cno"by (simp add: list_conj_qf) + with qe have cno_qf:"qfree (qe ?cno )" + and noE: "Ifm bs (qe ?cno) = Ifm bs (E ?cno)" by blast+ + from cno_qf yes_qf have qf: "qfree (CJNB qe p)" + by (simp add: CJNB_def Let_def conj_qf split_def) + {fix bs + from conjuncts have "Ifm bs p = (\q\ set ?cjs. Ifm bs q)" by blast + also have "\ = ((\q\ set ?yes. Ifm bs q) \ (\q\ set ?no. Ifm bs q))" + using partition_set[OF part] by auto + finally have "Ifm bs p = ((Ifm bs ?cyes) \ (Ifm bs ?cno))" using list_conj by simp} + hence "Ifm bs (E p) = (\x. (Ifm (x#bs) ?cyes) \ (Ifm (x#bs) ?cno))" by simp + also fix y have "\ = (\x. (Ifm (y#bs) ?cyes) \ (Ifm (x#bs) ?cno))" + using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast + also have "\ = (Ifm bs (decr ?cyes) \ Ifm bs (E ?cno))" + by (auto simp add: decr[OF yes_nb]) + also have "\ = (Ifm bs (conj (decr ?cyes) (qe ?cno)))" + using qe[rule_format, OF no_qf] by auto + finally have "Ifm bs (E p) = Ifm bs (CJNB qe p)" + by (simp add: Let_def CJNB_def split_def) + with qf show "qfree (CJNB qe p) \ Ifm bs (CJNB qe p) = Ifm bs (E p)" by blast +qed + +consts qelim :: "fm \ (fm \ fm) \ fm" +recdef qelim "measure fmsize" + "qelim (E p) = (\ qe. DJ (CJNB qe) (qelim p qe))" + "qelim (A p) = (\ qe. not (qe ((qelim (NOT p) qe))))" + "qelim (NOT p) = (\ qe. not (qelim p qe))" + "qelim (And p q) = (\ qe. conj (qelim p qe) (qelim q qe))" + "qelim (Or p q) = (\ qe. disj (qelim p qe) (qelim q qe))" + "qelim (Imp p q) = (\ qe. disj (qelim (NOT p) qe) (qelim q qe))" + "qelim (Iff p q) = (\ qe. iff (qelim p qe) (qelim q qe))" + "qelim p = (\ y. simpfm p)" + +lemma qelim_ci: + assumes qe_inv: "\ bs p. qfree p \ qfree (qe p) \ (Ifm bs (qe p) = Ifm bs (E p))" + shows "\ bs. qfree (qelim p qe) \ (Ifm bs (qelim p qe) = Ifm bs p)" +using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]] +by(induct p rule: qelim.induct) +(auto simp del: simpfm.simps) + + +text {* The @{text "\"} Part *} +text{* Linearity for fm where Bound 0 ranges over @{text "\"} *} +consts + zsplit0 :: "num \ int \ num" (* splits the bounded from the unbounded part*) +recdef zsplit0 "measure num_size" + "zsplit0 (C c) = (0,C c)" + "zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))" + "zsplit0 (CN n c a) = zsplit0 (Add (Mul c (Bound n)) a)" + "zsplit0 (CF c a b) = zsplit0 (Add (Mul c (Floor a)) b)" + "zsplit0 (Neg a) = (let (i',a') = zsplit0 a in (-i', Neg a'))" + "zsplit0 (Add a b) = (let (ia,a') = zsplit0 a ; + (ib,b') = zsplit0 b + in (ia+ib, Add a' b'))" + "zsplit0 (Sub a b) = (let (ia,a') = zsplit0 a ; + (ib,b') = zsplit0 b + in (ia-ib, Sub a' b'))" + "zsplit0 (Mul i a) = (let (i',a') = zsplit0 a in (i*i', Mul i a'))" + "zsplit0 (Floor a) = (let (i',a') = zsplit0 a in (i',Floor a'))" +(hints simp add: Let_def) + +lemma zsplit0_I: + shows "\ n a. zsplit0 t = (n,a) \ (Inum ((real (x::int)) #bs) (CN 0 n a) = Inum (real x #bs) t) \ numbound0 a" + (is "\ n a. ?S t = (n,a) \ (?I x (CN 0 n a) = ?I x t) \ ?N a") +proof(induct t rule: zsplit0.induct) + case (1 c n a) thus ?case by auto +next + case (2 m n a) thus ?case by (cases "m=0") auto +next + case (3 n i a n a') thus ?case by auto +next + case (4 c a b n a') thus ?case by auto +next + case (5 t n a) + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Neg ?at \ n=-?nt" using prems + by (simp add: Let_def split_def) + from abj prems have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \ ?N ?at" by blast + from th2[simplified] th[simplified] show ?case by simp +next + case (6 s t n a) + let ?ns = "fst (zsplit0 s)" + let ?as = "snd (zsplit0 s)" + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abjs: "zsplit0 s = (?ns,?as)" by simp + moreover have abjt: "zsplit0 t = (?nt,?at)" by simp + ultimately have th: "a=Add ?as ?at \ n=?ns + ?nt" using prems + by (simp add: Let_def split_def) + from abjs[symmetric] have bluddy: "\ x y. (x,y) = zsplit0 s" by blast + from prems have "(\ x y. (x,y) = zsplit0 s) \ (\xa xb. zsplit0 t = (xa, xb) \ Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \ numbound0 xb)" by simp + with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \ ?N ?at" by blast + from abjs prems have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \ ?N ?as" by blast + from th3[simplified] th2[simplified] th[simplified] show ?case + by (simp add: left_distrib) +next + case (7 s t n a) + let ?ns = "fst (zsplit0 s)" + let ?as = "snd (zsplit0 s)" + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abjs: "zsplit0 s = (?ns,?as)" by simp + moreover have abjt: "zsplit0 t = (?nt,?at)" by simp + ultimately have th: "a=Sub ?as ?at \ n=?ns - ?nt" using prems + by (simp add: Let_def split_def) + from abjs[symmetric] have bluddy: "\ x y. (x,y) = zsplit0 s" by blast + from prems have "(\ x y. (x,y) = zsplit0 s) \ (\xa xb. zsplit0 t = (xa, xb) \ Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \ numbound0 xb)" by simp + with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \ ?N ?at" by blast + from abjs prems have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \ ?N ?as" by blast + from th3[simplified] th2[simplified] th[simplified] show ?case + by (simp add: left_diff_distrib) +next + case (8 i t n a) + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Mul i ?at \ n=i*?nt" using prems + by (simp add: Let_def split_def) + from abj prems have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \ ?N ?at" by blast + hence " ?I x (Mul i t) = (real i) * ?I x (CN 0 ?nt ?at)" by simp + also have "\ = ?I x (CN 0 (i*?nt) (Mul i ?at))" by (simp add: right_distrib) + finally show ?case using th th2 by simp +next + case (9 t n a) + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a= Floor ?at \ n=?nt" using prems + by (simp add: Let_def split_def) + from abj prems have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \ ?N ?at" by blast + hence na: "?N a" using th by simp + have th': "(real ?nt)*(real x) = real (?nt * x)" by simp + have "?I x (Floor t) = ?I x (Floor (CN 0 ?nt ?at))" using th2 by simp + also have "\ = real (floor ((real ?nt)* real(x) + ?I x ?at))" by simp + also have "\ = real (floor (?I x ?at + real (?nt* x)))" by (simp add: add_ac) + also have "\ = real (floor (?I x ?at) + (?nt* x))" + using floor_add[where x="?I x ?at" and a="?nt* x"] by simp + also have "\ = real (?nt)*(real x) + real (floor (?I x ?at))" by (simp add: add_ac) + finally have "?I x (Floor t) = ?I x (CN 0 n a)" using th by simp + with na show ?case by simp +qed + +consts + iszlfm :: "fm \ real list \ bool" (* Linearity test for fm *) + zlfm :: "fm \ fm" (* Linearity transformation for fm *) +recdef iszlfm "measure size" + "iszlfm (And p q) = (\ bs. iszlfm p bs \ iszlfm q bs)" + "iszlfm (Or p q) = (\ bs. iszlfm p bs \ iszlfm q bs)" + "iszlfm (Eq (CN 0 c e)) = (\ bs. c>0 \ numbound0 e \ isint e bs)" + "iszlfm (NEq (CN 0 c e)) = (\ bs. c>0 \ numbound0 e \ isint e bs)" + "iszlfm (Lt (CN 0 c e)) = (\ bs. c>0 \ numbound0 e \ isint e bs)" + "iszlfm (Le (CN 0 c e)) = (\ bs. c>0 \ numbound0 e \ isint e bs)" + "iszlfm (Gt (CN 0 c e)) = (\ bs. c>0 \ numbound0 e \ isint e bs)" + "iszlfm (Ge (CN 0 c e)) = (\ bs. c>0 \ numbound0 e \ isint e bs)" + "iszlfm (Dvd i (CN 0 c e)) = + (\ bs. c>0 \ i>0 \ numbound0 e \ isint e bs)" + "iszlfm (NDvd i (CN 0 c e))= + (\ bs. c>0 \ i>0 \ numbound0 e \ isint e bs)" + "iszlfm p = (\ bs. isatom p \ (bound0 p))" + +lemma zlin_qfree: "iszlfm p bs \ qfree p" + by (induct p rule: iszlfm.induct) auto + +lemma iszlfm_gen: + assumes lp: "iszlfm p (x#bs)" + shows "\ y. iszlfm p (y#bs)" +proof + fix y + show "iszlfm p (y#bs)" + using lp + by(induct p rule: iszlfm.induct, simp_all add: numbound0_gen[rule_format, where x="x" and y="y"]) +qed + +lemma conj_zl[simp]: "iszlfm p bs \ iszlfm q bs \ iszlfm (conj p q) bs" + using conj_def by (cases p,auto) +lemma disj_zl[simp]: "iszlfm p bs \ iszlfm q bs \ iszlfm (disj p q) bs" + using disj_def by (cases p,auto) +lemma not_zl[simp]: "iszlfm p bs \ iszlfm (not p) bs" + by (induct p rule:iszlfm.induct ,auto) + +recdef zlfm "measure fmsize" + "zlfm (And p q) = conj (zlfm p) (zlfm q)" + "zlfm (Or p q) = disj (zlfm p) (zlfm q)" + "zlfm (Imp p q) = disj (zlfm (NOT p)) (zlfm q)" + "zlfm (Iff p q) = disj (conj (zlfm p) (zlfm q)) (conj (zlfm (NOT p)) (zlfm (NOT q)))" + "zlfm (Lt a) = (let (c,r) = zsplit0 a in + if c=0 then Lt r else + if c>0 then Or (Lt (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r))) + else Or (Gt (CN 0 (-c) (Floor(Neg r)))) (And (Eq(CN 0 (-c) (Floor(Neg r)))) (Lt (Add (Floor (Neg r)) r))))" + "zlfm (Le a) = (let (c,r) = zsplit0 a in + if c=0 then Le r else + if c>0 then Or (Le (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r))) + else Or (Ge (CN 0 (-c) (Floor(Neg r)))) (And (Eq(CN 0 (-c) (Floor(Neg r)))) (Lt (Add (Floor (Neg r)) r))))" + "zlfm (Gt a) = (let (c,r) = zsplit0 a in + if c=0 then Gt r else + if c>0 then Or (Gt (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r))) + else Or (Lt (CN 0 (-c) (Neg (Floor r)))) (And (Eq(CN 0 (-c) (Neg (Floor r)))) (Lt (Sub (Floor r) r))))" + "zlfm (Ge a) = (let (c,r) = zsplit0 a in + if c=0 then Ge r else + if c>0 then Or (Ge (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r))) + else Or (Le (CN 0 (-c) (Neg (Floor r)))) (And (Eq(CN 0 (-c) (Neg (Floor r)))) (Lt (Sub (Floor r) r))))" + "zlfm (Eq a) = (let (c,r) = zsplit0 a in + if c=0 then Eq r else + if c>0 then (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Eq (Add (Floor (Neg r)) r))) + else (And (Eq (CN 0 (-c) (Floor (Neg r)))) (Eq (Add (Floor (Neg r)) r))))" + "zlfm (NEq a) = (let (c,r) = zsplit0 a in + if c=0 then NEq r else + if c>0 then (Or (NEq (CN 0 c (Neg (Floor (Neg r))))) (NEq (Add (Floor (Neg r)) r))) + else (Or (NEq (CN 0 (-c) (Floor (Neg r)))) (NEq (Add (Floor (Neg r)) r))))" + "zlfm (Dvd i a) = (if i=0 then zlfm (Eq a) + else (let (c,r) = zsplit0 a in + if c=0 then Dvd (abs i) r else + if c>0 then And (Eq (Sub (Floor r) r)) (Dvd (abs i) (CN 0 c (Floor r))) + else And (Eq (Sub (Floor r) r)) (Dvd (abs i) (CN 0 (-c) (Neg (Floor r))))))" + "zlfm (NDvd i a) = (if i=0 then zlfm (NEq a) + else (let (c,r) = zsplit0 a in + if c=0 then NDvd (abs i) r else + if c>0 then Or (NEq (Sub (Floor r) r)) (NDvd (abs i) (CN 0 c (Floor r))) + else Or (NEq (Sub (Floor r) r)) (NDvd (abs i) (CN 0 (-c) (Neg (Floor r))))))" + "zlfm (NOT (And p q)) = disj (zlfm (NOT p)) (zlfm (NOT q))" + "zlfm (NOT (Or p q)) = conj (zlfm (NOT p)) (zlfm (NOT q))" + "zlfm (NOT (Imp p q)) = conj (zlfm p) (zlfm (NOT q))" + "zlfm (NOT (Iff p q)) = disj (conj(zlfm p) (zlfm(NOT q))) (conj (zlfm(NOT p)) (zlfm q))" + "zlfm (NOT (NOT p)) = zlfm p" + "zlfm (NOT T) = F" + "zlfm (NOT F) = T" + "zlfm (NOT (Lt a)) = zlfm (Ge a)" + "zlfm (NOT (Le a)) = zlfm (Gt a)" + "zlfm (NOT (Gt a)) = zlfm (Le a)" + "zlfm (NOT (Ge a)) = zlfm (Lt a)" + "zlfm (NOT (Eq a)) = zlfm (NEq a)" + "zlfm (NOT (NEq a)) = zlfm (Eq a)" + "zlfm (NOT (Dvd i a)) = zlfm (NDvd i a)" + "zlfm (NOT (NDvd i a)) = zlfm (Dvd i a)" + "zlfm p = p" (hints simp add: fmsize_pos) + +lemma split_int_less_real: + "(real (a::int) < b) = (a < floor b \ (a = floor b \ real (floor b) < b))" +proof( auto) + assume alb: "real a < b" and agb: "\ a < floor b" + from agb have "floor b \ a" by simp hence th: "b < real a + 1" by (simp only: floor_le_eq) + from floor_eq[OF alb th] show "a= floor b" by simp +next + assume alb: "a < floor b" + hence "real a < real (floor b)" by simp + moreover have "real (floor b) \ b" by simp ultimately show "real a < b" by arith +qed + +lemma split_int_less_real': + "(real (a::int) + b < 0) = (real a - real (floor(-b)) < 0 \ (real a - real (floor (-b)) = 0 \ real (floor (-b)) + b < 0))" +proof- + have "(real a + b <0) = (real a < -b)" by arith + with split_int_less_real[where a="a" and b="-b"] show ?thesis by arith +qed + +lemma split_int_gt_real': + "(real (a::int) + b > 0) = (real a + real (floor b) > 0 \ (real a + real (floor b) = 0 \ real (floor b) - b < 0))" +proof- + have th: "(real a + b >0) = (real (-a) + (-b)< 0)" by arith + show ?thesis using myless[rule_format, where b="real (floor b)"] + by (simp only:th split_int_less_real'[where a="-a" and b="-b"]) + (simp add: algebra_simps diff_def[symmetric],arith) +qed + +lemma split_int_le_real: + "(real (a::int) \ b) = (a \ floor b \ (a = floor b \ real (floor b) < b))" +proof( auto) + assume alb: "real a \ b" and agb: "\ a \ floor b" + from alb have "floor (real a) \ floor b " by (simp only: floor_mono2) + hence "a \ floor b" by simp with agb show "False" by simp +next + assume alb: "a \ floor b" + hence "real a \ real (floor b)" by (simp only: floor_mono2) + also have "\\ b" by simp finally show "real a \ b" . +qed + +lemma split_int_le_real': + "(real (a::int) + b \ 0) = (real a - real (floor(-b)) \ 0 \ (real a - real (floor (-b)) = 0 \ real (floor (-b)) + b < 0))" +proof- + have "(real a + b \0) = (real a \ -b)" by arith + with split_int_le_real[where a="a" and b="-b"] show ?thesis by arith +qed + +lemma split_int_ge_real': + "(real (a::int) + b \ 0) = (real a + real (floor b) \ 0 \ (real a + real (floor b) = 0 \ real (floor b) - b < 0))" +proof- + have th: "(real a + b \0) = (real (-a) + (-b) \ 0)" by arith + show ?thesis by (simp only: th split_int_le_real'[where a="-a" and b="-b"]) + (simp add: algebra_simps diff_def[symmetric],arith) +qed + +lemma split_int_eq_real: "(real (a::int) = b) = ( a = floor b \ b = real (floor b))" (is "?l = ?r") +by auto + +lemma split_int_eq_real': "(real (a::int) + b = 0) = ( a - floor (-b) = 0 \ real (floor (-b)) + b = 0)" (is "?l = ?r") +proof- + have "?l = (real a = -b)" by arith + with split_int_eq_real[where a="a" and b="-b"] show ?thesis by simp arith +qed + +lemma zlfm_I: + assumes qfp: "qfree p" + shows "(Ifm (real i #bs) (zlfm p) = Ifm (real i# bs) p) \ iszlfm (zlfm p) (real (i::int) #bs)" + (is "(?I (?l p) = ?I p) \ ?L (?l p)") +using qfp +proof(induct p rule: zlfm.induct) + case (5 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (real i#bs) t" + have "?c = 0 \ (?c >0 \ ?c\0) \ (?c<0 \ ?c\0)" by arith + moreover + {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] + by (cases "?r", simp_all add: Let_def split_def,case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\0" hence l: "?L (?l (Lt a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Lt a) = (real (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def) + also have "\ = (?I (?l (Lt a)))" apply (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) by (simp add: Ia cp cnz Let_def split_def diff_def) + finally have ?case using l by simp} + moreover + {assume cn: "?c < 0" and cnz: "?c\0" hence l: "?L (?l (Lt a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Lt a) = (real (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def) + also from cn cnz have "\ = (?I (?l (Lt a)))" by (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac, arith) + finally have ?case using l by simp} + ultimately show ?case by blast +next + case (6 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (real i#bs) t" + have "?c = 0 \ (?c >0 \ ?c\0) \ (?c<0 \ ?c\0)" by arith + moreover + {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat",simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\0" hence l: "?L (?l (Le a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Le a) = (real (?c * i) + (?N ?r) \ 0)" using Ia by (simp add: Let_def split_def) + also have "\ = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def diff_def) + finally have ?case using l by simp} + moreover + {assume cn: "?c < 0" and cnz: "?c\0" hence l: "?L (?l (Le a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Le a) = (real (?c * i) + (?N ?r) \ 0)" using Ia by (simp add: Let_def split_def) + also from cn cnz have "\ = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac ,arith) + finally have ?case using l by simp} + ultimately show ?case by blast +next + case (7 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (real i#bs) t" + have "?c = 0 \ (?c >0 \ ?c\0) \ (?c<0 \ ?c\0)" by arith + moreover + {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\0" hence l: "?L (?l (Gt a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Gt a) = (real (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def) + also have "\ = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def diff_def) + finally have ?case using l by simp} + moreover + {assume cn: "?c < 0" and cnz: "?c\0" hence l: "?L (?l (Gt a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Gt a) = (real (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def) + also from cn cnz have "\ = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac, arith) + finally have ?case using l by simp} + ultimately show ?case by blast +next + case (8 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (real i#bs) t" + have "?c = 0 \ (?c >0 \ ?c\0) \ (?c<0 \ ?c\0)" by arith + moreover + {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\0" hence l: "?L (?l (Ge a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Ge a) = (real (?c * i) + (?N ?r) \ 0)" using Ia by (simp add: Let_def split_def) + also have "\ = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def diff_def) + finally have ?case using l by simp} + moreover + {assume cn: "?c < 0" and cnz: "?c\0" hence l: "?L (?l (Ge a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Ge a) = (real (?c * i) + (?N ?r) \ 0)" using Ia by (simp add: Let_def split_def) + also from cn cnz have "\ = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac, arith) + finally have ?case using l by simp} + ultimately show ?case by blast +next + case (9 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (real i#bs) t" + have "?c = 0 \ (?c >0 \ ?c\0) \ (?c<0 \ ?c\0)" by arith + moreover + {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\0" hence l: "?L (?l (Eq a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Eq a) = (real (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def) + also have "\ = (?I (?l (Eq a)))" using cp cnz by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult) + finally have ?case using l by simp} + moreover + {assume cn: "?c < 0" and cnz: "?c\0" hence l: "?L (?l (Eq a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Eq a) = (real (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def) + also from cn cnz have "\ = (?I (?l (Eq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult,arith) + finally have ?case using l by simp} + ultimately show ?case by blast +next + case (10 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (real i#bs) t" + have "?c = 0 \ (?c >0 \ ?c\0) \ (?c<0 \ ?c\0)" by arith + moreover + {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\0" hence l: "?L (?l (NEq a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (NEq a) = (real (?c * i) + (?N ?r) \ 0)" using Ia by (simp add: Let_def split_def) + also have "\ = (?I (?l (NEq a)))" using cp cnz by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult) + finally have ?case using l by simp} + moreover + {assume cn: "?c < 0" and cnz: "?c\0" hence l: "?L (?l (NEq a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (NEq a) = (real (?c * i) + (?N ?r) \ 0)" using Ia by (simp add: Let_def split_def) + also from cn cnz have "\ = (?I (?l (NEq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult,arith) + finally have ?case using l by simp} + ultimately show ?case by blast +next + case (11 j a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (real i#bs) t" + have "j=0 \ (j\0 \ ?c = 0) \ (j\0 \ ?c >0 \ ?c\0) \ (j\ 0 \ ?c<0 \ ?c\0)" by arith + moreover + {assume "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def) + hence ?case using prems by (simp del: zlfm.simps add: rdvd_left_0_eq)} + moreover + {assume "?c=0" and "j\0" hence ?case + using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\0" and jnz: "j\0" hence l: "?L (?l (Dvd j a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Dvd j a) = (real j rdvd (real (?c * i) + (?N ?r)))" + using Ia by (simp add: Let_def split_def) + also have "\ = (real (abs j) rdvd real (?c*i) + (?N ?r))" + by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp + also have "\ = ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \ + (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r))))" + by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac) + also have "\ = (?I (?l (Dvd j a)))" using cp cnz jnz + by (simp add: Let_def split_def int_rdvd_iff[symmetric] + del: real_of_int_mult) (auto simp add: add_ac) + finally have ?case using l jnz by simp } + moreover + {assume cn: "?c < 0" and cnz: "?c\0" and jnz: "j\0" hence l: "?L (?l (Dvd j a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Dvd j a) = (real j rdvd (real (?c * i) + (?N ?r)))" + using Ia by (simp add: Let_def split_def) + also have "\ = (real (abs j) rdvd real (?c*i) + (?N ?r))" + by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp + also have "\ = ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \ + (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r))))" + by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac) + also have "\ = (?I (?l (Dvd j a)))" using cn cnz jnz + using rdvd_minus [where d="abs j" and t="real (?c*i + floor (?N ?r))", simplified, symmetric] + by (simp add: Let_def split_def int_rdvd_iff[symmetric] + del: real_of_int_mult) (auto simp add: add_ac) + finally have ?case using l jnz by blast } + ultimately show ?case by blast +next + case (12 j a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\ t. Inum (real i#bs) t" + have "j=0 \ (j\0 \ ?c = 0) \ (j\0 \ ?c >0 \ ?c\0) \ (j\ 0 \ ?c<0 \ ?c\0)" by arith + moreover + {assume "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def) + hence ?case using prems by (simp del: zlfm.simps add: rdvd_left_0_eq)} + moreover + {assume "?c=0" and "j\0" hence ?case + using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\0" and jnz: "j\0" hence l: "?L (?l (NDvd j a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (NDvd j a) = (\ (real j rdvd (real (?c * i) + (?N ?r))))" + using Ia by (simp add: Let_def split_def) + also have "\ = (\ (real (abs j) rdvd real (?c*i) + (?N ?r)))" + by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp + also have "\ = (\ ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \ + (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r)))))" + by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac) + also have "\ = (?I (?l (NDvd j a)))" using cp cnz jnz + by (simp add: Let_def split_def int_rdvd_iff[symmetric] + del: real_of_int_mult) (auto simp add: add_ac) + finally have ?case using l jnz by simp } + moreover + {assume cn: "?c < 0" and cnz: "?c\0" and jnz: "j\0" hence l: "?L (?l (NDvd j a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (NDvd j a) = (\ (real j rdvd (real (?c * i) + (?N ?r))))" + using Ia by (simp add: Let_def split_def) + also have "\ = (\ (real (abs j) rdvd real (?c*i) + (?N ?r)))" + by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp + also have "\ = (\ ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \ + (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r)))))" + by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac) + also have "\ = (?I (?l (NDvd j a)))" using cn cnz jnz + using rdvd_minus [where d="abs j" and t="real (?c*i + floor (?N ?r))", simplified, symmetric] + by (simp add: Let_def split_def int_rdvd_iff[symmetric] + del: real_of_int_mult) (auto simp add: add_ac) + finally have ?case using l jnz by blast } + ultimately show ?case by blast +qed auto + +text{* plusinf : Virtual substitution of @{text "+\"} + minusinf: Virtual substitution of @{text "-\"} + @{text "\"} Compute lcm @{text "d| Dvd d c*x+t \ p"} + @{text "d\"} checks if a given l divides all the ds above*} + +consts + plusinf:: "fm \ fm" + minusinf:: "fm \ fm" + \ :: "fm \ int" + d\ :: "fm \ int \ bool" + +recdef minusinf "measure size" + "minusinf (And p q) = conj (minusinf p) (minusinf q)" + "minusinf (Or p q) = disj (minusinf p) (minusinf q)" + "minusinf (Eq (CN 0 c e)) = F" + "minusinf (NEq (CN 0 c e)) = T" + "minusinf (Lt (CN 0 c e)) = T" + "minusinf (Le (CN 0 c e)) = T" + "minusinf (Gt (CN 0 c e)) = F" + "minusinf (Ge (CN 0 c e)) = F" + "minusinf p = p" + +lemma minusinf_qfree: "qfree p \ qfree (minusinf p)" + by (induct p rule: minusinf.induct, auto) + +recdef plusinf "measure size" + "plusinf (And p q) = conj (plusinf p) (plusinf q)" + "plusinf (Or p q) = disj (plusinf p) (plusinf q)" + "plusinf (Eq (CN 0 c e)) = F" + "plusinf (NEq (CN 0 c e)) = T" + "plusinf (Lt (CN 0 c e)) = F" + "plusinf (Le (CN 0 c e)) = F" + "plusinf (Gt (CN 0 c e)) = T" + "plusinf (Ge (CN 0 c e)) = T" + "plusinf p = p" + +recdef \ "measure size" + "\ (And p q) = zlcm (\ p) (\ q)" + "\ (Or p q) = zlcm (\ p) (\ q)" + "\ (Dvd i (CN 0 c e)) = i" + "\ (NDvd i (CN 0 c e)) = i" + "\ p = 1" + +recdef d\ "measure size" + "d\ (And p q) = (\ d. d\ p d \ d\ q d)" + "d\ (Or p q) = (\ d. d\ p d \ d\ q d)" + "d\ (Dvd i (CN 0 c e)) = (\ d. i dvd d)" + "d\ (NDvd i (CN 0 c e)) = (\ d. i dvd d)" + "d\ p = (\ d. True)" + +lemma delta_mono: + assumes lin: "iszlfm p bs" + and d: "d dvd d'" + and ad: "d\ p d" + shows "d\ p d'" + using lin ad d +proof(induct p rule: iszlfm.induct) + case (9 i c e) thus ?case using d + by (simp add: zdvd_trans[where m="i" and n="d" and k="d'"]) +next + case (10 i c e) thus ?case using d + by (simp add: zdvd_trans[where m="i" and n="d" and k="d'"]) +qed simp_all + +lemma \ : assumes lin:"iszlfm p bs" + shows "d\ p (\ p) \ \ p >0" +using lin +proof (induct p rule: iszlfm.induct) + case (1 p q) + let ?d = "\ (And p q)" + from prems zlcm_pos have dp: "?d >0" by simp + have d1: "\ p dvd \ (And p q)" using prems by simp + hence th: "d\ p ?d" + using delta_mono prems by (auto simp del: dvd_zlcm_self1) + have "\ q dvd \ (And p q)" using prems by simp + hence th': "d\ q ?d" using delta_mono prems by (auto simp del: dvd_zlcm_self2) + from th th' dp show ?case by simp +next + case (2 p q) + let ?d = "\ (And p q)" + from prems zlcm_pos have dp: "?d >0" by simp + have "\ p dvd \ (And p q)" using prems by simp hence th: "d\ p ?d" using delta_mono prems + by (auto simp del: dvd_zlcm_self1) + have "\ q dvd \ (And p q)" using prems by simp hence th': "d\ q ?d" using delta_mono prems by (auto simp del: dvd_zlcm_self2) + from th th' dp show ?case by simp +qed simp_all + + +lemma minusinf_inf: + assumes linp: "iszlfm p (a # bs)" + shows "\ (z::int). \ x < z. Ifm ((real x)#bs) (minusinf p) = Ifm ((real x)#bs) p" + (is "?P p" is "\ (z::int). \ x < z. ?I x (?M p) = ?I x p") +using linp +proof (induct p rule: minusinf.induct) + case (1 f g) + from prems have "?P f" by simp + then obtain z1 where z1_def: "\ x < z1. ?I x (?M f) = ?I x f" by blast + from prems have "?P g" by simp + then obtain z2 where z2_def: "\ x < z2. ?I x (?M g) = ?I x g" by blast + let ?z = "min z1 z2" + from z1_def z2_def have "\ x < ?z. ?I x (?M (And f g)) = ?I x (And f g)" by simp + thus ?case by blast +next + case (2 f g) from prems have "?P f" by simp + then obtain z1 where z1_def: "\ x < z1. ?I x (?M f) = ?I x f" by blast + from prems have "?P g" by simp + then obtain z2 where z2_def: "\ x < z2. ?I x (?M g) = ?I x g" by blast + let ?z = "min z1 z2" + from z1_def z2_def have "\ x < ?z. ?I x (?M (Or f g)) = ?I x (Or f g)" by simp + thus ?case by blast +next + case (3 c e) + from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp + from prems have nbe: "numbound0 e" by simp + fix y + have "\ x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Eq (CN 0 c e))) = ?I x (Eq (CN 0 c e))" + proof (simp add: less_floor_eq , rule allI, rule impI) + fix x + assume A: "real x + (1\real) \ - (Inum (y # bs) e / real c)" + hence th1:"real x < - (Inum (y # bs) e / real c)" by simp + with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))" + by (simp only: real_mult_less_mono2[OF rcpos th1]) + hence "real c * real x + Inum (y # bs) e \ 0"using rcpos by simp + thus "real c * real x + Inum (real x # bs) e \ 0" + using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] by simp + qed + thus ?case by blast +next + case (4 c e) + from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp + from prems have nbe: "numbound0 e" by simp + fix y + have "\ x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (NEq (CN 0 c e))) = ?I x (NEq (CN 0 c e))" + proof (simp add: less_floor_eq , rule allI, rule impI) + fix x + assume A: "real x + (1\real) \ - (Inum (y # bs) e / real c)" + hence th1:"real x < - (Inum (y # bs) e / real c)" by simp + with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))" + by (simp only: real_mult_less_mono2[OF rcpos th1]) + hence "real c * real x + Inum (y # bs) e \ 0"using rcpos by simp + thus "real c * real x + Inum (real x # bs) e \ 0" + using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] by simp + qed + thus ?case by blast +next + case (5 c e) + from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp + from prems have nbe: "numbound0 e" by simp + fix y + have "\ x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Lt (CN 0 c e))) = ?I x (Lt (CN 0 c e))" + proof (simp add: less_floor_eq , rule allI, rule impI) + fix x + assume A: "real x + (1\real) \ - (Inum (y # bs) e / real c)" + hence th1:"real x < - (Inum (y # bs) e / real c)" by simp + with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))" + by (simp only: real_mult_less_mono2[OF rcpos th1]) + thus "real c * real x + Inum (real x # bs) e < 0" + using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp + qed + thus ?case by blast +next + case (6 c e) + from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp + from prems have nbe: "numbound0 e" by simp + fix y + have "\ x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Le (CN 0 c e))) = ?I x (Le (CN 0 c e))" + proof (simp add: less_floor_eq , rule allI, rule impI) + fix x + assume A: "real x + (1\real) \ - (Inum (y # bs) e / real c)" + hence th1:"real x < - (Inum (y # bs) e / real c)" by simp + with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))" + by (simp only: real_mult_less_mono2[OF rcpos th1]) + thus "real c * real x + Inum (real x # bs) e \ 0" + using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp + qed + thus ?case by blast +next + case (7 c e) + from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp + from prems have nbe: "numbound0 e" by simp + fix y + have "\ x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Gt (CN 0 c e))) = ?I x (Gt (CN 0 c e))" + proof (simp add: less_floor_eq , rule allI, rule impI) + fix x + assume A: "real x + (1\real) \ - (Inum (y # bs) e / real c)" + hence th1:"real x < - (Inum (y # bs) e / real c)" by simp + with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))" + by (simp only: real_mult_less_mono2[OF rcpos th1]) + thus "\ (real c * real x + Inum (real x # bs) e>0)" + using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp + qed + thus ?case by blast +next + case (8 c e) + from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp + from prems have nbe: "numbound0 e" by simp + fix y + have "\ x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Ge (CN 0 c e))) = ?I x (Ge (CN 0 c e))" + proof (simp add: less_floor_eq , rule allI, rule impI) + fix x + assume A: "real x + (1\real) \ - (Inum (y # bs) e / real c)" + hence th1:"real x < - (Inum (y # bs) e / real c)" by simp + with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))" + by (simp only: real_mult_less_mono2[OF rcpos th1]) + thus "\ real c * real x + Inum (real x # bs) e \ 0" + using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp + qed + thus ?case by blast +qed simp_all + +lemma minusinf_repeats: + assumes d: "d\ p d" and linp: "iszlfm p (a # bs)" + shows "Ifm ((real(x - k*d))#bs) (minusinf p) = Ifm (real x #bs) (minusinf p)" +using linp d +proof(induct p rule: iszlfm.induct) + case (9 i c e) hence nbe: "numbound0 e" and id: "i dvd d" by simp+ + hence "\ k. d=i*k" by (simp add: dvd_def) + then obtain "di" where di_def: "d=i*di" by blast + show ?case + proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real x - real k * real d" and b'="real x"] right_diff_distrib, rule iffI) + assume + "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" + (is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt") + hence "\ (l::int). ?rt = ?ri * (real l)" by (simp add: rdvd_def) + hence "\ (l::int). ?rc*?rx+ ?I x e = ?ri*(real l)+?rc*(?rk * (real i) * (real di))" + by (simp add: algebra_simps di_def) + hence "\ (l::int). ?rc*?rx+ ?I x e = ?ri*(real (l + c*k*di))" + by (simp add: algebra_simps) + hence "\ (l::int). ?rc*?rx+ ?I x e = ?ri* (real l)" by blast + thus "real i rdvd real c * real x + Inum (real x # bs) e" using rdvd_def by simp + next + assume + "real i rdvd real c * real x + Inum (real x # bs) e" (is "?ri rdvd ?rc*?rx+?e") + hence "\ (l::int). ?rc*?rx+?e = ?ri * (real l)" by (simp add: rdvd_def) + hence "\ (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real d)" by simp + hence "\ (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real i * real di)" by (simp add: di_def) + hence "\ (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: algebra_simps) + hence "\ (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l)" + by blast + thus "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" using rdvd_def by simp + qed +next + case (10 i c e) hence nbe: "numbound0 e" and id: "i dvd d" by simp+ + hence "\ k. d=i*k" by (simp add: dvd_def) + then obtain "di" where di_def: "d=i*di" by blast + show ?case + proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real x - real k * real d" and b'="real x"] right_diff_distrib, rule iffI) + assume + "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" + (is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt") + hence "\ (l::int). ?rt = ?ri * (real l)" by (simp add: rdvd_def) + hence "\ (l::int). ?rc*?rx+ ?I x e = ?ri*(real l)+?rc*(?rk * (real i) * (real di))" + by (simp add: algebra_simps di_def) + hence "\ (l::int). ?rc*?rx+ ?I x e = ?ri*(real (l + c*k*di))" + by (simp add: algebra_simps) + hence "\ (l::int). ?rc*?rx+ ?I x e = ?ri* (real l)" by blast + thus "real i rdvd real c * real x + Inum (real x # bs) e" using rdvd_def by simp + next + assume + "real i rdvd real c * real x + Inum (real x # bs) e" (is "?ri rdvd ?rc*?rx+?e") + hence "\ (l::int). ?rc*?rx+?e = ?ri * (real l)" by (simp add: rdvd_def) + hence "\ (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real d)" by simp + hence "\ (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real i * real di)" by (simp add: di_def) + hence "\ (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: algebra_simps) + hence "\ (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l)" + by blast + thus "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" using rdvd_def by simp + qed +qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="real(x - k*d)" and b'="real x"] simp del: real_of_int_mult real_of_int_diff) + +lemma minusinf_ex: + assumes lin: "iszlfm p (real (a::int) #bs)" + and exmi: "\ (x::int). Ifm (real x#bs) (minusinf p)" (is "\ x. ?P1 x") + shows "\ (x::int). Ifm (real x#bs) p" (is "\ x. ?P x") +proof- + let ?d = "\ p" + from \ [OF lin] have dpos: "?d >0" by simp + from \ [OF lin] have alld: "d\ p ?d" by simp + from minusinf_repeats[OF alld lin] have th1:"\ x k. ?P1 x = ?P1 (x - (k * ?d))" by simp + from minusinf_inf[OF lin] have th2:"\ z. \ x. x (?P x = ?P1 x)" by blast + from minusinfinity [OF dpos th1 th2] exmi show ?thesis by blast +qed + +lemma minusinf_bex: + assumes lin: "iszlfm p (real (a::int) #bs)" + shows "(\ (x::int). Ifm (real x#bs) (minusinf p)) = + (\ (x::int)\ {1..\ p}. Ifm (real x#bs) (minusinf p))" + (is "(\ x. ?P x) = _") +proof- + let ?d = "\ p" + from \ [OF lin] have dpos: "?d >0" by simp + from \ [OF lin] have alld: "d\ p ?d" by simp + from minusinf_repeats[OF alld lin] have th1:"\ x k. ?P x = ?P (x - (k * ?d))" by simp + from periodic_finite_ex[OF dpos th1] show ?thesis by blast +qed + +lemma dvd1_eq1: "x >0 \ (x::int) dvd 1 = (x = 1)" by auto + +consts + a\ :: "fm \ int \ fm" (* adjusts the coeffitients of a formula *) + d\ :: "fm \ int \ bool" (* tests if all coeffs c of c divide a given l*) + \ :: "fm \ int" (* computes the lcm of all coefficients of x*) + \ :: "fm \ num list" + \ :: "fm \ num list" + +recdef a\ "measure size" + "a\ (And p q) = (\ k. And (a\ p k) (a\ q k))" + "a\ (Or p q) = (\ k. Or (a\ p k) (a\ q k))" + "a\ (Eq (CN 0 c e)) = (\ k. Eq (CN 0 1 (Mul (k div c) e)))" + "a\ (NEq (CN 0 c e)) = (\ k. NEq (CN 0 1 (Mul (k div c) e)))" + "a\ (Lt (CN 0 c e)) = (\ k. Lt (CN 0 1 (Mul (k div c) e)))" + "a\ (Le (CN 0 c e)) = (\ k. Le (CN 0 1 (Mul (k div c) e)))" + "a\ (Gt (CN 0 c e)) = (\ k. Gt (CN 0 1 (Mul (k div c) e)))" + "a\ (Ge (CN 0 c e)) = (\ k. Ge (CN 0 1 (Mul (k div c) e)))" + "a\ (Dvd i (CN 0 c e)) =(\ k. Dvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))" + "a\ (NDvd i (CN 0 c e))=(\ k. NDvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))" + "a\ p = (\ k. p)" + +recdef d\ "measure size" + "d\ (And p q) = (\ k. (d\ p k) \ (d\ q k))" + "d\ (Or p q) = (\ k. (d\ p k) \ (d\ q k))" + "d\ (Eq (CN 0 c e)) = (\ k. c dvd k)" + "d\ (NEq (CN 0 c e)) = (\ k. c dvd k)" + "d\ (Lt (CN 0 c e)) = (\ k. c dvd k)" + "d\ (Le (CN 0 c e)) = (\ k. c dvd k)" + "d\ (Gt (CN 0 c e)) = (\ k. c dvd k)" + "d\ (Ge (CN 0 c e)) = (\ k. c dvd k)" + "d\ (Dvd i (CN 0 c e)) =(\ k. c dvd k)" + "d\ (NDvd i (CN 0 c e))=(\ k. c dvd k)" + "d\ p = (\ k. True)" + +recdef \ "measure size" + "\ (And p q) = zlcm (\ p) (\ q)" + "\ (Or p q) = zlcm (\ p) (\ q)" + "\ (Eq (CN 0 c e)) = c" + "\ (NEq (CN 0 c e)) = c" + "\ (Lt (CN 0 c e)) = c" + "\ (Le (CN 0 c e)) = c" + "\ (Gt (CN 0 c e)) = c" + "\ (Ge (CN 0 c e)) = c" + "\ (Dvd i (CN 0 c e)) = c" + "\ (NDvd i (CN 0 c e))= c" + "\ p = 1" + +recdef \ "measure size" + "\ (And p q) = (\ p @ \ q)" + "\ (Or p q) = (\ p @ \ q)" + "\ (Eq (CN 0 c e)) = [Sub (C -1) e]" + "\ (NEq (CN 0 c e)) = [Neg e]" + "\ (Lt (CN 0 c e)) = []" + "\ (Le (CN 0 c e)) = []" + "\ (Gt (CN 0 c e)) = [Neg e]" + "\ (Ge (CN 0 c e)) = [Sub (C -1) e]" + "\ p = []" + +recdef \ "measure size" + "\ (And p q) = (\ p @ \ q)" + "\ (Or p q) = (\ p @ \ q)" + "\ (Eq (CN 0 c e)) = [Add (C -1) e]" + "\ (NEq (CN 0 c e)) = [e]" + "\ (Lt (CN 0 c e)) = [e]" + "\ (Le (CN 0 c e)) = [Add (C -1) e]" + "\ (Gt (CN 0 c e)) = []" + "\ (Ge (CN 0 c e)) = []" + "\ p = []" +consts mirror :: "fm \ fm" +recdef mirror "measure size" + "mirror (And p q) = And (mirror p) (mirror q)" + "mirror (Or p q) = Or (mirror p) (mirror q)" + "mirror (Eq (CN 0 c e)) = Eq (CN 0 c (Neg e))" + "mirror (NEq (CN 0 c e)) = NEq (CN 0 c (Neg e))" + "mirror (Lt (CN 0 c e)) = Gt (CN 0 c (Neg e))" + "mirror (Le (CN 0 c e)) = Ge (CN 0 c (Neg e))" + "mirror (Gt (CN 0 c e)) = Lt (CN 0 c (Neg e))" + "mirror (Ge (CN 0 c e)) = Le (CN 0 c (Neg e))" + "mirror (Dvd i (CN 0 c e)) = Dvd i (CN 0 c (Neg e))" + "mirror (NDvd i (CN 0 c e)) = NDvd i (CN 0 c (Neg e))" + "mirror p = p" + +lemma mirror\\: + assumes lp: "iszlfm p (a#bs)" + shows "(Inum (real (i::int)#bs)) ` set (\ p) = (Inum (real i#bs)) ` set (\ (mirror p))" +using lp +by (induct p rule: mirror.induct, auto) + +lemma mirror: + assumes lp: "iszlfm p (a#bs)" + shows "Ifm (real (x::int)#bs) (mirror p) = Ifm (real (- x)#bs) p" +using lp +proof(induct p rule: iszlfm.induct) + case (9 j c e) + have th: "(real j rdvd real c * real x - Inum (real x # bs) e) = + (real j rdvd - (real c * real x - Inum (real x # bs) e))" + by (simp only: rdvd_minus[symmetric]) + from prems th show ?case + by (simp add: algebra_simps + numbound0_I[where bs="bs" and b'="real x" and b="- real x"]) +next + case (10 j c e) + have th: "(real j rdvd real c * real x - Inum (real x # bs) e) = + (real j rdvd - (real c * real x - Inum (real x # bs) e))" + by (simp only: rdvd_minus[symmetric]) + from prems th show ?case + by (simp add: algebra_simps + numbound0_I[where bs="bs" and b'="real x" and b="- real x"]) +qed (auto simp add: numbound0_I[where bs="bs" and b="real x" and b'="- real x"] nth_pos2) + +lemma mirror_l: "iszlfm p (a#bs) \ iszlfm (mirror p) (a#bs)" +by (induct p rule: mirror.induct, auto simp add: isint_neg) + +lemma mirror_d\: "iszlfm p (a#bs) \ d\ p 1 + \ iszlfm (mirror p) (a#bs) \ d\ (mirror p) 1" +by (induct p rule: mirror.induct, auto simp add: isint_neg) + +lemma mirror_\: "iszlfm p (a#bs) \ \ (mirror p) = \ p" +by (induct p rule: mirror.induct,auto) + + +lemma mirror_ex: + assumes lp: "iszlfm p (real (i::int)#bs)" + shows "(\ (x::int). Ifm (real x#bs) (mirror p)) = (\ (x::int). Ifm (real x#bs) p)" + (is "(\ x. ?I x ?mp) = (\ x. ?I x p)") +proof(auto) + fix x assume "?I x ?mp" hence "?I (- x) p" using mirror[OF lp] by blast + thus "\ x. ?I x p" by blast +next + fix x assume "?I x p" hence "?I (- x) ?mp" + using mirror[OF lp, where x="- x", symmetric] by auto + thus "\ x. ?I x ?mp" by blast +qed + +lemma \_numbound0: assumes lp: "iszlfm p bs" + shows "\ b\ set (\ p). numbound0 b" + using lp by (induct p rule: \.induct,auto) + +lemma d\_mono: + assumes linp: "iszlfm p (a #bs)" + and dr: "d\ p l" + and d: "l dvd l'" + shows "d\ p l'" +using dr linp zdvd_trans[where n="l" and k="l'", simplified d] +by (induct p rule: iszlfm.induct) simp_all + +lemma \_l: assumes lp: "iszlfm p (a#bs)" + shows "\ b\ set (\ p). numbound0 b \ isint b (a#bs)" +using lp +by(induct p rule: \.induct, auto simp add: isint_add isint_c) + +lemma \: + assumes linp: "iszlfm p (a #bs)" + shows "\ p > 0 \ d\ p (\ p)" +using linp +proof(induct p rule: iszlfm.induct) + case (1 p q) + from prems have dl1: "\ p dvd zlcm (\ p) (\ q)" by simp + from prems have dl2: "\ q dvd zlcm (\ p) (\ q)" by simp + from prems d\_mono[where p = "p" and l="\ p" and l'="zlcm (\ p) (\ q)"] + d\_mono[where p = "q" and l="\ q" and l'="zlcm (\ p) (\ q)"] + dl1 dl2 show ?case by (auto simp add: zlcm_pos) +next + case (2 p q) + from prems have dl1: "\ p dvd zlcm (\ p) (\ q)" by simp + from prems have dl2: "\ q dvd zlcm (\ p) (\ q)" by simp + from prems d\_mono[where p = "p" and l="\ p" and l'="zlcm (\ p) (\ q)"] + d\_mono[where p = "q" and l="\ q" and l'="zlcm (\ p) (\ q)"] + dl1 dl2 show ?case by (auto simp add: zlcm_pos) +qed (auto simp add: zlcm_pos) + +lemma a\: assumes linp: "iszlfm p (a #bs)" and d: "d\ p l" and lp: "l > 0" + shows "iszlfm (a\ p l) (a #bs) \ d\ (a\ p l) 1 \ (Ifm (real (l * x) #bs) (a\ p l) = Ifm ((real x)#bs) p)" +using linp d +proof (induct p rule: iszlfm.induct) + case (5 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(real l * real x + real (l div c) * Inum (real x # bs) e < (0\real)) = + (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e < 0)" + by simp + also have "\ = (real (l div c) * (real c * real x + Inum (real x # bs) e) < (real (l div c)) * 0)" by (simp add: algebra_simps) + also have "\ = (real c * real x + Inum (real x # bs) e < 0)" + using mult_less_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp +next + case (6 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(real l * real x + real (l div c) * Inum (real x # bs) e \ (0\real)) = + (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \ 0)" + by simp + also have "\ = (real (l div c) * (real c * real x + Inum (real x # bs) e) \ (real (l div c)) * 0)" by (simp add: algebra_simps) + also have "\ = (real c * real x + Inum (real x # bs) e \ 0)" + using mult_le_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp +next + case (7 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(real l * real x + real (l div c) * Inum (real x # bs) e > (0\real)) = + (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e > 0)" + by simp + also have "\ = (real (l div c) * (real c * real x + Inum (real x # bs) e) > (real (l div c)) * 0)" by (simp add: algebra_simps) + also have "\ = (real c * real x + Inum (real x # bs) e > 0)" + using zero_less_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp +next + case (8 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(real l * real x + real (l div c) * Inum (real x # bs) e \ (0\real)) = + (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \ 0)" + by simp + also have "\ = (real (l div c) * (real c * real x + Inum (real x # bs) e) \ (real (l div c)) * 0)" by (simp add: algebra_simps) + also have "\ = (real c * real x + Inum (real x # bs) e \ 0)" + using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp +next + case (3 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(real l * real x + real (l div c) * Inum (real x # bs) e = (0\real)) = + (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = 0)" + by simp + also have "\ = (real (l div c) * (real c * real x + Inum (real x # bs) e) = (real (l div c)) * 0)" by (simp add: algebra_simps) + also have "\ = (real c * real x + Inum (real x # bs) e = 0)" + using mult_eq_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp +next + case (4 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(real l * real x + real (l div c) * Inum (real x # bs) e \ (0\real)) = + (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \ 0)" + by simp + also have "\ = (real (l div c) * (real c * real x + Inum (real x # bs) e) \ (real (l div c)) * 0)" by (simp add: algebra_simps) + also have "\ = (real c * real x + Inum (real x # bs) e \ 0)" + using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp +next + case (9 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(\ (k::int). real l * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k) = (\ (k::int). real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k)" by simp + also have "\ = (\ (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: algebra_simps) + also fix k have "\ = (\ (k::int). real c * real x + Inum (real x # bs) e - real j * real k = 0)" + using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e - real j * real k"] ldcp by simp + also have "\ = (\ (k::int). real c * real x + Inum (real x # bs) e = real j * real k)" by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] rdvd_def be isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp +next + case (10 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+ + from lp cp have clel: "c\l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \ 0" by simp + have "c div c\ l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(\ (k::int). real l * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k) = (\ (k::int). real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k)" by simp + also have "\ = (\ (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: algebra_simps) + also fix k have "\ = (\ (k::int). real c * real x + Inum (real x # bs) e - real j * real k = 0)" + using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e - real j * real k"] ldcp by simp + also have "\ = (\ (k::int). real c * real x + Inum (real x # bs) e = real j * real k)" by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] rdvd_def be isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp +qed (simp_all add: nth_pos2 numbound0_I[where bs="bs" and b="real (l * x)" and b'="real x"] isint_Mul del: real_of_int_mult) + +lemma a\_ex: assumes linp: "iszlfm p (a#bs)" and d: "d\ p l" and lp: "l>0" + shows "(\ x. l dvd x \ Ifm (real x #bs) (a\ p l)) = (\ (x::int). Ifm (real x#bs) p)" + (is "(\ x. l dvd x \ ?P x) = (\ x. ?P' x)") +proof- + have "(\ x. l dvd x \ ?P x) = (\ (x::int). ?P (l*x))" + using unity_coeff_ex[where l="l" and P="?P", simplified] by simp + also have "\ = (\ (x::int). ?P' x)" using a\[OF linp d lp] by simp + finally show ?thesis . +qed + +lemma \: + assumes lp: "iszlfm p (a#bs)" + and u: "d\ p 1" + and d: "d\ p d" + and dp: "d > 0" + and nob: "\(\(j::int) \ {1 .. d}. \ b\ (Inum (a#bs)) ` set(\ p). real x = b + real j)" + and p: "Ifm (real x#bs) p" (is "?P x") + shows "?P (x - d)" +using lp u d dp nob p +proof(induct p rule: iszlfm.induct) + case (5 c e) hence c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+ + with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] prems + show ?case by (simp del: real_of_int_minus) +next + case (6 c e) hence c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+ + with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] prems + show ?case by (simp del: real_of_int_minus) +next + case (7 c e) hence p: "Ifm (real x #bs) (Gt (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" and ie1:"isint e (a#bs)" using dvd1_eq1[where x="c"] by simp+ + let ?e = "Inum (real x # bs) e" + from ie1 have ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="a#bs"] + numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"] + by (simp add: isint_iff) + {assume "real (x-d) +?e > 0" hence ?case using c1 + numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] + by (simp del: real_of_int_minus)} + moreover + {assume H: "\ real (x-d) + ?e > 0" + let ?v="Neg e" + have vb: "?v \ set (\ (Gt (CN 0 c e)))" by simp + from prems(11)[simplified simp_thms Inum.simps \.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]] + have nob: "\ (\ j\ {1 ..d}. real x = - ?e + real j)" by auto + from H p have "real x + ?e > 0 \ real x + ?e \ real d" by (simp add: c1) + hence "real (x + floor ?e) > real (0::int) \ real (x + floor ?e) \ real d" + using ie by simp + hence "x + floor ?e \ 1 \ x + floor ?e \ d" by simp + hence "\ (j::int) \ {1 .. d}. j = x + floor ?e" by simp + hence "\ (j::int) \ {1 .. d}. real x = real (- floor ?e + j)" + by (simp only: real_of_int_inject) (simp add: algebra_simps) + hence "\ (j::int) \ {1 .. d}. real x = - ?e + real j" + by (simp add: ie[simplified isint_iff]) + with nob have ?case by auto} + ultimately show ?case by blast +next + case (8 c e) hence p: "Ifm (real x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" + and ie1:"isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+ + let ?e = "Inum (real x # bs) e" + from ie1 have ie: "real (floor ?e) = ?e" using numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real x)#bs"] + by (simp add: isint_iff) + {assume "real (x-d) +?e \ 0" hence ?case using c1 + numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] + by (simp del: real_of_int_minus)} + moreover + {assume H: "\ real (x-d) + ?e \ 0" + let ?v="Sub (C -1) e" + have vb: "?v \ set (\ (Ge (CN 0 c e)))" by simp + from prems(11)[simplified simp_thms Inum.simps \.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]] + have nob: "\ (\ j\ {1 ..d}. real x = - ?e - 1 + real j)" by auto + from H p have "real x + ?e \ 0 \ real x + ?e < real d" by (simp add: c1) + hence "real (x + floor ?e) \ real (0::int) \ real (x + floor ?e) < real d" + using ie by simp + hence "x + floor ?e +1 \ 1 \ x + floor ?e + 1 \ d" by simp + hence "\ (j::int) \ {1 .. d}. j = x + floor ?e + 1" by simp + hence "\ (j::int) \ {1 .. d}. x= - floor ?e - 1 + j" by (simp add: algebra_simps) + hence "\ (j::int) \ {1 .. d}. real x= real (- floor ?e - 1 + j)" + by (simp only: real_of_int_inject) + hence "\ (j::int) \ {1 .. d}. real x= - ?e - 1 + real j" + by (simp add: ie[simplified isint_iff]) + with nob have ?case by simp } + ultimately show ?case by blast +next + case (3 c e) hence p: "Ifm (real x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1" + and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+ + let ?e = "Inum (real x # bs) e" + let ?v="(Sub (C -1) e)" + have vb: "?v \ set (\ (Eq (CN 0 c e)))" by simp + from p have "real x= - ?e" by (simp add: c1) with prems(11) show ?case using dp + by simp (erule ballE[where x="1"], + simp_all add:algebra_simps numbound0_I[OF bn,where b="real x"and b'="a"and bs="bs"]) +next + case (4 c e)hence p: "Ifm (real x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1" + and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+ + let ?e = "Inum (real x # bs) e" + let ?v="Neg e" + have vb: "?v \ set (\ (NEq (CN 0 c e)))" by simp + {assume "real x - real d + Inum ((real (x -d)) # bs) e \ 0" + hence ?case by (simp add: c1)} + moreover + {assume H: "real x - real d + Inum ((real (x -d)) # bs) e = 0" + hence "real x = - Inum ((real (x -d)) # bs) e + real d" by simp + hence "real x = - Inum (a # bs) e + real d" + by (simp add: numbound0_I[OF bn,where b="real x - real d"and b'="a"and bs="bs"]) + with prems(11) have ?case using dp by simp} + ultimately show ?case by blast +next + case (9 j c e) hence p: "Ifm (real x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1" + and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+ + let ?e = "Inum (real x # bs) e" + from prems have "isint e (a #bs)" by simp + hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real x)#bs"] numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] + by (simp add: isint_iff) + from prems have id: "j dvd d" by simp + from c1 ie[symmetric] have "?p x = (real j rdvd real (x+ floor ?e))" by simp + also have "\ = (j dvd x + floor ?e)" + using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp + also have "\ = (j dvd x - d + floor ?e)" + using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp + also have "\ = (real j rdvd real (x - d + floor ?e))" + using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified] + ie by simp + also have "\ = (real j rdvd real x - real d + ?e)" + using ie by simp + finally show ?case + using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp +next + case (10 j c e) hence p: "Ifm (real x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+ + let ?e = "Inum (real x # bs) e" + from prems have "isint e (a#bs)" by simp + hence ie: "real (floor ?e) = ?e" using numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real x)#bs"] + by (simp add: isint_iff) + from prems have id: "j dvd d" by simp + from c1 ie[symmetric] have "?p x = (\ real j rdvd real (x+ floor ?e))" by simp + also have "\ = (\ j dvd x + floor ?e)" + using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp + also have "\ = (\ j dvd x - d + floor ?e)" + using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp + also have "\ = (\ real j rdvd real (x - d + floor ?e))" + using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified] + ie by simp + also have "\ = (\ real j rdvd real x - real d + ?e)" + using ie by simp + finally show ?case using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp +qed (auto simp add: numbound0_I[where bs="bs" and b="real (x - d)" and b'="real x"] nth_pos2 simp del: real_of_int_diff) + +lemma \': + assumes lp: "iszlfm p (a #bs)" + and u: "d\ p 1" + and d: "d\ p d" + and dp: "d > 0" + shows "\ x. \(\(j::int) \ {1 .. d}. \ b\ set(\ p). Ifm ((Inum (a#bs) b + real j) #bs) p) \ Ifm (real x#bs) p \ Ifm (real (x - d)#bs) p" (is "\ x. ?b \ ?P x \ ?P (x - d)") +proof(clarify) + fix x + assume nb:"?b" and px: "?P x" + hence nb2: "\(\(j::int) \ {1 .. d}. \ b\ (Inum (a#bs)) ` set(\ p). real x = b + real j)" + by auto + from \[OF lp u d dp nb2 px] show "?P (x -d )" . +qed + +lemma \_int: assumes lp: "iszlfm p bs" + shows "\ b\ set (\ p). isint b bs" +using lp by (induct p rule: iszlfm.induct) (auto simp add: isint_neg isint_sub) + +lemma cpmi_eq: "0 < D \ (EX z::int. ALL x. x < z --> (P x = P1 x)) +==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D) +==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D)))) +==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))" +apply(rule iffI) +prefer 2 +apply(drule minusinfinity) +apply assumption+ +apply(fastsimp) +apply clarsimp +apply(subgoal_tac "!!k. 0<=k \ !x. P x \ P (x - k*D)") +apply(frule_tac x = x and z=z in decr_lemma) +apply(subgoal_tac "P1(x - (\x - z\ + 1) * D)") +prefer 2 +apply(subgoal_tac "0 <= (\x - z\ + 1)") +prefer 2 apply arith + apply fastsimp +apply(drule (1) periodic_finite_ex) +apply blast +apply(blast dest:decr_mult_lemma) +done + + +theorem cp_thm: + assumes lp: "iszlfm p (a #bs)" + and u: "d\ p 1" + and d: "d\ p d" + and dp: "d > 0" + shows "(\ (x::int). Ifm (real x #bs) p) = (\ j\ {1.. d}. Ifm (real j #bs) (minusinf p) \ (\ b \ set (\ p). Ifm ((Inum (a#bs) b + real j) #bs) p))" + (is "(\ (x::int). ?P (real x)) = (\ j\ ?D. ?M j \ (\ b\ ?B. ?P (?I b + real j)))") +proof- + from minusinf_inf[OF lp] + have th: "\(z::int). \x_int[OF lp] isint_iff[where bs="a # bs"] have B: "\ b\ ?B. real (floor (?I b)) = ?I b" by simp + from B[rule_format] + have "(\j\?D. \b\ ?B. ?P (?I b + real j)) = (\j\?D. \b\ ?B. ?P (real (floor (?I b)) + real j))" + by simp + also have "\ = (\j\?D. \b\ ?B. ?P (real (floor (?I b) + j)))" by simp + also have"\ = (\ j \ ?D. \ b \ ?B'. ?P (real (b + j)))" by blast + finally have BB': + "(\j\?D. \b\ ?B. ?P (?I b + real j)) = (\ j \ ?D. \ b \ ?B'. ?P (real (b + j)))" + by blast + hence th2: "\ x. \ (\ j \ ?D. \ b \ ?B'. ?P (real (b + j))) \ ?P (real x) \ ?P (real (x - d))" using \'[OF lp u d dp] by blast + from minusinf_repeats[OF d lp] + have th3: "\ x k. ?M x = ?M (x-k*d)" by simp + from cpmi_eq[OF dp th th2 th3] BB' show ?thesis by blast +qed + + (* Reddy and Loveland *) + + +consts + \ :: "fm \ (num \ int) list" (* Compute the Reddy and Loveland Bset*) + \\:: "fm \ num \ int \ fm" (* Performs the modified substitution of Reddy and Loveland*) + \\ :: "fm \ (num\int) list" + a\ :: "fm \ int \ fm" +recdef \ "measure size" + "\ (And p q) = (\ p @ \ q)" + "\ (Or p q) = (\ p @ \ q)" + "\ (Eq (CN 0 c e)) = [(Sub (C -1) e,c)]" + "\ (NEq (CN 0 c e)) = [(Neg e,c)]" + "\ (Lt (CN 0 c e)) = []" + "\ (Le (CN 0 c e)) = []" + "\ (Gt (CN 0 c e)) = [(Neg e, c)]" + "\ (Ge (CN 0 c e)) = [(Sub (C (-1)) e, c)]" + "\ p = []" + +recdef \\ "measure size" + "\\ (And p q) = (\ (t,k). And (\\ p (t,k)) (\\ q (t,k)))" + "\\ (Or p q) = (\ (t,k). Or (\\ p (t,k)) (\\ q (t,k)))" + "\\ (Eq (CN 0 c e)) = (\ (t,k). if k dvd c then (Eq (Add (Mul (c div k) t) e)) + else (Eq (Add (Mul c t) (Mul k e))))" + "\\ (NEq (CN 0 c e)) = (\ (t,k). if k dvd c then (NEq (Add (Mul (c div k) t) e)) + else (NEq (Add (Mul c t) (Mul k e))))" + "\\ (Lt (CN 0 c e)) = (\ (t,k). if k dvd c then (Lt (Add (Mul (c div k) t) e)) + else (Lt (Add (Mul c t) (Mul k e))))" + "\\ (Le (CN 0 c e)) = (\ (t,k). if k dvd c then (Le (Add (Mul (c div k) t) e)) + else (Le (Add (Mul c t) (Mul k e))))" + "\\ (Gt (CN 0 c e)) = (\ (t,k). if k dvd c then (Gt (Add (Mul (c div k) t) e)) + else (Gt (Add (Mul c t) (Mul k e))))" + "\\ (Ge (CN 0 c e)) = (\ (t,k). if k dvd c then (Ge (Add (Mul (c div k) t) e)) + else (Ge (Add (Mul c t) (Mul k e))))" + "\\ (Dvd i (CN 0 c e)) =(\ (t,k). if k dvd c then (Dvd i (Add (Mul (c div k) t) e)) + else (Dvd (i*k) (Add (Mul c t) (Mul k e))))" + "\\ (NDvd i (CN 0 c e))=(\ (t,k). if k dvd c then (NDvd i (Add (Mul (c div k) t) e)) + else (NDvd (i*k) (Add (Mul c t) (Mul k e))))" + "\\ p = (\ (t,k). p)" + +recdef \\ "measure size" + "\\ (And p q) = (\\ p @ \\ q)" + "\\ (Or p q) = (\\ p @ \\ q)" + "\\ (Eq (CN 0 c e)) = [(Add (C -1) e,c)]" + "\\ (NEq (CN 0 c e)) = [(e,c)]" + "\\ (Lt (CN 0 c e)) = [(e,c)]" + "\\ (Le (CN 0 c e)) = [(Add (C -1) e,c)]" + "\\ p = []" + + (* Simulates normal substituion by modifying the formula see correctness theorem *) + +recdef a\ "measure size" + "a\ (And p q) = (\ k. And (a\ p k) (a\ q k))" + "a\ (Or p q) = (\ k. Or (a\ p k) (a\ q k))" + "a\ (Eq (CN 0 c e)) = (\ k. if k dvd c then (Eq (CN 0 (c div k) e)) + else (Eq (CN 0 c (Mul k e))))" + "a\ (NEq (CN 0 c e)) = (\ k. if k dvd c then (NEq (CN 0 (c div k) e)) + else (NEq (CN 0 c (Mul k e))))" + "a\ (Lt (CN 0 c e)) = (\ k. if k dvd c then (Lt (CN 0 (c div k) e)) + else (Lt (CN 0 c (Mul k e))))" + "a\ (Le (CN 0 c e)) = (\ k. if k dvd c then (Le (CN 0 (c div k) e)) + else (Le (CN 0 c (Mul k e))))" + "a\ (Gt (CN 0 c e)) = (\ k. if k dvd c then (Gt (CN 0 (c div k) e)) + else (Gt (CN 0 c (Mul k e))))" + "a\ (Ge (CN 0 c e)) = (\ k. if k dvd c then (Ge (CN 0 (c div k) e)) + else (Ge (CN 0 c (Mul k e))))" + "a\ (Dvd i (CN 0 c e)) = (\ k. if k dvd c then (Dvd i (CN 0 (c div k) e)) + else (Dvd (i*k) (CN 0 c (Mul k e))))" + "a\ (NDvd i (CN 0 c e)) = (\ k. if k dvd c then (NDvd i (CN 0 (c div k) e)) + else (NDvd (i*k) (CN 0 c (Mul k e))))" + "a\ p = (\ k. p)" + +constdefs \ :: "fm \ int \ num \ fm" + "\ p k t \ And (Dvd k t) (\\ p (t,k))" + +lemma \\: + assumes linp: "iszlfm p (real (x::int)#bs)" + and kpos: "real k > 0" + and tnb: "numbound0 t" + and tint: "isint t (real x#bs)" + and kdt: "k dvd floor (Inum (b'#bs) t)" + shows "Ifm (real x#bs) (\\ p (t,k)) = + (Ifm ((real ((floor (Inum (b'#bs) t)) div k))#bs) p)" + (is "?I (real x) (?s p) = (?I (real ((floor (?N b' t)) div k)) p)" is "_ = (?I ?tk p)") +using linp kpos tnb +proof(induct p rule: \\.induct) + case (3 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\ k dvd c" + from kpos have knz: "k\0" by simp hence knz': "real k \ 0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (Eq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k = 0)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps) + also have "\ = (?I ?tk (Eq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (4 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\ k dvd c" + from kpos have knz: "k\0" by simp hence knz': "real k \ 0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (NEq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \ 0)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps) + also have "\ = (?I ?tk (NEq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (5 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\ k dvd c" + from kpos have knz: "k\0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (Lt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k < 0)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps) + also have "\ = (?I ?tk (Lt (CN 0 c e)))" using pos_less_divide_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (6 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\ k dvd c" + from kpos have knz: "k\0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (Le (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \ 0)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps) + also have "\ = (?I ?tk (Le (CN 0 c e)))" using pos_le_divide_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (7 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\ k dvd c" + from kpos have knz: "k\0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (Gt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k > 0)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps) + also have "\ = (?I ?tk (Gt (CN 0 c e)))" using pos_divide_less_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (8 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\ k dvd c" + from kpos have knz: "k\0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (Ge (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \ 0)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps) + also have "\ = (?I ?tk (Ge (CN 0 c e)))" using pos_divide_le_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (9 i c e) from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\ k dvd c" + from kpos have knz: "k\0" by simp hence knz': "real k \ 0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (Dvd i (CN 0 c e))) = (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps) + also have "\ = (?I ?tk (Dvd i (CN 0 c e)))" using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (10 i c e) from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\ k dvd c" + from kpos have knz: "k\0" by simp hence knz': "real k \ 0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (NDvd i (CN 0 c e))) = (\ (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k))" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps) + also have "\ = (?I ?tk (NDvd i (CN 0 c e)))" using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +qed (simp_all add: nth_pos2 bound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"] numbound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"]) + + +lemma a\: + assumes lp: "iszlfm p (real (x::int)#bs)" and kp: "real k > 0" + shows "Ifm (real (x*k)#bs) (a\ p k) = Ifm (real x#bs) p" (is "?I (x*k) (?f p k) = ?I x p") +using lp bound0_I[where bs="bs" and b="real (x*k)" and b'="real x"] numbound0_I[where bs="bs" and b="real (x*k)" and b'="real x"] +proof(induct p rule: a\.induct) + case (3 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\0" by simp hence knz': "real k \ 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume nkdc: "\ k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] nonzero_eq_divide_eq[OF knz', where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)} + ultimately show ?case by blast +next + case (4 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\0" by simp hence knz': "real k \ 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume nkdc: "\ k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] nonzero_eq_divide_eq[OF knz', where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)} + ultimately show ?case by blast +next + case (5 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\0" by simp hence knz': "real k \ 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume nkdc: "\ k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_less_divide_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)} + ultimately show ?case by blast +next + case (6 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\0" by simp hence knz': "real k \ 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume nkdc: "\ k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_le_divide_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)} + ultimately show ?case by blast +next + case (7 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\0" by simp hence knz': "real k \ 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume nkdc: "\ k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_divide_less_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)} + ultimately show ?case by blast +next + case (8 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\0" by simp hence knz': "real k \ 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume nkdc: "\ k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_divide_le_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)} + ultimately show ?case by blast +next + case (9 i c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\0" by simp hence knz': "real k \ 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume "\ k dvd c" + hence "Ifm (real (x*k)#bs) (a\ (Dvd i (CN 0 c e)) k) = + (real i * real k rdvd (real c * real x + Inum (real x#bs) e) * real k)" + using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] + by (simp add: algebra_simps) + also have "\ = (Ifm (real x#bs) (Dvd i (CN 0 c e)))" by (simp add: rdvd_mult[OF knz, where n="i"]) + finally have ?case . } + ultimately show ?case by blast +next + case (10 i c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\0" by simp hence knz': "real k \ 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume "\ k dvd c" + hence "Ifm (real (x*k)#bs) (a\ (NDvd i (CN 0 c e)) k) = + (\ (real i * real k rdvd (real c * real x + Inum (real x#bs) e) * real k))" + using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] + by (simp add: algebra_simps) + also have "\ = (Ifm (real x#bs) (NDvd i (CN 0 c e)))" by (simp add: rdvd_mult[OF knz, where n="i"]) + finally have ?case . } + ultimately show ?case by blast +qed (simp_all add: nth_pos2) + +lemma a\_ex: + assumes lp: "iszlfm p (real (x::int)#bs)" and kp: "k > 0" + shows "(\ (x::int). real k rdvd real x \ Ifm (real x#bs) (a\ p k)) = + (\ (x::int). Ifm (real x#bs) p)" (is "(\ x. ?D x \ ?P' x) = (\ x. ?P x)") +proof- + have "(\ x. ?D x \ ?P' x) = (\ x. k dvd x \ ?P' x)" using int_rdvd_iff by simp + also have "\ = (\x. ?P' (x*k))" using unity_coeff_ex[where P="?P'" and l="k", simplified] + by (simp add: algebra_simps) + also have "\ = (\ x. ?P x)" using a\ iszlfm_gen[OF lp] kp by auto + finally show ?thesis . +qed + +lemma \\': assumes lp: "iszlfm p (real (x::int)#bs)" and kp: "k > 0" and nb: "numbound0 t" + shows "Ifm (real x#bs) (\\ p (t,k)) = Ifm ((Inum (real x#bs) t)#bs) (a\ p k)" +using lp +by(induct p rule: \\.induct, simp_all add: + numbound0_I[OF nb, where bs="bs" and b="Inum (real x#bs) t" and b'="real x"] + numbound0_I[where bs="bs" and b="Inum (real x#bs) t" and b'="real x"] + bound0_I[where bs="bs" and b="Inum (real x#bs) t" and b'="real x"] nth_pos2 cong: imp_cong) + +lemma \\_nb: assumes lp:"iszlfm p (a#bs)" and nb: "numbound0 t" + shows "bound0 (\\ p (t,k))" + using lp + by (induct p rule: iszlfm.induct, auto simp add: nb) + +lemma \_l: + assumes lp: "iszlfm p (real (i::int)#bs)" + shows "\ (b,k) \ set (\ p). k >0 \ numbound0 b \ isint b (real i#bs)" +using lp by (induct p rule: \.induct, auto simp add: isint_sub isint_neg) + +lemma \\_l: + assumes lp: "iszlfm p (real (i::int)#bs)" + shows "\ (b,k) \ set (\\ p). k >0 \ numbound0 b \ isint b (real i#bs)" +using lp isint_add [OF isint_c[where j="- 1"],where bs="real i#bs"] + by (induct p rule: \\.induct, auto) + +lemma zminusinf_\: + assumes lp: "iszlfm p (real (i::int)#bs)" + and nmi: "\ (Ifm (real i#bs) (minusinf p))" (is "\ (Ifm (real i#bs) (?M p))") + and ex: "Ifm (real i#bs) p" (is "?I i p") + shows "\ (e,c) \ set (\ p). real (c*i) > Inum (real i#bs) e" (is "\ (e,c) \ ?R p. real (c*i) > ?N i e") + using lp nmi ex +by (induct p rule: minusinf.induct, auto) + + +lemma \_And: "Ifm bs (\ (And p q) k t) = Ifm bs (And (\ p k t) (\ q k t))" +using \_def by auto +lemma \_Or: "Ifm bs (\ (Or p q) k t) = Ifm bs (Or (\ p k t) (\ q k t))" +using \_def by auto + +lemma \: assumes lp: "iszlfm p (real (i::int) #bs)" + and pi: "Ifm (real i#bs) p" + and d: "d\ p d" + and dp: "d > 0" + and nob: "\(e,c) \ set (\ p). \ j\ {1 .. c*d}. real (c*i) \ Inum (real i#bs) e + real j" + (is "\(e,c) \ set (\ p). \ j\ {1 .. c*d}. _ \ ?N i e + _") + shows "Ifm (real(i - d)#bs) p" + using lp pi d nob +proof(induct p rule: iszlfm.induct) + case (3 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)" + and pi: "real (c*i) = - 1 - ?N i e + real (1::int)" and nob: "\ j\ {1 .. c*d}. real (c*i) \ -1 - ?N i e + real j" + by simp+ + from mult_strict_left_mono[OF dp cp] have one:"1 \ {1 .. c*d}" by auto + from nob[rule_format, where j="1", OF one] pi show ?case by simp +next + case (4 c e) + hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)" + and nob: "\ j\ {1 .. c*d}. real (c*i) \ - ?N i e + real j" + by simp+ + {assume "real (c*i) \ - ?N i e + real (c*d)" + with numbound0_I[OF nb, where bs="bs" and b="real i - real d" and b'="real i"] + have ?case by (simp add: algebra_simps)} + moreover + {assume pi: "real (c*i) = - ?N i e + real (c*d)" + from mult_strict_left_mono[OF dp cp] have d: "(c*d) \ {1 .. c*d}" by simp + from nob[rule_format, where j="c*d", OF d] pi have ?case by simp } + ultimately show ?case by blast +next + case (5 c e) hence cp: "c > 0" by simp + from prems mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric] + real_of_int_mult] + show ?case using prems dp + by (simp add: add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] + algebra_simps) +next + case (6 c e) hence cp: "c > 0" by simp + from prems mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric] + real_of_int_mult] + show ?case using prems dp + by (simp add: add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] + algebra_simps) +next + case (7 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)" + and nob: "\ j\ {1 .. c*d}. real (c*i) \ - ?N i e + real j" + and pi: "real (c*i) + ?N i e > 0" and cp': "real c >0" + by simp+ + let ?fe = "floor (?N i e)" + from pi cp have th:"(real i +?N i e / real c)*real c > 0" by (simp add: algebra_simps) + from pi ei[simplified isint_iff] have "real (c*i + ?fe) > real (0::int)" by simp + hence pi': "c*i + ?fe > 0" by (simp only: real_of_int_less_iff[symmetric]) + have "real (c*i) + ?N i e > real (c*d) \ real (c*i) + ?N i e \ real (c*d)" by auto + moreover + {assume "real (c*i) + ?N i e > real (c*d)" hence ?case + by (simp add: algebra_simps + numbound0_I[OF nb,where bs="bs" and b="real i - real d" and b'="real i"])} + moreover + {assume H:"real (c*i) + ?N i e \ real (c*d)" + with ei[simplified isint_iff] have "real (c*i + ?fe) \ real (c*d)" by simp + hence pid: "c*i + ?fe \ c*d" by (simp only: real_of_int_le_iff) + with pi' have "\ j1\ {1 .. c*d}. c*i + ?fe = j1" by auto + hence "\ j1\ {1 .. c*d}. real (c*i) = - ?N i e + real j1" + by (simp only: diff_def[symmetric] real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] algebra_simps) + with nob have ?case by blast } + ultimately show ?case by blast +next + case (8 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)" + and nob: "\ j\ {1 .. c*d}. real (c*i) \ - 1 - ?N i e + real j" + and pi: "real (c*i) + ?N i e \ 0" and cp': "real c >0" + by simp+ + let ?fe = "floor (?N i e)" + from pi cp have th:"(real i +?N i e / real c)*real c \ 0" by (simp add: algebra_simps) + from pi ei[simplified isint_iff] have "real (c*i + ?fe) \ real (0::int)" by simp + hence pi': "c*i + 1 + ?fe \ 1" by (simp only: real_of_int_le_iff[symmetric]) + have "real (c*i) + ?N i e \ real (c*d) \ real (c*i) + ?N i e < real (c*d)" by auto + moreover + {assume "real (c*i) + ?N i e \ real (c*d)" hence ?case + by (simp add: algebra_simps + numbound0_I[OF nb,where bs="bs" and b="real i - real d" and b'="real i"])} + moreover + {assume H:"real (c*i) + ?N i e < real (c*d)" + with ei[simplified isint_iff] have "real (c*i + ?fe) < real (c*d)" by simp + hence pid: "c*i + 1 + ?fe \ c*d" by (simp only: real_of_int_le_iff) + with pi' have "\ j1\ {1 .. c*d}. c*i + 1+ ?fe = j1" by auto + hence "\ j1\ {1 .. c*d}. real (c*i) + 1= - ?N i e + real j1" + by (simp only: diff_def[symmetric] real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] algebra_simps real_of_one) + hence "\ j1\ {1 .. c*d}. real (c*i) = (- ?N i e + real j1) - 1" + by (simp only: algebra_simps diff_def[symmetric]) + hence "\ j1\ {1 .. c*d}. real (c*i) = - 1 - ?N i e + real j1" + by (simp only: add_ac diff_def) + with nob have ?case by blast } + ultimately show ?case by blast +next + case (9 j c e) hence p: "real j rdvd real (c*i) + ?N i e" (is "?p x") and cp: "c > 0" and bn:"numbound0 e" by simp+ + let ?e = "Inum (real i # bs) e" + from prems have "isint e (real i #bs)" by simp + hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"] + by (simp add: isint_iff) + from prems have id: "j dvd d" by simp + from ie[symmetric] have "?p i = (real j rdvd real (c*i+ floor ?e))" by simp + also have "\ = (j dvd c*i + floor ?e)" + using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp + also have "\ = (j dvd c*i - c*d + floor ?e)" + using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp + also have "\ = (real j rdvd real (c*i - c*d + floor ?e))" + using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified] + ie by simp + also have "\ = (real j rdvd real (c*(i - d)) + ?e)" + using ie by (simp add:algebra_simps) + finally show ?case + using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p + by (simp add: algebra_simps) +next + case (10 j c e) hence p: "\ (real j rdvd real (c*i) + ?N i e)" (is "?p x") and cp: "c > 0" and bn:"numbound0 e" by simp+ + let ?e = "Inum (real i # bs) e" + from prems have "isint e (real i #bs)" by simp + hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"] + by (simp add: isint_iff) + from prems have id: "j dvd d" by simp + from ie[symmetric] have "?p i = (\ (real j rdvd real (c*i+ floor ?e)))" by simp + also have "\ = Not (j dvd c*i + floor ?e)" + using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp + also have "\ = Not (j dvd c*i - c*d + floor ?e)" + using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp + also have "\ = Not (real j rdvd real (c*i - c*d + floor ?e))" + using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified] + ie by simp + also have "\ = Not (real j rdvd real (c*(i - d)) + ?e)" + using ie by (simp add:algebra_simps) + finally show ?case + using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p + by (simp add: algebra_simps) +qed(auto simp add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] nth_pos2) + +lemma \_nb: assumes lp: "iszlfm p (a#bs)" and nb: "numbound0 t" + shows "bound0 (\ p k t)" + using \\_nb[OF lp nb] nb by (simp add: \_def) + +lemma \': assumes lp: "iszlfm p (a #bs)" + and d: "d\ p d" + and dp: "d > 0" + shows "\ x. \(\ (e,c) \ set(\ p). \(j::int) \ {1 .. c*d}. Ifm (a #bs) (\ p c (Add e (C j)))) \ Ifm (real x#bs) p \ Ifm (real (x - d)#bs) p" (is "\ x. ?b x \ ?P x \ ?P (x - d)") +proof(clarify) + fix x + assume nob1:"?b x" and px: "?P x" + from iszlfm_gen[OF lp, rule_format, where y="real x"] have lp': "iszlfm p (real x#bs)". + have nob: "\(e, c)\set (\ p). \j\{1..c * d}. real (c * x) \ Inum (real x # bs) e + real j" + proof(clarify) + fix e c j assume ecR: "(e,c) \ set (\ p)" and jD: "j\ {1 .. c*d}" + and cx: "real (c*x) = Inum (real x#bs) e + real j" + let ?e = "Inum (real x#bs) e" + let ?fe = "floor ?e" + from \_l[OF lp'] ecR have ei:"isint e (real x#bs)" and cp:"c>0" and nb:"numbound0 e" + by auto + from numbound0_gen [OF nb ei, rule_format,where y="a"] have "isint e (a#bs)" . + from cx ei[simplified isint_iff] have "real (c*x) = real (?fe + j)" by simp + hence cx: "c*x = ?fe + j" by (simp only: real_of_int_inject) + hence cdej:"c dvd ?fe + j" by (simp add: dvd_def) (rule_tac x="x" in exI, simp) + hence "real c rdvd real (?fe + j)" by (simp only: int_rdvd_iff) + hence rcdej: "real c rdvd ?e + real j" by (simp add: ei[simplified isint_iff]) + from cx have "(c*x) div c = (?fe + j) div c" by simp + with cp have "x = (?fe + j) div c" by simp + with px have th: "?P ((?fe + j) div c)" by auto + from cp have cp': "real c > 0" by simp + from cdej have cdej': "c dvd floor (Inum (real x#bs) (Add e (C j)))" by simp + from nb have nb': "numbound0 (Add e (C j))" by simp + have ji: "isint (C j) (real x#bs)" by (simp add: isint_def) + from isint_add[OF ei ji] have ei':"isint (Add e (C j)) (real x#bs)" . + from th \\[where b'="real x", OF lp' cp' nb' ei' cdej',symmetric] + have "Ifm (real x#bs) (\\ p (Add e (C j), c))" by simp + with rcdej have th: "Ifm (real x#bs) (\ p c (Add e (C j)))" by (simp add: \_def) + from th bound0_I[OF \_nb[OF lp nb', where k="c"],where bs="bs" and b="real x" and b'="a"] + have "Ifm (a#bs) (\ p c (Add e (C j)))" by blast + with ecR jD nob1 show "False" by blast + qed + from \[OF lp' px d dp nob] show "?P (x -d )" . +qed + + +lemma rl_thm: + assumes lp: "iszlfm p (real (i::int)#bs)" + shows "(\ (x::int). Ifm (real x#bs) p) = ((\ j\ {1 .. \ p}. Ifm (real j#bs) (minusinf p)) \ (\ (e,c) \ set (\ p). \ j\ {1 .. c*(\ p)}. Ifm (a#bs) (\ p c (Add e (C j)))))" + (is "(\(x::int). ?P x) = ((\ j\ {1.. \ p}. ?MP j)\(\ (e,c) \ ?R. \ j\ _. ?SP c e j))" + is "?lhs = (?MD \ ?RD)" is "?lhs = ?rhs") +proof- + let ?d= "\ p" + from \[OF lp] have d:"d\ p ?d" and dp: "?d > 0" by auto + { assume H:"?MD" hence th:"\ (x::int). ?MP x" by blast + from H minusinf_ex[OF lp th] have ?thesis by blast} + moreover + { fix e c j assume exR:"(e,c) \ ?R" and jD:"j\ {1 .. c*?d}" and spx:"?SP c e j" + from exR \_l[OF lp] have nb: "numbound0 e" and ei:"isint e (real i#bs)" and cp: "c > 0" + by auto + have "isint (C j) (real i#bs)" by (simp add: isint_iff) + with isint_add[OF numbound0_gen[OF nb ei,rule_format, where y="real i"]] + have eji:"isint (Add e (C j)) (real i#bs)" by simp + from nb have nb': "numbound0 (Add e (C j))" by simp + from spx bound0_I[OF \_nb[OF lp nb', where k="c"], where bs="bs" and b="a" and b'="real i"] + have spx': "Ifm (real i # bs) (\ p c (Add e (C j)))" by blast + from spx' have rcdej:"real c rdvd (Inum (real i#bs) (Add e (C j)))" + and sr:"Ifm (real i#bs) (\\ p (Add e (C j),c))" by (simp add: \_def)+ + from rcdej eji[simplified isint_iff] + have "real c rdvd real (floor (Inum (real i#bs) (Add e (C j))))" by simp + hence cdej:"c dvd floor (Inum (real i#bs) (Add e (C j)))" by (simp only: int_rdvd_iff) + from cp have cp': "real c > 0" by simp + from \\[OF lp cp' nb' eji cdej] spx' have "?P (\Inum (real i # bs) (Add e (C j))\ div c)" + by (simp add: \_def) + hence ?lhs by blast + with exR jD spx have ?thesis by blast} + moreover + { fix x assume px: "?P x" and nob: "\ ?RD" + from iszlfm_gen [OF lp,rule_format, where y="a"] have lp':"iszlfm p (a#bs)" . + from \'[OF lp' d dp, rule_format, OF nob] have th:"\ x. ?P x \ ?P (x - ?d)" by blast + from minusinf_inf[OF lp] obtain z where z:"\ x 0" by arith + from decr_lemma[OF dp,where x="x" and z="z"] + decr_mult_lemma[OF dp th zp, rule_format, OF px] z have th:"\ x. ?MP x" by auto + with minusinf_bex[OF lp] px nob have ?thesis by blast} + ultimately show ?thesis by blast +qed + +lemma mirror_\\: assumes lp: "iszlfm p (a#bs)" + shows "(\ (t,k). (Inum (a#bs) t, k)) ` set (\\ p) = (\ (t,k). (Inum (a#bs) t,k)) ` set (\ (mirror p))" +using lp +by (induct p rule: mirror.induct, simp_all add: split_def image_Un ) + +text {* The @{text "\"} part*} + +text{* Linearity for fm where Bound 0 ranges over @{text "\"}*} +consts + isrlfm :: "fm \ bool" (* Linearity test for fm *) +recdef isrlfm "measure size" + "isrlfm (And p q) = (isrlfm p \ isrlfm q)" + "isrlfm (Or p q) = (isrlfm p \ isrlfm q)" + "isrlfm (Eq (CN 0 c e)) = (c>0 \ numbound0 e)" + "isrlfm (NEq (CN 0 c e)) = (c>0 \ numbound0 e)" + "isrlfm (Lt (CN 0 c e)) = (c>0 \ numbound0 e)" + "isrlfm (Le (CN 0 c e)) = (c>0 \ numbound0 e)" + "isrlfm (Gt (CN 0 c e)) = (c>0 \ numbound0 e)" + "isrlfm (Ge (CN 0 c e)) = (c>0 \ numbound0 e)" + "isrlfm p = (isatom p \ (bound0 p))" + +constdefs fp :: "fm \ int \ num \ int \ fm" + "fp p n s j \ (if n > 0 then + (And p (And (Ge (CN 0 n (Sub s (Add (Floor s) (C j))))) + (Lt (CN 0 n (Sub s (Add (Floor s) (C (j+1)))))))) + else + (And p (And (Le (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C j))))) + (Gt (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C (j + 1)))))))))" + + (* splits the bounded from the unbounded part*) +consts rsplit0 :: "num \ (fm \ int \ num) list" +recdef rsplit0 "measure num_size" + "rsplit0 (Bound 0) = [(T,1,C 0)]" + "rsplit0 (Add a b) = (let acs = rsplit0 a ; bcs = rsplit0 b + in map (\ ((p,n,t),(q,m,s)). (And p q, n+m, Add t s)) [(a,b). a\acs,b\bcs])" + "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))" + "rsplit0 (Neg a) = map (\ (p,n,s). (p,-n,Neg s)) (rsplit0 a)" + "rsplit0 (Floor a) = foldl (op @) [] (map + (\ (p,n,s). if n=0 then [(p,0,Floor s)] + else (map (\ j. (fp p n s j, 0, Add (Floor s) (C j))) (if n > 0 then iupt (0,n) else iupt(n,0)))) + (rsplit0 a))" + "rsplit0 (CN 0 c a) = map (\ (p,n,s). (p,n+c,s)) (rsplit0 a)" + "rsplit0 (CN m c a) = map (\ (p,n,s). (p,n,CN m c s)) (rsplit0 a)" + "rsplit0 (CF c t s) = rsplit0 (Add (Mul c (Floor t)) s)" + "rsplit0 (Mul c a) = map (\ (p,n,s). (p,c*n,Mul c s)) (rsplit0 a)" + "rsplit0 t = [(T,0,t)]" + +lemma not_rl[simp]: "isrlfm p \ isrlfm (not p)" + by (induct p rule: isrlfm.induct, auto) +lemma conj_rl[simp]: "isrlfm p \ isrlfm q \ isrlfm (conj p q)" + using conj_def by (cases p, auto) +lemma disj_rl[simp]: "isrlfm p \ isrlfm q \ isrlfm (disj p q)" + using disj_def by (cases p, auto) + + +lemma rsplit0_cs: + shows "\ (p,n,s) \ set (rsplit0 t). + (Ifm (x#bs) p \ (Inum (x#bs) t = Inum (x#bs) (CN 0 n s))) \ numbound0 s \ isrlfm p" + (is "\ (p,n,s) \ ?SS t. (?I p \ ?N t = ?N (CN 0 n s)) \ _ \ _ ") +proof(induct t rule: rsplit0.induct) + case (5 a) + let ?p = "\ (p,n,s) j. fp p n s j" + let ?f = "(\ (p,n,s) j. (?p (p,n,s) j, (0::int),Add (Floor s) (C j)))" + let ?J = "\ n. if n>0 then iupt (0,n) else iupt (n,0)" + let ?ff=" (\ (p,n,s). if n= 0 then [(p,0,Floor s)] else map (?f (p,n,s)) (?J n))" + have int_cases: "\ (i::int). i= 0 \ i < 0 \ i > 0" by arith + have U1: "(UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). set (?ff (p,n,s)))) = + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). set [(p,0,Floor s)]))" by auto + have U2': "\ (p,n,s) \ {(p,n,s). (p,n,s) \ ?SS a \ n>0}. + ?ff (p,n,s) = map (?f(p,n,s)) (iupt(0,n))" by auto + hence U2: "(UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). set (?ff (p,n,s)))) = + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). + set (map (?f(p,n,s)) (iupt(0,n)))))" + proof- + fix M :: "('a\'b\'c) set" and f :: "('a\'b\'c) \ 'd list" and g + assume "\ (a,b,c) \ M. f (a,b,c) = g a b c" + thus "(UNION M (\ (a,b,c). set (f (a,b,c)))) = (UNION M (\ (a,b,c). set (g a b c)))" + by (auto simp add: split_def) + qed + have U3': "\ (p,n,s) \ {(p,n,s). (p,n,s) \ ?SS a \ n<0}. ?ff (p,n,s) = map (?f(p,n,s)) (iupt(n,0))" + by auto + hence U3: "(UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). set (?ff (p,n,s)))) = + (UNION {(p,n,s). (p,n,s)\ ?SS a\n<0} (\(p,n,s). set (map (?f(p,n,s)) (iupt(n,0)))))" + proof- + fix M :: "('a\'b\'c) set" and f :: "('a\'b\'c) \ 'd list" and g + assume "\ (a,b,c) \ M. f (a,b,c) = g a b c" + thus "(UNION M (\ (a,b,c). set (f (a,b,c)))) = (UNION M (\ (a,b,c). set (g a b c)))" + by (auto simp add: split_def) + qed + have "?SS (Floor a) = UNION (?SS a) (\x. set (?ff x))" + by (auto simp add: foldl_conv_concat) + also have "\ = UNION (?SS a) (\ (p,n,s). set (?ff (p,n,s)))" by auto + also have "\ = + ((UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). set (?ff (p,n,s)))) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). set (?ff (p,n,s)))) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). set (?ff (p,n,s)))))" + using int_cases[rule_format] by blast + also have "\ = + ((UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). set [(p,0,Floor s)])) Un + (UNION {(p,n,s). (p,n,s)\ ?SS a\n>0} (\(p,n,s). set(map(?f(p,n,s)) (iupt(0,n))))) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). + set (map (?f(p,n,s)) (iupt(n,0))))))" by (simp only: U1 U2 U3) + also have "\ = + ((UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). {(p,0,Floor s)})) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). (?f(p,n,s)) ` {n .. 0})))" + by (simp only: set_map iupt_set set.simps) + also have "\ = + ((UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). {(p,0,Floor s)})) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). {?f(p,n,s) j| j. j\ {0 .. n}})) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). {?f(p,n,s) j| j. j\ {n .. 0}})))" by blast + finally + have FS: "?SS (Floor a) = + ((UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). {(p,0,Floor s)})) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). {?f(p,n,s) j| j. j\ {0 .. n}})) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). {?f(p,n,s) j| j. j\ {n .. 0}})))" by blast + show ?case + proof(simp only: FS, clarsimp simp del: Ifm.simps Inum.simps, -) + fix p n s + let ?ths = "(?I p \ (?N (Floor a) = ?N (CN 0 n s))) \ numbound0 s \ isrlfm p" + assume "(\ba. (p, 0, ba) \ set (rsplit0 a) \ n = 0 \ s = Floor ba) \ + (\ab ac ba. + (ab, ac, ba) \ set (rsplit0 a) \ + 0 < ac \ + (\j. p = fp ab ac ba j \ + n = 0 \ s = Add (Floor ba) (C j) \ 0 \ j \ j \ ac)) \ + (\ab ac ba. + (ab, ac, ba) \ set (rsplit0 a) \ + ac < 0 \ + (\j. p = fp ab ac ba j \ + n = 0 \ s = Add (Floor ba) (C j) \ ac \ j \ j \ 0))" + moreover + {fix s' + assume "(p, 0, s') \ ?SS a" and "n = 0" and "s = Floor s'" + hence ?ths using prems by auto} + moreover + { fix p' n' s' j + assume pns: "(p', n', s') \ ?SS a" + and np: "0 < n'" + and p_def: "p = ?p (p',n',s') j" + and n0: "n = 0" + and s_def: "s = (Add (Floor s') (C j))" + and jp: "0 \ j" and jn: "j \ n'" + from prems pns have H:"(Ifm ((x\real) # (bs\real list)) p' \ + Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \ + numbound0 s' \ isrlfm p'" by blast + hence nb: "numbound0 s'" by simp + from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by (simp add: numsub_nb) + let ?nxs = "CN 0 n' s'" + let ?l = "floor (?N s') + j" + from H + have "?I (?p (p',n',s') j) \ + (((?N ?nxs \ real ?l) \ (?N ?nxs < real (?l + 1))) \ (?N a = ?N ?nxs ))" + by (simp add: fp_def np algebra_simps numsub numadd numfloor) + also have "\ \ ((floor (?N ?nxs) = ?l) \ (?N a = ?N ?nxs ))" + using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp + moreover + have "\ \ (?N (Floor a) = ?N ((Add (Floor s') (C j))))" by simp + ultimately have "?I (?p (p',n',s') j) \ (?N (Floor a) = ?N ((Add (Floor s') (C j))))" + by blast + with s_def n0 p_def nb nf have ?ths by auto} + moreover + {fix p' n' s' j + assume pns: "(p', n', s') \ ?SS a" + and np: "n' < 0" + and p_def: "p = ?p (p',n',s') j" + and n0: "n = 0" + and s_def: "s = (Add (Floor s') (C j))" + and jp: "n' \ j" and jn: "j \ 0" + from prems pns have H:"(Ifm ((x\real) # (bs\real list)) p' \ + Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \ + numbound0 s' \ isrlfm p'" by blast + hence nb: "numbound0 s'" by simp + from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by (simp add: numneg_nb) + let ?nxs = "CN 0 n' s'" + let ?l = "floor (?N s') + j" + from H + have "?I (?p (p',n',s') j) \ + (((?N ?nxs \ real ?l) \ (?N ?nxs < real (?l + 1))) \ (?N a = ?N ?nxs ))" + by (simp add: np fp_def algebra_simps numneg numfloor numadd numsub) + also have "\ \ ((floor (?N ?nxs) = ?l) \ (?N a = ?N ?nxs ))" + using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp + moreover + have "\ \ (?N (Floor a) = ?N ((Add (Floor s') (C j))))" by simp + ultimately have "?I (?p (p',n',s') j) \ (?N (Floor a) = ?N ((Add (Floor s') (C j))))" + by blast + with s_def n0 p_def nb nf have ?ths by auto} + ultimately show ?ths by auto + qed +next + case (3 a b) then show ?case + apply auto + apply (erule_tac x = "(aa, aaa, ba)" in ballE) apply simp_all + apply (erule_tac x = "(ab, ac, baa)" in ballE) apply simp_all + done +qed (auto simp add: Let_def split_def algebra_simps conj_rl) + +lemma real_in_int_intervals: + assumes xb: "real m \ x \ x < real ((n::int) + 1)" + shows "\ j\ {m.. n}. real j \ x \ x < real (j+1)" (is "\ j\ ?N. ?P j") +by (rule bexI[where P="?P" and x="floor x" and A="?N"]) +(auto simp add: floor_less_eq[where x="x" and a="n+1", simplified] xb[simplified] floor_mono2[where x="real m" and y="x", OF conjunct1[OF xb], simplified floor_real_of_int[where n="m"]]) + +lemma rsplit0_complete: + assumes xp:"0 \ x" and x1:"x < 1" + shows "\ (p,n,s) \ set (rsplit0 t). Ifm (x#bs) p" (is "\ (p,n,s) \ ?SS t. ?I p") +proof(induct t rule: rsplit0.induct) + case (2 a b) + from prems have "\ (pa,na,sa) \ ?SS a. ?I pa" by auto + then obtain "pa" "na" "sa" where pa: "(pa,na,sa)\ ?SS a \ ?I pa" by blast + from prems have "\ (pb,nb,sb) \ ?SS b. ?I pb" by auto + then obtain "pb" "nb" "sb" where pb: "(pb,nb,sb)\ ?SS b \ ?I pb" by blast + from pa pb have th: "((pa,na,sa),(pb,nb,sb)) \ set[(x,y). x\rsplit0 a, y\rsplit0 b]" + by (auto) + let ?f="(\ ((p,n,t),(q,m,s)). (And p q, n+m, Add t s))" + from imageI[OF th, where f="?f"] have "?f ((pa,na,sa),(pb,nb,sb)) \ ?SS (Add a b)" + by (simp add: Let_def) + hence "(And pa pb, na +nb, Add sa sb) \ ?SS (Add a b)" by simp + moreover from pa pb have "?I (And pa pb)" by simp + ultimately show ?case by blast +next + case (5 a) + let ?p = "\ (p,n,s) j. fp p n s j" + let ?f = "(\ (p,n,s) j. (?p (p,n,s) j, (0::int),(Add (Floor s) (C j))))" + let ?J = "\ n. if n>0 then iupt (0,n) else iupt (n,0)" + let ?ff=" (\ (p,n,s). if n= 0 then [(p,0,Floor s)] else map (?f (p,n,s)) (?J n))" + have int_cases: "\ (i::int). i= 0 \ i < 0 \ i > 0" by arith + have U1: "(UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). set [(p,0,Floor s)]))" by auto + have U2': "\ (p,n,s) \ {(p,n,s). (p,n,s) \ ?SS a \ n>0}. ?ff (p,n,s) = map (?f(p,n,s)) (iupt(0,n))" + by auto + hence U2: "(UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). set (map (?f(p,n,s)) (iupt(0,n)))))" + proof- + fix M :: "('a\'b\'c) set" and f :: "('a\'b\'c) \ 'd list" and g + assume "\ (a,b,c) \ M. f (a,b,c) = g a b c" + thus "(UNION M (\ (a,b,c). set (f (a,b,c)))) = (UNION M (\ (a,b,c). set (g a b c)))" + by (auto simp add: split_def) + qed + have U3': "\ (p,n,s) \ {(p,n,s). (p,n,s) \ ?SS a \ n<0}. ?ff (p,n,s) = map (?f(p,n,s)) (iupt(n,0))" + by auto + hence U3: "(UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). set (map (?f(p,n,s)) (iupt(n,0)))))" + proof- + fix M :: "('a\'b\'c) set" and f :: "('a\'b\'c) \ 'd list" and g + assume "\ (a,b,c) \ M. f (a,b,c) = g a b c" + thus "(UNION M (\ (a,b,c). set (f (a,b,c)))) = (UNION M (\ (a,b,c). set (g a b c)))" + by (auto simp add: split_def) + qed + + have "?SS (Floor a) = UNION (?SS a) (\x. set (?ff x))" by (auto simp add: foldl_conv_concat) + also have "\ = UNION (?SS a) (\ (p,n,s). set (?ff (p,n,s)))" by auto + also have "\ = + ((UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). set (?ff (p,n,s)))) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). set (?ff (p,n,s)))) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). set (?ff (p,n,s)))))" + using int_cases[rule_format] by blast + also have "\ = + ((UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). set [(p,0,Floor s)])) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). set (map (?f(p,n,s)) (iupt(0,n))))) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). set (map (?f(p,n,s)) (iupt(n,0))))))" by (simp only: U1 U2 U3) + also have "\ = + ((UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). {(p,0,Floor s)})) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). (?f(p,n,s)) ` {n .. 0})))" + by (simp only: set_map iupt_set set.simps) + also have "\ = + ((UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). {(p,0,Floor s)})) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). {?f(p,n,s) j| j. j\ {0 .. n}})) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). {?f(p,n,s) j| j. j\ {n .. 0}})))" by blast + finally + have FS: "?SS (Floor a) = + ((UNION {(p,n,s). (p,n,s) \ ?SS a \ n=0} (\ (p,n,s). {(p,0,Floor s)})) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n>0} (\ (p,n,s). {?f(p,n,s) j| j. j\ {0 .. n}})) Un + (UNION {(p,n,s). (p,n,s) \ ?SS a \ n<0} (\ (p,n,s). {?f(p,n,s) j| j. j\ {n .. 0}})))" by blast + from prems have "\ (p,n,s) \ ?SS a. ?I p" by auto + then obtain "p" "n" "s" where pns: "(p,n,s) \ ?SS a \ ?I p" by blast + let ?N = "\ t. Inum (x#bs) t" + from rsplit0_cs[rule_format] pns have ans:"(?N a = ?N (CN 0 n s)) \ numbound0 s \ isrlfm p" + by auto + + have "n=0 \ n >0 \ n <0" by arith + moreover {assume "n=0" hence ?case using pns by (simp only: FS) auto } + moreover + { + assume np: "n > 0" + from real_of_int_floor_le[where r="?N s"] have "?N (Floor s) \ ?N s" by simp + also from mult_left_mono[OF xp] np have "?N s \ real n * x + ?N s" by simp + finally have "?N (Floor s) \ real n * x + ?N s" . + moreover + {from mult_strict_left_mono[OF x1] np + have "real n *x + ?N s < real n + ?N s" by simp + also from real_of_int_floor_add_one_gt[where r="?N s"] + have "\ < real n + ?N (Floor s) + 1" by simp + finally have "real n *x + ?N s < ?N (Floor s) + real (n+1)" by simp} + ultimately have "?N (Floor s) \ real n *x + ?N s\ real n *x + ?N s < ?N (Floor s) + real (n+1)" by simp + hence th: "0 \ real n *x + ?N s - ?N (Floor s) \ real n *x + ?N s - ?N (Floor s) < real (n+1)" by simp + from real_in_int_intervals th have "\ j\ {0 .. n}. real j \ real n *x + ?N s - ?N (Floor s)\ real n *x + ?N s - ?N (Floor s) < real (j+1)" by simp + + hence "\ j\ {0 .. n}. 0 \ real n *x + ?N s - ?N (Floor s) - real j \ real n *x + ?N s - ?N (Floor s) - real (j+1) < 0" + by(simp only: myl[rule_format, where b="real n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real n *x + ?N s - ?N (Floor s)"]) + hence "\ j\ {0.. n}. ?I (?p (p,n,s) j)" + using pns by (simp add: fp_def np algebra_simps numsub numadd) + then obtain "j" where j_def: "j\ {0 .. n} \ ?I (?p (p,n,s) j)" by blast + hence "\x \ {?p (p,n,s) j |j. 0\ j \ j \ n }. ?I x" by auto + hence ?case using pns + by (simp only: FS,simp add: bex_Un) + (rule disjI2, rule disjI1,rule exI [where x="p"], + rule exI [where x="n"],rule exI [where x="s"],simp_all add: np) + } + moreover + { assume nn: "n < 0" hence np: "-n >0" by simp + from real_of_int_floor_le[where r="?N s"] have "?N (Floor s) + 1 > ?N s" by simp + moreover from mult_left_mono_neg[OF xp] nn have "?N s \ real n * x + ?N s" by simp + ultimately have "?N (Floor s) + 1 > real n * x + ?N s" by arith + moreover + {from mult_strict_left_mono_neg[OF x1, where c="real n"] nn + have "real n *x + ?N s \ real n + ?N s" by simp + moreover from real_of_int_floor_le[where r="?N s"] have "real n + ?N s \ real n + ?N (Floor s)" by simp + ultimately have "real n *x + ?N s \ ?N (Floor s) + real n" + by (simp only: algebra_simps)} + ultimately have "?N (Floor s) + real n \ real n *x + ?N s\ real n *x + ?N s < ?N (Floor s) + real (1::int)" by simp + hence th: "real n \ real n *x + ?N s - ?N (Floor s) \ real n *x + ?N s - ?N (Floor s) < real (1::int)" by simp + have th1: "\ (a::real). (- a > 0) = (a < 0)" by auto + have th2: "\ (a::real). (0 \ - a) = (a \ 0)" by auto + from real_in_int_intervals th have "\ j\ {n .. 0}. real j \ real n *x + ?N s - ?N (Floor s)\ real n *x + ?N s - ?N (Floor s) < real (j+1)" by simp + + hence "\ j\ {n .. 0}. 0 \ real n *x + ?N s - ?N (Floor s) - real j \ real n *x + ?N s - ?N (Floor s) - real (j+1) < 0" + by(simp only: myl[rule_format, where b="real n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real n *x + ?N s - ?N (Floor s)"]) + hence "\ j\ {n .. 0}. 0 \ - (real n *x + ?N s - ?N (Floor s) - real j) \ - (real n *x + ?N s - ?N (Floor s) - real (j+1)) > 0" by (simp only: th1[rule_format] th2[rule_format]) + hence "\ j\ {n.. 0}. ?I (?p (p,n,s) j)" + using pns by (simp add: fp_def nn diff_def add_ac mult_ac numfloor numadd numneg + del: diff_less_0_iff_less diff_le_0_iff_le) + then obtain "j" where j_def: "j\ {n .. 0} \ ?I (?p (p,n,s) j)" by blast + hence "\x \ {?p (p,n,s) j |j. n\ j \ j \ 0 }. ?I x" by auto + hence ?case using pns + by (simp only: FS,simp add: bex_Un) + (rule disjI2, rule disjI2,rule exI [where x="p"], + rule exI [where x="n"],rule exI [where x="s"],simp_all add: nn) + } + ultimately show ?case by blast +qed (auto simp add: Let_def split_def) + + (* Linearize a formula where Bound 0 ranges over [0,1) *) + +constdefs rsplit :: "(int \ num \ fm) \ num \ fm" + "rsplit f a \ foldr disj (map (\ (\, n, s). conj \ (f n s)) (rsplit0 a)) F" + +lemma foldr_disj_map: "Ifm bs (foldr disj (map f xs) F) = (\ x \ set xs. Ifm bs (f x))" +by(induct xs, simp_all) + +lemma foldr_conj_map: "Ifm bs (foldr conj (map f xs) T) = (\ x \ set xs. Ifm bs (f x))" +by(induct xs, simp_all) + +lemma foldr_disj_map_rlfm: + assumes lf: "\ n s. numbound0 s \ isrlfm (f n s)" + and \: "\ (\,n,s) \ set xs. numbound0 s \ isrlfm \" + shows "isrlfm (foldr disj (map (\ (\, n, s). conj \ (f n s)) xs) F)" +using lf \ by (induct xs, auto) + +lemma rsplit_ex: "Ifm bs (rsplit f a) = (\ (\,n,s) \ set (rsplit0 a). Ifm bs (conj \ (f n s)))" +using foldr_disj_map[where xs="rsplit0 a"] rsplit_def by (simp add: split_def) + +lemma rsplit_l: assumes lf: "\ n s. numbound0 s \ isrlfm (f n s)" + shows "isrlfm (rsplit f a)" +proof- + from rsplit0_cs[where t="a"] have th: "\ (\,n,s) \ set (rsplit0 a). numbound0 s \ isrlfm \" by blast + from foldr_disj_map_rlfm[OF lf th] rsplit_def show ?thesis by simp +qed + +lemma rsplit: + assumes xp: "x \ 0" and x1: "x < 1" + and f: "\ a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \ numbound0 s \ (Ifm (x#bs) (f n s) = Ifm (x#bs) (g a))" + shows "Ifm (x#bs) (rsplit f a) = Ifm (x#bs) (g a)" +proof(auto) + let ?I = "\x p. Ifm (x#bs) p" + let ?N = "\ x t. Inum (x#bs) t" + assume "?I x (rsplit f a)" + hence "\ (\,n,s) \ set (rsplit0 a). ?I x (And \ (f n s))" using rsplit_ex by simp + then obtain "\" "n" "s" where fnsS:"(\,n,s) \ set (rsplit0 a)" and "?I x (And \ (f n s))" by blast + hence \: "?I x \" and fns: "?I x (f n s)" by auto + from rsplit0_cs[where t="a" and bs="bs" and x="x", rule_format, OF fnsS] \ + have th: "(?N x a = ?N x (CN 0 n s)) \ numbound0 s" by auto + from f[rule_format, OF th] fns show "?I x (g a)" by simp +next + let ?I = "\x p. Ifm (x#bs) p" + let ?N = "\ x t. Inum (x#bs) t" + assume ga: "?I x (g a)" + from rsplit0_complete[OF xp x1, where bs="bs" and t="a"] + obtain "\" "n" "s" where fnsS:"(\,n,s) \ set (rsplit0 a)" and fx: "?I x \" by blast + from rsplit0_cs[where t="a" and x="x" and bs="bs"] fnsS fx + have ans: "?N x a = ?N x (CN 0 n s)" and nb: "numbound0 s" by auto + with ga f have "?I x (f n s)" by auto + with rsplit_ex fnsS fx show "?I x (rsplit f a)" by auto +qed + +definition lt :: "int \ num \ fm" where + lt_def: "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) + else (Gt (CN 0 (-c) (Neg t))))" + +definition le :: "int \ num \ fm" where + le_def: "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) + else (Ge (CN 0 (-c) (Neg t))))" + +definition gt :: "int \ num \ fm" where + gt_def: "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) + else (Lt (CN 0 (-c) (Neg t))))" + +definition ge :: "int \ num \ fm" where + ge_def: "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) + else (Le (CN 0 (-c) (Neg t))))" + +definition eq :: "int \ num \ fm" where + eq_def: "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) + else (Eq (CN 0 (-c) (Neg t))))" + +definition neq :: "int \ num \ fm" where + neq_def: "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) + else (NEq (CN 0 (-c) (Neg t))))" + +lemma lt_mono: "\ a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \ numbound0 s \ Ifm (x#bs) (lt n s) = Ifm (x#bs) (Lt a)" + (is "\ a n s . ?N a = ?N (CN 0 n s) \ _\ ?I (lt n s) = ?I (Lt a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" + show "?I (lt n s) = ?I (Lt a)" using H by (cases "n=0", (simp add: lt_def)) + (cases "n > 0", simp_all add: lt_def algebra_simps myless[rule_format, where b="0"]) +qed + +lemma lt_l: "isrlfm (rsplit lt a)" + by (rule rsplit_l[where f="lt" and a="a"], auto simp add: lt_def, + case_tac s, simp_all, case_tac "nat", simp_all) + +lemma le_mono: "\ a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \ numbound0 s \ Ifm (x#bs) (le n s) = Ifm (x#bs) (Le a)" (is "\ a n s. ?N a = ?N (CN 0 n s) \ _ \ ?I (le n s) = ?I (Le a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" + show "?I (le n s) = ?I (Le a)" using H by (cases "n=0", (simp add: le_def)) + (cases "n > 0", simp_all add: le_def algebra_simps myl[rule_format, where b="0"]) +qed + +lemma le_l: "isrlfm (rsplit le a)" + by (rule rsplit_l[where f="le" and a="a"], auto simp add: le_def) +(case_tac s, simp_all, case_tac "nat",simp_all) + +lemma gt_mono: "\ a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \ numbound0 s \ Ifm (x#bs) (gt n s) = Ifm (x#bs) (Gt a)" (is "\ a n s. ?N a = ?N (CN 0 n s) \ _ \ ?I (gt n s) = ?I (Gt a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" + show "?I (gt n s) = ?I (Gt a)" using H by (cases "n=0", (simp add: gt_def)) + (cases "n > 0", simp_all add: gt_def algebra_simps myless[rule_format, where b="0"]) +qed +lemma gt_l: "isrlfm (rsplit gt a)" + by (rule rsplit_l[where f="gt" and a="a"], auto simp add: gt_def) +(case_tac s, simp_all, case_tac "nat", simp_all) + +lemma ge_mono: "\ a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \ numbound0 s \ Ifm (x#bs) (ge n s) = Ifm (x#bs) (Ge a)" (is "\ a n s . ?N a = ?N (CN 0 n s) \ _ \ ?I (ge n s) = ?I (Ge a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" + show "?I (ge n s) = ?I (Ge a)" using H by (cases "n=0", (simp add: ge_def)) + (cases "n > 0", simp_all add: ge_def algebra_simps myl[rule_format, where b="0"]) +qed +lemma ge_l: "isrlfm (rsplit ge a)" + by (rule rsplit_l[where f="ge" and a="a"], auto simp add: ge_def) +(case_tac s, simp_all, case_tac "nat", simp_all) + +lemma eq_mono: "\ a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \ numbound0 s \ Ifm (x#bs) (eq n s) = Ifm (x#bs) (Eq a)" (is "\ a n s. ?N a = ?N (CN 0 n s) \ _ \ ?I (eq n s) = ?I (Eq a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" + show "?I (eq n s) = ?I (Eq a)" using H by (auto simp add: eq_def algebra_simps) +qed +lemma eq_l: "isrlfm (rsplit eq a)" + by (rule rsplit_l[where f="eq" and a="a"], auto simp add: eq_def) +(case_tac s, simp_all, case_tac"nat", simp_all) + +lemma neq_mono: "\ a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \ numbound0 s \ Ifm (x#bs) (neq n s) = Ifm (x#bs) (NEq a)" (is "\ a n s. ?N a = ?N (CN 0 n s) \ _ \ ?I (neq n s) = ?I (NEq a)") +proof(clarify) + fix a n s bs + assume H: "?N a = ?N (CN 0 n s)" + show "?I (neq n s) = ?I (NEq a)" using H by (auto simp add: neq_def algebra_simps) +qed + +lemma neq_l: "isrlfm (rsplit neq a)" + by (rule rsplit_l[where f="neq" and a="a"], auto simp add: neq_def) +(case_tac s, simp_all, case_tac"nat", simp_all) + +lemma small_le: + assumes u0:"0 \ u" and u1: "u < 1" + shows "(-u \ real (n::int)) = (0 \ n)" +using u0 u1 by auto + +lemma small_lt: + assumes u0:"0 \ u" and u1: "u < 1" + shows "(real (n::int) < real (m::int) - u) = (n < m)" +using u0 u1 by auto + +lemma rdvd01_cs: + assumes up: "u \ 0" and u1: "u<1" and np: "real n > 0" + shows "(real (i::int) rdvd real (n::int) * u - s) = (\ j\ {0 .. n - 1}. real n * u = s - real (floor s) + real j \ real i rdvd real (j - floor s))" (is "?lhs = ?rhs") +proof- + let ?ss = "s - real (floor s)" + from real_of_int_floor_add_one_gt[where r="s", simplified myless[rule_format,where a="s"]] + real_of_int_floor_le[where r="s"] have ss0:"?ss \ 0" and ss1:"?ss < 1" + by (auto simp add: myl[rule_format, where b="s", symmetric] myless[rule_format, where a="?ss"]) + from np have n0: "real n \ 0" by simp + from mult_left_mono[OF up n0] mult_strict_left_mono[OF u1 np] + have nu0:"real n * u - s \ -s" and nun:"real n * u -s < real n - s" by auto + from int_rdvd_real[where i="i" and x="real (n::int) * u - s"] + have "real i rdvd real n * u - s = + (i dvd floor (real n * u -s) \ (real (floor (real n * u - s)) = real n * u - s ))" + (is "_ = (?DE)" is "_ = (?D \ ?E)") by simp + also have "\ = (?DE \ real(floor (real n * u - s) + floor s)\ -?ss + \ real(floor (real n * u - s) + floor s)< real n - ?ss)" (is "_=(?DE \real ?a \ _ \ real ?a < _)") + using nu0 nun by auto + also have "\ = (?DE \ ?a \ 0 \ ?a < n)" by(simp only: small_le[OF ss0 ss1] small_lt[OF ss0 ss1]) + also have "\ = (?DE \ (\ j\ {0 .. (n - 1)}. ?a = j))" by simp + also have "\ = (?DE \ (\ j\ {0 .. (n - 1)}. real (\real n * u - s\) = real j - real \s\ ))" + by (simp only: algebra_simps real_of_int_diff[symmetric] real_of_int_inject del: real_of_int_diff) + also have "\ = ((\ j\ {0 .. (n - 1)}. real n * u - s = real j - real \s\ \ real i rdvd real n * u - s))" using int_rdvd_iff[where i="i" and t="\real n * u - s\"] + by (auto cong: conj_cong) + also have "\ = ?rhs" by(simp cong: conj_cong) (simp add: algebra_simps ) + finally show ?thesis . +qed + +definition + DVDJ:: "int \ int \ num \ fm" +where + DVDJ_def: "DVDJ i n s = (foldr disj (map (\ j. conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))) (iupt(0,n - 1))) F)" + +definition + NDVDJ:: "int \ int \ num \ fm" +where + NDVDJ_def: "NDVDJ i n s = (foldr conj (map (\ j. disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor (Neg s))))) (iupt(0,n - 1))) T)" + +lemma DVDJ_DVD: + assumes xp:"x\ 0" and x1: "x < 1" and np:"real n > 0" + shows "Ifm (x#bs) (DVDJ i n s) = Ifm (x#bs) (Dvd i (CN 0 n s))" +proof- + let ?f = "\ j. conj (Eq(CN 0 n (Add s (Sub(Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))" + let ?s= "Inum (x#bs) s" + from foldr_disj_map[where xs="iupt(0,n - 1)" and bs="x#bs" and f="?f"] + have "Ifm (x#bs) (DVDJ i n s) = (\ j\ {0 .. (n - 1)}. Ifm (x#bs) (?f j))" + by (simp add: iupt_set np DVDJ_def del: iupt.simps) + also have "\ = (\ j\ {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \ real i rdvd real (j - floor (- ?s)))" by (simp add: algebra_simps diff_def[symmetric]) + also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] + have "\ = (real i rdvd real n * x - (-?s))" by simp + finally show ?thesis by simp +qed + +lemma NDVDJ_NDVD: + assumes xp:"x\ 0" and x1: "x < 1" and np:"real n > 0" + shows "Ifm (x#bs) (NDVDJ i n s) = Ifm (x#bs) (NDvd i (CN 0 n s))" +proof- + let ?f = "\ j. disj(NEq(CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor(Neg s))))" + let ?s= "Inum (x#bs) s" + from foldr_conj_map[where xs="iupt(0,n - 1)" and bs="x#bs" and f="?f"] + have "Ifm (x#bs) (NDVDJ i n s) = (\ j\ {0 .. (n - 1)}. Ifm (x#bs) (?f j))" + by (simp add: iupt_set np NDVDJ_def del: iupt.simps) + also have "\ = (\ (\ j\ {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \ real i rdvd real (j - floor (- ?s))))" by (simp add: algebra_simps diff_def[symmetric]) + also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] + have "\ = (\ (real i rdvd real n * x - (-?s)))" by simp + finally show ?thesis by simp +qed + +lemma foldr_disj_map_rlfm2: + assumes lf: "\ n . isrlfm (f n)" + shows "isrlfm (foldr disj (map f xs) F)" +using lf by (induct xs, auto) +lemma foldr_And_map_rlfm2: + assumes lf: "\ n . isrlfm (f n)" + shows "isrlfm (foldr conj (map f xs) T)" +using lf by (induct xs, auto) + +lemma DVDJ_l: assumes ip: "i >0" and np: "n>0" and nb: "numbound0 s" + shows "isrlfm (DVDJ i n s)" +proof- + let ?f="\j. conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) + (Dvd i (Sub (C j) (Floor (Neg s))))" + have th: "\ j. isrlfm (?f j)" using nb np by auto + from DVDJ_def foldr_disj_map_rlfm2[OF th] show ?thesis by simp +qed + +lemma NDVDJ_l: assumes ip: "i >0" and np: "n>0" and nb: "numbound0 s" + shows "isrlfm (NDVDJ i n s)" +proof- + let ?f="\j. disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) + (NDvd i (Sub (C j) (Floor (Neg s))))" + have th: "\ j. isrlfm (?f j)" using nb np by auto + from NDVDJ_def foldr_And_map_rlfm2[OF th] show ?thesis by auto +qed + +definition DVD :: "int \ int \ num \ fm" where + DVD_def: "DVD i c t = + (if i=0 then eq c t else + if c = 0 then (Dvd i t) else if c >0 then DVDJ (abs i) c t else DVDJ (abs i) (-c) (Neg t))" + +definition NDVD :: "int \ int \ num \ fm" where + "NDVD i c t = + (if i=0 then neq c t else + if c = 0 then (NDvd i t) else if c >0 then NDVDJ (abs i) c t else NDVDJ (abs i) (-c) (Neg t))" + +lemma DVD_mono: + assumes xp: "0\ x" and x1: "x < 1" + shows "\ a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \ numbound0 s \ Ifm (x#bs) (DVD i n s) = Ifm (x#bs) (Dvd i a)" + (is "\ a n s. ?N a = ?N (CN 0 n s) \ _ \ ?I (DVD i n s) = ?I (Dvd i a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" and nb: "numbound0 s" + let ?th = "?I (DVD i n s) = ?I (Dvd i a)" + have "i=0 \ (i\0 \ n=0) \ (i\0 \ n < 0) \ (i\0 \ n > 0)" by arith + moreover {assume iz: "i=0" hence ?th using eq_mono[rule_format, OF conjI[OF H nb]] + by (simp add: DVD_def rdvd_left_0_eq)} + moreover {assume inz: "i\0" and "n=0" hence ?th by (simp add: H DVD_def) } + moreover {assume inz: "i\0" and "n<0" hence ?th + by (simp add: DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1 + rdvd_minus[where d="i" and t="real n * x + Inum (x # bs) s"]) } + moreover {assume inz: "i\0" and "n>0" hence ?th by (simp add:DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1)} + ultimately show ?th by blast +qed + +lemma NDVD_mono: assumes xp: "0\ x" and x1: "x < 1" + shows "\ a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \ numbound0 s \ Ifm (x#bs) (NDVD i n s) = Ifm (x#bs) (NDvd i a)" + (is "\ a n s. ?N a = ?N (CN 0 n s) \ _ \ ?I (NDVD i n s) = ?I (NDvd i a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" and nb: "numbound0 s" + let ?th = "?I (NDVD i n s) = ?I (NDvd i a)" + have "i=0 \ (i\0 \ n=0) \ (i\0 \ n < 0) \ (i\0 \ n > 0)" by arith + moreover {assume iz: "i=0" hence ?th using neq_mono[rule_format, OF conjI[OF H nb]] + by (simp add: NDVD_def rdvd_left_0_eq)} + moreover {assume inz: "i\0" and "n=0" hence ?th by (simp add: H NDVD_def) } + moreover {assume inz: "i\0" and "n<0" hence ?th + by (simp add: NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1 + rdvd_minus[where d="i" and t="real n * x + Inum (x # bs) s"]) } + moreover {assume inz: "i\0" and "n>0" hence ?th + by (simp add:NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1)} + ultimately show ?th by blast +qed + +lemma DVD_l: "isrlfm (rsplit (DVD i) a)" + by (rule rsplit_l[where f="DVD i" and a="a"], auto simp add: DVD_def eq_def DVDJ_l) +(case_tac s, simp_all, case_tac "nat", simp_all) + +lemma NDVD_l: "isrlfm (rsplit (NDVD i) a)" + by (rule rsplit_l[where f="NDVD i" and a="a"], auto simp add: NDVD_def neq_def NDVDJ_l) +(case_tac s, simp_all, case_tac "nat", simp_all) + +consts rlfm :: "fm \ fm" +recdef rlfm "measure fmsize" + "rlfm (And p q) = conj (rlfm p) (rlfm q)" + "rlfm (Or p q) = disj (rlfm p) (rlfm q)" + "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)" + "rlfm (Iff p q) = disj (conj(rlfm p) (rlfm q)) (conj(rlfm (NOT p)) (rlfm (NOT q)))" + "rlfm (Lt a) = rsplit lt a" + "rlfm (Le a) = rsplit le a" + "rlfm (Gt a) = rsplit gt a" + "rlfm (Ge a) = rsplit ge a" + "rlfm (Eq a) = rsplit eq a" + "rlfm (NEq a) = rsplit neq a" + "rlfm (Dvd i a) = rsplit (\ t. DVD i t) a" + "rlfm (NDvd i a) = rsplit (\ t. NDVD i t) a" + "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))" + "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))" + "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))" + "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))" + "rlfm (NOT (NOT p)) = rlfm p" + "rlfm (NOT T) = F" + "rlfm (NOT F) = T" + "rlfm (NOT (Lt a)) = simpfm (rlfm (Ge a))" + "rlfm (NOT (Le a)) = simpfm (rlfm (Gt a))" + "rlfm (NOT (Gt a)) = simpfm (rlfm (Le a))" + "rlfm (NOT (Ge a)) = simpfm (rlfm (Lt a))" + "rlfm (NOT (Eq a)) = simpfm (rlfm (NEq a))" + "rlfm (NOT (NEq a)) = simpfm (rlfm (Eq a))" + "rlfm (NOT (Dvd i a)) = simpfm (rlfm (NDvd i a))" + "rlfm (NOT (NDvd i a)) = simpfm (rlfm (Dvd i a))" + "rlfm p = p" (hints simp add: fmsize_pos) + +lemma bound0at_l : "\isatom p ; bound0 p\ \ isrlfm p" + by (induct p rule: isrlfm.induct, auto) +lemma zgcd_le1: assumes ip: "0 < i" shows "zgcd i j \ i" +proof- + from zgcd_zdvd1 have th: "zgcd i j dvd i" by blast + from zdvd_imp_le[OF th ip] show ?thesis . +qed + + +lemma simpfm_rl: "isrlfm p \ isrlfm (simpfm p)" +proof (induct p) + case (Lt a) + hence "bound0 (Lt a) \ (\ c e. a = CN 0 c e \ c > 0 \ numbound0 e)" + by (cases a,simp_all, case_tac "nat", simp_all) + moreover + {assume "bound0 (Lt a)" hence bn:"bound0 (simpfm (Lt a))" + using simpfm_bound0 by blast + have "isatom (simpfm (Lt a))" by (cases "simpnum a", auto simp add: Let_def) + with bn bound0at_l have ?case by blast} + moreover + {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e" + { + assume cn1:"numgcd (CN 0 c (simpnum e)) \ 1" and cnz:"numgcd (CN 0 c (simpnum e)) \ 0" + with numgcd_pos[where t="CN 0 c (simpnum e)"] + have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp + from prems have th:"numgcd (CN 0 c (simpnum e)) \ c" + by (simp add: numgcd_def zgcd_le1) + from prems have th': "c\0" by auto + from prems have cp: "c \ 0" by simp + from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']] + have "0 < c div numgcd (CN 0 c (simpnum e))" by simp + } + with prems have ?case + by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)} + ultimately show ?case by blast +next + case (Le a) + hence "bound0 (Le a) \ (\ c e. a = CN 0 c e \ c > 0 \ numbound0 e)" + by (cases a,simp_all, case_tac "nat", simp_all) + moreover + {assume "bound0 (Le a)" hence bn:"bound0 (simpfm (Le a))" + using simpfm_bound0 by blast + have "isatom (simpfm (Le a))" by (cases "simpnum a", auto simp add: Let_def) + with bn bound0at_l have ?case by blast} + moreover + {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e" + { + assume cn1:"numgcd (CN 0 c (simpnum e)) \ 1" and cnz:"numgcd (CN 0 c (simpnum e)) \ 0" + with numgcd_pos[where t="CN 0 c (simpnum e)"] + have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp + from prems have th:"numgcd (CN 0 c (simpnum e)) \ c" + by (simp add: numgcd_def zgcd_le1) + from prems have th': "c\0" by auto + from prems have cp: "c \ 0" by simp + from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']] + have "0 < c div numgcd (CN 0 c (simpnum e))" by simp + } + with prems have ?case + by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)} + ultimately show ?case by blast +next + case (Gt a) + hence "bound0 (Gt a) \ (\ c e. a = CN 0 c e \ c > 0 \ numbound0 e)" + by (cases a,simp_all, case_tac "nat", simp_all) + moreover + {assume "bound0 (Gt a)" hence bn:"bound0 (simpfm (Gt a))" + using simpfm_bound0 by blast + have "isatom (simpfm (Gt a))" by (cases "simpnum a", auto simp add: Let_def) + with bn bound0at_l have ?case by blast} + moreover + {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e" + { + assume cn1:"numgcd (CN 0 c (simpnum e)) \ 1" and cnz:"numgcd (CN 0 c (simpnum e)) \ 0" + with numgcd_pos[where t="CN 0 c (simpnum e)"] + have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp + from prems have th:"numgcd (CN 0 c (simpnum e)) \ c" + by (simp add: numgcd_def zgcd_le1) + from prems have th': "c\0" by auto + from prems have cp: "c \ 0" by simp + from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']] + have "0 < c div numgcd (CN 0 c (simpnum e))" by simp + } + with prems have ?case + by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)} + ultimately show ?case by blast +next + case (Ge a) + hence "bound0 (Ge a) \ (\ c e. a = CN 0 c e \ c > 0 \ numbound0 e)" + by (cases a,simp_all, case_tac "nat", simp_all) + moreover + {assume "bound0 (Ge a)" hence bn:"bound0 (simpfm (Ge a))" + using simpfm_bound0 by blast + have "isatom (simpfm (Ge a))" by (cases "simpnum a", auto simp add: Let_def) + with bn bound0at_l have ?case by blast} + moreover + {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e" + { + assume cn1:"numgcd (CN 0 c (simpnum e)) \ 1" and cnz:"numgcd (CN 0 c (simpnum e)) \ 0" + with numgcd_pos[where t="CN 0 c (simpnum e)"] + have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp + from prems have th:"numgcd (CN 0 c (simpnum e)) \ c" + by (simp add: numgcd_def zgcd_le1) + from prems have th': "c\0" by auto + from prems have cp: "c \ 0" by simp + from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']] + have "0 < c div numgcd (CN 0 c (simpnum e))" by simp + } + with prems have ?case + by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)} + ultimately show ?case by blast +next + case (Eq a) + hence "bound0 (Eq a) \ (\ c e. a = CN 0 c e \ c > 0 \ numbound0 e)" + by (cases a,simp_all, case_tac "nat", simp_all) + moreover + {assume "bound0 (Eq a)" hence bn:"bound0 (simpfm (Eq a))" + using simpfm_bound0 by blast + have "isatom (simpfm (Eq a))" by (cases "simpnum a", auto simp add: Let_def) + with bn bound0at_l have ?case by blast} + moreover + {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e" + { + assume cn1:"numgcd (CN 0 c (simpnum e)) \ 1" and cnz:"numgcd (CN 0 c (simpnum e)) \ 0" + with numgcd_pos[where t="CN 0 c (simpnum e)"] + have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp + from prems have th:"numgcd (CN 0 c (simpnum e)) \ c" + by (simp add: numgcd_def zgcd_le1) + from prems have th': "c\0" by auto + from prems have cp: "c \ 0" by simp + from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']] + have "0 < c div numgcd (CN 0 c (simpnum e))" by simp + } + with prems have ?case + by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)} + ultimately show ?case by blast +next + case (NEq a) + hence "bound0 (NEq a) \ (\ c e. a = CN 0 c e \ c > 0 \ numbound0 e)" + by (cases a,simp_all, case_tac "nat", simp_all) + moreover + {assume "bound0 (NEq a)" hence bn:"bound0 (simpfm (NEq a))" + using simpfm_bound0 by blast + have "isatom (simpfm (NEq a))" by (cases "simpnum a", auto simp add: Let_def) + with bn bound0at_l have ?case by blast} + moreover + {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e" + { + assume cn1:"numgcd (CN 0 c (simpnum e)) \ 1" and cnz:"numgcd (CN 0 c (simpnum e)) \ 0" + with numgcd_pos[where t="CN 0 c (simpnum e)"] + have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp + from prems have th:"numgcd (CN 0 c (simpnum e)) \ c" + by (simp add: numgcd_def zgcd_le1) + from prems have th': "c\0" by auto + from prems have cp: "c \ 0" by simp + from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']] + have "0 < c div numgcd (CN 0 c (simpnum e))" by simp + } + with prems have ?case + by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)} + ultimately show ?case by blast +next + case (Dvd i a) hence "bound0 (Dvd i a)" by auto hence bn:"bound0 (simpfm (Dvd i a))" + using simpfm_bound0 by blast + have "isatom (simpfm (Dvd i a))" by (cases "simpnum a", auto simp add: Let_def split_def) + with bn bound0at_l show ?case by blast +next + case (NDvd i a) hence "bound0 (NDvd i a)" by auto hence bn:"bound0 (simpfm (NDvd i a))" + using simpfm_bound0 by blast + have "isatom (simpfm (NDvd i a))" by (cases "simpnum a", auto simp add: Let_def split_def) + with bn bound0at_l show ?case by blast +qed(auto simp add: conj_def imp_def disj_def iff_def Let_def simpfm_bound0 numadd_nb numneg_nb) + +lemma rlfm_I: + assumes qfp: "qfree p" + and xp: "0 \ x" and x1: "x < 1" + shows "(Ifm (x#bs) (rlfm p) = Ifm (x# bs) p) \ isrlfm (rlfm p)" + using qfp +by (induct p rule: rlfm.induct) +(auto simp add: rsplit[OF xp x1 lt_mono] lt_l rsplit[OF xp x1 le_mono] le_l rsplit[OF xp x1 gt_mono] gt_l + rsplit[OF xp x1 ge_mono] ge_l rsplit[OF xp x1 eq_mono] eq_l rsplit[OF xp x1 neq_mono] neq_l + rsplit[OF xp x1 DVD_mono[OF xp x1]] DVD_l rsplit[OF xp x1 NDVD_mono[OF xp x1]] NDVD_l simpfm_rl) +lemma rlfm_l: + assumes qfp: "qfree p" + shows "isrlfm (rlfm p)" + using qfp lt_l gt_l ge_l le_l eq_l neq_l DVD_l NDVD_l +by (induct p rule: rlfm.induct,auto simp add: simpfm_rl) + + (* Operations needed for Ferrante and Rackoff *) +lemma rminusinf_inf: + assumes lp: "isrlfm p" + shows "\ z. \ x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\ z. \ x. ?P z x p") +using lp +proof (induct p rule: minusinf.induct) + case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto +next + case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto +next + case (3 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + hence "real c * x + ?e \ 0" by simp + with xz have "?P ?z x (Eq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp + thus ?case by blast +next + case (4 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + hence "real c * x + ?e \ 0" by simp + with xz have "?P ?z x (NEq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp + thus ?case by blast +next + case (5 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Lt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp + thus ?case by blast +next + case (6 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Le (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x < ?z. ?P ?z x (Le (CN 0 c e))" by simp + thus ?case by blast +next + case (7 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Gt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp + thus ?case by blast +next + case (8 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Ge (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp + thus ?case by blast +qed simp_all + +lemma rplusinf_inf: + assumes lp: "isrlfm p" + shows "\ z. \ x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\ z. \ x. ?P z x p") +using lp +proof (induct p rule: isrlfm.induct) + case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto +next + case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto +next + case (3 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + hence "real c * x + ?e \ 0" by simp + with xz have "?P ?z x (Eq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp + thus ?case by blast +next + case (4 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + hence "real c * x + ?e \ 0" by simp + with xz have "?P ?z x (NEq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp + thus ?case by blast +next + case (5 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Lt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp + thus ?case by blast +next + case (6 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Le (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x > ?z. ?P ?z x (Le (CN 0 c e))" by simp + thus ?case by blast +next + case (7 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Gt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp + thus ?case by blast +next + case (8 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + fix a + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Ge (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\ x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp + thus ?case by blast +qed simp_all + +lemma rminusinf_bound0: + assumes lp: "isrlfm p" + shows "bound0 (minusinf p)" + using lp + by (induct p rule: minusinf.induct) simp_all + +lemma rplusinf_bound0: + assumes lp: "isrlfm p" + shows "bound0 (plusinf p)" + using lp + by (induct p rule: plusinf.induct) simp_all + +lemma rminusinf_ex: + assumes lp: "isrlfm p" + and ex: "Ifm (a#bs) (minusinf p)" + shows "\ x. Ifm (x#bs) p" +proof- + from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex + have th: "\ x. Ifm (x#bs) (minusinf p)" by auto + from rminusinf_inf[OF lp, where bs="bs"] + obtain z where z_def: "\x x. Ifm (x#bs) p" +proof- + from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex + have th: "\ x. Ifm (x#bs) (plusinf p)" by auto + from rplusinf_inf[OF lp, where bs="bs"] + obtain z where z_def: "\x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast + from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp + moreover have "z + 1 > z" by simp + ultimately show ?thesis using z_def by auto +qed + +consts + \:: "fm \ (num \ int) list" + \ :: "fm \ (num \ int) \ fm " +recdef \ "measure size" + "\ (And p q) = (\ p @ \ q)" + "\ (Or p q) = (\ p @ \ q)" + "\ (Eq (CN 0 c e)) = [(Neg e,c)]" + "\ (NEq (CN 0 c e)) = [(Neg e,c)]" + "\ (Lt (CN 0 c e)) = [(Neg e,c)]" + "\ (Le (CN 0 c e)) = [(Neg e,c)]" + "\ (Gt (CN 0 c e)) = [(Neg e,c)]" + "\ (Ge (CN 0 c e)) = [(Neg e,c)]" + "\ p = []" + +recdef \ "measure size" + "\ (And p q) = (\ (t,n). And (\ p (t,n)) (\ q (t,n)))" + "\ (Or p q) = (\ (t,n). Or (\ p (t,n)) (\ q (t,n)))" + "\ (Eq (CN 0 c e)) = (\ (t,n). Eq (Add (Mul c t) (Mul n e)))" + "\ (NEq (CN 0 c e)) = (\ (t,n). NEq (Add (Mul c t) (Mul n e)))" + "\ (Lt (CN 0 c e)) = (\ (t,n). Lt (Add (Mul c t) (Mul n e)))" + "\ (Le (CN 0 c e)) = (\ (t,n). Le (Add (Mul c t) (Mul n e)))" + "\ (Gt (CN 0 c e)) = (\ (t,n). Gt (Add (Mul c t) (Mul n e)))" + "\ (Ge (CN 0 c e)) = (\ (t,n). Ge (Add (Mul c t) (Mul n e)))" + "\ p = (\ (t,n). p)" + +lemma \_I: assumes lp: "isrlfm p" + and np: "real n > 0" and nbt: "numbound0 t" + shows "(Ifm (x#bs) (\ p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \ bound0 (\ p (t,n))" (is "(?I x (\ p (t,n)) = ?I ?u p) \ ?B p" is "(_ = ?I (?t/?n) p) \ _" is "(_ = ?I (?N x t /_) p) \ _") + using lp +proof(induct p rule: \.induct) + case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\ = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)" + by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: algebra_simps) + also have "\ = (real c *?t + ?n* (?N x e) < 0)" + using np by simp + finally show ?case using nbt nb by (simp add: algebra_simps) +next + case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \ 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\ = (?n*(real c *(?t/?n)) + ?n*(?N x e) \ 0)" + by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: algebra_simps) + also have "\ = (real c *?t + ?n* (?N x e) \ 0)" + using np by simp + finally show ?case using nbt nb by (simp add: algebra_simps) +next + case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\ = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)" + by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: algebra_simps) + also have "\ = (real c *?t + ?n* (?N x e) > 0)" + using np by simp + finally show ?case using nbt nb by (simp add: algebra_simps) +next + case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \ 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\ = (?n*(real c *(?t/?n)) + ?n*(?N x e) \ 0)" + by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: algebra_simps) + also have "\ = (real c *?t + ?n* (?N x e) \ 0)" + using np by simp + finally show ?case using nbt nb by (simp add: algebra_simps) +next + case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + from np have np: "real n \ 0" by simp + have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\ = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)" + by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: algebra_simps) + also have "\ = (real c *?t + ?n* (?N x e) = 0)" + using np by simp + finally show ?case using nbt nb by (simp add: algebra_simps) +next + case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + from np have np: "real n \ 0" by simp + have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \ 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\ = (?n*(real c *(?t/?n)) + ?n*(?N x e) \ 0)" + by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: algebra_simps) + also have "\ = (real c *?t + ?n* (?N x e) \ 0)" + using np by simp + finally show ?case using nbt nb by (simp add: algebra_simps) +qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2) + +lemma \_l: + assumes lp: "isrlfm p" + shows "\ (t,k) \ set (\ p). numbound0 t \ k >0" +using lp +by(induct p rule: \.induct) auto + +lemma rminusinf_\: + assumes lp: "isrlfm p" + and nmi: "\ (Ifm (a#bs) (minusinf p))" (is "\ (Ifm (a#bs) (?M p))") + and ex: "Ifm (x#bs) p" (is "?I x p") + shows "\ (s,m) \ set (\ p). x \ Inum (a#bs) s / real m" (is "\ (s,m) \ ?U p. x \ ?N a s / real m") +proof- + have "\ (s,m) \ set (\ p). real m * x \ Inum (a#bs) s " (is "\ (s,m) \ ?U p. real m *x \ ?N a s") + using lp nmi ex + by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2) + then obtain s m where smU: "(s,m) \ set (\ p)" and mx: "real m * x \ ?N a s" by blast + from \_l[OF lp] smU have mp: "real m > 0" by auto + from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \ ?N a s / real m" + by (auto simp add: mult_commute) + thus ?thesis using smU by auto +qed + +lemma rplusinf_\: + assumes lp: "isrlfm p" + and nmi: "\ (Ifm (a#bs) (plusinf p))" (is "\ (Ifm (a#bs) (?M p))") + and ex: "Ifm (x#bs) p" (is "?I x p") + shows "\ (s,m) \ set (\ p). x \ Inum (a#bs) s / real m" (is "\ (s,m) \ ?U p. x \ ?N a s / real m") +proof- + have "\ (s,m) \ set (\ p). real m * x \ Inum (a#bs) s " (is "\ (s,m) \ ?U p. real m *x \ ?N a s") + using lp nmi ex + by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2) + then obtain s m where smU: "(s,m) \ set (\ p)" and mx: "real m * x \ ?N a s" by blast + from \_l[OF lp] smU have mp: "real m > 0" by auto + from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \ ?N a s / real m" + by (auto simp add: mult_commute) + thus ?thesis using smU by auto +qed + +lemma lin_dense: + assumes lp: "isrlfm p" + and noS: "\ t. l < t \ t< u \ t \ (\ (t,n). Inum (x#bs) t / real n) ` set (\ p)" + (is "\ t. _ \ _ \ t \ (\ (t,n). ?N x t / real n ) ` (?U p)") + and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p" + and ly: "l < y" and yu: "y < u" + shows "Ifm (y#bs) p" +using lp px noS +proof (induct p rule: isrlfm.induct) + case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from prems have "x * real c + ?N x e < 0" by (simp add: algebra_simps) + hence pxc: "x < (- ?N x e) / real c" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\ t. l < t \ t < u \ t \ (- ?N x e) / real c" by auto + with ly yu have yne: "y \ - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \ y > (-?N x e) / real c" by auto + moreover {assume y: "y < (-?N x e)/ real c" + hence "y * real c < - ?N x e" + by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e < 0" by (simp add: algebra_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y > (- ?N x e) / real c" + with yu have eu: "u > (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \ l" by (cases "(- ?N x e) / real c > l", auto) + with lx pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp + + from prems have "x * real c + ?N x e \ 0" by (simp add: algebra_simps) + hence pxc: "x \ (- ?N x e) / real c" + by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\ t. l < t \ t < u \ t \ (- ?N x e) / real c" by auto + with ly yu have yne: "y \ - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \ y > (-?N x e) / real c" by auto + moreover {assume y: "y < (-?N x e)/ real c" + hence "y * real c < - ?N x e" + by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e < 0" by (simp add: algebra_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y > (- ?N x e) / real c" + with yu have eu: "u > (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \ l" by (cases "(- ?N x e) / real c > l", auto) + with lx pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from prems have "x * real c + ?N x e > 0" by (simp add: algebra_simps) + hence pxc: "x > (- ?N x e) / real c" + by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\ t. l < t \ t < u \ t \ (- ?N x e) / real c" by auto + with ly yu have yne: "y \ - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \ y > (-?N x e) / real c" by auto + moreover {assume y: "y > (-?N x e)/ real c" + hence "y * real c > - ?N x e" + by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e > 0" by (simp add: algebra_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y < (- ?N x e) / real c" + with ly have eu: "l < (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \ u" by (cases "(- ?N x e) / real c > l", auto) + with xu pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from prems have "x * real c + ?N x e \ 0" by (simp add: algebra_simps) + hence pxc: "x \ (- ?N x e) / real c" + by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\ t. l < t \ t < u \ t \ (- ?N x e) / real c" by auto + with ly yu have yne: "y \ - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \ y > (-?N x e) / real c" by auto + moreover {assume y: "y > (-?N x e)/ real c" + hence "y * real c > - ?N x e" + by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e > 0" by (simp add: algebra_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y < (- ?N x e) / real c" + with ly have eu: "l < (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \ u" by (cases "(- ?N x e) / real c > l", auto) + with xu pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from cp have cnz: "real c \ 0" by simp + from prems have "x * real c + ?N x e = 0" by (simp add: algebra_simps) + hence pxc: "x = (- ?N x e) / real c" + by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"]) + from prems have noSc:"\ t. l < t \ t < u \ t \ (- ?N x e) / real c" by auto + with lx xu have yne: "x \ - ?N x e / real c" by auto + with pxc show ?case by simp +next + case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from cp have cnz: "real c \ 0" by simp + from prems have noSc:"\ t. l < t \ t < u \ t \ (- ?N x e) / real c" by auto + with ly yu have yne: "y \ - ?N x e / real c" by auto + hence "y* real c \ -?N x e" + by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp + hence "y* real c + ?N x e \ 0" by (simp add: algebra_simps) + thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] + by (simp add: algebra_simps) +qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"]) + +lemma finite_set_intervals: + assumes px: "P (x::real)" + and lx: "l \ x" and xu: "x \ u" + and linS: "l\ S" and uinS: "u \ S" + and fS:"finite S" and lS: "\ x\ S. l \ x" and Su: "\ x\ S. x \ u" + shows "\ a \ S. \ b \ S. (\ y. a < y \ y < b \ y \ S) \ a \ x \ x \ b \ P x" +proof- + let ?Mx = "{y. y\ S \ y \ x}" + let ?xM = "{y. y\ S \ x \ y}" + let ?a = "Max ?Mx" + let ?b = "Min ?xM" + have MxS: "?Mx \ S" by blast + hence fMx: "finite ?Mx" using fS finite_subset by auto + from lx linS have linMx: "l \ ?Mx" by blast + hence Mxne: "?Mx \ {}" by blast + have xMS: "?xM \ S" by blast + hence fxM: "finite ?xM" using fS finite_subset by auto + from xu uinS have linxM: "u \ ?xM" by blast + hence xMne: "?xM \ {}" by blast + have ax:"?a \ x" using Mxne fMx by auto + have xb:"x \ ?b" using xMne fxM by auto + have "?a \ ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \ S" using MxS by blast + have "?b \ ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \ S" using xMS by blast + have noy:"\ y. ?a < y \ y < ?b \ y \ S" + proof(clarsimp) + fix y + assume ay: "?a < y" and yb: "y < ?b" and yS: "y \ S" + from yS have "y\ ?Mx \ y\ ?xM" by auto + moreover {assume "y \ ?Mx" hence "y \ ?a" using Mxne fMx by auto with ay have "False" by simp} + moreover {assume "y \ ?xM" hence "y \ ?b" using xMne fxM by auto with yb have "False" by simp} + ultimately show "False" by blast + qed + from ainS binS noy ax xb px show ?thesis by blast +qed + +lemma finite_set_intervals2: + assumes px: "P (x::real)" + and lx: "l \ x" and xu: "x \ u" + and linS: "l\ S" and uinS: "u \ S" + and fS:"finite S" and lS: "\ x\ S. l \ x" and Su: "\ x\ S. x \ u" + shows "(\ s\ S. P s) \ (\ a \ S. \ b \ S. (\ y. a < y \ y < b \ y \ S) \ a < x \ x < b \ P x)" +proof- + from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su] + obtain a and b where + as: "a\ S" and bs: "b\ S" and noS:"\y. a < y \ y < b \ y \ S" and axb: "a \ x \ x \ b \ P x" by auto + from axb have "x= a \ x= b \ (a < x \ x < b)" by auto + thus ?thesis using px as bs noS by blast +qed + +lemma rinf_\: + assumes lp: "isrlfm p" + and nmi: "\ (Ifm (x#bs) (minusinf p))" (is "\ (Ifm (x#bs) (?M p))") + and npi: "\ (Ifm (x#bs) (plusinf p))" (is "\ (Ifm (x#bs) (?P p))") + and ex: "\ x. Ifm (x#bs) p" (is "\ x. ?I x p") + shows "\ (l,n) \ set (\ p). \ (s,m) \ set (\ p). ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p" +proof- + let ?N = "\ x t. Inum (x#bs) t" + let ?U = "set (\ p)" + from ex obtain a where pa: "?I a p" by blast + from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi + have nmi': "\ (?I a (?M p))" by simp + from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi + have npi': "\ (?I a (?P p))" by simp + have "\ (l,n) \ set (\ p). \ (s,m) \ set (\ p). ?I ((?N a l/real n + ?N a s /real m) / 2) p" + proof- + let ?M = "(\ (t,c). ?N a t / real c) ` ?U" + have fM: "finite ?M" by auto + from rminusinf_\[OF lp nmi pa] rplusinf_\[OF lp npi pa] + have "\ (l,n) \ set (\ p). \ (s,m) \ set (\ p). a \ ?N x l / real n \ a \ ?N x s / real m" by blast + then obtain "t" "n" "s" "m" where + tnU: "(t,n) \ ?U" and smU: "(s,m) \ ?U" + and xs1: "a \ ?N x s / real m" and tx1: "a \ ?N x t / real n" by blast + from \_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \ ?N a s / real m" and tx: "a \ ?N a t / real n" by auto + from tnU have Mne: "?M \ {}" by auto + hence Une: "?U \ {}" by simp + let ?l = "Min ?M" + let ?u = "Max ?M" + have linM: "?l \ ?M" using fM Mne by simp + have uinM: "?u \ ?M" using fM Mne by simp + have tnM: "?N a t / real n \ ?M" using tnU by auto + have smM: "?N a s / real m \ ?M" using smU by auto + have lM: "\ t\ ?M. ?l \ t" using Mne fM by auto + have Mu: "\ t\ ?M. t \ ?u" using Mne fM by auto + have "?l \ ?N a t / real n" using tnM Mne by simp hence lx: "?l \ a" using tx by simp + have "?N a s / real m \ ?u" using smM Mne by simp hence xu: "a \ ?u" using xs by simp + from finite_set_intervals2[where P="\ x. ?I x p",OF pa lx xu linM uinM fM lM Mu] + have "(\ s\ ?M. ?I s p) \ + (\ t1\ ?M. \ t2 \ ?M. (\ y. t1 < y \ y < t2 \ y \ ?M) \ t1 < a \ a < t2 \ ?I a p)" . + moreover { fix u assume um: "u\ ?M" and pu: "?I u p" + hence "\ (tu,nu) \ ?U. u = ?N a tu / real nu" by auto + then obtain "tu" "nu" where tuU: "(tu,nu) \ ?U" and tuu:"u= ?N a tu / real nu" by blast + have "(u + u) / 2 = u" by auto with pu tuu + have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp + with tuU have ?thesis by blast} + moreover{ + assume "\ t1\ ?M. \ t2 \ ?M. (\ y. t1 < y \ y < t2 \ y \ ?M) \ t1 < a \ a < t2 \ ?I a p" + then obtain t1 and t2 where t1M: "t1 \ ?M" and t2M: "t2\ ?M" + and noM: "\ y. t1 < y \ y < t2 \ y \ ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p" + by blast + from t1M have "\ (t1u,t1n) \ ?U. t1 = ?N a t1u / real t1n" by auto + then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \ ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast + from t2M have "\ (t2u,t2n) \ ?U. t2 = ?N a t2u / real t2n" by auto + then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \ ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast + from t1x xt2 have t1t2: "t1 < t2" by simp + let ?u = "(t1 + t2) / 2" + from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto + from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" . + with t1uU t2uU t1u t2u have ?thesis by blast} + ultimately show ?thesis by blast + qed + then obtain "l" "n" "s" "m" where lnU: "(l,n) \ ?U" and smU:"(s,m) \ ?U" + and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast + from lnU smU \_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto + from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] + numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu + have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp + with lnU smU + show ?thesis by auto +qed + (* The Ferrante - Rackoff Theorem *) + +theorem fr_eq: + assumes lp: "isrlfm p" + shows "(\ x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \ (Ifm (x#bs) (plusinf p)) \ (\ (t,n) \ set (\ p). \ (s,m) \ set (\ p). Ifm ((((Inum (x#bs) t)/ real n + (Inum (x#bs) s) / real m) /2)#bs) p))" + (is "(\ x. ?I x p) = (?M \ ?P \ ?F)" is "?E = ?D") +proof + assume px: "\ x. ?I x p" + have "?M \ ?P \ (\ ?M \ \ ?P)" by blast + moreover {assume "?M \ ?P" hence "?D" by blast} + moreover {assume nmi: "\ ?M" and npi: "\ ?P" + from rinf_\[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast} + ultimately show "?D" by blast +next + assume "?D" + moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .} + moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . } + moreover {assume f:"?F" hence "?E" by blast} + ultimately show "?E" by blast +qed + + +lemma fr_eq\: + assumes lp: "isrlfm p" + shows "(\ x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \ (Ifm (x#bs) (plusinf p)) \ (\ (t,k) \ set (\ p). \ (s,l) \ set (\ p). Ifm (x#bs) (\ p (Add(Mul l t) (Mul k s) , 2*k*l))))" + (is "(\ x. ?I x p) = (?M \ ?P \ ?F)" is "?E = ?D") +proof + assume px: "\ x. ?I x p" + have "?M \ ?P \ (\ ?M \ \ ?P)" by blast + moreover {assume "?M \ ?P" hence "?D" by blast} + moreover {assume nmi: "\ ?M" and npi: "\ ?P" + let ?f ="\ (t,n). Inum (x#bs) t / real n" + let ?N = "\ t. Inum (x#bs) t" + {fix t n s m assume "(t,n)\ set (\ p)" and "(s,m) \ set (\ p)" + with \_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0" + by auto + let ?st = "Add (Mul m t) (Mul n s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute) + from tnb snb have st_nb: "numbound0 ?st" by simp + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mnp mp np by (simp add: algebra_simps add_divide_distrib) + from \_I[OF lp mnp st_nb, where x="x" and bs="bs"] + have "?I x (\ p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])} + with rinf_\[OF lp nmi npi px] have "?F" by blast hence "?D" by blast} + ultimately show "?D" by blast +next + assume "?D" + moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .} + moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . } + moreover {fix t k s l assume "(t,k) \ set (\ p)" and "(s,l) \ set (\ p)" + and px:"?I x (\ p (Add (Mul l t) (Mul k s), 2*k*l))" + with \_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto + let ?st = "Add (Mul l t) (Mul k s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0" + by (simp add: mult_commute) + from tnb snb have st_nb: "numbound0 ?st" by simp + from \_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto} + ultimately show "?E" by blast +qed + +text{* The overall Part *} + +lemma real_ex_int_real01: + shows "(\ (x::real). P x) = (\ (i::int) (u::real). 0\ u \ u< 1 \ P (real i + u))" +proof(auto) + fix x + assume Px: "P x" + let ?i = "floor x" + let ?u = "x - real ?i" + have "x = real ?i + ?u" by simp + hence "P (real ?i + ?u)" using Px by simp + moreover have "real ?i \ x" using real_of_int_floor_le by simp hence "0 \ ?u" by arith + moreover have "?u < 1" using real_of_int_floor_add_one_gt[where r="x"] by arith + ultimately show "(\ (i::int) (u::real). 0\ u \ u< 1 \ P (real i + u))" by blast +qed + +consts exsplitnum :: "num \ num" + exsplit :: "fm \ fm" +recdef exsplitnum "measure size" + "exsplitnum (C c) = (C c)" + "exsplitnum (Bound 0) = Add (Bound 0) (Bound 1)" + "exsplitnum (Bound n) = Bound (n+1)" + "exsplitnum (Neg a) = Neg (exsplitnum a)" + "exsplitnum (Add a b) = Add (exsplitnum a) (exsplitnum b) " + "exsplitnum (Sub a b) = Sub (exsplitnum a) (exsplitnum b) " + "exsplitnum (Mul c a) = Mul c (exsplitnum a)" + "exsplitnum (Floor a) = Floor (exsplitnum a)" + "exsplitnum (CN 0 c a) = CN 0 c (Add (Mul c (Bound 1)) (exsplitnum a))" + "exsplitnum (CN n c a) = CN (n+1) c (exsplitnum a)" + "exsplitnum (CF c s t) = CF c (exsplitnum s) (exsplitnum t)" + +recdef exsplit "measure size" + "exsplit (Lt a) = Lt (exsplitnum a)" + "exsplit (Le a) = Le (exsplitnum a)" + "exsplit (Gt a) = Gt (exsplitnum a)" + "exsplit (Ge a) = Ge (exsplitnum a)" + "exsplit (Eq a) = Eq (exsplitnum a)" + "exsplit (NEq a) = NEq (exsplitnum a)" + "exsplit (Dvd i a) = Dvd i (exsplitnum a)" + "exsplit (NDvd i a) = NDvd i (exsplitnum a)" + "exsplit (And p q) = And (exsplit p) (exsplit q)" + "exsplit (Or p q) = Or (exsplit p) (exsplit q)" + "exsplit (Imp p q) = Imp (exsplit p) (exsplit q)" + "exsplit (Iff p q) = Iff (exsplit p) (exsplit q)" + "exsplit (NOT p) = NOT (exsplit p)" + "exsplit p = p" + +lemma exsplitnum: + "Inum (x#y#bs) (exsplitnum t) = Inum ((x+y) #bs) t" + by(induct t rule: exsplitnum.induct) (simp_all add: algebra_simps) + +lemma exsplit: + assumes qfp: "qfree p" + shows "Ifm (x#y#bs) (exsplit p) = Ifm ((x+y)#bs) p" +using qfp exsplitnum[where x="x" and y="y" and bs="bs"] +by(induct p rule: exsplit.induct) simp_all + +lemma splitex: + assumes qf: "qfree p" + shows "(Ifm bs (E p)) = (\ (i::int). Ifm (real i#bs) (E (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (exsplit p))))" (is "?lhs = ?rhs") +proof- + have "?rhs = (\ (i::int). \ x. 0\ x \ x < 1 \ Ifm (x#(real i)#bs) (exsplit p))" + by (simp add: myless[rule_format, where b="1"] myless[rule_format, where b="0"] add_ac diff_def) + also have "\ = (\ (i::int). \ x. 0\ x \ x < 1 \ Ifm ((real i + x) #bs) p)" + by (simp only: exsplit[OF qf] add_ac) + also have "\ = (\ x. Ifm (x#bs) p)" + by (simp only: real_ex_int_real01[where P="\ x. Ifm (x#bs) p"]) + finally show ?thesis by simp +qed + + (* Implement the right hand sides of Cooper's theorem and Ferrante and Rackoff. *) + +constdefs ferrack01:: "fm \ fm" + "ferrack01 p \ (let p' = rlfm(And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p); + U = remdups(map simp_num_pair + (map (\ ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) + (alluopairs (\ p')))) + in decr (evaldjf (\ p') U ))" + +lemma fr_eq_01: + assumes qf: "qfree p" + shows "(\ x. Ifm (x#bs) (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = (\ (t,n) \ set (\ (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). \ (s,m) \ set (\ (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). Ifm (x#bs) (\ (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) (Add (Mul m t) (Mul n s), 2*n*m)))" + (is "(\ x. ?I x ?q) = ?F") +proof- + let ?rq = "rlfm ?q" + let ?M = "?I x (minusinf ?rq)" + let ?P = "?I x (plusinf ?rq)" + have MF: "?M = False" + apply (simp add: Let_def reducecoeff_def numgcd_def zgcd_def rsplit_def ge_def lt_def conj_def disj_def) + by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C -1)))", simp_all) + have PF: "?P = False" apply (simp add: Let_def reducecoeff_def numgcd_def zgcd_def rsplit_def ge_def lt_def conj_def disj_def) + by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C -1)))", simp_all) + have "(\ x. ?I x ?q ) = + ((?I x (minusinf ?rq)) \ (?I x (plusinf ?rq )) \ (\ (t,n) \ set (\ ?rq). \ (s,m) \ set (\ ?rq ). ?I x (\ ?rq (Add (Mul m t) (Mul n s), 2*n*m))))" + (is "(\ x. ?I x ?q) = (?M \ ?P \ ?F)" is "?E = ?D") + proof + assume "\ x. ?I x ?q" + then obtain x where qx: "?I x ?q" by blast + hence xp: "0\ x" and x1: "x< 1" and px: "?I x p" + by (auto simp add: rsplit_def lt_def ge_def rlfm_I[OF qf]) + from qx have "?I x ?rq " + by (simp add: rsplit_def lt_def ge_def rlfm_I[OF qf xp x1]) + hence lqx: "?I x ?rq " using simpfm[where p="?rq" and bs="x#bs"] by auto + from qf have qfq:"isrlfm ?rq" + by (auto simp add: rsplit_def lt_def ge_def rlfm_I[OF qf xp x1]) + with lqx fr_eq\[OF qfq] show "?M \ ?P \ ?F" by blast + next + assume D: "?D" + let ?U = "set (\ ?rq )" + from MF PF D have "?F" by auto + then obtain t n s m where aU:"(t,n) \ ?U" and bU:"(s,m)\ ?U" and rqx: "?I x (\ ?rq (Add (Mul m t) (Mul n s), 2*n*m))" by blast + from qf have lrq:"isrlfm ?rq"using rlfm_l[OF qf] + by (auto simp add: rsplit_def lt_def ge_def) + from aU bU \_l[OF lrq] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0" by (auto simp add: split_def) + let ?st = "Add (Mul m t) (Mul n s)" + from tnb snb have stnb: "numbound0 ?st" by simp + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute) + from conjunct1[OF \_I[OF lrq mnp stnb, where bs="bs" and x="x"], symmetric] rqx + have "\ x. ?I x ?rq" by auto + thus "?E" + using rlfm_I[OF qf] by (auto simp add: rsplit_def lt_def ge_def) + qed + with MF PF show ?thesis by blast +qed + +lemma \_cong_aux: + assumes Ul: "\ (t,n) \ set U. numbound0 t \ n >0" + shows "((\ (t,n). Inum (x#bs) t /real n) ` (set (map (\ ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\ ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \ set U))" + (is "?lhs = ?rhs") +proof(auto) + fix t n s m + assume "((t,n),(s,m)) \ set (alluopairs U)" + hence th: "((t,n),(s,m)) \ (set U \ set U)" + using alluopairs_set1[where xs="U"] by blast + let ?N = "\ t. Inum (x#bs) t" + let ?st= "Add (Mul m t) (Mul n s)" + from Ul th have mnz: "m \ 0" by auto + from Ul th have nnz: "n \ 0" by auto + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mnz nnz by (simp add: algebra_simps add_divide_distrib) + + thus "(real m * Inum (x # bs) t + real n * Inum (x # bs) s) / + (2 * real n * real m) + \ (\((t, n), s, m). + (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) ` + (set U \ set U)"using mnz nnz th + apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def) + by (rule_tac x="(s,m)" in bexI,simp_all) + (rule_tac x="(t,n)" in bexI,simp_all) +next + fix t n s m + assume tnU: "(t,n) \ set U" and smU:"(s,m) \ set U" + let ?N = "\ t. Inum (x#bs) t" + let ?st= "Add (Mul m t) (Mul n s)" + from Ul smU have mnz: "m \ 0" by auto + from Ul tnU have nnz: "n \ 0" by auto + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mnz nnz by (simp add: algebra_simps add_divide_distrib) + let ?P = "\ (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" + have Pc:"\ a b. ?P a b = ?P b a" + by auto + from Ul alluopairs_set1 have Up:"\ ((t,n),(s,m)) \ set (alluopairs U). n \ 0 \ m \ 0" by blast + from alluopairs_ex[OF Pc, where xs="U"] tnU smU + have th':"\ ((t',n'),(s',m')) \ set (alluopairs U). ?P (t',n') (s',m')" + by blast + then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \ set (alluopairs U)" + and Pts': "?P (t',n') (s',m')" by blast + from ts'_U Up have mnz': "m' \ 0" and nnz': "n'\ 0" by auto + let ?st' = "Add (Mul m' t') (Mul n' s')" + have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')" + using mnz' nnz' by (simp add: algebra_simps add_divide_distrib) + from Pts' have + "(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp + also have "\ = ((\(t, n). Inum (x # bs) t / real n) ((\((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st') + finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2 + \ (\(t, n). Inum (x # bs) t / real n) ` + (\((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ` + set (alluopairs U)" + using ts'_U by blast +qed + +lemma \_cong: + assumes lp: "isrlfm p" + and UU': "((\ (t,n). Inum (x#bs) t /real n) ` U') = ((\ ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (U \ U))" (is "?f ` U' = ?g ` (U\U)") + and U: "\ (t,n) \ U. numbound0 t \ n > 0" + and U': "\ (t,n) \ U'. numbound0 t \ n > 0" + shows "(\ (t,n) \ U. \ (s,m) \ U. Ifm (x#bs) (\ p (Add (Mul m t) (Mul n s),2*n*m))) = (\ (t,n) \ U'. Ifm (x#bs) (\ p (t,n)))" + (is "?lhs = ?rhs") +proof + assume ?lhs + then obtain t n s m where tnU: "(t,n) \ U" and smU:"(s,m) \ U" and + Pst: "Ifm (x#bs) (\ p (Add (Mul m t) (Mul n s),2*n*m))" by blast + let ?N = "\ t. Inum (x#bs) t" + from tnU smU U have tnb: "numbound0 t" and np: "n > 0" + and snb: "numbound0 s" and mp:"m > 0" by auto + let ?st= "Add (Mul m t) (Mul n s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult) + from tnb snb have stnb: "numbound0 ?st" by simp + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mp np by (simp add: algebra_simps add_divide_distrib) + from tnU smU UU' have "?g ((t,n),(s,m)) \ ?f ` U'" by blast + hence "\ (t',n') \ U'. ?g ((t,n),(s,m)) = ?f (t',n')" + by auto (rule_tac x="(a,b)" in bexI, auto) + then obtain t' n' where tnU': "(t',n') \ U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast + from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto + from \_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst + have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp + from conjunct1[OF \_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]] + have "Ifm (x # bs) (\ p (t', n')) " by (simp only: st) + then show ?rhs using tnU' by auto +next + assume ?rhs + then obtain t' n' where tnU': "(t',n') \ U'" and Pt': "Ifm (x # bs) (\ p (t', n'))" + by blast + from tnU' UU' have "?f (t',n') \ ?g ` (U\U)" by blast + hence "\ ((t,n),(s,m)) \ (U\U). ?f (t',n') = ?g ((t,n),(s,m))" + by auto (rule_tac x="(a,b)" in bexI, auto) + then obtain t n s m where tnU: "(t,n) \ U" and smU:"(s,m) \ U" and + th: "?f (t',n') = ?g((t,n),(s,m)) "by blast + let ?N = "\ t. Inum (x#bs) t" + from tnU smU U have tnb: "numbound0 t" and np: "n > 0" + and snb: "numbound0 s" and mp:"m > 0" by auto + let ?st= "Add (Mul m t) (Mul n s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult) + from tnb snb have stnb: "numbound0 ?st" by simp + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mp np by (simp add: algebra_simps add_divide_distrib) + from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto + from \_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt' + have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp + with \_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast +qed + +lemma ferrack01: + assumes qf: "qfree p" + shows "((\ x. Ifm (x#bs) (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = (Ifm bs (ferrack01 p))) \ qfree (ferrack01 p)" (is "(?lhs = ?rhs) \ _") +proof- + let ?I = "\ x p. Ifm (x#bs) p" + fix x + let ?N = "\ t. Inum (x#bs) t" + let ?q = "rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)" + let ?U = "\ ?q" + let ?Up = "alluopairs ?U" + let ?g = "\ ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)" + let ?S = "map ?g ?Up" + let ?SS = "map simp_num_pair ?S" + let ?Y = "remdups ?SS" + let ?f= "(\ (t,n). ?N t / real n)" + let ?h = "\ ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2" + let ?F = "\ p. \ a \ set (\ p). \ b \ set (\ p). ?I x (\ p (?g(a,b)))" + let ?ep = "evaldjf (\ ?q) ?Y" + from rlfm_l[OF qf] have lq: "isrlfm ?q" + by (simp add: rsplit_def lt_def ge_def conj_def disj_def Let_def reducecoeff_def numgcd_def zgcd_def) + from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \ (set ?U \ set ?U)" by simp + from \_l[OF lq] have U_l: "\ (t,n) \ set ?U. numbound0 t \ n > 0" . + from U_l UpU + have Up_: "\ ((t,n),(s,m)) \ set ?Up. numbound0 t \ n> 0 \ numbound0 s \ m > 0" by auto + hence Snb: "\ (t,n) \ set ?S. numbound0 t \ n > 0 " + by (auto simp add: mult_pos_pos) + have Y_l: "\ (t,n) \ set ?Y. numbound0 t \ n > 0" + proof- + { fix t n assume tnY: "(t,n) \ set ?Y" + hence "(t,n) \ set ?SS" by simp + hence "\ (t',n') \ set ?S. simp_num_pair (t',n') = (t,n)" + by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all) + then obtain t' n' where tn'S: "(t',n') \ set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast + from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto + from simp_num_pair_l[OF tnb np tns] + have "numbound0 t \ n > 0" . } + thus ?thesis by blast + qed + + have YU: "(?f ` set ?Y) = (?h ` (set ?U \ set ?U))" + proof- + from simp_num_pair_ci[where bs="x#bs"] have + "\x. (?f o simp_num_pair) x = ?f x" by auto + hence th: "?f o simp_num_pair = ?f" using ext by blast + have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose) + also have "\ = (?f ` set ?S)" by (simp add: th) + also have "\ = ((?f o ?g) ` set ?Up)" + by (simp only: set_map o_def image_compose[symmetric]) + also have "\ = (?h ` (set ?U \ set ?U))" + using \_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast + finally show ?thesis . + qed + have "\ (t,n) \ set ?Y. bound0 (\ ?q (t,n))" + proof- + { fix t n assume tnY: "(t,n) \ set ?Y" + with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto + from \_I[OF lq np tnb] + have "bound0 (\ ?q (t,n))" by simp} + thus ?thesis by blast + qed + hence ep_nb: "bound0 ?ep" using evaldjf_bound0[where xs="?Y" and f="\ ?q"] + by auto + + from fr_eq_01[OF qf, where bs="bs" and x="x"] have "?lhs = ?F ?q" + by (simp only: split_def fst_conv snd_conv) + also have "\ = (\ (t,n) \ set ?Y. ?I x (\ ?q (t,n)))" using \_cong[OF lq YU U_l Y_l] + by (simp only: split_def fst_conv snd_conv) + also have "\ = (Ifm (x#bs) ?ep)" + using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="\ ?q",symmetric] + by (simp only: split_def pair_collapse) + also have "\ = (Ifm bs (decr ?ep))" using decr[OF ep_nb] by blast + finally have lr: "?lhs = ?rhs" by (simp only: ferrack01_def Let_def) + from decr_qf[OF ep_nb] have "qfree (ferrack01 p)" by (simp only: Let_def ferrack01_def) + with lr show ?thesis by blast +qed + +lemma cp_thm': + assumes lp: "iszlfm p (real (i::int)#bs)" + and up: "d\ p 1" and dd: "d\ p d" and dp: "d > 0" + shows "(\ (x::int). Ifm (real x#bs) p) = ((\ j\ {1 .. d}. Ifm (real j#bs) (minusinf p)) \ (\ j\ {1.. d}. \ b\ (Inum (real i#bs)) ` set (\ p). Ifm ((b+real j)#bs) p))" + using cp_thm[OF lp up dd dp] by auto + +constdefs unit:: "fm \ fm \ num list \ int" + "unit p \ (let p' = zlfm p ; l = \ p' ; q = And (Dvd l (CN 0 1 (C 0))) (a\ p' l); d = \ q; + B = remdups (map simpnum (\ q)) ; a = remdups (map simpnum (\ q)) + in if length B \ length a then (q,B,d) else (mirror q, a,d))" + +lemma unit: assumes qf: "qfree p" + shows "\ q B d. unit p = (q,B,d) \ ((\ (x::int). Ifm (real x#bs) p) = (\ (x::int). Ifm (real x#bs) q)) \ (Inum (real i#bs)) ` set B = (Inum (real i#bs)) ` set (\ q) \ d\ q 1 \ d\ q d \ d >0 \ iszlfm q (real (i::int)#bs) \ (\ b\ set B. numbound0 b)" +proof- + fix q B d + assume qBd: "unit p = (q,B,d)" + let ?thes = "((\ (x::int). Ifm (real x#bs) p) = (\ (x::int). Ifm (real x#bs) q)) \ + Inum (real i#bs) ` set B = Inum (real i#bs) ` set (\ q) \ + d\ q 1 \ d\ q d \ 0 < d \ iszlfm q (real i # bs) \ (\ b\ set B. numbound0 b)" + let ?I = "\ (x::int) p. Ifm (real x#bs) p" + let ?p' = "zlfm p" + let ?l = "\ ?p'" + let ?q = "And (Dvd ?l (CN 0 1 (C 0))) (a\ ?p' ?l)" + let ?d = "\ ?q" + let ?B = "set (\ ?q)" + let ?B'= "remdups (map simpnum (\ ?q))" + let ?A = "set (\ ?q)" + let ?A'= "remdups (map simpnum (\ ?q))" + from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] + have pp': "\ i. ?I i ?p' = ?I i p" by auto + from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]]] + have lp': "\ (i::int). iszlfm ?p' (real i#bs)" by simp + hence lp'': "iszlfm ?p' (real (i::int)#bs)" by simp + from lp' \[where p="?p'" and bs="bs"] have lp: "?l >0" and dl: "d\ ?p' ?l" by auto + from a\_ex[where p="?p'" and l="?l" and bs="bs", OF lp'' dl lp] pp' + have pq_ex:"(\ (x::int). ?I x p) = (\ x. ?I x ?q)" by (simp add: int_rdvd_iff) + from lp'' lp a\[OF lp'' dl lp] have lq:"iszlfm ?q (real i#bs)" and uq: "d\ ?q 1" + by (auto simp add: isint_def) + from \[OF lq] have dp:"?d >0" and dd: "d\ ?q ?d" by blast+ + let ?N = "\ t. Inum (real (i::int)#bs) t" + have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by (simp add:image_compose) + also have "\ = ?N ` ?B" using simpnum_ci[where bs="real i #bs"] by auto + finally have BB': "?N ` set ?B' = ?N ` ?B" . + have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by (simp add:image_compose) + also have "\ = ?N ` ?A" using simpnum_ci[where bs="real i #bs"] by auto + finally have AA': "?N ` set ?A' = ?N ` ?A" . + from \_numbound0[OF lq] have B_nb:"\ b\ set ?B'. numbound0 b" + by (simp add: simpnum_numbound0) + from \_l[OF lq] have A_nb: "\ b\ set ?A'. numbound0 b" + by (simp add: simpnum_numbound0) + {assume "length ?B' \ length ?A'" + hence q:"q=?q" and "B = ?B'" and d:"d = ?d" + using qBd by (auto simp add: Let_def unit_def) + with BB' B_nb have b: "?N ` (set B) = ?N ` set (\ q)" + and bn: "\b\ set B. numbound0 b" by simp+ + with pq_ex dp uq dd lq q d have ?thes by simp} + moreover + {assume "\ (length ?B' \ length ?A')" + hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d" + using qBd by (auto simp add: Let_def unit_def) + with AA' mirror\\[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\ q)" + and bn: "\b\ set B. numbound0 b" by simp+ + from mirror_ex[OF lq] pq_ex q + have pqm_eq:"(\ (x::int). ?I x p) = (\ (x::int). ?I x q)" by simp + from lq uq q mirror_d\ [where p="?q" and bs="bs" and a="real i"] + have lq': "iszlfm q (real i#bs)" and uq: "d\ q 1" by auto + from \[OF lq'] mirror_\[OF lq] q d have dq:"d\ q d " by auto + from pqm_eq b bn uq lq' dp dq q dp d have ?thes by simp + } + ultimately show ?thes by blast +qed + (* Cooper's Algorithm *) + +constdefs cooper :: "fm \ fm" + "cooper p \ + (let (q,B,d) = unit p; js = iupt (1,d); + mq = simpfm (minusinf q); + md = evaldjf (\ j. simpfm (subst0 (C j) mq)) js + in if md = T then T else + (let qd = evaldjf (\ t. simpfm (subst0 t q)) + (remdups (map (\ (b,j). simpnum (Add b (C j))) + [(b,j). b\B,j\js])) + in decr (disj md qd)))" +lemma cooper: assumes qf: "qfree p" + shows "((\ (x::int). Ifm (real x#bs) p) = (Ifm bs (cooper p))) \ qfree (cooper p)" + (is "(?lhs = ?rhs) \ _") +proof- + + let ?I = "\ (x::int) p. Ifm (real x#bs) p" + let ?q = "fst (unit p)" + let ?B = "fst (snd(unit p))" + let ?d = "snd (snd (unit p))" + let ?js = "iupt (1,?d)" + let ?mq = "minusinf ?q" + let ?smq = "simpfm ?mq" + let ?md = "evaldjf (\ j. simpfm (subst0 (C j) ?smq)) ?js" + fix i + let ?N = "\ t. Inum (real (i::int)#bs) t" + let ?bjs = "[(b,j). b\?B,j\?js]" + let ?sbjs = "map (\ (b,j). simpnum (Add b (C j))) ?bjs" + let ?qd = "evaldjf (\ t. simpfm (subst0 t ?q)) (remdups ?sbjs)" + have qbf:"unit p = (?q,?B,?d)" by simp + from unit[OF qf qbf] have pq_ex: "(\(x::int). ?I x p) = (\ (x::int). ?I x ?q)" and + B:"?N ` set ?B = ?N ` set (\ ?q)" and + uq:"d\ ?q 1" and dd: "d\ ?q ?d" and dp: "?d > 0" and + lq: "iszlfm ?q (real i#bs)" and + Bn: "\ b\ set ?B. numbound0 b" by auto + from zlin_qfree[OF lq] have qfq: "qfree ?q" . + from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq". + have jsnb: "\ j \ set ?js. numbound0 (C j)" by simp + hence "\ j\ set ?js. bound0 (subst0 (C j) ?smq)" + by (auto simp only: subst0_bound0[OF qfmq]) + hence th: "\ j\ set ?js. bound0 (simpfm (subst0 (C j) ?smq))" + by (auto simp add: simpfm_bound0) + from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp + from Bn jsnb have "\ (b,j) \ set ?bjs. numbound0 (Add b (C j))" + by simp + hence "\ (b,j) \ set ?bjs. numbound0 (simpnum (Add b (C j)))" + using simpnum_numbound0 by blast + hence "\ t \ set ?sbjs. numbound0 t" by simp + hence "\ t \ set (remdups ?sbjs). bound0 (subst0 t ?q)" + using subst0_bound0[OF qfq] by auto + hence th': "\ t \ set (remdups ?sbjs). bound0 (simpfm (subst0 t ?q))" + using simpfm_bound0 by blast + from evaldjf_bound0 [OF th'] have qdb: "bound0 ?qd" by simp + from mdb qdb + have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \ ?qd=T", simp_all) + from trans [OF pq_ex cp_thm'[OF lq uq dd dp]] B + have "?lhs = (\ j\ {1.. ?d}. ?I j ?mq \ (\ b\ ?N ` set ?B. Ifm ((b+ real j)#bs) ?q))" by auto + also have "\ = ((\ j\ set ?js. ?I j ?smq) \ (\ (b,j) \ (?N ` set ?B \ set ?js). Ifm ((b+ real j)#bs) ?q))" apply (simp only: iupt_set simpfm) by auto + also have "\= ((\ j\ set ?js. ?I j ?smq) \ (\ t \ (\ (b,j). ?N (Add b (C j))) ` set ?bjs. Ifm (t #bs) ?q))" by simp + also have "\= ((\ j\ set ?js. ?I j ?smq) \ (\ t \ (\ (b,j). ?N (simpnum (Add b (C j)))) ` set ?bjs. Ifm (t #bs) ?q))" by (simp only: simpnum_ci) + also have "\= ((\ j\ set ?js. ?I j ?smq) \ (\ t \ set ?sbjs. Ifm (?N t #bs) ?q))" + by (auto simp add: split_def) + also have "\ = ((\ j\ set ?js. (\ j. ?I i (simpfm (subst0 (C j) ?smq))) j) \ (\ t \ set (remdups ?sbjs). (\ t. ?I i (simpfm (subst0 t ?q))) t))" by (simp only: simpfm subst0_I[OF qfq] simpfm Inum.simps subst0_I[OF qfmq] set_remdups) + also have "\ = ((?I i (evaldjf (\ j. simpfm (subst0 (C j) ?smq)) ?js)) \ (?I i (evaldjf (\ t. simpfm (subst0 t ?q)) (remdups ?sbjs))))" by (simp only: evaldjf_ex) + finally have mdqd: "?lhs = (?I i (disj ?md ?qd))" by (simp add: disj) + hence mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" using decr [OF mdqdb] by simp + {assume mdT: "?md = T" + hence cT:"cooper p = T" + by (simp only: cooper_def unit_def split_def Let_def if_True) simp + from mdT mdqd have lhs:"?lhs" by (auto simp add: disj) + from mdT have "?rhs" by (simp add: cooper_def unit_def split_def) + with lhs cT have ?thesis by simp } + moreover + {assume mdT: "?md \ T" hence "cooper p = decr (disj ?md ?qd)" + by (simp only: cooper_def unit_def split_def Let_def if_False) + with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp } + ultimately show ?thesis by blast +qed + +lemma DJcooper: + assumes qf: "qfree p" + shows "((\ (x::int). Ifm (real x#bs) p) = (Ifm bs (DJ cooper p))) \ qfree (DJ cooper p)" +proof- + from cooper have cqf: "\ p. qfree p \ qfree (cooper p)" by blast + from DJ_qf[OF cqf] qf have thqf:"qfree (DJ cooper p)" by blast + have "Ifm bs (DJ cooper p) = (\ q\ set (disjuncts p). Ifm bs (cooper q))" + by (simp add: DJ_def evaldjf_ex) + also have "\ = (\ q \ set(disjuncts p). \ (x::int). Ifm (real x#bs) q)" + using cooper disjuncts_qf[OF qf] by blast + also have "\ = (\ (x::int). Ifm (real x#bs) p)" by (induct p rule: disjuncts.induct, auto) + finally show ?thesis using thqf by blast +qed + + (* Redy and Loveland *) + +lemma \\_cong: assumes lp: "iszlfm p (a#bs)" and tt': "Inum (a#bs) t = Inum (a#bs) t'" + shows "Ifm (a#bs) (\\ p (t,c)) = Ifm (a#bs) (\\ p (t',c))" + using lp + by (induct p rule: iszlfm.induct, auto simp add: tt') + +lemma \_cong: assumes lp: "iszlfm p (a#bs)" and tt': "Inum (a#bs) t = Inum (a#bs) t'" + shows "Ifm (a#bs) (\ p c t) = Ifm (a#bs) (\ p c t')" + by (simp add: \_def tt' \\_cong[OF lp tt']) + +lemma \_cong: assumes lp: "iszlfm p (a#bs)" + and RR: "(\(b,k). (Inum (a#bs) b,k)) ` R = (\(b,k). (Inum (a#bs) b,k)) ` set (\ p)" + shows "(\ (e,c) \ R. \ j\ {1.. c*(\ p)}. Ifm (a#bs) (\ p c (Add e (C j)))) = (\ (e,c) \ set (\ p). \ j\ {1.. c*(\ p)}. Ifm (a#bs) (\ p c (Add e (C j))))" + (is "?lhs = ?rhs") +proof + let ?d = "\ p" + assume ?lhs then obtain e c j where ecR: "(e,c) \ R" and jD:"j \ {1 .. c*?d}" + and px: "Ifm (a#bs) (\ p c (Add e (C j)))" (is "?sp c e j") by blast + from ecR have "(Inum (a#bs) e,c) \ (\(b,k). (Inum (a#bs) b,k)) ` R" by auto + hence "(Inum (a#bs) e,c) \ (\(b,k). (Inum (a#bs) b,k)) ` set (\ p)" using RR by simp + hence "\ (e',c') \ set (\ p). Inum (a#bs) e = Inum (a#bs) e' \ c = c'" by auto + then obtain e' c' where ecRo:"(e',c') \ set (\ p)" and ee':"Inum (a#bs) e = Inum (a#bs) e'" + and cc':"c = c'" by blast + from ee' have tt': "Inum (a#bs) (Add e (C j)) = Inum (a#bs) (Add e' (C j))" by simp + + from \_cong[OF lp tt', where c="c"] px have px':"?sp c e' j" by simp + from ecRo jD px' cc' show ?rhs apply auto + by (rule_tac x="(e', c')" in bexI,simp_all) + (rule_tac x="j" in bexI, simp_all add: cc'[symmetric]) +next + let ?d = "\ p" + assume ?rhs then obtain e c j where ecR: "(e,c) \ set (\ p)" and jD:"j \ {1 .. c*?d}" + and px: "Ifm (a#bs) (\ p c (Add e (C j)))" (is "?sp c e j") by blast + from ecR have "(Inum (a#bs) e,c) \ (\(b,k). (Inum (a#bs) b,k)) ` set (\ p)" by auto + hence "(Inum (a#bs) e,c) \ (\(b,k). (Inum (a#bs) b,k)) ` R" using RR by simp + hence "\ (e',c') \ R. Inum (a#bs) e = Inum (a#bs) e' \ c = c'" by auto + then obtain e' c' where ecRo:"(e',c') \ R" and ee':"Inum (a#bs) e = Inum (a#bs) e'" + and cc':"c = c'" by blast + from ee' have tt': "Inum (a#bs) (Add e (C j)) = Inum (a#bs) (Add e' (C j))" by simp + from \_cong[OF lp tt', where c="c"] px have px':"?sp c e' j" by simp + from ecRo jD px' cc' show ?lhs apply auto + by (rule_tac x="(e', c')" in bexI,simp_all) + (rule_tac x="j" in bexI, simp_all add: cc'[symmetric]) +qed + +lemma rl_thm': + assumes lp: "iszlfm p (real (i::int)#bs)" + and R: "(\(b,k). (Inum (a#bs) b,k)) ` R = (\(b,k). (Inum (a#bs) b,k)) ` set (\ p)" + shows "(\ (x::int). Ifm (real x#bs) p) = ((\ j\ {1 .. \ p}. Ifm (real j#bs) (minusinf p)) \ (\ (e,c) \ R. \ j\ {1.. c*(\ p)}. Ifm (a#bs) (\ p c (Add e (C j)))))" + using rl_thm[OF lp] \_cong[OF iszlfm_gen[OF lp, rule_format, where y="a"] R] by simp + +constdefs chooset:: "fm \ fm \ ((num\int) list) \ int" + "chooset p \ (let q = zlfm p ; d = \ q; + B = remdups (map (\ (t,k). (simpnum t,k)) (\ q)) ; + a = remdups (map (\ (t,k). (simpnum t,k)) (\\ q)) + in if length B \ length a then (q,B,d) else (mirror q, a,d))" + +lemma chooset: assumes qf: "qfree p" + shows "\ q B d. chooset p = (q,B,d) \ ((\ (x::int). Ifm (real x#bs) p) = (\ (x::int). Ifm (real x#bs) q)) \ ((\(t,k). (Inum (real i#bs) t,k)) ` set B = (\(t,k). (Inum (real i#bs) t,k)) ` set (\ q)) \ (\ q = d) \ d >0 \ iszlfm q (real (i::int)#bs) \ (\ (e,c)\ set B. numbound0 e \ c>0)" +proof- + fix q B d + assume qBd: "chooset p = (q,B,d)" + let ?thes = "((\ (x::int). Ifm (real x#bs) p) = (\ (x::int). Ifm (real x#bs) q)) \ ((\(t,k). (Inum (real i#bs) t,k)) ` set B = (\(t,k). (Inum (real i#bs) t,k)) ` set (\ q)) \ (\ q = d) \ d >0 \ iszlfm q (real (i::int)#bs) \ (\ (e,c)\ set B. numbound0 e \ c>0)" + let ?I = "\ (x::int) p. Ifm (real x#bs) p" + let ?q = "zlfm p" + let ?d = "\ ?q" + let ?B = "set (\ ?q)" + let ?f = "\ (t,k). (simpnum t,k)" + let ?B'= "remdups (map ?f (\ ?q))" + let ?A = "set (\\ ?q)" + let ?A'= "remdups (map ?f (\\ ?q))" + from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] + have pp': "\ i. ?I i ?q = ?I i p" by auto + hence pq_ex:"(\ (x::int). ?I x p) = (\ x. ?I x ?q)" by simp + from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]], rule_format, where y="real i"] + have lq: "iszlfm ?q (real (i::int)#bs)" . + from \[OF lq] have dp:"?d >0" by blast + let ?N = "\ (t,c). (Inum (real (i::int)#bs) t,c)" + have "?N ` set ?B' = ((?N o ?f) ` ?B)" by (simp add: split_def image_compose) + also have "\ = ?N ` ?B" + by(simp add: split_def image_compose simpnum_ci[where bs="real i #bs"] image_def) + finally have BB': "?N ` set ?B' = ?N ` ?B" . + have "?N ` set ?A' = ((?N o ?f) ` ?A)" by (simp add: split_def image_compose) + also have "\ = ?N ` ?A" using simpnum_ci[where bs="real i #bs"] + by(simp add: split_def image_compose simpnum_ci[where bs="real i #bs"] image_def) + finally have AA': "?N ` set ?A' = ?N ` ?A" . + from \_l[OF lq] have B_nb:"\ (e,c)\ set ?B'. numbound0 e \ c > 0" + by (simp add: simpnum_numbound0 split_def) + from \\_l[OF lq] have A_nb: "\ (e,c)\ set ?A'. numbound0 e \ c > 0" + by (simp add: simpnum_numbound0 split_def) + {assume "length ?B' \ length ?A'" + hence q:"q=?q" and "B = ?B'" and d:"d = ?d" + using qBd by (auto simp add: Let_def chooset_def) + with BB' B_nb have b: "?N ` (set B) = ?N ` set (\ q)" + and bn: "\(e,c)\ set B. numbound0 e \ c > 0" by auto + with pq_ex dp lq q d have ?thes by simp} + moreover + {assume "\ (length ?B' \ length ?A')" + hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d" + using qBd by (auto simp add: Let_def chooset_def) + with AA' mirror_\\[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\ q)" + and bn: "\(e,c)\ set B. numbound0 e \ c > 0" by auto + from mirror_ex[OF lq] pq_ex q + have pqm_eq:"(\ (x::int). ?I x p) = (\ (x::int). ?I x q)" by simp + from lq q mirror_l [where p="?q" and bs="bs" and a="real i"] + have lq': "iszlfm q (real i#bs)" by auto + from mirror_\[OF lq] pqm_eq b bn lq' dp q dp d have ?thes by simp + } + ultimately show ?thes by blast +qed + +constdefs stage:: "fm \ int \ (num \ int) \ fm" + "stage p d \ (\ (e,c). evaldjf (\ j. simpfm (\ p c (Add e (C j)))) (iupt (1,c*d)))" +lemma stage: + shows "Ifm bs (stage p d (e,c)) = (\ j\{1 .. c*d}. Ifm bs (\ p c (Add e (C j))))" + by (unfold stage_def split_def ,simp only: evaldjf_ex iupt_set simpfm) simp + +lemma stage_nb: assumes lp: "iszlfm p (a#bs)" and cp: "c >0" and nb:"numbound0 e" + shows "bound0 (stage p d (e,c))" +proof- + let ?f = "\ j. simpfm (\ p c (Add e (C j)))" + have th: "\ j\ set (iupt(1,c*d)). bound0 (?f j)" + proof + fix j + from nb have nb':"numbound0 (Add e (C j))" by simp + from simpfm_bound0[OF \_nb[OF lp nb', where k="c"]] + show "bound0 (simpfm (\ p c (Add e (C j))))" . + qed + from evaldjf_bound0[OF th] show ?thesis by (unfold stage_def split_def) simp +qed + +constdefs redlove:: "fm \ fm" + "redlove p \ + (let (q,B,d) = chooset p; + mq = simpfm (minusinf q); + md = evaldjf (\ j. simpfm (subst0 (C j) mq)) (iupt (1,d)) + in if md = T then T else + (let qd = evaldjf (stage q d) B + in decr (disj md qd)))" + +lemma redlove: assumes qf: "qfree p" + shows "((\ (x::int). Ifm (real x#bs) p) = (Ifm bs (redlove p))) \ qfree (redlove p)" + (is "(?lhs = ?rhs) \ _") +proof- + + let ?I = "\ (x::int) p. Ifm (real x#bs) p" + let ?q = "fst (chooset p)" + let ?B = "fst (snd(chooset p))" + let ?d = "snd (snd (chooset p))" + let ?js = "iupt (1,?d)" + let ?mq = "minusinf ?q" + let ?smq = "simpfm ?mq" + let ?md = "evaldjf (\ j. simpfm (subst0 (C j) ?smq)) ?js" + fix i + let ?N = "\ (t,k). (Inum (real (i::int)#bs) t,k)" + let ?qd = "evaldjf (stage ?q ?d) ?B" + have qbf:"chooset p = (?q,?B,?d)" by simp + from chooset[OF qf qbf] have pq_ex: "(\(x::int). ?I x p) = (\ (x::int). ?I x ?q)" and + B:"?N ` set ?B = ?N ` set (\ ?q)" and dd: "\ ?q = ?d" and dp: "?d > 0" and + lq: "iszlfm ?q (real i#bs)" and + Bn: "\ (e,c)\ set ?B. numbound0 e \ c > 0" by auto + from zlin_qfree[OF lq] have qfq: "qfree ?q" . + from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq". + have jsnb: "\ j \ set ?js. numbound0 (C j)" by simp + hence "\ j\ set ?js. bound0 (subst0 (C j) ?smq)" + by (auto simp only: subst0_bound0[OF qfmq]) + hence th: "\ j\ set ?js. bound0 (simpfm (subst0 (C j) ?smq))" + by (auto simp add: simpfm_bound0) + from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp + from Bn stage_nb[OF lq] have th:"\ x \ set ?B. bound0 (stage ?q ?d x)" by auto + from evaldjf_bound0[OF th] have qdb: "bound0 ?qd" . + from mdb qdb + have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \ ?qd=T", simp_all) + from trans [OF pq_ex rl_thm'[OF lq B]] dd + have "?lhs = ((\ j\ {1.. ?d}. ?I j ?mq) \ (\ (e,c)\ set ?B. \ j\ {1 .. c*?d}. Ifm (real i#bs) (\ ?q c (Add e (C j)))))" by auto + also have "\ = ((\ j\ {1.. ?d}. ?I j ?smq) \ (\ (e,c)\ set ?B. ?I i (stage ?q ?d (e,c) )))" + by (simp add: simpfm stage split_def) + also have "\ = ((\ j\ {1 .. ?d}. ?I i (subst0 (C j) ?smq)) \ ?I i ?qd)" + by (simp add: evaldjf_ex subst0_I[OF qfmq]) + finally have mdqd:"?lhs = (?I i ?md \ ?I i ?qd)" by (simp only: evaldjf_ex iupt_set simpfm) + also have "\ = (?I i (disj ?md ?qd))" by (simp add: disj) + also have "\ = (Ifm bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb]) + finally have mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" . + {assume mdT: "?md = T" + hence cT:"redlove p = T" by (simp add: redlove_def Let_def chooset_def split_def) + from mdT have lhs:"?lhs" using mdqd by simp + from mdT have "?rhs" by (simp add: redlove_def chooset_def split_def) + with lhs cT have ?thesis by simp } + moreover + {assume mdT: "?md \ T" hence "redlove p = decr (disj ?md ?qd)" + by (simp add: redlove_def chooset_def split_def Let_def) + with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp } + ultimately show ?thesis by blast +qed + +lemma DJredlove: + assumes qf: "qfree p" + shows "((\ (x::int). Ifm (real x#bs) p) = (Ifm bs (DJ redlove p))) \ qfree (DJ redlove p)" +proof- + from redlove have cqf: "\ p. qfree p \ qfree (redlove p)" by blast + from DJ_qf[OF cqf] qf have thqf:"qfree (DJ redlove p)" by blast + have "Ifm bs (DJ redlove p) = (\ q\ set (disjuncts p). Ifm bs (redlove q))" + by (simp add: DJ_def evaldjf_ex) + also have "\ = (\ q \ set(disjuncts p). \ (x::int). Ifm (real x#bs) q)" + using redlove disjuncts_qf[OF qf] by blast + also have "\ = (\ (x::int). Ifm (real x#bs) p)" by (induct p rule: disjuncts.induct, auto) + finally show ?thesis using thqf by blast +qed + + +lemma exsplit_qf: assumes qf: "qfree p" + shows "qfree (exsplit p)" +using qf by (induct p rule: exsplit.induct, auto) + +definition mircfr :: "fm \ fm" where + "mircfr = DJ cooper o ferrack01 o simpfm o exsplit" + +definition mirlfr :: "fm \ fm" where + "mirlfr = DJ redlove o ferrack01 o simpfm o exsplit" + +lemma mircfr: "\ bs p. qfree p \ qfree (mircfr p) \ Ifm bs (mircfr p) = Ifm bs (E p)" +proof(clarsimp simp del: Ifm.simps) + fix bs p + assume qf: "qfree p" + show "qfree (mircfr p)\(Ifm bs (mircfr p) = Ifm bs (E p))" (is "_ \ (?lhs = ?rhs)") + proof- + let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))" + have "?rhs = (\ (i::int). \ x. Ifm (x#real i#bs) ?es)" + using splitex[OF qf] by simp + with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\ (i::int). Ifm (real i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+ + with DJcooper[OF qf'] show ?thesis by (simp add: mircfr_def) + qed +qed + +lemma mirlfr: "\ bs p. qfree p \ qfree(mirlfr p) \ Ifm bs (mirlfr p) = Ifm bs (E p)" +proof(clarsimp simp del: Ifm.simps) + fix bs p + assume qf: "qfree p" + show "qfree (mirlfr p)\(Ifm bs (mirlfr p) = Ifm bs (E p))" (is "_ \ (?lhs = ?rhs)") + proof- + let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))" + have "?rhs = (\ (i::int). \ x. Ifm (x#real i#bs) ?es)" + using splitex[OF qf] by simp + with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\ (i::int). Ifm (real i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+ + with DJredlove[OF qf'] show ?thesis by (simp add: mirlfr_def) + qed +qed + +definition mircfrqe:: "fm \ fm" where + "mircfrqe p = qelim (prep p) mircfr" + +definition mirlfrqe:: "fm \ fm" where + "mirlfrqe p = qelim (prep p) mirlfr" + +theorem mircfrqe: "(Ifm bs (mircfrqe p) = Ifm bs p) \ qfree (mircfrqe p)" + using qelim_ci[OF mircfr] prep by (auto simp add: mircfrqe_def) + +theorem mirlfrqe: "(Ifm bs (mirlfrqe p) = Ifm bs p) \ qfree (mirlfrqe p)" + using qelim_ci[OF mirlfr] prep by (auto simp add: mirlfrqe_def) + +definition + "test1 (u\unit) = mircfrqe (A (And (Le (Sub (Floor (Bound 0)) (Bound 0))) (Le (Add (Bound 0) (Floor (Neg (Bound 0)))))))" + +definition + "test2 (u\unit) = mircfrqe (A (Iff (Eq (Add (Floor (Bound 0)) (Floor (Neg (Bound 0))))) (Eq (Sub (Floor (Bound 0)) (Bound 0)))))" + +definition + "test3 (u\unit) = mirlfrqe (A (And (Le (Sub (Floor (Bound 0)) (Bound 0))) (Le (Add (Bound 0) (Floor (Neg (Bound 0)))))))" + +definition + "test4 (u\unit) = mirlfrqe (A (Iff (Eq (Add (Floor (Bound 0)) (Floor (Neg (Bound 0))))) (Eq (Sub (Floor (Bound 0)) (Bound 0)))))" + +definition + "test5 (u\unit) = mircfrqe (A(E(And (Ge(Sub (Bound 1) (Bound 0))) (Eq (Add (Floor (Bound 1)) (Floor (Neg(Bound 0))))))))" + +ML {* @{code test1} () *} +ML {* @{code test2} () *} +ML {* @{code test3} () *} +ML {* @{code test4} () *} +ML {* @{code test5} () *} + +(*export_code mircfrqe mirlfrqe + in SML module_name Mir file "raw_mir.ML"*) + +oracle mirfr_oracle = {* fn (proofs, ct) => +let + +fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t + of NONE => error "Variable not found in the list!" + | SOME n => @{code Bound} n) + | num_of_term vs @{term "real (0::int)"} = @{code C} 0 + | num_of_term vs @{term "real (1::int)"} = @{code C} 1 + | num_of_term vs @{term "0::real"} = @{code C} 0 + | num_of_term vs @{term "1::real"} = @{code C} 1 + | num_of_term vs (Bound i) = @{code Bound} i + | num_of_term vs (@{term "uminus :: real \ real"} $ t') = @{code Neg} (num_of_term vs t') + | num_of_term vs (@{term "op + :: real \ real \ real"} $ t1 $ t2) = + @{code Add} (num_of_term vs t1, num_of_term vs t2) + | num_of_term vs (@{term "op - :: real \ real \ real"} $ t1 $ t2) = + @{code Sub} (num_of_term vs t1, num_of_term vs t2) + | num_of_term vs (@{term "op * :: real \ real \ real"} $ t1 $ t2) = + (case (num_of_term vs t1) + of @{code C} i => @{code Mul} (i, num_of_term vs t2) + | _ => error "num_of_term: unsupported Multiplication") + | num_of_term vs (@{term "real :: int \ real"} $ (@{term "number_of :: int \ int"} $ t')) = + @{code C} (HOLogic.dest_numeral t') + | num_of_term vs (@{term "real :: int \ real"} $ (@{term "floor :: real \ int"} $ t')) = + @{code Floor} (num_of_term vs t') + | num_of_term vs (@{term "real :: int \ real"} $ (@{term "ceiling :: real \ int"} $ t')) = + @{code Neg} (@{code Floor} (@{code Neg} (num_of_term vs t'))) + | num_of_term vs (@{term "number_of :: int \ real"} $ t') = + @{code C} (HOLogic.dest_numeral t') + | num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t); + +fun fm_of_term vs @{term True} = @{code T} + | fm_of_term vs @{term False} = @{code F} + | fm_of_term vs (@{term "op < :: real \ real \ bool"} $ t1 $ t2) = + @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) + | fm_of_term vs (@{term "op \ :: real \ real \ bool"} $ t1 $ t2) = + @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) + | fm_of_term vs (@{term "op = :: real \ real \ bool"} $ t1 $ t2) = + @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) + | fm_of_term vs (@{term "op rdvd"} $ (@{term "real :: int \ real"} $ (@{term "number_of :: int \ int"} $ t1)) $ t2) = + @{code Dvd} (HOLogic.dest_numeral t1, num_of_term vs t2) + | fm_of_term vs (@{term "op = :: bool \ bool \ bool"} $ t1 $ t2) = + @{code Iff} (fm_of_term vs t1, fm_of_term vs t2) + | fm_of_term vs (@{term "op &"} $ t1 $ t2) = + @{code And} (fm_of_term vs t1, fm_of_term vs t2) + | fm_of_term vs (@{term "op |"} $ t1 $ t2) = + @{code Or} (fm_of_term vs t1, fm_of_term vs t2) + | fm_of_term vs (@{term "op -->"} $ t1 $ t2) = + @{code Imp} (fm_of_term vs t1, fm_of_term vs t2) + | fm_of_term vs (@{term "Not"} $ t') = + @{code NOT} (fm_of_term vs t') + | fm_of_term vs (Const ("Ex", _) $ Abs (xn, xT, p)) = + @{code E} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p) + | fm_of_term vs (Const ("All", _) $ Abs (xn, xT, p)) = + @{code A} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p) + | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t); + +fun term_of_num vs (@{code C} i) = @{term "real :: int \ real"} $ HOLogic.mk_number HOLogic.intT i + | term_of_num vs (@{code Bound} n) = fst (the (find_first (fn (_, m) => n = m) vs)) + | term_of_num vs (@{code Neg} (@{code Floor} (@{code Neg} t'))) = + @{term "real :: int \ real"} $ (@{term "ceiling :: real \ int"} $ term_of_num vs t') + | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \ real"} $ term_of_num vs t' + | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \ real \ real"} $ + term_of_num vs t1 $ term_of_num vs t2 + | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \ real \ real"} $ + term_of_num vs t1 $ term_of_num vs t2 + | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \ real \ real"} $ + term_of_num vs (@{code C} i) $ term_of_num vs t2 + | term_of_num vs (@{code Floor} t) = @{term "real :: int \ real"} $ (@{term "floor :: real \ int"} $ term_of_num vs t) + | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t)) + | term_of_num vs (@{code CF} (c, t, s)) = term_of_num vs (@{code Add} (@{code Mul} (c, @{code Floor} t), s)); + +fun term_of_fm vs @{code T} = HOLogic.true_const + | term_of_fm vs @{code F} = HOLogic.false_const + | term_of_fm vs (@{code Lt} t) = + @{term "op < :: real \ real \ bool"} $ term_of_num vs t $ @{term "0::real"} + | term_of_fm vs (@{code Le} t) = + @{term "op \ :: real \ real \ bool"} $ term_of_num vs t $ @{term "0::real"} + | term_of_fm vs (@{code Gt} t) = + @{term "op < :: real \ real \ bool"} $ @{term "0::real"} $ term_of_num vs t + | term_of_fm vs (@{code Ge} t) = + @{term "op \ :: real \ real \ bool"} $ @{term "0::real"} $ term_of_num vs t + | term_of_fm vs (@{code Eq} t) = + @{term "op = :: real \ real \ bool"} $ term_of_num vs t $ @{term "0::real"} + | term_of_fm vs (@{code NEq} t) = + term_of_fm vs (@{code NOT} (@{code Eq} t)) + | term_of_fm vs (@{code Dvd} (i, t)) = + @{term "op rdvd"} $ term_of_num vs (@{code C} i) $ term_of_num vs t + | term_of_fm vs (@{code NDvd} (i, t)) = + term_of_fm vs (@{code NOT} (@{code Dvd} (i, t))) + | term_of_fm vs (@{code NOT} t') = + HOLogic.Not $ term_of_fm vs t' + | term_of_fm vs (@{code And} (t1, t2)) = + HOLogic.conj $ term_of_fm vs t1 $ term_of_fm vs t2 + | term_of_fm vs (@{code Or} (t1, t2)) = + HOLogic.disj $ term_of_fm vs t1 $ term_of_fm vs t2 + | term_of_fm vs (@{code Imp} (t1, t2)) = + HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2 + | term_of_fm vs (@{code Iff} (t1, t2)) = + @{term "op = :: bool \ bool \ bool"} $ term_of_fm vs t1 $ term_of_fm vs t2; + +in + let + val thy = Thm.theory_of_cterm ct; + val t = Thm.term_of ct; + val fs = OldTerm.term_frees t; + val vs = fs ~~ (0 upto (length fs - 1)); + val qe = if proofs then @{code mirlfrqe} else @{code mircfrqe}; + val t' = (term_of_fm vs o qe o fm_of_term vs) t; + in (cterm_of thy o HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, t') end +end; +*} + +use "mir_tac.ML" +setup "Mir_Tac.setup" + +lemma "ALL (x::real). (\x\ = \x\ = (x = real \x\))" +apply mir +done + +lemma "ALL (x::real). real (2::int)*x - (real (1::int)) < real \x\ + real \x\ \ real \x\ + real \x\ \ real (2::int)*x + (real (1::int))" +apply mir +done + +lemma "ALL (x::real). 2*\x\ \ \2*x\ \ \2*x\ \ 2*\x+1\" +apply mir +done + +lemma "ALL (x::real). \y \ x. (\x\ = \y\)" +apply mir +done + +lemma "ALL x y. \x\ = \y\ \ 0 \ abs (y - x) \ abs (y - x) \ 1" +apply mir +done + +end diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Decision_Procs/ROOT.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Decision_Procs/ROOT.ML Fri Feb 06 15:15:46 2009 +0100 @@ -0,0 +1,2 @@ + +use_thy "Decision_Procs"; diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Decision_Procs/cooper_tac.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Decision_Procs/cooper_tac.ML Fri Feb 06 15:15:46 2009 +0100 @@ -0,0 +1,139 @@ +(* Title: HOL/Reflection/cooper_tac.ML + Author: Amine Chaieb, TU Muenchen +*) + +structure Cooper_Tac = +struct + +val trace = ref false; +fun trace_msg s = if !trace then tracing s else (); + +val cooper_ss = @{simpset}; + +val nT = HOLogic.natT; +val binarith = @{thms normalize_bin_simps}; +val comp_arith = binarith @ simp_thms + +val zdvd_int = @{thm zdvd_int}; +val zdiff_int_split = @{thm zdiff_int_split}; +val all_nat = @{thm all_nat}; +val ex_nat = @{thm ex_nat}; +val number_of1 = @{thm number_of1}; +val number_of2 = @{thm number_of2}; +val split_zdiv = @{thm split_zdiv}; +val split_zmod = @{thm split_zmod}; +val mod_div_equality' = @{thm mod_div_equality'}; +val split_div' = @{thm split_div'}; +val Suc_plus1 = @{thm Suc_plus1}; +val imp_le_cong = @{thm imp_le_cong}; +val conj_le_cong = @{thm conj_le_cong}; +val nat_mod_add_eq = @{thm mod_add1_eq} RS sym; +val nat_mod_add_left_eq = @{thm mod_add_left_eq} RS sym; +val nat_mod_add_right_eq = @{thm mod_add_right_eq} RS sym; +val int_mod_add_eq = @{thm zmod_zadd1_eq} RS sym; +val int_mod_add_left_eq = @{thm zmod_zadd_left_eq} RS sym; +val int_mod_add_right_eq = @{thm zmod_zadd_right_eq} RS sym; +val nat_div_add_eq = @{thm div_add1_eq} RS sym; +val int_div_add_eq = @{thm zdiv_zadd1_eq} RS sym; + +fun prepare_for_linz q fm = + let + val ps = Logic.strip_params fm + val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm) + val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm) + fun mk_all ((s, T), (P,n)) = + if 0 mem loose_bnos P then + (HOLogic.all_const T $ Abs (s, T, P), n) + else (incr_boundvars ~1 P, n-1) + fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t; + val rhs = hs + val np = length ps + val (fm',np) = foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n))) + (foldr HOLogic.mk_imp c rhs, np) ps + val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT) + (OldTerm.term_frees fm' @ OldTerm.term_vars fm'); + val fm2 = foldr mk_all2 fm' vs + in (fm2, np + length vs, length rhs) end; + +(*Object quantifier to meta --*) +fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ; + +(* object implication to meta---*) +fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp; + + +fun linz_tac ctxt q i = ObjectLogic.atomize_prems_tac i THEN (fn st => + let + val g = List.nth (prems_of st, i - 1) + val thy = ProofContext.theory_of ctxt + (* Transform the term*) + val (t,np,nh) = prepare_for_linz q g + (* Some simpsets for dealing with mod div abs and nat*) + val mod_div_simpset = HOL_basic_ss + addsimps [refl,nat_mod_add_eq, nat_mod_add_left_eq, + nat_mod_add_right_eq, int_mod_add_eq, + int_mod_add_right_eq, int_mod_add_left_eq, + nat_div_add_eq, int_div_add_eq, + @{thm mod_self}, @{thm "zmod_self"}, + @{thm mod_by_0}, @{thm div_by_0}, + @{thm "zdiv_zero"}, @{thm "zmod_zero"}, @{thm "div_0"}, @{thm "mod_0"}, + @{thm "zdiv_1"}, @{thm "zmod_1"}, @{thm "div_1"}, @{thm "mod_1"}, + Suc_plus1] + addsimps @{thms add_ac} + addsimprocs [cancel_div_mod_proc] + val simpset0 = HOL_basic_ss + addsimps [mod_div_equality', Suc_plus1] + addsimps comp_arith + addsplits [split_zdiv, split_zmod, split_div', @{thm "split_min"}, @{thm "split_max"}] + (* Simp rules for changing (n::int) to int n *) + val simpset1 = HOL_basic_ss + addsimps [nat_number_of_def, zdvd_int] @ map (fn r => r RS sym) + [@{thm int_int_eq}, @{thm zle_int}, @{thm zless_int}, @{thm zadd_int}, @{thm zmult_int}] + addsplits [zdiff_int_split] + (*simp rules for elimination of int n*) + + val simpset2 = HOL_basic_ss + addsimps [@{thm nat_0_le}, @{thm all_nat}, @{thm ex_nat}, @{thm number_of1}, @{thm number_of2}, @{thm int_0}, @{thm int_1}] + addcongs [@{thm conj_le_cong}, @{thm imp_le_cong}] + (* simp rules for elimination of abs *) + val simpset3 = HOL_basic_ss addsplits [@{thm abs_split}] + val ct = cterm_of thy (HOLogic.mk_Trueprop t) + (* Theorem for the nat --> int transformation *) + val pre_thm = Seq.hd (EVERY + [simp_tac mod_div_simpset 1, simp_tac simpset0 1, + TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1), + TRY (simp_tac simpset3 1), TRY (simp_tac cooper_ss 1)] + (trivial ct)) + fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i) + (* The result of the quantifier elimination *) + val (th, tac) = case (prop_of pre_thm) of + Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ => + let val pth = linzqe_oracle (cterm_of thy (Pattern.eta_long [] t1)) + in + ((pth RS iffD2) RS pre_thm, + assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i)) + end + | _ => (pre_thm, assm_tac i) + in (rtac (((mp_step nh) o (spec_step np)) th) i + THEN tac) st + end handle Subscript => no_tac st); + +fun linz_args meth = + let val parse_flag = + Args.$$$ "no_quantify" >> (K (K false)); + in + Method.simple_args + (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >> + curry (Library.foldl op |>) true) + (fn q => fn ctxt => meth ctxt q 1) + end; + +fun linz_method ctxt q i = Method.METHOD (fn facts => + Method.insert_tac facts 1 THEN linz_tac ctxt q i); + +val setup = + Method.add_method ("cooper", + linz_args linz_method, + "decision procedure for linear integer arithmetic"); + +end diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Decision_Procs/ferrack_tac.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Decision_Procs/ferrack_tac.ML Fri Feb 06 15:15:46 2009 +0100 @@ -0,0 +1,113 @@ +(* Title: HOL/Reflection/ferrack_tac.ML + Author: Amine Chaieb, TU Muenchen +*) + +structure Ferrack_Tac = +struct + +val trace = ref false; +fun trace_msg s = if !trace then tracing s else (); + +val ferrack_ss = let val ths = [@{thm real_of_int_inject}, @{thm real_of_int_less_iff}, + @{thm real_of_int_le_iff}] + in @{simpset} delsimps ths addsimps (map (fn th => th RS sym) ths) + end; + +val binarith = + @{thms normalize_bin_simps} @ @{thms pred_bin_simps} @ @{thms succ_bin_simps} @ + @{thms add_bin_simps} @ @{thms minus_bin_simps} @ @{thms mult_bin_simps}; +val comp_arith = binarith @ simp_thms + +val zdvd_int = @{thm zdvd_int}; +val zdiff_int_split = @{thm zdiff_int_split}; +val all_nat = @{thm all_nat}; +val ex_nat = @{thm ex_nat}; +val number_of1 = @{thm number_of1}; +val number_of2 = @{thm number_of2}; +val split_zdiv = @{thm split_zdiv}; +val split_zmod = @{thm split_zmod}; +val mod_div_equality' = @{thm mod_div_equality'}; +val split_div' = @{thm split_div'}; +val Suc_plus1 = @{thm Suc_plus1}; +val imp_le_cong = @{thm imp_le_cong}; +val conj_le_cong = @{thm conj_le_cong}; +val nat_mod_add_eq = @{thm mod_add1_eq} RS sym; +val nat_mod_add_left_eq = @{thm mod_add_left_eq} RS sym; +val nat_mod_add_right_eq = @{thm mod_add_right_eq} RS sym; +val int_mod_add_eq = @{thm zmod_zadd1_eq} RS sym; +val int_mod_add_left_eq = @{thm zmod_zadd_left_eq} RS sym; +val int_mod_add_right_eq = @{thm zmod_zadd_right_eq} RS sym; +val nat_div_add_eq = @{thm div_add1_eq} RS sym; +val int_div_add_eq = @{thm zdiv_zadd1_eq} RS sym; +val ZDIVISION_BY_ZERO_MOD = @{thm DIVISION_BY_ZERO} RS conjunct2; +val ZDIVISION_BY_ZERO_DIV = @{thm DIVISION_BY_ZERO} RS conjunct1; + +fun prepare_for_linr sg q fm = + let + val ps = Logic.strip_params fm + val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm) + val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm) + fun mk_all ((s, T), (P,n)) = + if 0 mem loose_bnos P then + (HOLogic.all_const T $ Abs (s, T, P), n) + else (incr_boundvars ~1 P, n-1) + fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t; + val rhs = hs +(* val (rhs,irhs) = List.partition (relevant (rev ps)) hs *) + val np = length ps + val (fm',np) = foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n))) + (foldr HOLogic.mk_imp c rhs, np) ps + val (vs, _) = List.partition (fn t => q orelse (type_of t) = HOLogic.natT) + (OldTerm.term_frees fm' @ OldTerm.term_vars fm'); + val fm2 = foldr mk_all2 fm' vs + in (fm2, np + length vs, length rhs) end; + +(*Object quantifier to meta --*) +fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ; + +(* object implication to meta---*) +fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp; + + +fun linr_tac ctxt q i = + (ObjectLogic.atomize_prems_tac i) + THEN (REPEAT_DETERM (split_tac [@{thm split_min}, @{thm split_max}, @{thm abs_split}] i)) + THEN (fn st => + let + val g = List.nth (prems_of st, i - 1) + val thy = ProofContext.theory_of ctxt + (* Transform the term*) + val (t,np,nh) = prepare_for_linr thy q g + (* Some simpsets for dealing with mod div abs and nat*) + val simpset0 = Simplifier.theory_context thy HOL_basic_ss addsimps comp_arith + val ct = cterm_of thy (HOLogic.mk_Trueprop t) + (* Theorem for the nat --> int transformation *) + val pre_thm = Seq.hd (EVERY + [simp_tac simpset0 1, + TRY (simp_tac (Simplifier.theory_context thy ferrack_ss) 1)] + (trivial ct)) + fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i) + (* The result of the quantifier elimination *) + val (th, tac) = case (prop_of pre_thm) of + Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ => + let val pth = linr_oracle (cterm_of thy (Pattern.eta_long [] t1)) + in + (trace_msg ("calling procedure with term:\n" ^ + Syntax.string_of_term ctxt t1); + ((pth RS iffD2) RS pre_thm, + assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))) + end + | _ => (pre_thm, assm_tac i) + in (rtac (((mp_step nh) o (spec_step np)) th) i + THEN tac) st + end handle Subscript => no_tac st); + +fun linr_meth src = + Method.syntax (Args.mode "no_quantify") src + #> (fn (q, ctxt) => Method.SIMPLE_METHOD' (linr_tac ctxt (not q))); + +val setup = + Method.add_method ("rferrack", linr_meth, + "decision procedure for linear real arithmetic"); + +end diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Decision_Procs/mir_tac.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Decision_Procs/mir_tac.ML Fri Feb 06 15:15:46 2009 +0100 @@ -0,0 +1,168 @@ +(* Title: HOL/Reflection/mir_tac.ML + Author: Amine Chaieb, TU Muenchen +*) + +structure Mir_Tac = +struct + +val trace = ref false; +fun trace_msg s = if !trace then tracing s else (); + +val mir_ss = +let val ths = map thm ["real_of_int_inject", "real_of_int_less_iff", "real_of_int_le_iff"] +in @{simpset} delsimps ths addsimps (map (fn th => th RS sym) ths) +end; + +val nT = HOLogic.natT; + val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of", + "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"]; + + val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0", + "add_Suc", "add_number_of_left", "mult_number_of_left", + "Suc_eq_add_numeral_1"])@ + (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"]) + @ @{thms arith_simps} @ nat_arith @ @{thms rel_simps} + val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, + @{thm "real_of_nat_number_of"}, + @{thm "real_of_nat_Suc"}, @{thm "real_of_nat_one"}, @{thm "real_of_one"}, + @{thm "real_of_int_zero"}, @{thm "real_of_nat_zero"}, + @{thm "Ring_and_Field.divide_zero"}, + @{thm "divide_divide_eq_left"}, @{thm "times_divide_eq_right"}, + @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"}, + @{thm "diff_def"}, @{thm "minus_divide_left"}] +val comp_ths = ths @ comp_arith @ simp_thms + + +val zdvd_int = @{thm "zdvd_int"}; +val zdiff_int_split = @{thm "zdiff_int_split"}; +val all_nat = @{thm "all_nat"}; +val ex_nat = @{thm "ex_nat"}; +val number_of1 = @{thm "number_of1"}; +val number_of2 = @{thm "number_of2"}; +val split_zdiv = @{thm "split_zdiv"}; +val split_zmod = @{thm "split_zmod"}; +val mod_div_equality' = @{thm "mod_div_equality'"}; +val split_div' = @{thm "split_div'"}; +val Suc_plus1 = @{thm "Suc_plus1"}; +val imp_le_cong = @{thm "imp_le_cong"}; +val conj_le_cong = @{thm "conj_le_cong"}; +val nat_mod_add_eq = @{thm "mod_add1_eq"} RS sym; +val nat_mod_add_left_eq = @{thm "mod_add_left_eq"} RS sym; +val nat_mod_add_right_eq = @{thm "mod_add_right_eq"} RS sym; +val int_mod_add_eq = @{thm "zmod_zadd1_eq"} RS sym; +val int_mod_add_left_eq = @{thm "zmod_zadd_left_eq"} RS sym; +val int_mod_add_right_eq = @{thm "zmod_zadd_right_eq"} RS sym; +val nat_div_add_eq = @{thm "div_add1_eq"} RS sym; +val int_div_add_eq = @{thm "zdiv_zadd1_eq"} RS sym; +val ZDIVISION_BY_ZERO_MOD = @{thm "DIVISION_BY_ZERO"} RS conjunct2; +val ZDIVISION_BY_ZERO_DIV = @{thm "DIVISION_BY_ZERO"} RS conjunct1; + +fun prepare_for_mir thy q fm = + let + val ps = Logic.strip_params fm + val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm) + val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm) + fun mk_all ((s, T), (P,n)) = + if 0 mem loose_bnos P then + (HOLogic.all_const T $ Abs (s, T, P), n) + else (incr_boundvars ~1 P, n-1) + fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t; + val rhs = hs +(* val (rhs,irhs) = List.partition (relevant (rev ps)) hs *) + val np = length ps + val (fm',np) = foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n))) + (foldr HOLogic.mk_imp c rhs, np) ps + val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT) + (OldTerm.term_frees fm' @ OldTerm.term_vars fm'); + val fm2 = foldr mk_all2 fm' vs + in (fm2, np + length vs, length rhs) end; + +(*Object quantifier to meta --*) +fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ; + +(* object implication to meta---*) +fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp; + + +fun mir_tac ctxt q i = + (ObjectLogic.atomize_prems_tac i) + THEN (simp_tac (HOL_basic_ss addsimps [@{thm "abs_ge_zero"}] addsimps simp_thms) i) + THEN (REPEAT_DETERM (split_tac [@{thm "split_min"}, @{thm "split_max"},@{thm "abs_split"}] i)) + THEN (fn st => + let + val g = List.nth (prems_of st, i - 1) + val thy = ProofContext.theory_of ctxt + (* Transform the term*) + val (t,np,nh) = prepare_for_mir thy q g + (* Some simpsets for dealing with mod div abs and nat*) + val mod_div_simpset = HOL_basic_ss + addsimps [refl,nat_mod_add_eq, + @{thm "mod_self"}, @{thm "zmod_self"}, + @{thm "zdiv_zero"},@{thm "zmod_zero"},@{thm "div_0"}, @{thm "mod_0"}, + @{thm "zdiv_1"}, @{thm "zmod_1"}, @{thm "div_1"}, @{thm "mod_1"}, + @{thm "Suc_plus1"}] + addsimps @{thms add_ac} + addsimprocs [cancel_div_mod_proc] + val simpset0 = HOL_basic_ss + addsimps [mod_div_equality', Suc_plus1] + addsimps comp_ths + addsplits [@{thm "split_zdiv"}, @{thm "split_zmod"}, @{thm "split_div'"}, @{thm "split_min"}, @{thm "split_max"}] + (* Simp rules for changing (n::int) to int n *) + val simpset1 = HOL_basic_ss + addsimps [@{thm "nat_number_of_def"}, @{thm "zdvd_int"}] @ map (fn r => r RS sym) + [@{thm "int_int_eq"}, @{thm "zle_int"}, @{thm "zless_int"}, @{thm "zadd_int"}, + @{thm "zmult_int"}] + addsplits [@{thm "zdiff_int_split"}] + (*simp rules for elimination of int n*) + + val simpset2 = HOL_basic_ss + addsimps [@{thm "nat_0_le"}, @{thm "all_nat"}, @{thm "ex_nat"}, @{thm "number_of1"}, + @{thm "number_of2"}, @{thm "int_0"}, @{thm "int_1"}] + addcongs [@{thm "conj_le_cong"}, @{thm "imp_le_cong"}] + (* simp rules for elimination of abs *) + val ct = cterm_of thy (HOLogic.mk_Trueprop t) + (* Theorem for the nat --> int transformation *) + val pre_thm = Seq.hd (EVERY + [simp_tac mod_div_simpset 1, simp_tac simpset0 1, + TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1), TRY (simp_tac mir_ss 1)] + (trivial ct)) + fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i) + (* The result of the quantifier elimination *) + val (th, tac) = case (prop_of pre_thm) of + Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ => + let val pth = + (* If quick_and_dirty then run without proof generation as oracle*) + if !quick_and_dirty + then mirfr_oracle (false, cterm_of thy (Pattern.eta_long [] t1)) + else mirfr_oracle (true, cterm_of thy (Pattern.eta_long [] t1)) + in + (trace_msg ("calling procedure with term:\n" ^ + Syntax.string_of_term ctxt t1); + ((pth RS iffD2) RS pre_thm, + assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))) + end + | _ => (pre_thm, assm_tac i) + in (rtac (((mp_step nh) o (spec_step np)) th) i + THEN tac) st + end handle Subscript => no_tac st); + +fun mir_args meth = + let val parse_flag = + Args.$$$ "no_quantify" >> (K (K false)); + in + Method.simple_args + (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >> + curry (Library.foldl op |>) true) + (fn q => fn ctxt => meth ctxt q 1) + end; + +fun mir_method ctxt q i = Method.METHOD (fn facts => + Method.insert_tac facts 1 THEN mir_tac ctxt q i); + +val setup = + Method.add_method ("mir", + mir_args mir_method, + "decision procedure for MIR arithmetic"); + + +end diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Extraction/Pigeonhole.thy --- a/src/HOL/Extraction/Pigeonhole.thy Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/Extraction/Pigeonhole.thy Fri Feb 06 15:15:46 2009 +0100 @@ -1,5 +1,4 @@ (* Title: HOL/Extraction/Pigeonhole.thy - ID: $Id$ Author: Stefan Berghofer, TU Muenchen *) diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Extraction/Warshall.thy --- a/src/HOL/Extraction/Warshall.thy Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/Extraction/Warshall.thy Fri Feb 06 15:15:46 2009 +0100 @@ -1,5 +1,4 @@ (* Title: HOL/Extraction/Warshall.thy - ID: $Id$ Author: Stefan Berghofer, TU Muenchen *) diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Imperative_HOL/Array.thy --- a/src/HOL/Imperative_HOL/Array.thy Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/Imperative_HOL/Array.thy Fri Feb 06 15:15:46 2009 +0100 @@ -32,15 +32,6 @@ else raise (''array lookup: index out of range'')) done)" --- {* FIXME adjustion for List theory *} -no_syntax - nth :: "'a list \ nat \ 'a" (infixl "!" 100) - -abbreviation - nth_list :: "'a list \ nat \ 'a" (infixl "!" 100) -where - "nth_list \ List.nth" - definition upd :: "nat \ 'a \ 'a\heap array \ 'a\heap array Heap" where diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Imperative_HOL/Imperative_HOL.thy --- a/src/HOL/Imperative_HOL/Imperative_HOL.thy Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/Imperative_HOL/Imperative_HOL.thy Fri Feb 06 15:15:46 2009 +0100 @@ -1,5 +1,4 @@ -(* Title: HOL/Library/Imperative_HOL.thy - ID: $Id$ +(* Title: HOL/Imperative_HOL/Imperative_HOL.thy Author: John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen *) diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/IsaMakefile --- a/src/HOL/IsaMakefile Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/IsaMakefile Fri Feb 06 15:15:46 2009 +0100 @@ -15,6 +15,7 @@ HOL-Auth \ HOL-AxClasses \ HOL-Bali \ + HOL-Decision_Procs \ HOL-Extraction \ HOL-HahnBanach \ HOL-Hoare \ @@ -36,7 +37,6 @@ HOL-Nominal-Examples \ HOL-NumberTheory \ HOL-Prolog \ - HOL-Reflection \ HOL-SET-Protocol \ HOL-SizeChange \ HOL-Statespace \ @@ -315,7 +315,7 @@ Library/Abstract_Rat.thy \ Library/BigO.thy Library/ContNotDenum.thy Library/Efficient_Nat.thy \ Library/Executable_Set.thy Library/Infinite_Set.thy \ - Library/FuncSet.thy Library/Dense_Linear_Order.thy \ + Library/FuncSet.thy \ Library/Library.thy Library/List_Prefix.thy Library/State_Monad.thy \ Library/Multiset.thy Library/Permutation.thy \ Library/Primes.thy Library/Pocklington.thy Library/Quotient.thy \ @@ -658,6 +658,24 @@ @$(ISABELLE_TOOL) usedir $(OUT)/HOL SizeChange +## HOL-Decision_Procs + +HOL-Decision_Procs: HOL $(LOG)/HOL-Decision_Procs.gz + +$(LOG)/HOL-Decision_Procs.gz: $(OUT)/HOL \ + Decision_Procs/Approximation.thy \ + Decision_Procs/Cooper.thy \ + Decision_Procs/cooper_tac.ML \ + Decision_Procs/Dense_Linear_Order.thy \ + Decision_Procs/Ferrack.thy \ + Decision_Procs/ferrack_tac.ML \ + Decision_Procs/MIR.thy \ + Decision_Procs/mir_tac.ML \ + Decision_Procs/Decision_Procs.thy \ + Decision_Procs/ROOT.ML + @$(ISABELLE_TOOL) usedir $(OUT)/HOL Decision_Procs + + ## HOL-Lambda HOL-Lambda: HOL $(LOG)/HOL-Lambda.gz @@ -680,22 +698,6 @@ @$(ISABELLE_TOOL) usedir $(OUT)/HOL Prolog -## HOL-Reflection - -HOL-Reflection: HOL $(LOG)/HOL-Reflection.gz - -$(LOG)/HOL-Reflection.gz: $(OUT)/HOL \ - Reflection/Approximation.thy \ - Reflection/Cooper.thy \ - Reflection/cooper_tac.ML \ - Reflection/Ferrack.thy \ - Reflection/ferrack_tac.ML \ - Reflection/MIR.thy \ - Reflection/mir_tac.ML \ - Reflection/ROOT.ML - @$(ISABELLE_TOOL) usedir $(OUT)/HOL Reflection - - ## HOL-W0 HOL-W0: HOL $(LOG)/HOL-W0.gz diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Library/Code_Index.thy --- a/src/HOL/Library/Code_Index.thy Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/Library/Code_Index.thy Fri Feb 06 15:15:46 2009 +0100 @@ -302,8 +302,8 @@ (Haskell infixl 7 "*") code_const div_mod_index - (SML "(fn n => fn m =>/ (n div m, n mod m))") - (OCaml "(fun n -> fun m ->/ (n '/ m, n mod m))") + (SML "(fn n => fn m =>/ if m = 0/ then (0, n) else/ (n div m, n mod m))") + (OCaml "(fun n -> fun m ->/ if m = 0/ then (0, n) else/ (n '/ m, n mod m))") (Haskell "divMod") code_const "eq_class.eq \ index \ index \ bool" diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Library/Dense_Linear_Order.thy --- a/src/HOL/Library/Dense_Linear_Order.thy Fri Feb 06 14:36:58 2009 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,879 +0,0 @@ -(* Title : HOL/Dense_Linear_Order.thy - Author : Amine Chaieb, TU Muenchen -*) - -header {* Dense linear order without endpoints - and a quantifier elimination procedure in Ferrante and Rackoff style *} - -theory Dense_Linear_Order -imports Plain Groebner_Basis Main -uses - "~~/src/HOL/Tools/Qelim/langford_data.ML" - "~~/src/HOL/Tools/Qelim/ferrante_rackoff_data.ML" - ("~~/src/HOL/Tools/Qelim/langford.ML") - ("~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML") -begin - -setup {* Langford_Data.setup #> Ferrante_Rackoff_Data.setup *} - -context linorder -begin - -lemma less_not_permute[noatp]: "\ (x < y \ y < x)" by (simp add: not_less linear) - -lemma gather_simps[noatp]: - shows - "(\x. (\y \ L. y < x) \ (\y \ U. x < y) \ x < u \ P x) \ (\x. (\y \ L. y < x) \ (\y \ (insert u U). x < y) \ P x)" - and "(\x. (\y \ L. y < x) \ (\y \ U. x < y) \ l < x \ P x) \ (\x. (\y \ (insert l L). y < x) \ (\y \ U. x < y) \ P x)" - "(\x. (\y \ L. y < x) \ (\y \ U. x < y) \ x < u) \ (\x. (\y \ L. y < x) \ (\y \ (insert u U). x < y))" - and "(\x. (\y \ L. y < x) \ (\y \ U. x < y) \ l < x) \ (\x. (\y \ (insert l L). y < x) \ (\y \ U. x < y))" by auto - -lemma - gather_start[noatp]: "(\x. P x) \ (\x. (\y \ {}. y < x) \ (\y\ {}. x < y) \ P x)" - by simp - -text{* Theorems for @{text "\z. \x. x < z \ (P x \ P\<^bsub>-\\<^esub>)"}*} -lemma minf_lt[noatp]: "\z . \x. x < z \ (x < t \ True)" by auto -lemma minf_gt[noatp]: "\z . \x. x < z \ (t < x \ False)" - by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le) - -lemma minf_le[noatp]: "\z. \x. x < z \ (x \ t \ True)" by (auto simp add: less_le) -lemma minf_ge[noatp]: "\z. \x. x < z \ (t \ x \ False)" - by (auto simp add: less_le not_less not_le) -lemma minf_eq[noatp]: "\z. \x. x < z \ (x = t \ False)" by auto -lemma minf_neq[noatp]: "\z. \x. x < z \ (x \ t \ True)" by auto -lemma minf_P[noatp]: "\z. \x. x < z \ (P \ P)" by blast - -text{* Theorems for @{text "\z. \x. x < z \ (P x \ P\<^bsub>+\\<^esub>)"}*} -lemma pinf_gt[noatp]: "\z . \x. z < x \ (t < x \ True)" by auto -lemma pinf_lt[noatp]: "\z . \x. z < x \ (x < t \ False)" - by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le) - -lemma pinf_ge[noatp]: "\z. \x. z < x \ (t \ x \ True)" by (auto simp add: less_le) -lemma pinf_le[noatp]: "\z. \x. z < x \ (x \ t \ False)" - by (auto simp add: less_le not_less not_le) -lemma pinf_eq[noatp]: "\z. \x. z < x \ (x = t \ False)" by auto -lemma pinf_neq[noatp]: "\z. \x. z < x \ (x \ t \ True)" by auto -lemma pinf_P[noatp]: "\z. \x. z < x \ (P \ P)" by blast - -lemma nmi_lt[noatp]: "t \ U \ \x. \True \ x < t \ (\ u\ U. u \ x)" by auto -lemma nmi_gt[noatp]: "t \ U \ \x. \False \ t < x \ (\ u\ U. u \ x)" - by (auto simp add: le_less) -lemma nmi_le[noatp]: "t \ U \ \x. \True \ x\ t \ (\ u\ U. u \ x)" by auto -lemma nmi_ge[noatp]: "t \ U \ \x. \False \ t\ x \ (\ u\ U. u \ x)" by auto -lemma nmi_eq[noatp]: "t \ U \ \x. \False \ x = t \ (\ u\ U. u \ x)" by auto -lemma nmi_neq[noatp]: "t \ U \\x. \True \ x \ t \ (\ u\ U. u \ x)" by auto -lemma nmi_P[noatp]: "\ x. ~P \ P \ (\ u\ U. u \ x)" by auto -lemma nmi_conj[noatp]: "\\x. \P1' \ P1 x \ (\ u\ U. u \ x) ; - \x. \P2' \ P2 x \ (\ u\ U. u \ x)\ \ - \x. \(P1' \ P2') \ (P1 x \ P2 x) \ (\ u\ U. u \ x)" by auto -lemma nmi_disj[noatp]: "\\x. \P1' \ P1 x \ (\ u\ U. u \ x) ; - \x. \P2' \ P2 x \ (\ u\ U. u \ x)\ \ - \x. \(P1' \ P2') \ (P1 x \ P2 x) \ (\ u\ U. u \ x)" by auto - -lemma npi_lt[noatp]: "t \ U \ \x. \False \ x < t \ (\ u\ U. x \ u)" by (auto simp add: le_less) -lemma npi_gt[noatp]: "t \ U \ \x. \True \ t < x \ (\ u\ U. x \ u)" by auto -lemma npi_le[noatp]: "t \ U \ \x. \False \ x \ t \ (\ u\ U. x \ u)" by auto -lemma npi_ge[noatp]: "t \ U \ \x. \True \ t \ x \ (\ u\ U. x \ u)" by auto -lemma npi_eq[noatp]: "t \ U \ \x. \False \ x = t \ (\ u\ U. x \ u)" by auto -lemma npi_neq[noatp]: "t \ U \ \x. \True \ x \ t \ (\ u\ U. x \ u )" by auto -lemma npi_P[noatp]: "\ x. ~P \ P \ (\ u\ U. x \ u)" by auto -lemma npi_conj[noatp]: "\\x. \P1' \ P1 x \ (\ u\ U. x \ u) ; \x. \P2' \ P2 x \ (\ u\ U. x \ u)\ - \ \x. \(P1' \ P2') \ (P1 x \ P2 x) \ (\ u\ U. x \ u)" by auto -lemma npi_disj[noatp]: "\\x. \P1' \ P1 x \ (\ u\ U. x \ u) ; \x. \P2' \ P2 x \ (\ u\ U. x \ u)\ - \ \x. \(P1' \ P2') \ (P1 x \ P2 x) \ (\ u\ U. x \ u)" by auto - -lemma lin_dense_lt[noatp]: "t \ U \ \x l u. (\ t. l < t \ t < u \ t \ U) \ l< x \ x < u \ x < t \ (\ y. l < y \ y < u \ y < t)" -proof(clarsimp) - fix x l u y assume tU: "t \ U" and noU: "\t. l < t \ t < u \ t \ U" and lx: "l < x" - and xu: "xy" by auto - {assume H: "t < y" - from less_trans[OF lx px] less_trans[OF H yu] - have "l < t \ t < u" by simp - with tU noU have "False" by auto} - hence "\ t < y" by auto hence "y \ t" by (simp add: not_less) - thus "y < t" using tny by (simp add: less_le) -qed - -lemma lin_dense_gt[noatp]: "t \ U \ \x l u. (\ t. l < t \ t< u \ t \ U) \ l < x \ x < u \ t < x \ (\ y. l < y \ y < u \ t < y)" -proof(clarsimp) - fix x l u y - assume tU: "t \ U" and noU: "\t. l < t \ t < u \ t \ U" and lx: "l < x" and xu: "xy" by auto - {assume H: "y< t" - from less_trans[OF ly H] less_trans[OF px xu] have "l < t \ t < u" by simp - with tU noU have "False" by auto} - hence "\ y y" by (auto simp add: not_less) - thus "t < y" using tny by (simp add:less_le) -qed - -lemma lin_dense_le[noatp]: "t \ U \ \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ x \ t \ (\ y. l < y \ y < u \ y\ t)" -proof(clarsimp) - fix x l u y - assume tU: "t \ U" and noU: "\t. l < t \ t < u \ t \ U" and lx: "l < x" and xu: "x t" and ly: "ly" by auto - {assume H: "t < y" - from less_le_trans[OF lx px] less_trans[OF H yu] - have "l < t \ t < u" by simp - with tU noU have "False" by auto} - hence "\ t < y" by auto thus "y \ t" by (simp add: not_less) -qed - -lemma lin_dense_ge[noatp]: "t \ U \ \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ t \ x \ (\ y. l < y \ y < u \ t \ y)" -proof(clarsimp) - fix x l u y - assume tU: "t \ U" and noU: "\t. l < t \ t < u \ t \ U" and lx: "l < x" and xu: "x x" and ly: "ly" by auto - {assume H: "y< t" - from less_trans[OF ly H] le_less_trans[OF px xu] - have "l < t \ t < u" by simp - with tU noU have "False" by auto} - hence "\ y y" by (simp add: not_less) -qed -lemma lin_dense_eq[noatp]: "t \ U \ \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ x = t \ (\ y. l < y \ y < u \ y= t)" by auto -lemma lin_dense_neq[noatp]: "t \ U \ \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ x \ t \ (\ y. l < y \ y < u \ y\ t)" by auto -lemma lin_dense_P[noatp]: "\x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ P \ (\ y. l < y \ y < u \ P)" by auto - -lemma lin_dense_conj[noatp]: - "\\x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ P1 x - \ (\ y. l < y \ y < u \ P1 y) ; - \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ P2 x - \ (\ y. l < y \ y < u \ P2 y)\ \ - \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ (P1 x \ P2 x) - \ (\ y. l < y \ y < u \ (P1 y \ P2 y))" - by blast -lemma lin_dense_disj[noatp]: - "\\x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ P1 x - \ (\ y. l < y \ y < u \ P1 y) ; - \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ P2 x - \ (\ y. l < y \ y < u \ P2 y)\ \ - \x l u. (\ t. l < t \ t< u \ t \ U) \ l< x \ x < u \ (P1 x \ P2 x) - \ (\ y. l < y \ y < u \ (P1 y \ P2 y))" - by blast - -lemma npmibnd[noatp]: "\\x. \ MP \ P x \ (\ u\ U. u \ x); \x. \PP \ P x \ (\ u\ U. x \ u)\ - \ \x. \ MP \ \PP \ P x \ (\ u\ U. \ u' \ U. u \ x \ x \ u')" -by auto - -lemma finite_set_intervals[noatp]: - assumes px: "P x" and lx: "l \ x" and xu: "x \ u" and linS: "l\ S" - and uinS: "u \ S" and fS:"finite S" and lS: "\ x\ S. l \ x" and Su: "\ x\ S. x \ u" - shows "\ a \ S. \ b \ S. (\ y. a < y \ y < b \ y \ S) \ a \ x \ x \ b \ P x" -proof- - let ?Mx = "{y. y\ S \ y \ x}" - let ?xM = "{y. y\ S \ x \ y}" - let ?a = "Max ?Mx" - let ?b = "Min ?xM" - have MxS: "?Mx \ S" by blast - hence fMx: "finite ?Mx" using fS finite_subset by auto - from lx linS have linMx: "l \ ?Mx" by blast - hence Mxne: "?Mx \ {}" by blast - have xMS: "?xM \ S" by blast - hence fxM: "finite ?xM" using fS finite_subset by auto - from xu uinS have linxM: "u \ ?xM" by blast - hence xMne: "?xM \ {}" by blast - have ax:"?a \ x" using Mxne fMx by auto - have xb:"x \ ?b" using xMne fxM by auto - have "?a \ ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \ S" using MxS by blast - have "?b \ ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \ S" using xMS by blast - have noy:"\ y. ?a < y \ y < ?b \ y \ S" - proof(clarsimp) - fix y assume ay: "?a < y" and yb: "y < ?b" and yS: "y \ S" - from yS have "y\ ?Mx \ y\ ?xM" by (auto simp add: linear) - moreover {assume "y \ ?Mx" hence "y \ ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])} - moreover {assume "y \ ?xM" hence "?b \ y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])} - ultimately show "False" by blast - qed - from ainS binS noy ax xb px show ?thesis by blast -qed - -lemma finite_set_intervals2[noatp]: - assumes px: "P x" and lx: "l \ x" and xu: "x \ u" and linS: "l\ S" - and uinS: "u \ S" and fS:"finite S" and lS: "\ x\ S. l \ x" and Su: "\ x\ S. x \ u" - shows "(\ s\ S. P s) \ (\ a \ S. \ b \ S. (\ y. a < y \ y < b \ y \ S) \ a < x \ x < b \ P x)" -proof- - from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su] - obtain a and b where - as: "a\ S" and bs: "b\ S" and noS:"\y. a < y \ y < b \ y \ S" - and axb: "a \ x \ x \ b \ P x" by auto - from axb have "x= a \ x= b \ (a < x \ x < b)" by (auto simp add: le_less) - thus ?thesis using px as bs noS by blast -qed - -end - -section {* The classical QE after Langford for dense linear orders *} - -context dense_linear_order -begin - -lemma interval_empty_iff: - "{y. x < y \ y < z} = {} \ \ x < z" - by (auto dest: dense) - -lemma dlo_qe_bnds[noatp]: - assumes ne: "L \ {}" and neU: "U \ {}" and fL: "finite L" and fU: "finite U" - shows "(\x. (\y \ L. y < x) \ (\y \ U. x < y)) \ (\ l \ L. \u \ U. l < u)" -proof (simp only: atomize_eq, rule iffI) - assume H: "\x. (\y\L. y < x) \ (\y\U. x < y)" - then obtain x where xL: "\y\L. y < x" and xU: "\y\U. x < y" by blast - {fix l u assume l: "l \ L" and u: "u \ U" - have "l < x" using xL l by blast - also have "x < u" using xU u by blast - finally (less_trans) have "l < u" .} - thus "\l\L. \u\U. l < u" by blast -next - assume H: "\l\L. \u\U. l < u" - let ?ML = "Max L" - let ?MU = "Min U" - from fL ne have th1: "?ML \ L" and th1': "\l\L. l \ ?ML" by auto - from fU neU have th2: "?MU \ U" and th2': "\u\U. ?MU \ u" by auto - from th1 th2 H have "?ML < ?MU" by auto - with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" by blast - from th3 th1' have "\l \ L. l < w" by auto - moreover from th4 th2' have "\u \ U. w < u" by auto - ultimately show "\x. (\y\L. y < x) \ (\y\U. x < y)" by auto -qed - -lemma dlo_qe_noub[noatp]: - assumes ne: "L \ {}" and fL: "finite L" - shows "(\x. (\y \ L. y < x) \ (\y \ {}. x < y)) \ True" -proof(simp add: atomize_eq) - from gt_ex[of "Max L"] obtain M where M: "Max L < M" by blast - from ne fL have "\x \ L. x \ Max L" by simp - with M have "\x\L. x < M" by (auto intro: le_less_trans) - thus "\x. \y\L. y < x" by blast -qed - -lemma dlo_qe_nolb[noatp]: - assumes ne: "U \ {}" and fU: "finite U" - shows "(\x. (\y \ {}. y < x) \ (\y \ U. x < y)) \ True" -proof(simp add: atomize_eq) - from lt_ex[of "Min U"] obtain M where M: "M < Min U" by blast - from ne fU have "\x \ U. Min U \ x" by simp - with M have "\x\U. M < x" by (auto intro: less_le_trans) - thus "\x. \y\U. x < y" by blast -qed - -lemma exists_neq[noatp]: "\(x::'a). x \ t" "\(x::'a). t \ x" - using gt_ex[of t] by auto - -lemmas dlo_simps[noatp] = order_refl less_irrefl not_less not_le exists_neq - le_less neq_iff linear less_not_permute - -lemma axiom[noatp]: "dense_linear_order (op \) (op <)" by (rule dense_linear_order_axioms) -lemma atoms[noatp]: - shows "TERM (less :: 'a \ _)" - and "TERM (less_eq :: 'a \ _)" - and "TERM (op = :: 'a \ _)" . - -declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms] -declare dlo_simps[langfordsimp] - -end - -(* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *) -lemma dnf[noatp]: - "(P & (Q | R)) = ((P&Q) | (P&R))" - "((Q | R) & P) = ((Q&P) | (R&P))" - by blast+ - -lemmas weak_dnf_simps[noatp] = simp_thms dnf - -lemma nnf_simps[noatp]: - "(\(P \ Q)) = (\P \ \Q)" "(\(P \ Q)) = (\P \ \Q)" "(P \ Q) = (\P \ Q)" - "(P = Q) = ((P \ Q) \ (\P \ \ Q))" "(\ \(P)) = P" - by blast+ - -lemma ex_distrib[noatp]: "(\x. P x \ Q x) \ ((\x. P x) \ (\x. Q x))" by blast - -lemmas dnf_simps[noatp] = weak_dnf_simps nnf_simps ex_distrib - -use "~~/src/HOL/Tools/Qelim/langford.ML" -method_setup dlo = {* - Method.ctxt_args (Method.SIMPLE_METHOD' o LangfordQE.dlo_tac) -*} "Langford's algorithm for quantifier elimination in dense linear orders" - - -section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields -- see @{text "Arith_Tools.thy"} *} - -text {* Linear order without upper bounds *} - -locale linorder_stupid_syntax = linorder -begin -notation - less_eq ("op \") and - less_eq ("(_/ \ _)" [51, 51] 50) and - less ("op \") and - less ("(_/ \ _)" [51, 51] 50) - -end - -locale linorder_no_ub = linorder_stupid_syntax + - assumes gt_ex: "\y. less x y" -begin -lemma ge_ex[noatp]: "\y. x \ y" using gt_ex by auto - -text {* Theorems for @{text "\z. \x. z \ x \ (P x \ P\<^bsub>+\\<^esub>)"} *} -lemma pinf_conj[noatp]: - assumes ex1: "\z1. \x. z1 \ x \ (P1 x \ P1')" - and ex2: "\z2. \x. z2 \ x \ (P2 x \ P2')" - shows "\z. \x. z \ x \ ((P1 x \ P2 x) \ (P1' \ P2'))" -proof- - from ex1 ex2 obtain z1 and z2 where z1: "\x. z1 \ x \ (P1 x \ P1')" - and z2: "\x. z2 \ x \ (P2 x \ P2')" by blast - from gt_ex obtain z where z:"ord.max less_eq z1 z2 \ z" by blast - from z have zz1: "z1 \ z" and zz2: "z2 \ z" by simp_all - {fix x assume H: "z \ x" - from less_trans[OF zz1 H] less_trans[OF zz2 H] - have "(P1 x \ P2 x) \ (P1' \ P2')" using z1 zz1 z2 zz2 by auto - } - thus ?thesis by blast -qed - -lemma pinf_disj[noatp]: - assumes ex1: "\z1. \x. z1 \ x \ (P1 x \ P1')" - and ex2: "\z2. \x. z2 \ x \ (P2 x \ P2')" - shows "\z. \x. z \ x \ ((P1 x \ P2 x) \ (P1' \ P2'))" -proof- - from ex1 ex2 obtain z1 and z2 where z1: "\x. z1 \ x \ (P1 x \ P1')" - and z2: "\x. z2 \ x \ (P2 x \ P2')" by blast - from gt_ex obtain z where z:"ord.max less_eq z1 z2 \ z" by blast - from z have zz1: "z1 \ z" and zz2: "z2 \ z" by simp_all - {fix x assume H: "z \ x" - from less_trans[OF zz1 H] less_trans[OF zz2 H] - have "(P1 x \ P2 x) \ (P1' \ P2')" using z1 zz1 z2 zz2 by auto - } - thus ?thesis by blast -qed - -lemma pinf_ex[noatp]: assumes ex:"\z. \x. z \ x \ (P x \ P1)" and p1: P1 shows "\ x. P x" -proof- - from ex obtain z where z: "\x. z \ x \ (P x \ P1)" by blast - from gt_ex obtain x where x: "z \ x" by blast - from z x p1 show ?thesis by blast -qed - -end - -text {* Linear order without upper bounds *} - -locale linorder_no_lb = linorder_stupid_syntax + - assumes lt_ex: "\y. less y x" -begin -lemma le_ex[noatp]: "\y. y \ x" using lt_ex by auto - - -text {* Theorems for @{text "\z. \x. x \ z \ (P x \ P\<^bsub>-\\<^esub>)"} *} -lemma minf_conj[noatp]: - assumes ex1: "\z1. \x. x \ z1 \ (P1 x \ P1')" - and ex2: "\z2. \x. x \ z2 \ (P2 x \ P2')" - shows "\z. \x. x \ z \ ((P1 x \ P2 x) \ (P1' \ P2'))" -proof- - from ex1 ex2 obtain z1 and z2 where z1: "\x. x \ z1 \ (P1 x \ P1')"and z2: "\x. x \ z2 \ (P2 x \ P2')" by blast - from lt_ex obtain z where z:"z \ ord.min less_eq z1 z2" by blast - from z have zz1: "z \ z1" and zz2: "z \ z2" by simp_all - {fix x assume H: "x \ z" - from less_trans[OF H zz1] less_trans[OF H zz2] - have "(P1 x \ P2 x) \ (P1' \ P2')" using z1 zz1 z2 zz2 by auto - } - thus ?thesis by blast -qed - -lemma minf_disj[noatp]: - assumes ex1: "\z1. \x. x \ z1 \ (P1 x \ P1')" - and ex2: "\z2. \x. x \ z2 \ (P2 x \ P2')" - shows "\z. \x. x \ z \ ((P1 x \ P2 x) \ (P1' \ P2'))" -proof- - from ex1 ex2 obtain z1 and z2 where z1: "\x. x \ z1 \ (P1 x \ P1')"and z2: "\x. x \ z2 \ (P2 x \ P2')" by blast - from lt_ex obtain z where z:"z \ ord.min less_eq z1 z2" by blast - from z have zz1: "z \ z1" and zz2: "z \ z2" by simp_all - {fix x assume H: "x \ z" - from less_trans[OF H zz1] less_trans[OF H zz2] - have "(P1 x \ P2 x) \ (P1' \ P2')" using z1 zz1 z2 zz2 by auto - } - thus ?thesis by blast -qed - -lemma minf_ex[noatp]: assumes ex:"\z. \x. x \ z \ (P x \ P1)" and p1: P1 shows "\ x. P x" -proof- - from ex obtain z where z: "\x. x \ z \ (P x \ P1)" by blast - from lt_ex obtain x where x: "x \ z" by blast - from z x p1 show ?thesis by blast -qed - -end - - -locale constr_dense_linear_order = linorder_no_lb + linorder_no_ub + - fixes between - assumes between_less: "less x y \ less x (between x y) \ less (between x y) y" - and between_same: "between x x = x" - -sublocale constr_dense_linear_order < dense_linear_order - apply unfold_locales - using gt_ex lt_ex between_less - by (auto, rule_tac x="between x y" in exI, simp) - -context constr_dense_linear_order -begin - -lemma rinf_U[noatp]: - assumes fU: "finite U" - and lin_dense: "\x l u. (\ t. l \ t \ t\ u \ t \ U) \ l\ x \ x \ u \ P x - \ (\ y. l \ y \ y \ u \ P y )" - and nmpiU: "\x. \ MP \ \PP \ P x \ (\ u\ U. \ u' \ U. u \ x \ x \ u')" - and nmi: "\ MP" and npi: "\ PP" and ex: "\ x. P x" - shows "\ u\ U. \ u' \ U. P (between u u')" -proof- - from ex obtain x where px: "P x" by blast - from px nmi npi nmpiU have "\ u\ U. \ u' \ U. u \ x \ x \ u'" by auto - then obtain u and u' where uU:"u\ U" and uU': "u' \ U" and ux:"u \ x" and xu':"x \ u'" by auto - from uU have Une: "U \ {}" by auto - term "linorder.Min less_eq" - let ?l = "linorder.Min less_eq U" - let ?u = "linorder.Max less_eq U" - have linM: "?l \ U" using fU Une by simp - have uinM: "?u \ U" using fU Une by simp - have lM: "\ t\ U. ?l \ t" using Une fU by auto - have Mu: "\ t\ U. t \ ?u" using Une fU by auto - have th:"?l \ u" using uU Une lM by auto - from order_trans[OF th ux] have lx: "?l \ x" . - have th: "u' \ ?u" using uU' Une Mu by simp - from order_trans[OF xu' th] have xu: "x \ ?u" . - from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu] - have "(\ s\ U. P s) \ - (\ t1\ U. \ t2 \ U. (\ y. t1 \ y \ y \ t2 \ y \ U) \ t1 \ x \ x \ t2 \ P x)" . - moreover { fix u assume um: "u\U" and pu: "P u" - have "between u u = u" by (simp add: between_same) - with um pu have "P (between u u)" by simp - with um have ?thesis by blast} - moreover{ - assume "\ t1\ U. \ t2 \ U. (\ y. t1 \ y \ y \ t2 \ y \ U) \ t1 \ x \ x \ t2 \ P x" - then obtain t1 and t2 where t1M: "t1 \ U" and t2M: "t2\ U" - and noM: "\ y. t1 \ y \ y \ t2 \ y \ U" and t1x: "t1 \ x" and xt2: "x \ t2" and px: "P x" - by blast - from less_trans[OF t1x xt2] have t1t2: "t1 \ t2" . - let ?u = "between t1 t2" - from between_less t1t2 have t1lu: "t1 \ ?u" and ut2: "?u \ t2" by auto - from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast - with t1M t2M have ?thesis by blast} - ultimately show ?thesis by blast - qed - -theorem fr_eq[noatp]: - assumes fU: "finite U" - and lin_dense: "\x l u. (\ t. l \ t \ t\ u \ t \ U) \ l\ x \ x \ u \ P x - \ (\ y. l \ y \ y \ u \ P y )" - and nmibnd: "\x. \ MP \ P x \ (\ u\ U. u \ x)" - and npibnd: "\x. \PP \ P x \ (\ u\ U. x \ u)" - and mi: "\z. \x. x \ z \ (P x = MP)" and pi: "\z. \x. z \ x \ (P x = PP)" - shows "(\ x. P x) \ (MP \ PP \ (\ u \ U. \ u'\ U. P (between u u')))" - (is "_ \ (_ \ _ \ ?F)" is "?E \ ?D") -proof- - { - assume px: "\ x. P x" - have "MP \ PP \ (\ MP \ \ PP)" by blast - moreover {assume "MP \ PP" hence "?D" by blast} - moreover {assume nmi: "\ MP" and npi: "\ PP" - from npmibnd[OF nmibnd npibnd] - have nmpiU: "\x. \ MP \ \PP \ P x \ (\ u\ U. \ u' \ U. u \ x \ x \ u')" . - from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast} - ultimately have "?D" by blast} - moreover - { assume "?D" - moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .} - moreover {assume p: "PP" from pinf_ex[OF pi p] have "?E" . } - moreover {assume f:"?F" hence "?E" by blast} - ultimately have "?E" by blast} - ultimately have "?E = ?D" by blast thus "?E \ ?D" by simp -qed - -lemmas minf_thms[noatp] = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P -lemmas pinf_thms[noatp] = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P - -lemmas nmi_thms[noatp] = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P -lemmas npi_thms[noatp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P -lemmas lin_dense_thms[noatp] = lin_dense_conj lin_dense_disj lin_dense_eq lin_dense_neq lin_dense_lt lin_dense_le lin_dense_gt lin_dense_ge lin_dense_P - -lemma ferrack_axiom[noatp]: "constr_dense_linear_order less_eq less between" - by (rule constr_dense_linear_order_axioms) -lemma atoms[noatp]: - shows "TERM (less :: 'a \ _)" - and "TERM (less_eq :: 'a \ _)" - and "TERM (op = :: 'a \ _)" . - -declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms - nmi: nmi_thms npi: npi_thms lindense: - lin_dense_thms qe: fr_eq atoms: atoms] - -declaration {* -let -fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}] -fun generic_whatis phi = - let - val [lt, le] = map (Morphism.term phi) [@{term "op \"}, @{term "op \"}] - fun h x t = - case term_of t of - Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq - else Ferrante_Rackoff_Data.Nox - | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq - else Ferrante_Rackoff_Data.Nox - | b$y$z => if Term.could_unify (b, lt) then - if term_of x aconv y then Ferrante_Rackoff_Data.Lt - else if term_of x aconv z then Ferrante_Rackoff_Data.Gt - else Ferrante_Rackoff_Data.Nox - else if Term.could_unify (b, le) then - if term_of x aconv y then Ferrante_Rackoff_Data.Le - else if term_of x aconv z then Ferrante_Rackoff_Data.Ge - else Ferrante_Rackoff_Data.Nox - else Ferrante_Rackoff_Data.Nox - | _ => Ferrante_Rackoff_Data.Nox - in h end - fun ss phi = HOL_ss addsimps (simps phi) -in - Ferrante_Rackoff_Data.funs @{thm "ferrack_axiom"} - {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss} -end -*} - -end - -use "~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML" - -method_setup ferrack = {* - Method.ctxt_args (Method.SIMPLE_METHOD' o FerranteRackoff.dlo_tac) -*} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders" - -subsection {* Ferrante and Rackoff algorithm over ordered fields *} - -lemma neg_prod_lt:"(c\'a\ordered_field) < 0 \ ((c*x < 0) == (x > 0))" -proof- - assume H: "c < 0" - have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps) - also have "\ = (0 < x)" by simp - finally show "(c*x < 0) == (x > 0)" by simp -qed - -lemma pos_prod_lt:"(c\'a\ordered_field) > 0 \ ((c*x < 0) == (x < 0))" -proof- - assume H: "c > 0" - hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps) - also have "\ = (0 > x)" by simp - finally show "(c*x < 0) == (x < 0)" by simp -qed - -lemma neg_prod_sum_lt: "(c\'a\ordered_field) < 0 \ ((c*x + t< 0) == (x > (- 1/c)*t))" -proof- - assume H: "c < 0" - have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp) - also have "\ = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps) - also have "\ = ((- 1/c)*t < x)" by simp - finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simp -qed - -lemma pos_prod_sum_lt:"(c\'a\ordered_field) > 0 \ ((c*x + t < 0) == (x < (- 1/c)*t))" -proof- - assume H: "c > 0" - have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp) - also have "\ = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps) - also have "\ = ((- 1/c)*t > x)" by simp - finally show "(c*x + t < 0) == (x < (- 1/c)*t)" by simp -qed - -lemma sum_lt:"((x::'a::pordered_ab_group_add) + t < 0) == (x < - t)" - using less_diff_eq[where a= x and b=t and c=0] by simp - -lemma neg_prod_le:"(c\'a\ordered_field) < 0 \ ((c*x <= 0) == (x >= 0))" -proof- - assume H: "c < 0" - have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps) - also have "\ = (0 <= x)" by simp - finally show "(c*x <= 0) == (x >= 0)" by simp -qed - -lemma pos_prod_le:"(c\'a\ordered_field) > 0 \ ((c*x <= 0) == (x <= 0))" -proof- - assume H: "c > 0" - hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps) - also have "\ = (0 >= x)" by simp - finally show "(c*x <= 0) == (x <= 0)" by simp -qed - -lemma neg_prod_sum_le: "(c\'a\ordered_field) < 0 \ ((c*x + t <= 0) == (x >= (- 1/c)*t))" -proof- - assume H: "c < 0" - have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp) - also have "\ = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps) - also have "\ = ((- 1/c)*t <= x)" by simp - finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp -qed - -lemma pos_prod_sum_le:"(c\'a\ordered_field) > 0 \ ((c*x + t <= 0) == (x <= (- 1/c)*t))" -proof- - assume H: "c > 0" - have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp) - also have "\ = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps) - also have "\ = ((- 1/c)*t >= x)" by simp - finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp -qed - -lemma sum_le:"((x::'a::pordered_ab_group_add) + t <= 0) == (x <= - t)" - using le_diff_eq[where a= x and b=t and c=0] by simp - -lemma nz_prod_eq:"(c\'a\ordered_field) \ 0 \ ((c*x = 0) == (x = 0))" by simp -lemma nz_prod_sum_eq: "(c\'a\ordered_field) \ 0 \ ((c*x + t = 0) == (x = (- 1/c)*t))" -proof- - assume H: "c \ 0" - have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp) - also have "\ = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] algebra_simps) - finally show "(c*x + t = 0) == (x = (- 1/c)*t)" by simp -qed -lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)" - using eq_diff_eq[where a= x and b=t and c=0] by simp - - -interpretation class_ordered_field_dense_linear_order!: constr_dense_linear_order - "op <=" "op <" - "\ x y. 1/2 * ((x::'a::{ordered_field,recpower,number_ring}) + y)" -proof (unfold_locales, dlo, dlo, auto) - fix x y::'a assume lt: "x < y" - from less_half_sum[OF lt] show "x < (x + y) /2" by simp -next - fix x y::'a assume lt: "x < y" - from gt_half_sum[OF lt] show "(x + y) /2 < y" by simp -qed - -declaration{* -let -fun earlier [] x y = false - | earlier (h::t) x y = - if h aconvc y then false else if h aconvc x then true else earlier t x y; - -fun dest_frac ct = case term_of ct of - Const (@{const_name "HOL.divide"},_) $ a $ b=> - Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b)) - | t => Rat.rat_of_int (snd (HOLogic.dest_number t)) - -fun mk_frac phi cT x = - let val (a, b) = Rat.quotient_of_rat x - in if b = 1 then Numeral.mk_cnumber cT a - else Thm.capply - (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"}) - (Numeral.mk_cnumber cT a)) - (Numeral.mk_cnumber cT b) - end - -fun whatis x ct = case term_of ct of - Const(@{const_name "HOL.plus"}, _)$(Const(@{const_name "HOL.times"},_)$_$y)$_ => - if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct]) - else ("Nox",[]) -| Const(@{const_name "HOL.plus"}, _)$y$_ => - if y aconv term_of x then ("x+t",[Thm.dest_arg ct]) - else ("Nox",[]) -| Const(@{const_name "HOL.times"}, _)$_$y => - if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct]) - else ("Nox",[]) -| t => if t aconv term_of x then ("x",[]) else ("Nox",[]); - -fun xnormalize_conv ctxt [] ct = reflexive ct -| xnormalize_conv ctxt (vs as (x::_)) ct = - case term_of ct of - Const(@{const_name HOL.less},_)$_$Const(@{const_name "HOL.zero"},_) => - (case whatis x (Thm.dest_arg1 ct) of - ("c*x+t",[c,t]) => - let - val cr = dest_frac c - val clt = Thm.dest_fun2 ct - val cz = Thm.dest_arg ct - val neg = cr - let - val T = ctyp_of_term x - val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"} - val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv - (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th - in rth end - | ("c*x",[c]) => - let - val cr = dest_frac c - val clt = Thm.dest_fun2 ct - val cz = Thm.dest_arg ct - val neg = cr reflexive ct) - - -| Const(@{const_name HOL.less_eq},_)$_$Const(@{const_name "HOL.zero"},_) => - (case whatis x (Thm.dest_arg1 ct) of - ("c*x+t",[c,t]) => - let - val T = ctyp_of_term x - val cr = dest_frac c - val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"} - val cz = Thm.dest_arg ct - val neg = cr - let - val T = ctyp_of_term x - val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"} - val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv - (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th - in rth end - | ("c*x",[c]) => - let - val T = ctyp_of_term x - val cr = dest_frac c - val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"} - val cz = Thm.dest_arg ct - val neg = cr reflexive ct) - -| Const("op =",_)$_$Const(@{const_name "HOL.zero"},_) => - (case whatis x (Thm.dest_arg1 ct) of - ("c*x+t",[c,t]) => - let - val T = ctyp_of_term x - val cr = dest_frac c - val ceq = Thm.dest_fun2 ct - val cz = Thm.dest_arg ct - val cthp = Simplifier.rewrite (local_simpset_of ctxt) - (Thm.capply @{cterm "Trueprop"} - (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz))) - val cth = equal_elim (symmetric cthp) TrueI - val th = implies_elim - (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth - val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv - (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th - in rth end - | ("x+t",[t]) => - let - val T = ctyp_of_term x - val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"} - val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv - (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th - in rth end - | ("c*x",[c]) => - let - val T = ctyp_of_term x - val cr = dest_frac c - val ceq = Thm.dest_fun2 ct - val cz = Thm.dest_arg ct - val cthp = Simplifier.rewrite (local_simpset_of ctxt) - (Thm.capply @{cterm "Trueprop"} - (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz))) - val cth = equal_elim (symmetric cthp) TrueI - val rth = implies_elim - (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth - in rth end - | _ => reflexive ct); - -local - val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"} - val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"} - val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"} -in -fun field_isolate_conv phi ctxt vs ct = case term_of ct of - Const(@{const_name HOL.less},_)$a$b => - let val (ca,cb) = Thm.dest_binop ct - val T = ctyp_of_term ca - val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0 - val nth = Conv.fconv_rule - (Conv.arg_conv (Conv.arg1_conv - (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th - val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) - in rth end -| Const(@{const_name HOL.less_eq},_)$a$b => - let val (ca,cb) = Thm.dest_binop ct - val T = ctyp_of_term ca - val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0 - val nth = Conv.fconv_rule - (Conv.arg_conv (Conv.arg1_conv - (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th - val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) - in rth end - -| Const("op =",_)$a$b => - let val (ca,cb) = Thm.dest_binop ct - val T = ctyp_of_term ca - val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0 - val nth = Conv.fconv_rule - (Conv.arg_conv (Conv.arg1_conv - (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th - val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) - in rth end -| @{term "Not"} $(Const("op =",_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct -| _ => reflexive ct -end; - -fun classfield_whatis phi = - let - fun h x t = - case term_of t of - Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq - else Ferrante_Rackoff_Data.Nox - | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq - else Ferrante_Rackoff_Data.Nox - | Const(@{const_name HOL.less},_)$y$z => - if term_of x aconv y then Ferrante_Rackoff_Data.Lt - else if term_of x aconv z then Ferrante_Rackoff_Data.Gt - else Ferrante_Rackoff_Data.Nox - | Const (@{const_name HOL.less_eq},_)$y$z => - if term_of x aconv y then Ferrante_Rackoff_Data.Le - else if term_of x aconv z then Ferrante_Rackoff_Data.Ge - else Ferrante_Rackoff_Data.Nox - | _ => Ferrante_Rackoff_Data.Nox - in h end; -fun class_field_ss phi = - HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}]) - addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}] - -in -Ferrante_Rackoff_Data.funs @{thm "class_ordered_field_dense_linear_order.ferrack_axiom"} - {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss} -end -*} - - -end diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Library/Library.thy --- a/src/HOL/Library/Library.thy Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/Library/Library.thy Fri Feb 06 15:15:46 2009 +0100 @@ -15,7 +15,6 @@ Continuity ContNotDenum Countable - Dense_Linear_Order Efficient_Nat Enum Eval_Witness diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Library/Quickcheck.thy --- a/src/HOL/Library/Quickcheck.thy Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/Library/Quickcheck.thy Fri Feb 06 15:15:46 2009 +0100 @@ -17,7 +17,7 @@ begin definition - "random _ = return (TYPE('a), \u. Code_Eval.Const (STR ''TYPE'') TYPEREP('a))" + "random _ = Pair (TYPE('a), \u. Code_Eval.Const (STR ''TYPE'') TYPEREP('a))" instance .. diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Library/Random.thy --- a/src/HOL/Library/Random.thy Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/Library/Random.thy Fri Feb 06 15:15:46 2009 +0100 @@ -3,33 +3,29 @@ header {* A HOL random engine *} theory Random -imports State_Monad Code_Index +imports Code_Index begin +notation fcomp (infixl "o>" 60) +notation scomp (infixl "o\" 60) + + subsection {* Auxiliary functions *} -definition - inc_shift :: "index \ index \ index" -where +definition inc_shift :: "index \ index \ index" where "inc_shift v k = (if v = k then 1 else k + 1)" -definition - minus_shift :: "index \ index \ index \ index" -where +definition minus_shift :: "index \ index \ index \ index" where "minus_shift r k l = (if k < l then r + k - l else k - l)" -fun - log :: "index \ index \ index" -where +fun log :: "index \ index \ index" where "log b i = (if b \ 1 \ i < b then 1 else 1 + log b (i div b))" subsection {* Random seeds *} types seed = "index \ index" -primrec - "next" :: "seed \ index \ seed" -where +primrec "next" :: "seed \ index \ seed" where "next (v, w) = (let k = v div 53668; v' = minus_shift 2147483563 (40014 * (v mod 53668)) (k * 12211); @@ -40,22 +36,15 @@ lemma next_not_0: "fst (next s) \ 0" -apply (cases s) -apply (auto simp add: minus_shift_def Let_def) -done + by (cases s) (auto simp add: minus_shift_def Let_def) -primrec - seed_invariant :: "seed \ bool" -where +primrec seed_invariant :: "seed \ bool" where "seed_invariant (v, w) \ 0 < v \ v < 9438322952 \ 0 < w \ True" -lemma if_same: - "(if b then f x else f y) = f (if b then x else y)" +lemma if_same: "(if b then f x else f y) = f (if b then x else y)" by (cases b) simp_all -definition - split_seed :: "seed \ seed \ seed" -where +definition split_seed :: "seed \ seed \ seed" where "split_seed s = (let (v, w) = s; (v', w') = snd (next s); @@ -68,9 +57,7 @@ subsection {* Base selectors *} -function - range_aux :: "index \ index \ seed \ index \ seed" -where +function range_aux :: "index \ index \ seed \ index \ seed" where "range_aux k l s = (if k = 0 then (l, s) else let (v, s') = next s in range_aux (k - 1) (v + l * 2147483561) s')" @@ -79,13 +66,9 @@ by (relation "measure (Code_Index.nat_of o fst)") (auto simp add: index) -definition - range :: "index \ seed \ index \ seed" -where - "range k = (do - v \ range_aux (log 2147483561 k) 1; - return (v mod k) - done)" +definition range :: "index \ seed \ index \ seed" where + "range k = range_aux (log 2147483561 k) 1 + o\ (\v. Pair (v mod k))" lemma range: assumes "k > 0" @@ -95,17 +78,13 @@ "range_aux (log 2147483561 k) 1 s = (v, w)" by (cases "range_aux (log 2147483561 k) 1 s") with assms show ?thesis - by (simp add: monad_collapse range_def del: range_aux.simps log.simps) + by (simp add: scomp_apply range_def del: range_aux.simps log.simps) qed -definition - select :: "'a list \ seed \ 'a \ seed" -where - "select xs = (do - k \ range (Code_Index.of_nat (length xs)); - return (nth xs (Code_Index.nat_of k)) - done)" - +definition select :: "'a list \ seed \ 'a \ seed" where + "select xs = range (Code_Index.of_nat (length xs)) + o\ (\k. Pair (nth xs (Code_Index.nat_of k)))" + lemma select: assumes "xs \ []" shows "fst (select xs s) \ set xs" @@ -116,34 +95,29 @@ then have "Code_Index.nat_of (fst (range (Code_Index.of_nat (length xs)) s)) < length xs" by simp then show ?thesis - by (auto simp add: monad_collapse select_def) + by (simp add: scomp_apply split_beta select_def) qed -definition - select_default :: "index \ 'a \ 'a \ seed \ 'a \ seed" -where - [code del]: "select_default k x y = (do - l \ range k; - return (if l + 1 < k then x else y) - done)" +definition select_default :: "index \ 'a \ 'a \ seed \ 'a \ seed" where + [code del]: "select_default k x y = range k + o\ (\l. Pair (if l + 1 < k then x else y))" lemma select_default_zero: "fst (select_default 0 x y s) = y" - by (simp add: monad_collapse select_default_def) + by (simp add: scomp_apply split_beta select_default_def) lemma select_default_code [code]: - "select_default k x y = (if k = 0 then do - _ \ range 1; - return y - done else do - l \ range k; - return (if l + 1 < k then x else y) - done)" -proof (cases "k = 0") - case False then show ?thesis by (simp add: select_default_def) -next - case True then show ?thesis - by (simp add: monad_collapse select_default_def range_def) + "select_default k x y = (if k = 0 + then range 1 o\ (\_. Pair y) + else range k o\ (\l. Pair (if l + 1 < k then x else y)))" +proof + fix s + have "snd (range (Code_Index.of_nat 0) s) = snd (range (Code_Index.of_nat 1) s)" + by (simp add: range_def scomp_Pair scomp_apply split_beta) + then show "select_default k x y s = (if k = 0 + then range 1 o\ (\_. Pair y) + else range k o\ (\l. Pair (if l + 1 < k then x else y))) s" + by (cases "k = 0") (simp_all add: select_default_def scomp_apply split_beta) qed @@ -177,5 +151,8 @@ end; *} +no_notation fcomp (infixl "o>" 60) +no_notation scomp (infixl "o\" 60) + end diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/List.thy --- a/src/HOL/List.thy Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/List.thy Fri Feb 06 15:15:46 2009 +0100 @@ -27,7 +27,6 @@ set :: "'a list => 'a set" map :: "('a=>'b) => ('a list => 'b list)" listsum :: "'a list => 'a::monoid_add" - nth :: "'a list => nat => 'a" (infixl "!" 100) list_update :: "'a list => nat => 'a => 'a list" take:: "nat => 'a list => 'a list" drop:: "nat => 'a list => 'a list" @@ -146,8 +145,8 @@ -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} -primrec - nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)" +primrec nth :: "'a list => nat => 'a" (infixl "!" 100) where + nth_Cons: "(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/MetisExamples/BigO.thy --- a/src/HOL/MetisExamples/BigO.thy Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/MetisExamples/BigO.thy Fri Feb 06 15:15:46 2009 +0100 @@ -7,7 +7,7 @@ header {* Big O notation *} theory BigO -imports Dense_Linear_Order Main SetsAndFunctions +imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Main SetsAndFunctions begin subsection {* Definitions *} diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Orderings.thy --- a/src/HOL/Orderings.thy Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/Orderings.thy Fri Feb 06 15:15:46 2009 +0100 @@ -1033,7 +1033,6 @@ assumes gt_ex: "\y. x < y" and lt_ex: "\y. y < x" and dense: "x < y \ (\z. x < z \ z < y)" - (*see further theory Dense_Linear_Order*) subsection {* Wellorders *} diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/Tools/Qelim/generated_cooper.ML --- a/src/HOL/Tools/Qelim/generated_cooper.ML Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/Tools/Qelim/generated_cooper.ML Fri Feb 06 15:15:46 2009 +0100 @@ -1,6 +1,6 @@ (* Title: HOL/Tools/Qelim/generated_cooper.ML -This file is generated from HOL/Reflection/Cooper.thy. DO NOT EDIT. +This file is generated from HOL/Decision_Procs/Cooper.thy. DO NOT EDIT. *) structure GeneratedCooper = diff -r ab8c54355f2e -r 2cf979ed69b8 src/HOL/ex/Dense_Linear_Order_Ex.thy --- a/src/HOL/ex/Dense_Linear_Order_Ex.thy Fri Feb 06 14:36:58 2009 +0100 +++ b/src/HOL/ex/Dense_Linear_Order_Ex.thy Fri Feb 06 15:15:46 2009 +0100 @@ -1,12 +1,9 @@ -(* - ID: $Id$ - Author: Amine Chaieb, TU Muenchen -*) +(* Author: Amine Chaieb, TU Muenchen *) header {* Examples for Ferrante and Rackoff's quantifier elimination procedure *} theory Dense_Linear_Order_Ex -imports Dense_Linear_Order Main +imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Main begin lemma