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theory MacLaurin(* ID : $Id: MacLaurin.thy,v 1.31 2007/10/23 21:27:24 nipkow Exp $ Author : Jacques D. Fleuriot Copyright : 2001 University of Edinburgh Conversion to Isar and new proofs by Lawrence C Paulson, 2004 *) header{*MacLaurin Series*} theory MacLaurin imports Transcendental begin subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*} text{*This is a very long, messy proof even now that it's been broken down into lemmas.*} lemma Maclaurin_lemma: "0 < h ==> ∃B. f h = (∑m=0..<n. (j m / real (fact m)) * (h^m)) + (B * ((h^n) / real(fact n)))" apply (rule_tac x = "(f h - (∑m=0..<n. (j m / real (fact m)) * h^m)) * real(fact n) / (h^n)" in exI) apply (simp) done lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))" by arith text{*A crude tactic to differentiate by proof.*} lemmas deriv_rulesI = DERIV_ident DERIV_const DERIV_cos DERIV_cmult DERIV_sin DERIV_exp DERIV_inverse DERIV_pow DERIV_add DERIV_diff DERIV_mult DERIV_minus DERIV_inverse_fun DERIV_quotient DERIV_fun_pow DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos DERIV_ident DERIV_const DERIV_cos ML {* local exception DERIV_name; fun get_fun_name (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _)) = f | get_fun_name (_ $ (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _))) = f | get_fun_name _ = raise DERIV_name; in val deriv_tac = SUBGOAL (fn (prem,i) => (resolve_tac @{thms deriv_rulesI} i) ORELSE ((rtac (read_instantiate [("f",get_fun_name prem)] @{thm DERIV_chain2}) i) handle DERIV_name => no_tac));; val DERIV_tac = ALLGOALS(fn i => REPEAT(deriv_tac i)); end *} lemma Maclaurin_lemma2: "[| ∀m t. m < n ∧ 0≤t ∧ t≤h --> DERIV (diff m) t :> diff (Suc m) t; n = Suc k; difg = (λm t. diff m t - ((∑p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) + B * (t ^ (n - m) / real (fact (n - m)))))|] ==> ∀m t. m < n & 0 ≤ t & t ≤ h --> DERIV (difg m) t :> difg (Suc m) t" apply clarify apply (rule DERIV_diff) apply (simp (no_asm_simp)) apply (tactic DERIV_tac) apply (tactic DERIV_tac) apply (rule_tac [2] lemma_DERIV_subst) apply (rule_tac [2] DERIV_quotient) apply (rule_tac [3] DERIV_const) apply (rule_tac [2] DERIV_pow) prefer 3 apply (simp add: fact_diff_Suc) prefer 2 apply simp apply (frule_tac m = m in less_add_one, clarify) apply (simp del: setsum_op_ivl_Suc) apply (insert sumr_offset4 [of 1]) apply (simp del: setsum_op_ivl_Suc fact_Suc realpow_Suc) apply (rule lemma_DERIV_subst) apply (rule DERIV_add) apply (rule_tac [2] DERIV_const) apply (rule DERIV_sumr, clarify) prefer 2 apply simp apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc realpow_Suc) apply (rule DERIV_cmult) apply (rule lemma_DERIV_subst) apply (best intro: DERIV_chain2 intro!: DERIV_intros) apply (subst fact_Suc) apply (subst real_of_nat_mult) apply (simp add: mult_ac) done lemma Maclaurin_lemma3: fixes difg :: "nat => real => real" shows "[|∀k t. k < Suc m ∧ 0≤t & t≤h --> DERIV (difg k) t :> difg (Suc k) t; ∀k<Suc m. difg k 0 = 0; DERIV (difg n) t :> 0; n < m; 0 < t; t < h|] ==> ∃ta. 0 < ta & ta < t & DERIV (difg (Suc n)) ta :> 0" apply (rule Rolle, assumption, simp) apply (drule_tac x = n and P="%k. k<Suc m --> difg k 0 = 0" in spec) apply (rule DERIV_unique) prefer 2 apply assumption apply force apply (metis DERIV_isCont dlo_simps(4) dlo_simps(9) less_trans_Suc nat_less_le not_less_eq real_le_trans) apply (metis Suc_less_eq differentiableI dlo_simps(7) dlo_simps(8) dlo_simps(9) real_le_trans xt1(8)) done lemma Maclaurin: "[| 0 < h; n > 0; diff 0 = f; ∀m t. m < n & 0 ≤ t & t ≤ h --> DERIV (diff m) t :> diff (Suc m) t |] ==> ∃t. 0 < t & t < h & f h = setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} + (diff n t / real (fact n)) * h ^ n" apply (case_tac "n = 0", force) apply (drule not0_implies_Suc) apply (erule exE) apply (frule_tac f=f and n=n and j="%m. diff m 0" in Maclaurin_lemma) apply (erule exE) apply (subgoal_tac "∃g. g = (%t. f t - (setsum (%m. (diff m 0 / real(fact m)) * t^m) {0..<n} + (B * (t^n / real(fact n)))))") prefer 2 apply blast apply (erule exE) apply (subgoal_tac "g 0 = 0 & g h =0") prefer 2 apply (simp del: setsum_op_ivl_Suc) apply (cut_tac n = m and k = 1 in sumr_offset2) apply (simp add: eq_diff_eq' del: setsum_op_ivl_Suc) apply (subgoal_tac "∃difg. difg = (%m t. diff m t - (setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {0..<n-m} + (B * ((t ^ (n - m)) / real (fact (n - m))))))") prefer 2 apply blast apply (erule exE) apply (subgoal_tac "difg 0 = g") prefer 2 apply simp apply (frule Maclaurin_lemma2, assumption+) apply (subgoal_tac "∀ma. ma < n --> (∃t. 0 < t & t < h & difg (Suc ma) t = 0) ") apply (drule_tac x = m and P="%m. m<n --> (∃t. ?QQ m t)" in spec) apply (erule impE) apply (simp (no_asm_simp)) apply (erule exE) apply (rule_tac x = t in exI) apply (simp del: realpow_Suc fact_Suc) apply (subgoal_tac "∀m. m < n --> difg m 0 = 0") prefer 2 apply clarify apply simp apply (frule_tac m = ma in less_add_one, clarify) apply (simp del: setsum_op_ivl_Suc) apply (insert sumr_offset4 [of 1]) apply (simp del: setsum_op_ivl_Suc fact_Suc realpow_Suc) apply (subgoal_tac "∀m. m < n --> (∃t. 0 < t & t < h & DERIV (difg m) t :> 0) ") apply (rule allI, rule impI) apply (drule_tac x = ma and P="%m. m<n --> (∃t. ?QQ m t)" in spec) apply (erule impE, assumption) apply (erule exE) apply (rule_tac x = t in exI) (* do some tidying up *) apply (erule_tac [!] V= "difg = (%m t. diff m t - (setsum (%p. diff (m + p) 0 / real (fact p) * t ^ p) {0..<n-m} + B * (t ^ (n - m) / real (fact (n - m)))))" in thin_rl) apply (erule_tac [!] V="g = (%t. f t - (setsum (%m. diff m 0 / real (fact m) * t ^ m) {0..<n} + B * (t ^ n / real (fact n))))" in thin_rl) apply (erule_tac [!] V="f h = setsum (%m. diff m 0 / real (fact m) * h ^ m) {0..<n} + B * (h ^ n / real (fact n))" in thin_rl) (* back to business *) apply (simp (no_asm_simp)) apply (rule DERIV_unique) prefer 2 apply blast apply force apply (rule allI, induct_tac "ma") apply (rule impI, rule Rolle, assumption, simp, simp) apply (metis DERIV_isCont zero_less_Suc) apply (metis One_nat_def differentiableI dlo_simps(7)) apply safe apply force apply (frule Maclaurin_lemma3, assumption+, safe) apply (rule_tac x = ta in exI, force) done lemma Maclaurin_objl: "0 < h & n>0 & diff 0 = f & (∀m t. m < n & 0 ≤ t & t ≤ h --> DERIV (diff m) t :> diff (Suc m) t) --> (∃t. 0 < t & t < h & f h = (∑m=0..<n. diff m 0 / real (fact m) * h ^ m) + diff n t / real (fact n) * h ^ n)" by (blast intro: Maclaurin) lemma Maclaurin2: "[| 0 < h; diff 0 = f; ∀m t. m < n & 0 ≤ t & t ≤ h --> DERIV (diff m) t :> diff (Suc m) t |] ==> ∃t. 0 < t & t ≤ h & f h = (∑m=0..<n. diff m 0 / real (fact m) * h ^ m) + diff n t / real (fact n) * h ^ n" apply (case_tac "n", auto) apply (drule Maclaurin, auto) done lemma Maclaurin2_objl: "0 < h & diff 0 = f & (∀m t. m < n & 0 ≤ t & t ≤ h --> DERIV (diff m) t :> diff (Suc m) t) --> (∃t. 0 < t & t ≤ h & f h = (∑m=0..<n. diff m 0 / real (fact m) * h ^ m) + diff n t / real (fact n) * h ^ n)" by (blast intro: Maclaurin2) lemma Maclaurin_minus: "[| h < 0; n > 0; diff 0 = f; ∀m t. m < n & h ≤ t & t ≤ 0 --> DERIV (diff m) t :> diff (Suc m) t |] ==> ∃t. h < t & t < 0 & f h = (∑m=0..<n. diff m 0 / real (fact m) * h ^ m) + diff n t / real (fact n) * h ^ n" apply (cut_tac f = "%x. f (-x)" and diff = "%n x. (-1 ^ n) * diff n (-x)" and h = "-h" and n = n in Maclaurin_objl) apply (simp) apply safe apply (subst minus_mult_right) apply (rule DERIV_cmult) apply (rule lemma_DERIV_subst) apply (rule DERIV_chain2 [where g=uminus]) apply (rule_tac [2] DERIV_minus, rule_tac [2] DERIV_ident) prefer 2 apply force apply force apply (rule_tac x = "-t" in exI, auto) apply (subgoal_tac "(∑m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) = (∑m = 0..<n. diff m 0 * h ^ m / real(fact m))") apply (rule_tac [2] setsum_cong[OF refl]) apply (auto simp add: divide_inverse power_mult_distrib [symmetric]) done lemma Maclaurin_minus_objl: "(h < 0 & n > 0 & diff 0 = f & (∀m t. m < n & h ≤ t & t ≤ 0 --> DERIV (diff m) t :> diff (Suc m) t)) --> (∃t. h < t & t < 0 & f h = (∑m=0..<n. diff m 0 / real (fact m) * h ^ m) + diff n t / real (fact n) * h ^ n)" by (blast intro: Maclaurin_minus) subsection{*More Convenient "Bidirectional" Version.*} (* not good for PVS sin_approx, cos_approx *) lemma Maclaurin_bi_le_lemma [rule_format]: "n>0 --> diff 0 0 = (∑m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) + diff n 0 * 0 ^ n / real (fact n)" by (induct "n", auto) lemma Maclaurin_bi_le: "[| diff 0 = f; ∀m t. m < n & abs t ≤ abs x --> DERIV (diff m) t :> diff (Suc m) t |] ==> ∃t. abs t ≤ abs x & f x = (∑m=0..<n. diff m 0 / real (fact m) * x ^ m) + diff n t / real (fact n) * x ^ n" apply (case_tac "n = 0", force) apply (case_tac "x = 0") apply (rule_tac x = 0 in exI) apply (force simp add: Maclaurin_bi_le_lemma) apply (cut_tac x = x and y = 0 in linorder_less_linear, auto) txt{*Case 1, where @{term "x < 0"}*} apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe) apply (simp add: abs_if) apply (rule_tac x = t in exI) apply (simp add: abs_if) txt{*Case 2, where @{term "0 < x"}*} apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe) apply (simp add: abs_if) apply (rule_tac x = t in exI) apply (simp add: abs_if) done lemma Maclaurin_all_lt: "[| diff 0 = f; ∀m x. DERIV (diff m) x :> diff(Suc m) x; x ~= 0; n > 0 |] ==> ∃t. 0 < abs t & abs t < abs x & f x = (∑m=0..<n. (diff m 0 / real (fact m)) * x ^ m) + (diff n t / real (fact n)) * x ^ n" apply (rule_tac x = x and y = 0 in linorder_cases) prefer 2 apply blast apply (drule_tac [2] diff=diff in Maclaurin) apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe) apply (rule_tac [!] x = t in exI, auto) done lemma Maclaurin_all_lt_objl: "diff 0 = f & (∀m x. DERIV (diff m) x :> diff(Suc m) x) & x ~= 0 & n > 0 --> (∃t. 0 < abs t & abs t < abs x & f x = (∑m=0..<n. (diff m 0 / real (fact m)) * x ^ m) + (diff n t / real (fact n)) * x ^ n)" by (blast intro: Maclaurin_all_lt) lemma Maclaurin_zero [rule_format]: "x = (0::real) ==> n ≠ 0 --> (∑m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) = diff 0 0" by (induct n, auto) lemma Maclaurin_all_le: "[| diff 0 = f; ∀m x. DERIV (diff m) x :> diff (Suc m) x |] ==> ∃t. abs t ≤ abs x & f x = (∑m=0..<n. (diff m 0 / real (fact m)) * x ^ m) + (diff n t / real (fact n)) * x ^ n" apply(cases "n=0") apply (force) apply (case_tac "x = 0") apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption) apply (drule not0_implies_Suc) apply (rule_tac x = 0 in exI, force) apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto) apply (rule_tac x = t in exI, auto) done lemma Maclaurin_all_le_objl: "diff 0 = f & (∀m x. DERIV (diff m) x :> diff (Suc m) x) --> (∃t. abs t ≤ abs x & f x = (∑m=0..<n. (diff m 0 / real (fact m)) * x ^ m) + (diff n t / real (fact n)) * x ^ n)" by (blast intro: Maclaurin_all_le) subsection{*Version for Exponential Function*} lemma Maclaurin_exp_lt: "[| x ~= 0; n > 0 |] ==> (∃t. 0 < abs t & abs t < abs x & exp x = (∑m=0..<n. (x ^ m) / real (fact m)) + (exp t / real (fact n)) * x ^ n)" by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto) lemma Maclaurin_exp_le: "∃t. abs t ≤ abs x & exp x = (∑m=0..<n. (x ^ m) / real (fact m)) + (exp t / real (fact n)) * x ^ n" by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto) subsection{*Version for Sine Function*} lemma MVT2: "[| a < b; ∀x. a ≤ x & x ≤ b --> DERIV f x :> f'(x) |] ==> ∃z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))" apply (drule MVT) apply (blast intro: DERIV_isCont) apply (force dest: order_less_imp_le simp add: differentiable_def) apply (blast dest: DERIV_unique order_less_imp_le) done lemma mod_exhaust_less_4: "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)" by auto lemma Suc_Suc_mult_two_diff_two [rule_format, simp]: "n≠0 --> Suc (Suc (2 * n - 2)) = 2*n" by (induct "n", auto) lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]: "n≠0 --> Suc (Suc (4*n - 2)) = 4*n" by (induct "n", auto) lemma Suc_mult_two_diff_one [rule_format, simp]: "n≠0 --> Suc (2 * n - 1) = 2*n" by (induct "n", auto) text{*It is unclear why so many variant results are needed.*} lemma Maclaurin_sin_expansion2: "∃t. abs t ≤ abs x & sin x = (∑m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) * x ^ m) + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)" apply (cut_tac f = sin and n = n and x = x and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl) apply safe apply (simp (no_asm)) apply (simp (no_asm)) apply (case_tac "n", clarify, simp, simp add: lemma_STAR_sin) apply (rule ccontr, simp) apply (drule_tac x = x in spec, simp) apply (erule ssubst) apply (rule_tac x = t in exI, simp) apply (rule setsum_cong[OF refl]) apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex) done lemma Maclaurin_sin_expansion: "∃t. sin x = (∑m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) * x ^ m) + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)" apply (insert Maclaurin_sin_expansion2 [of x n]) apply (blast intro: elim:); done lemma Maclaurin_sin_expansion3: "[| n > 0; 0 < x |] ==> ∃t. 0 < t & t < x & sin x = (∑m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) * x ^ m) + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)" apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin_objl) apply safe apply simp apply (simp (no_asm)) apply (erule ssubst) apply (rule_tac x = t in exI, simp) apply (rule setsum_cong[OF refl]) apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex) done lemma Maclaurin_sin_expansion4: "0 < x ==> ∃t. 0 < t & t ≤ x & sin x = (∑m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) * x ^ m) + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)" apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin2_objl) apply safe apply simp apply (simp (no_asm)) apply (erule ssubst) apply (rule_tac x = t in exI, simp) apply (rule setsum_cong[OF refl]) apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex) done subsection{*Maclaurin Expansion for Cosine Function*} lemma sumr_cos_zero_one [simp]: "(∑m=0..<(Suc n). (if even m then -1 ^ (m div 2)/(real (fact m)) else 0) * 0 ^ m) = 1" by (induct "n", auto) lemma Maclaurin_cos_expansion: "∃t. abs t ≤ abs x & cos x = (∑m=0..<n. (if even m then -1 ^ (m div 2)/(real (fact m)) else 0) * x ^ m) + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)" apply (cut_tac f = cos and n = n and x = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_all_lt_objl) apply safe apply (simp (no_asm)) apply (simp (no_asm)) apply (case_tac "n", simp) apply (simp del: setsum_op_ivl_Suc) apply (rule ccontr, simp) apply (drule_tac x = x in spec, simp) apply (erule ssubst) apply (rule_tac x = t in exI, simp) apply (rule setsum_cong[OF refl]) apply (auto simp add: cos_zero_iff even_mult_two_ex) done lemma Maclaurin_cos_expansion2: "[| 0 < x; n > 0 |] ==> ∃t. 0 < t & t < x & cos x = (∑m=0..<n. (if even m then -1 ^ (m div 2)/(real (fact m)) else 0) * x ^ m) + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)" apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_objl) apply safe apply simp apply (simp (no_asm)) apply (erule ssubst) apply (rule_tac x = t in exI, simp) apply (rule setsum_cong[OF refl]) apply (auto simp add: cos_zero_iff even_mult_two_ex) done lemma Maclaurin_minus_cos_expansion: "[| x < 0; n > 0 |] ==> ∃t. x < t & t < 0 & cos x = (∑m=0..<n. (if even m then -1 ^ (m div 2)/(real (fact m)) else 0) * x ^ m) + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)" apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_minus_objl) apply safe apply simp apply (simp (no_asm)) apply (erule ssubst) apply (rule_tac x = t in exI, simp) apply (rule setsum_cong[OF refl]) apply (auto simp add: cos_zero_iff even_mult_two_ex) done (* ------------------------------------------------------------------------- *) (* Version for ln(1 +/- x). Where is it?? *) (* ------------------------------------------------------------------------- *) lemma sin_bound_lemma: "[|x = y; abs u ≤ (v::real) |] ==> ¦(x + u) - y¦ ≤ v" by auto lemma Maclaurin_sin_bound: "abs(sin x - (∑m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) * x ^ m)) ≤ inverse(real (fact n)) * ¦x¦ ^ n" proof - have "!! x (y::real). x ≤ 1 ==> 0 ≤ y ==> x * y ≤ 1 * y" by (rule_tac mult_right_mono,simp_all) note est = this[simplified] let ?diff = "λ(n::nat) x. if n mod 4 = 0 then sin(x) else if n mod 4 = 1 then cos(x) else if n mod 4 = 2 then -sin(x) else -cos(x)" have diff_0: "?diff 0 = sin" by simp have DERIV_diff: "∀m x. DERIV (?diff m) x :> ?diff (Suc m) x" apply (clarify) apply (subst (1 2 3) mod_Suc_eq_Suc_mod) apply (cut_tac m=m in mod_exhaust_less_4) apply (safe, simp_all) apply (rule DERIV_minus, simp) apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp) done from Maclaurin_all_le [OF diff_0 DERIV_diff] obtain t where t1: "¦t¦ ≤ ¦x¦" and t2: "sin x = (∑m = 0..<n. ?diff m 0 / real (fact m) * x ^ m) + ?diff n t / real (fact n) * x ^ n" by fast have diff_m_0: "!!m. ?diff m 0 = (if even m then 0 else -1 ^ ((m - Suc 0) div 2))" apply (subst even_even_mod_4_iff) apply (cut_tac m=m in mod_exhaust_less_4) apply (elim disjE, simp_all) apply (safe dest!: mod_eqD, simp_all) done show ?thesis apply (subst t2) apply (rule sin_bound_lemma) apply (rule setsum_cong[OF refl]) apply (subst diff_m_0, simp) apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono simp add: est mult_nonneg_nonneg mult_ac divide_inverse power_abs [symmetric] abs_mult) done qed end
lemma Maclaurin_lemma:
0 < h
==> ∃B. f h =
(∑m = 0..<n. j m / real (fact m) * h ^ m) + B * (h ^ n / real (fact n))
lemma eq_diff_eq':
(x = y - z) = (y = x + z)
lemma deriv_rulesI:
DERIV (λx. x) x :> (1::'a)
DERIV (λx. k) x :> (0::'a)
DERIV cos x :> - sin x
DERIV f x :> D ==> DERIV (λx. c * f x) x :> c * D
DERIV sin x :> cos x
DERIV exp x :> exp x
x ≠ (0::'a) ==> DERIV inverse x :> - (inverse x ^ Suc (Suc 0))
DERIV (λx. x ^ n) x :> real n * x ^ (n - Suc 0)
[| DERIV f x :> D; DERIV g x :> E |] ==> DERIV (λx. f x + g x) x :> D + E
[| DERIV f x :> D; DERIV g x :> E |] ==> DERIV (λx. f x - g x) x :> D - E
[| DERIV f x :> Da; DERIV g x :> Db |]
==> DERIV (λx. f x * g x) x :> Da * g x + Db * f x
DERIV f x :> D ==> DERIV (λx. - f x) x :> - D
[| DERIV f x :> d; f x ≠ (0::'a) |]
==> DERIV (λx. inverse (f x)) x :> - (d * inverse (f x ^ Suc (Suc 0)))
[| DERIV f x :> d; DERIV g x :> e; g x ≠ (0::'a) |]
==> DERIV (λy. f y / g y) x :> (d * g x - e * f x) / g x ^ Suc (Suc 0)
DERIV g x :> m ==> DERIV (λx. g x ^ n) x :> real n * g x ^ (n - 1) * m
DERIV g x :> m ==> DERIV (λx. exp (g x)) x :> exp (g x) * m
DERIV g x :> m ==> DERIV (λx. sin (g x)) x :> cos (g x) * m
DERIV g x :> m ==> DERIV (λx. cos (g x)) x :> - sin (g x) * m
DERIV (λx. x) x :> (1::'a)
DERIV (λx. k) x :> (0::'a)
DERIV cos x :> - sin x
lemma Maclaurin_lemma2:
[| ∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h --> DERIV (diff m) t :> diff (Suc m) t;
n = Suc k;
difg =
(λm t. diff m t -
((∑p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
B * (t ^ (n - m) / real (fact (n - m))))) |]
==> ∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h --> DERIV (difg m) t :> difg (Suc m) t
lemma Maclaurin_lemma3:
[| ∀k t. k < Suc m ∧ 0 ≤ t ∧ t ≤ h --> DERIV (difg k) t :> difg (Suc k) t;
∀k<Suc m. difg k 0 = 0; DERIV (difg n) t :> 0; n < m; 0 < t; t < h |]
==> ∃ta>0. ta < t ∧ DERIV (difg (Suc n)) ta :> 0
lemma Maclaurin:
[| 0 < h; 0 < n; diff 0 = f;
∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h --> DERIV (diff m) t :> diff (Suc m) t |]
==> ∃t>0. t < h ∧
f h =
(∑m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n
lemma Maclaurin_objl:
0 < h ∧
0 < n ∧
diff 0 = f ∧
(∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h --> DERIV (diff m) t :> diff (Suc m) t) -->
(∃t>0. t < h ∧
f h =
(∑m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n)
lemma Maclaurin2:
[| 0 < h; diff 0 = f;
∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h --> DERIV (diff m) t :> diff (Suc m) t |]
==> ∃t>0. t ≤ h ∧
f h =
(∑m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n
lemma Maclaurin2_objl:
0 < h ∧
diff 0 = f ∧
(∀m t. m < n ∧ 0 ≤ t ∧ t ≤ h --> DERIV (diff m) t :> diff (Suc m) t) -->
(∃t>0. t ≤ h ∧
f h =
(∑m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n)
lemma Maclaurin_minus:
[| h < 0; 0 < n; diff 0 = f;
∀m t. m < n ∧ h ≤ t ∧ t ≤ 0 --> DERIV (diff m) t :> diff (Suc m) t |]
==> ∃t>h. t < 0 ∧
f h =
(∑m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n
lemma Maclaurin_minus_objl:
h < 0 ∧
0 < n ∧
diff 0 = f ∧
(∀m t. m < n ∧ h ≤ t ∧ t ≤ 0 --> DERIV (diff m) t :> diff (Suc m) t) -->
(∃t>h. t < 0 ∧
f h =
(∑m = 0..<n. diff m 0 / real (fact m) * h ^ m) +
diff n t / real (fact n) * h ^ n)
lemma Maclaurin_bi_le_lemma:
0 < n
==> diff 0 (0::'a) =
(∑m = 0..<n. diff m (0::'a) * 0 ^ m / real (fact m)) +
diff n (0::'a) * 0 ^ n / real (fact n)
lemma Maclaurin_bi_le:
[| diff 0 = f; ∀m t. m < n ∧ ¦t¦ ≤ ¦x¦ --> DERIV (diff m) t :> diff (Suc m) t |]
==> ∃t. ¦t¦ ≤ ¦x¦ ∧
f x =
(∑m = 0..<n. diff m 0 / real (fact m) * x ^ m) +
diff n t / real (fact n) * x ^ n
lemma Maclaurin_all_lt:
[| diff 0 = f; ∀m x. DERIV (diff m) x :> diff (Suc m) x; x ≠ 0; 0 < n |]
==> ∃t. 0 < ¦t¦ ∧
¦t¦ < ¦x¦ ∧
f x =
(∑m = 0..<n. diff m 0 / real (fact m) * x ^ m) +
diff n t / real (fact n) * x ^ n
lemma Maclaurin_all_lt_objl:
diff 0 = f ∧ (∀m x. DERIV (diff m) x :> diff (Suc m) x) ∧ x ≠ 0 ∧ 0 < n -->
(∃t. 0 < ¦t¦ ∧
¦t¦ < ¦x¦ ∧
f x =
(∑m = 0..<n. diff m 0 / real (fact m) * x ^ m) +
diff n t / real (fact n) * x ^ n)
lemma Maclaurin_zero:
[| x = 0; n ≠ 0 |] ==> (∑m = 0..<n. diff m 0 / real (fact m) * x ^ m) = diff 0 0
lemma Maclaurin_all_le:
[| diff 0 = f; ∀m x. DERIV (diff m) x :> diff (Suc m) x |]
==> ∃t. ¦t¦ ≤ ¦x¦ ∧
f x =
(∑m = 0..<n. diff m 0 / real (fact m) * x ^ m) +
diff n t / real (fact n) * x ^ n
lemma Maclaurin_all_le_objl:
diff 0 = f ∧ (∀m x. DERIV (diff m) x :> diff (Suc m) x) -->
(∃t. ¦t¦ ≤ ¦x¦ ∧
f x =
(∑m = 0..<n. diff m 0 / real (fact m) * x ^ m) +
diff n t / real (fact n) * x ^ n)
lemma Maclaurin_exp_lt:
[| x ≠ 0; 0 < n |]
==> ∃t. 0 < ¦t¦ ∧
¦t¦ < ¦x¦ ∧
exp x =
(∑m = 0..<n. x ^ m / real (fact m)) + exp t / real (fact n) * x ^ n
lemma Maclaurin_exp_le:
∃t. ¦t¦ ≤ ¦x¦ ∧
exp x = (∑m = 0..<n. x ^ m / real (fact m)) + exp t / real (fact n) * x ^ n
lemma MVT2:
[| a < b; ∀x. a ≤ x ∧ x ≤ b --> DERIV f x :> f' x |]
==> ∃z>a. z < b ∧ f b - f a = (b - a) * f' z
lemma mod_exhaust_less_4:
m mod 4 = 0 ∨ m mod 4 = 1 ∨ m mod 4 = 2 ∨ m mod 4 = 3
lemma Suc_Suc_mult_two_diff_two:
n ≠ 0 ==> Suc (Suc (2 * n - 2)) = 2 * n
lemma lemma_Suc_Suc_4n_diff_2:
n ≠ 0 ==> Suc (Suc (4 * n - 2)) = 4 * n
lemma Suc_mult_two_diff_one:
n ≠ 0 ==> Suc (2 * n - 1) = 2 * n
lemma Maclaurin_sin_expansion2:
∃t. ¦t¦ ≤ ¦x¦ ∧
sin x =
(∑m = 0..<n.
(if even m then 0 else -1 ^ ((m - Suc 0) div 2) / real (fact m)) *
x ^ m) +
sin (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n
lemma Maclaurin_sin_expansion:
∃t. sin x =
(∑m = 0..<n.
(if even m then 0 else -1 ^ ((m - Suc 0) div 2) / real (fact m)) *
x ^ m) +
sin (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n
lemma Maclaurin_sin_expansion3:
[| 0 < n; 0 < x |]
==> ∃t>0. t < x ∧
sin x =
(∑m = 0..<n.
(if even m then 0 else -1 ^ ((m - Suc 0) div 2) / real (fact m)) *
x ^ m) +
sin (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n
lemma Maclaurin_sin_expansion4:
0 < x
==> ∃t>0. t ≤ x ∧
sin x =
(∑m = 0..<n.
(if even m then 0 else -1 ^ ((m - Suc 0) div 2) / real (fact m)) *
x ^ m) +
sin (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n
lemma sumr_cos_zero_one:
(∑m = 0..<Suc n.
(if even m then -1 ^ (m div 2) / real (fact m) else 0) * 0 ^ m) =
1
lemma Maclaurin_cos_expansion:
∃t. ¦t¦ ≤ ¦x¦ ∧
cos x =
(∑m = 0..<n.
(if even m then -1 ^ (m div 2) / real (fact m) else 0) * x ^ m) +
cos (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n
lemma Maclaurin_cos_expansion2:
[| 0 < x; 0 < n |]
==> ∃t>0. t < x ∧
cos x =
(∑m = 0..<n.
(if even m then -1 ^ (m div 2) / real (fact m) else 0) * x ^ m) +
cos (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n
lemma Maclaurin_minus_cos_expansion:
[| x < 0; 0 < n |]
==> ∃t>x. t < 0 ∧
cos x =
(∑m = 0..<n.
(if even m then -1 ^ (m div 2) / real (fact m) else 0) * x ^ m) +
cos (t + 1 / 2 * real n * pi) / real (fact n) * x ^ n
lemma sin_bound_lemma:
[| x = y; ¦u¦ ≤ v |] ==> ¦x + u - y¦ ≤ v
lemma Maclaurin_sin_bound:
¦sin x -
(∑m = 0..<n.
(if even m then 0 else -1 ^ ((m - Suc 0) div 2) / real (fact m)) * x ^ m)¦
≤ inverse (real (fact n)) * ¦x¦ ^ n