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Quellcode-Bibliothek© Kompilation durch diese Firma[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]Datei: sigma_3D.pvs Sprache: PVSOriginal von: PVS© |
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sigma_3D[T: TYPE FROM int]: THEORY
%------------------------------------------------------------------------------ % % Summation over Vector-valued Functions % % The summations theory introduces and defines properties of the sigma % function that sums an arbitrary function F: [T -> vector] over a range % from low to high % % high % ---- % sigma(low, high, F) = \ F(j) % / % ---- % j = low % % AUTHORS: % % Rick Butler NASA Langley Research Center % Paul Miner NASA Langley Research Center % %------------------------------------------------------------------------------ BEGIN ASSUMING connected_domain: ASSUMPTION (FORALL (x, y: T), (z: integer): x <= z AND z <= y IMPLIES T_pred(z)) ENDASSUMING IMPORTING vectors_3D T_low?: MACRO set[int] = {i:int | EXISTS (j:T): j <= i} T_low : TYPE = {i:int | T_pred(i) OR T_low?(i)} T_high?: MACRO set[int] = {i:int | EXISTS (j:T): i <= j} T_high : TYPE = {i:int | T_pred(i) OR T_high?(i)} low : VAR T_low high : VAR T_high l,h,n,m,i : VAR T rng, nn : VAR nat pn : VAR posnat z : VAR int a,x,B : VAR real v : VAR Vector T_pred_lem : LEMMA low <= z AND z <= high IMPLIES T_pred(z) high_low_rewrite: LEMMA (FORALL (p: pred[[T_high, T_low]]): (FORALL (high: T_high), (low: T_low): IF high < low THEN p(high, low) ELSE (FORALL (n: subrange(low, high)): p(n, low)) ENDIF) IMPLIES (FORALL high, low: p(high, low))) F, G: VAR function[T -> Vector] sigma(low, high, F): RECURSIVE Vector = IF low > high THEN zero ELSE F(high) + sigma(low, high-1, F) ENDIF MEASURE (LAMBDA low, high, F: abs(high+1-low)) % provides one unfolding analogous to old definition -- useful for upgrade sigma_rew : LEMMA sigma(low, high, F) = IF low > high THEN zero ELSIF high = low THEN F(low) ELSE F(high) + sigma(low, high-1, F) ENDIF sigma_spl : THEOREM T_pred(low + nn + rng) IMPLIES sigma(low, low+nn+rng, F) = sigma(low,low+nn,F) + sigma(low+nn+1,low+nn+rng,F) sigma_split : THEOREM low - 1 <= z AND z <= high IMPLIES sigma(low, high, F) = sigma(low, z, F) + sigma(z+1, high, F) sigma_diff : THEOREM low - 1 <= z AND z <= high IMPLIES sigma(low, high, F) - sigma(low, z, F) = sigma(z+1, high, F) sigma_diff_neg: THEOREM low - 1 <= z AND z <= high IMPLIES sigma(low, z, F) - sigma(low, high, F) = - sigma(z+1, high, F) sigma_eq_arg : THEOREM sigma(m, m, F) = F(m) sigma_first : THEOREM high >= low IMPLIES sigma(low, high, F) = F(low) + sigma(low+1, high, F) sigma_last : THEOREM high >= low IMPLIES sigma(low, high, F) = sigma(low, high-1, F) + F(high) sigma_middle : THEOREM high >= i AND i >= low IMPLIES sigma(low, high, F) = sigma(low, i-1, F) + F(i) + sigma(i+1, high, F) sigma_const : THEOREM sigma(low, high, (LAMBDA i: const_vec(x))) = IF high >= low THEN (high-low+1) * const_vec(x) ELSE zero ENDIF sigma_zero : THEOREM sigma(low, high, (LAMBDA i: zero)) = zero sigma_scal : THEOREM sigma(low, high, (LAMBDA i: a * F(i))) = a * sigma(low, high, F) % Lack of an order prevents proof of the following % % sigma_bound : THEOREM low - 1 <= high AND % (FORALL i: i >= low AND i <= high IMPLIES F(i) <= B) % IMPLIES sigma(low, high, F) <= B*(high-low+1) % % sigma_abs : THEOREM abs(sigma(low, high, F)) <= % sigma(low, high, LAMBDA (n: T): abs(F(n))) % % sigma_ge_0 : THEOREM (FORALL (n: subrange(low,high)): F(n) >= 0) % IMPLIES sigma(low,high,F) >= 0 % % sigma_gt_0 : THEOREM low <= high AND % (FORALL (n: subrange(low,high)): F(n) > 0) % IMPLIES sigma(low,high,F) > 0 sigma_shift_T : THEOREM (FORALL (i:T): T_pred(i+z)) IMPLIES sigma(low+z,high+z,F) = sigma(low,high, (LAMBDA (i:T): F(i+z))) sigma_shift_T2 : THEOREM T_pred(low+z) AND T_pred(high+z) IMPLIES sigma(low+z,high+z,F) = sigma(low,high, (LAMBDA (i:T): IF T_pred(i+z) THEN F(i+z) ELSE zero ENDIF)) % ------------------ Summation Involving Two Functions ------------------ sigma_sum : THEOREM sigma(low, high, F) + sigma(low, high, G) = sigma(low, high, (LAMBDA i: F(i) + G(i))) sigma_minus : THEOREM sigma(low, high, F) - sigma(low, high, G) = sigma(low, high, (LAMBDA i: F(i) - G(i))) % Lack of an order prevents proof of the following % % sigma_abs_bnd : THEOREM (FORALL (i: subrange(low,high)): abs(F(i)) <= G(i)) % IMPLIES sigma(low, high, LAMBDA (n: T): abs(F(n))) <= % sigma(low, high, G) restrict(F, low, high): function[T -> Vector] = (LAMBDA i: IF i < low OR i > high THEN zero ELSE F(i) ENDIF ) sigma_restrict : THEOREM l <= low AND h >= high IMPLIES sigma(low,high,F) = sigma(low,high,restrict(F,l,h)) sigma_restrict_to: THEOREM sigma(low,high,F) = sigma(low,high,restrict(F,low,high)) sigma_restrict_eq: THEOREM restrict(F,low,high) = restrict(G,low,high) IMPLIES sigma(low,high,F) = sigma(low,high,G) sigma_eq : THEOREM (FORALL (n: subrange(low,high)): F(n) = G(n)) IMPLIES sigma(low,high,F) = sigma(low,high,G) % Lack of an order prevents proof of the following % % sigma_le : THEOREM (FORALL (n: subrange(low,high)): F(n) <= G(n)) % IMPLIES sigma(low,high,F) <= sigma(low,high,G) % % sigma_lt : THEOREM low <= high AND % (FORALL (n: subrange(low,high)): F(n) < G(n)) % IMPLIES sigma(low,high,F) < sigma(low,high,G) sigma_with : THEOREM low <= i AND i <= high AND F = G WITH [i := v] IMPLIES sigma(low, high,F) = sigma(low,high,G) - G(i) + v % ------------------------ Special Arguments -------------------------------- % % sigma_nonneg : THEOREM (FORALL i: F(i) >= 0) % IMPLIES sigma(low, high, F) >= 0 % % sigma_nonpos : THEOREM (FORALL i: F(i) <= 0) % IMPLIES sigma(low, high, F) <= 0 % % % Fnnr: VAR function[T -> nonneg_real] % Fnpr: VAR function[T -> nonpos_real] % Fnat: VAR function[T -> nat] % Fnpi: VAR function[T -> nonpos_int] % Fposnat: VAR function[T -> posnat] % % % sigma_Fnnr : JUDGEMENT sigma(low,high,Fnnr) HAS_TYPE nonneg_real % % sigma_nonpos_real : JUDGEMENT sigma(low,high,Fnpr) HAS_TYPE nonpos_real % % sigma_nat : JUDGEMENT sigma(low,high,Fnat) HAS_TYPE nat % % sigma_nonpos_int : JUDGEMENT sigma(low,high,Fnpi) HAS_TYPE nonpos_int % % sigma_posnat : LEMMA low <= high IMPLIES sigma(low,high,Fposnat) > 0 % % sigma_nonneg_eq_0 : THEOREM sigma(low,high,Fnnr) = 0 % AND low <= i AND i <= high IMPLIES Fnnr(i) = 0 % % sigma_nn_ge_comps : THEOREM low <= i AND i <= high IMPLIES % sigma(low,high,Fnnr) >= Fnnr(i) % Lack of an order prevents proof of the following % % ------------------- Iterative Summation (tail recursion) ------------ sum_it_def(low, high, F,v): RECURSIVE Vector = IF low > high THEN v ELSE sum_it_def(low+1,high,F,v+F(low)) ENDIF MEASURE (LAMBDA low, high, F, v: abs(high+1-low)) sum_it(low,high,F) : Vector = sum_it_def(low,high,F,zero) sum_it_prop : LEMMA low - 1 <= i AND i <= high => sum_it_def(i+1,high,F,sigma(low,i,F)) = sigma(low,high,F) sum_it_sigma: LEMMA sum_it(low,high,F) = sigma(low,high,F) END sigma_3D ¤ Dauer der Verarbeitung: 0.45 Sekunden (vorverarbeitet) ¤ |
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