14738
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(* Title : MacLaurin.thy
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Author : Jacques D. Fleuriot
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Copyright : 2001 University of Edinburgh
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Description : MacLaurin series
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*)
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Goal "sumr 0 n (%m. f (m + k)) = sumr 0 (n + k) f - sumr 0 k f";
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by (induct_tac "n" 1);
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by Auto_tac;
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qed "sumr_offset";
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Goal "ALL f. sumr 0 n (%m. f (m + k)) = sumr 0 (n + k) f - sumr 0 k f";
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by (induct_tac "n" 1);
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by Auto_tac;
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qed "sumr_offset2";
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Goal "sumr 0 (n + k) f = sumr 0 n (%m. f (m + k)) + sumr 0 k f";
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by (simp_tac (simpset() addsimps [sumr_offset]) 1);
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qed "sumr_offset3";
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Goal "ALL n f. sumr 0 (n + k) f = sumr 0 n (%m. f (m + k)) + sumr 0 k f";
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by (simp_tac (simpset() addsimps [sumr_offset]) 1);
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qed "sumr_offset4";
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Goal "0 < n ==> \
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\ sumr (Suc 0) (Suc n) (%n. (if even(n) then 0 else \
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\ ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n) = \
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\ sumr 0 (Suc n) (%n. (if even(n) then 0 else \
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\ ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n)";
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by (res_inst_tac [("n1","1")] (sumr_split_add RS subst) 1);
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by Auto_tac;
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qed "sumr_from_1_from_0";
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(*---------------------------------------------------------------------------*)
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(* Maclaurin's theorem with Lagrange form of remainder *)
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(*---------------------------------------------------------------------------*)
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(* Annoying: Proof is now even longer due mostly to
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change in behaviour of simplifier since Isabelle99 *)
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Goal " [| 0 < h; 0 < n; diff 0 = f; \
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\ ALL m t. \
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\ m < n & 0 <= t & t <= h --> DERIV (diff m) t :> diff (Suc m) t |] \
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\ ==> EX t. 0 < t & \
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\ t < h & \
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\ f h = \
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\ sumr 0 n (%m. (diff m 0 / real (fact m)) * h ^ m) + \
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\ (diff n t / real (fact n)) * h ^ n";
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by (case_tac "n = 0" 1);
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by (Force_tac 1);
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by (dtac not0_implies_Suc 1);
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by (etac exE 1);
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by (subgoal_tac
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"EX B. f h = sumr 0 n (%m. (diff m 0 / real (fact m)) * (h ^ m)) \
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\ + (B * ((h ^ n) / real (fact n)))" 1);
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by (simp_tac (HOL_ss addsimps [real_add_commute, real_divide_def,
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ARITH_PROVE "(x = z + (y::real)) = (x - y = z)"]) 2);
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by (res_inst_tac
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[("x","(f(h) - sumr 0 n (%m. (diff(m)(0) / real (fact m)) * (h ^ m))) \
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\ * real (fact n) / (h ^ n)")] exI 2);
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by (simp_tac (HOL_ss addsimps [real_mult_assoc,real_divide_def]) 2);
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by (rtac (CLAIM "x = (1::real) ==> a = a * (x::real)") 2);
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by (asm_simp_tac (HOL_ss addsimps
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[CLAIM "(a::real) * (b * (c * d)) = (d * a) * (b * c)"]
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delsimps [realpow_Suc]) 2);
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by (stac left_inverse 2);
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by (stac left_inverse 3);
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by (rtac (real_not_refl2 RS not_sym) 2);
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by (etac zero_less_power 2);
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by (rtac real_of_nat_fact_not_zero 2);
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by (Simp_tac 2);
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by (etac exE 1);
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by (cut_inst_tac [("b","%t. f t - \
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\ (sumr 0 n (%m. (diff m 0 / real (fact m)) * (t ^ m)) + \
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\ (B * ((t ^ n) / real (fact n))))")]
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(CLAIM "EX g. g = b") 1);
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by (etac exE 1);
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by (subgoal_tac "g 0 = 0 & g h =0" 1);
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by (asm_simp_tac (simpset() addsimps
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[ARITH_PROVE "(x - y = z) = (x = z + (y::real))"]
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delsimps [sumr_Suc]) 2);
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by (cut_inst_tac [("n","m"),("k","1")] sumr_offset2 2);
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by (asm_full_simp_tac (simpset() addsimps
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[ARITH_PROVE "(x = y - z) = (y = x + (z::real))"]
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delsimps [sumr_Suc]) 2);
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by (cut_inst_tac [("b","%m t. diff m t - \
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\ (sumr 0 (n - m) (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) \
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\ + (B * ((t ^ (n - m)) / real (fact(n - m)))))")]
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(CLAIM "EX difg. difg = b") 1);
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by (etac exE 1);
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by (subgoal_tac "difg 0 = g" 1);
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by (asm_simp_tac (simpset() delsimps [realpow_Suc,fact_Suc]) 2);
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by (subgoal_tac "ALL m t. m < n & 0 <= t & t <= h --> \
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\ DERIV (difg m) t :> difg (Suc m) t" 1);
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by (Clarify_tac 2);
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by (rtac DERIV_diff 2);
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by (Asm_simp_tac 2);
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by DERIV_tac;
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by DERIV_tac;
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by (rtac lemma_DERIV_subst 3);
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by (rtac DERIV_quotient 3);
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by (rtac DERIV_const 4);
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by (rtac DERIV_pow 3);
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by (asm_simp_tac (simpset() addsimps [inverse_mult_distrib,
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CLAIM_SIMP "(a::real) * b * c * (d * e) = a * b * (c * d) * e"
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mult_ac,fact_diff_Suc]) 4);
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by (Asm_simp_tac 3);
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by (forw_inst_tac [("m","ma")] less_add_one 2);
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by (Clarify_tac 2);
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by (asm_simp_tac (simpset() addsimps
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[CLAIM "Suc m = ma + d + 1 ==> m - ma = d"]
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delsimps [sumr_Suc]) 2);
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by (asm_simp_tac (simpset() addsimps [(simplify (simpset() delsimps [sumr_Suc])
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(read_instantiate [("k","1")] sumr_offset4))]
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delsimps [sumr_Suc,fact_Suc,realpow_Suc]) 2);
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by (rtac lemma_DERIV_subst 2);
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by (rtac DERIV_add 2);
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by (rtac DERIV_const 3);
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by (rtac DERIV_sumr 2);
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by (Clarify_tac 2);
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by (Simp_tac 3);
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by (simp_tac (simpset() addsimps [real_divide_def,real_mult_assoc]
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delsimps [fact_Suc,realpow_Suc]) 2);
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by (rtac DERIV_cmult 2);
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by (rtac lemma_DERIV_subst 2);
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by DERIV_tac;
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by (stac fact_Suc 2);
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by (stac real_of_nat_mult 2);
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by (simp_tac (simpset() addsimps [inverse_mult_distrib] @
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mult_ac) 2);
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by (subgoal_tac "ALL ma. ma < n --> \
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\ (EX t. 0 < t & t < h & difg (Suc ma) t = 0)" 1);
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by (rotate_tac 11 1);
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by (dres_inst_tac [("x","m")] spec 1);
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by (etac impE 1);
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by (Asm_simp_tac 1);
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by (etac exE 1);
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by (res_inst_tac [("x","t")] exI 1);
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by (asm_full_simp_tac (simpset() addsimps
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[ARITH_PROVE "(x - y = 0) = (y = (x::real))"]
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delsimps [realpow_Suc,fact_Suc]) 1);
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by (subgoal_tac "ALL m. m < n --> difg m 0 = 0" 1);
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by (Clarify_tac 2);
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by (Asm_simp_tac 2);
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by (forw_inst_tac [("m","ma")] less_add_one 2);
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by (Clarify_tac 2);
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by (asm_simp_tac (simpset() delsimps [sumr_Suc]) 2);
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by (asm_simp_tac (simpset() addsimps [(simplify (simpset() delsimps [sumr_Suc])
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(read_instantiate [("k","1")] sumr_offset4))]
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delsimps [sumr_Suc,fact_Suc,realpow_Suc]) 2);
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by (subgoal_tac "ALL m. m < n --> (EX t. 0 < t & t < h & \
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\ DERIV (difg m) t :> 0)" 1);
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by (rtac allI 1 THEN rtac impI 1);
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by (rotate_tac 12 1);
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by (dres_inst_tac [("x","ma")] spec 1);
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by (etac impE 1 THEN assume_tac 1);
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by (etac exE 1);
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by (res_inst_tac [("x","t")] exI 1);
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(* do some tidying up *)
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by (ALLGOALS(thin_tac "difg = \
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\ (%m t. diff m t - \
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\ (sumr 0 (n - m) \
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\ (%p. diff (m + p) 0 / real (fact p) * t ^ p) + \
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\ B * (t ^ (n - m) / real (fact (n - m)))))"));
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by (ALLGOALS(thin_tac "g = \
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\ (%t. f t - \
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\ (sumr 0 n (%m. diff m 0 / real (fact m) * t ^ m) + \
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\ B * (t ^ n / real (fact n))))"));
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by (ALLGOALS(thin_tac "f h = \
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\ sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + \
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\ B * (h ^ n / real (fact n))"));
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(* back to business *)
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by (Asm_simp_tac 1);
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by (rtac DERIV_unique 1);
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by (Blast_tac 2);
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by (Force_tac 1);
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by (rtac allI 1 THEN induct_tac "ma" 1);
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by (rtac impI 1 THEN rtac Rolle 1);
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by (assume_tac 1);
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by (Asm_full_simp_tac 1);
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by (Asm_full_simp_tac 1);
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by (subgoal_tac "ALL t. 0 <= t & t <= h --> g differentiable t" 1);
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by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 1);
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by (blast_tac (claset() addDs [DERIV_isCont]) 1);
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by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 1);
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by (Clarify_tac 1);
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by (res_inst_tac [("x","difg (Suc 0) t")] exI 1);
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by (Force_tac 1);
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by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 1);
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by (Clarify_tac 1);
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by (res_inst_tac [("x","difg (Suc 0) x")] exI 1);
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by (Force_tac 1);
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by (Step_tac 1);
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by (Force_tac 1);
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by (subgoal_tac "EX ta. 0 < ta & ta < t & \
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\ DERIV difg (Suc n) ta :> 0" 1);
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by (rtac Rolle 2 THEN assume_tac 2);
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by (Asm_full_simp_tac 2);
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by (rotate_tac 2 2);
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by (dres_inst_tac [("x","n")] spec 2);
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by (ftac (ARITH_PROVE "n < m ==> n < Suc m") 2);
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by (rtac DERIV_unique 2);
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by (assume_tac 3);
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by (Force_tac 2);
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by (subgoal_tac
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"ALL ta. 0 <= ta & ta <= t --> (difg (Suc n)) differentiable ta" 2);
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by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 2);
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by (blast_tac (claset() addSDs [DERIV_isCont]) 2);
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by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 2);
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by (Clarify_tac 2);
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by (res_inst_tac [("x","difg (Suc (Suc n)) ta")] exI 2);
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by (Force_tac 2);
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by (asm_full_simp_tac (simpset() addsimps [differentiable_def]) 2);
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by (Clarify_tac 2);
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by (res_inst_tac [("x","difg (Suc (Suc n)) x")] exI 2);
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by (Force_tac 2);
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by (Step_tac 1);
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by (res_inst_tac [("x","ta")] exI 1);
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by (Force_tac 1);
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qed "Maclaurin";
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Goal "0 < h & 0 < n & diff 0 = f & \
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\ (ALL m t. \
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\ m < n & 0 <= t & t <= h --> DERIV (diff m) t :> diff (Suc m) t) \
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\ --> (EX t. 0 < t & \
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\ t < h & \
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\ f h = \
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\ sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + \
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\ diff n t / real (fact n) * h ^ n)";
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by (blast_tac (claset() addIs [Maclaurin]) 1);
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qed "Maclaurin_objl";
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Goal " [| 0 < h; diff 0 = f; \
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\ ALL m t. \
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\ m < n & 0 <= t & t <= h --> DERIV (diff m) t :> diff (Suc m) t |] \
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\ ==> EX t. 0 < t & \
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\ t <= h & \
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\ f h = \
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\ sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + \
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\ diff n t / real (fact n) * h ^ n";
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by (case_tac "n" 1);
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by Auto_tac;
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by (dtac Maclaurin 1 THEN Auto_tac);
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qed "Maclaurin2";
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Goal "0 < h & diff 0 = f & \
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\ (ALL m t. \
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\ m < n & 0 <= t & t <= h --> DERIV (diff m) t :> diff (Suc m) t) \
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\ --> (EX t. 0 < t & \
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\ t <= h & \
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\ f h = \
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\ sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + \
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\ diff n t / real (fact n) * h ^ n)";
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by (blast_tac (claset() addIs [Maclaurin2]) 1);
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qed "Maclaurin2_objl";
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Goal " [| h < 0; 0 < n; diff 0 = f; \
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\ ALL m t. \
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\ m < n & h <= t & t <= 0 --> DERIV (diff m) t :> diff (Suc m) t |] \
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\ ==> EX t. h < t & \
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\ t < 0 & \
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\ f h = \
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\ sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + \
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\ diff n t / real (fact n) * h ^ n";
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by (cut_inst_tac [("f","%x. f (-x)"),
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("diff","%n x. ((- 1) ^ n) * diff n (-x)"),
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("h","-h"),("n","n")] Maclaurin_objl 1);
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by (Asm_full_simp_tac 1);
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by (etac impE 1 THEN Step_tac 1);
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by (stac minus_mult_right 1);
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by (rtac DERIV_cmult 1);
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by (rtac lemma_DERIV_subst 1);
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by (rtac (read_instantiate [("g","uminus")] DERIV_chain2) 1);
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by (rtac DERIV_minus 2 THEN rtac DERIV_Id 2);
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by (Force_tac 2);
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by (Force_tac 1);
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by (res_inst_tac [("x","-t")] exI 1);
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by Auto_tac;
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by (rtac (CLAIM "[| x = x'; y = y' |] ==> x + y = x' + (y'::real)") 1);
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by (rtac sumr_fun_eq 1);
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by (Asm_full_simp_tac 1);
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by (auto_tac (claset(),simpset() addsimps [real_divide_def,
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CLAIM "((a * b) * c) * d = (b * c) * (a * (d::real))",
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power_mult_distrib RS sym]));
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qed "Maclaurin_minus";
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Goal "(h < 0 & 0 < n & diff 0 = f & \
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\ (ALL m t. \
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\ m < n & h <= t & t <= 0 --> DERIV (diff m) t :> diff (Suc m) t))\
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\ --> (EX t. h < t & \
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\ t < 0 & \
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\ f h = \
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\ sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + \
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\ diff n t / real (fact n) * h ^ n)";
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by (blast_tac (claset() addIs [Maclaurin_minus]) 1);
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qed "Maclaurin_minus_objl";
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(* ------------------------------------------------------------------------- *)
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(* More convenient "bidirectional" version. *)
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(* ------------------------------------------------------------------------- *)
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(* not good for PVS sin_approx, cos_approx *)
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Goal " [| diff 0 = f; \
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\ ALL m t. \
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|
305 |
\ m < n & abs t <= abs x --> DERIV (diff m) t :> diff (Suc m) t |] \
|
|
306 |
\ ==> EX t. abs t <= abs x & \
|
|
307 |
\ f x = \
|
|
308 |
\ sumr 0 n (%m. diff m 0 / real (fact m) * x ^ m) + \
|
|
309 |
\ diff n t / real (fact n) * x ^ n";
|
|
310 |
by (case_tac "n = 0" 1);
|
|
311 |
by (Force_tac 1);
|
|
312 |
by (case_tac "x = 0" 1);
|
|
313 |
by (res_inst_tac [("x","0")] exI 1);
|
|
314 |
by (Asm_full_simp_tac 1);
|
|
315 |
by (res_inst_tac [("P","0 < n")] impE 1);
|
|
316 |
by (assume_tac 2 THEN assume_tac 2);
|
|
317 |
by (induct_tac "n" 1);
|
|
318 |
by (Simp_tac 1);
|
|
319 |
by Auto_tac;
|
|
320 |
by (cut_inst_tac [("x","x"),("y","0")] linorder_less_linear 1);
|
|
321 |
by Auto_tac;
|
|
322 |
by (cut_inst_tac [("f","diff 0"),
|
|
323 |
("diff","diff"),
|
|
324 |
("h","x"),("n","n")] Maclaurin_objl 2);
|
|
325 |
by (Step_tac 2);
|
|
326 |
by (blast_tac (claset() addDs
|
|
327 |
[ARITH_PROVE "[|(0::real) <= t;t <= x |] ==> abs t <= abs x"]) 2);
|
|
328 |
by (res_inst_tac [("x","t")] exI 2);
|
|
329 |
by (force_tac (claset() addIs
|
|
330 |
[ARITH_PROVE "[| 0 < t; (t::real) < x|] ==> abs t <= abs x"],simpset()) 2);
|
|
331 |
by (cut_inst_tac [("f","diff 0"),
|
|
332 |
("diff","diff"),
|
|
333 |
("h","x"),("n","n")] Maclaurin_minus_objl 1);
|
|
334 |
by (Step_tac 1);
|
|
335 |
by (blast_tac (claset() addDs
|
|
336 |
[ARITH_PROVE "[|x <= t;t <= (0::real) |] ==> abs t <= abs x"]) 1);
|
|
337 |
by (res_inst_tac [("x","t")] exI 1);
|
|
338 |
by (force_tac (claset() addIs
|
|
339 |
[ARITH_PROVE "[| x < t; (t::real) < 0|] ==> abs t <= abs x"],simpset()) 1);
|
|
340 |
qed "Maclaurin_bi_le";
|
|
341 |
|
|
342 |
Goal "[| diff 0 = f; \
|
|
343 |
\ ALL m x. DERIV (diff m) x :> diff(Suc m) x; \
|
|
344 |
\ x ~= 0; 0 < n \
|
|
345 |
\ |] ==> EX t. 0 < abs t & abs t < abs x & \
|
|
346 |
\ f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) + \
|
|
347 |
\ (diff n t / real (fact n)) * x ^ n";
|
|
348 |
by (res_inst_tac [("x","x"),("y","0")] linorder_cases 1);
|
|
349 |
by (Blast_tac 2);
|
|
350 |
by (dtac Maclaurin_minus 1);
|
|
351 |
by (dtac Maclaurin 5);
|
|
352 |
by (TRYALL(assume_tac));
|
|
353 |
by (Blast_tac 1);
|
|
354 |
by (Blast_tac 2);
|
|
355 |
by (Step_tac 1);
|
|
356 |
by (ALLGOALS(res_inst_tac [("x","t")] exI));
|
|
357 |
by (Step_tac 1);
|
|
358 |
by (ALLGOALS(arith_tac));
|
|
359 |
qed "Maclaurin_all_lt";
|
|
360 |
|
|
361 |
Goal "diff 0 = f & \
|
|
362 |
\ (ALL m x. DERIV (diff m) x :> diff(Suc m) x) & \
|
|
363 |
\ x ~= 0 & 0 < n \
|
|
364 |
\ --> (EX t. 0 < abs t & abs t < abs x & \
|
|
365 |
\ f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) + \
|
|
366 |
\ (diff n t / real (fact n)) * x ^ n)";
|
|
367 |
by (blast_tac (claset() addIs [Maclaurin_all_lt]) 1);
|
|
368 |
qed "Maclaurin_all_lt_objl";
|
|
369 |
|
|
370 |
Goal "x = (0::real) \
|
|
371 |
\ ==> 0 < n --> \
|
|
372 |
\ sumr 0 n (%m. (diff m (0::real) / real (fact m)) * x ^ m) = \
|
|
373 |
\ diff 0 0";
|
|
374 |
by (Asm_simp_tac 1);
|
|
375 |
by (induct_tac "n" 1);
|
|
376 |
by Auto_tac;
|
|
377 |
qed_spec_mp "Maclaurin_zero";
|
|
378 |
|
|
379 |
Goal "[| diff 0 = f; \
|
|
380 |
\ ALL m x. DERIV (diff m) x :> diff (Suc m) x \
|
|
381 |
\ |] ==> EX t. abs t <= abs x & \
|
|
382 |
\ f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) + \
|
|
383 |
\ (diff n t / real (fact n)) * x ^ n";
|
|
384 |
by (cut_inst_tac [("n","n"),("m","0")]
|
|
385 |
(ARITH_PROVE "n <= m | m < (n::nat)") 1);
|
|
386 |
by (etac disjE 1);
|
|
387 |
by (Force_tac 1);
|
|
388 |
by (case_tac "x = 0" 1);
|
|
389 |
by (forw_inst_tac [("diff","diff"),("n","n")] Maclaurin_zero 1);
|
|
390 |
by (assume_tac 1);
|
|
391 |
by (dtac (gr_implies_not0 RS not0_implies_Suc) 1);
|
|
392 |
by (res_inst_tac [("x","0")] exI 1);
|
|
393 |
by (Force_tac 1);
|
|
394 |
by (forw_inst_tac [("diff","diff"),("n","n")] Maclaurin_all_lt 1);
|
|
395 |
by (TRYALL(assume_tac));
|
|
396 |
by (Step_tac 1);
|
|
397 |
by (res_inst_tac [("x","t")] exI 1);
|
|
398 |
by Auto_tac;
|
|
399 |
qed "Maclaurin_all_le";
|
|
400 |
|
|
401 |
Goal "diff 0 = f & \
|
|
402 |
\ (ALL m x. DERIV (diff m) x :> diff (Suc m) x) \
|
|
403 |
\ --> (EX t. abs t <= abs x & \
|
|
404 |
\ f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) + \
|
|
405 |
\ (diff n t / real (fact n)) * x ^ n)";
|
|
406 |
by (blast_tac (claset() addIs [Maclaurin_all_le]) 1);
|
|
407 |
qed "Maclaurin_all_le_objl";
|
|
408 |
|
|
409 |
(* ------------------------------------------------------------------------- *)
|
|
410 |
(* Version for exp. *)
|
|
411 |
(* ------------------------------------------------------------------------- *)
|
|
412 |
|
|
413 |
Goal "[| x ~= 0; 0 < n |] \
|
|
414 |
\ ==> (EX t. 0 < abs t & \
|
|
415 |
\ abs t < abs x & \
|
|
416 |
\ exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) + \
|
|
417 |
\ (exp t / real (fact n)) * x ^ n)";
|
|
418 |
by (cut_inst_tac [("diff","%n. exp"),("f","exp"),("x","x"),("n","n")]
|
|
419 |
Maclaurin_all_lt_objl 1);
|
|
420 |
by Auto_tac;
|
|
421 |
qed "Maclaurin_exp_lt";
|
|
422 |
|
|
423 |
Goal "EX t. abs t <= abs x & \
|
|
424 |
\ exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) + \
|
|
425 |
\ (exp t / real (fact n)) * x ^ n";
|
|
426 |
by (cut_inst_tac [("diff","%n. exp"),("f","exp"),("x","x"),("n","n")]
|
|
427 |
Maclaurin_all_le_objl 1);
|
|
428 |
by Auto_tac;
|
|
429 |
qed "Maclaurin_exp_le";
|
|
430 |
|
|
431 |
(* ------------------------------------------------------------------------- *)
|
|
432 |
(* Version for sin function *)
|
|
433 |
(* ------------------------------------------------------------------------- *)
|
|
434 |
|
|
435 |
Goal "[| a < b; ALL x. a <= x & x <= b --> DERIV f x :> f'(x) |] \
|
|
436 |
\ ==> EX z. a < z & z < b & (f b - f a = (b - a) * f'(z))";
|
|
437 |
by (dtac MVT 1);
|
|
438 |
by (blast_tac (claset() addIs [DERIV_isCont]) 1);
|
|
439 |
by (force_tac (claset() addDs [order_less_imp_le],
|
|
440 |
simpset() addsimps [differentiable_def]) 1);
|
|
441 |
by (blast_tac (claset() addDs [DERIV_unique,order_less_imp_le]) 1);
|
|
442 |
qed "MVT2";
|
|
443 |
|
|
444 |
Goal "d < (4::nat) ==> d = 0 | d = 1 | d = 2 | d = 3";
|
|
445 |
by (case_tac "d" 1 THEN Auto_tac);
|
|
446 |
qed "lemma_exhaust_less_4";
|
|
447 |
|
|
448 |
bind_thm ("real_mult_le_lemma",
|
|
449 |
simplify (simpset()) (inst "b" "1" mult_right_mono));
|
|
450 |
|
|
451 |
|
|
452 |
Goal "0 < n --> Suc (Suc (2 * n - 2)) = 2*n";
|
|
453 |
by (induct_tac "n" 1);
|
|
454 |
by Auto_tac;
|
|
455 |
qed_spec_mp "Suc_Suc_mult_two_diff_two";
|
|
456 |
Addsimps [Suc_Suc_mult_two_diff_two];
|
|
457 |
|
|
458 |
Goal "0 < n --> Suc (Suc (4*n - 2)) = 4*n";
|
|
459 |
by (induct_tac "n" 1);
|
|
460 |
by Auto_tac;
|
|
461 |
qed_spec_mp "lemma_Suc_Suc_4n_diff_2";
|
|
462 |
Addsimps [lemma_Suc_Suc_4n_diff_2];
|
|
463 |
|
|
464 |
Goal "0 < n --> Suc (2 * n - 1) = 2*n";
|
|
465 |
by (induct_tac "n" 1);
|
|
466 |
by Auto_tac;
|
|
467 |
qed_spec_mp "Suc_mult_two_diff_one";
|
|
468 |
Addsimps [Suc_mult_two_diff_one];
|
|
469 |
|
|
470 |
Goal "EX t. sin x = \
|
|
471 |
\ (sumr 0 n (%m. (if even m then 0 \
|
|
472 |
\ else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) * \
|
|
473 |
\ x ^ m)) \
|
|
474 |
\ + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)";
|
|
475 |
by (cut_inst_tac [("f","sin"),("n","n"),("x","x"),
|
|
476 |
("diff","%n x. sin(x + 1/2*real (n)*pi)")]
|
|
477 |
Maclaurin_all_lt_objl 1);
|
|
478 |
by (Safe_tac);
|
|
479 |
by (Simp_tac 1);
|
|
480 |
by (Simp_tac 1);
|
|
481 |
by (case_tac "n" 1);
|
|
482 |
by (Clarify_tac 1);
|
|
483 |
by (Asm_full_simp_tac 1);
|
|
484 |
by (dres_inst_tac [("x","0")] spec 1 THEN Asm_full_simp_tac 1);
|
|
485 |
by (Asm_full_simp_tac 1);
|
|
486 |
by (rtac ccontr 1);
|
|
487 |
by (Asm_full_simp_tac 1);
|
|
488 |
by (dres_inst_tac [("x","x")] spec 1 THEN Asm_full_simp_tac 1);
|
|
489 |
by (dtac ssubst 1 THEN assume_tac 2);
|
|
490 |
by (res_inst_tac [("x","t")] exI 1);
|
|
491 |
by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);
|
|
492 |
by (rtac sumr_fun_eq 1);
|
|
493 |
by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));
|
|
494 |
by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex] delsimps [fact_Suc,realpow_Suc]));
|
|
495 |
(*Could sin_zero_iff help?*)
|
|
496 |
qed "Maclaurin_sin_expansion";
|
|
497 |
|
|
498 |
Goal "EX t. abs t <= abs x & \
|
|
499 |
\ sin x = \
|
|
500 |
\ (sumr 0 n (%m. (if even m then 0 \
|
|
501 |
\ else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) * \
|
|
502 |
\ x ^ m)) \
|
|
503 |
\ + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)";
|
|
504 |
|
|
505 |
by (cut_inst_tac [("f","sin"),("n","n"),("x","x"),
|
|
506 |
("diff","%n x. sin(x + 1/2*real (n)*pi)")]
|
|
507 |
Maclaurin_all_lt_objl 1);
|
|
508 |
by (Step_tac 1);
|
|
509 |
by (Simp_tac 1);
|
|
510 |
by (Simp_tac 1);
|
|
511 |
by (case_tac "n" 1);
|
|
512 |
by (Clarify_tac 1);
|
|
513 |
by (Asm_full_simp_tac 1);
|
|
514 |
by (Asm_full_simp_tac 1);
|
|
515 |
by (rtac ccontr 1);
|
|
516 |
by (Asm_full_simp_tac 1);
|
|
517 |
by (dres_inst_tac [("x","x")] spec 1 THEN Asm_full_simp_tac 1);
|
|
518 |
by (dtac ssubst 1 THEN assume_tac 2);
|
|
519 |
by (res_inst_tac [("x","t")] exI 1);
|
|
520 |
by (rtac conjI 1);
|
|
521 |
by (arith_tac 1);
|
|
522 |
by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);
|
|
523 |
by (rtac sumr_fun_eq 1);
|
|
524 |
by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));
|
|
525 |
by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex] delsimps [fact_Suc,realpow_Suc]));
|
|
526 |
qed "Maclaurin_sin_expansion2";
|
|
527 |
|
|
528 |
Goal "[| 0 < n; 0 < x |] ==> \
|
|
529 |
\ EX t. 0 < t & t < x & \
|
|
530 |
\ sin x = \
|
|
531 |
\ (sumr 0 n (%m. (if even m then 0 \
|
|
532 |
\ else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) * \
|
|
533 |
\ x ^ m)) \
|
|
534 |
\ + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)";
|
|
535 |
by (cut_inst_tac [("f","sin"),("n","n"),("h","x"),
|
|
536 |
("diff","%n x. sin(x + 1/2*real (n)*pi)")]
|
|
537 |
Maclaurin_objl 1);
|
|
538 |
by (Step_tac 1);
|
|
539 |
by (Asm_full_simp_tac 1);
|
|
540 |
by (Simp_tac 1);
|
|
541 |
by (dtac ssubst 1 THEN assume_tac 2);
|
|
542 |
by (res_inst_tac [("x","t")] exI 1);
|
|
543 |
by (rtac conjI 1 THEN rtac conjI 2);
|
|
544 |
by (assume_tac 1 THEN assume_tac 1);
|
|
545 |
by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);
|
|
546 |
by (rtac sumr_fun_eq 1);
|
|
547 |
by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));
|
|
548 |
by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex] delsimps [fact_Suc,realpow_Suc]));
|
|
549 |
qed "Maclaurin_sin_expansion3";
|
|
550 |
|
|
551 |
Goal "0 < x ==> \
|
|
552 |
\ EX t. 0 < t & t <= x & \
|
|
553 |
\ sin x = \
|
|
554 |
\ (sumr 0 n (%m. (if even m then 0 \
|
|
555 |
\ else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) * \
|
|
556 |
\ x ^ m)) \
|
|
557 |
\ + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)";
|
|
558 |
by (cut_inst_tac [("f","sin"),("n","n"),("h","x"),
|
|
559 |
("diff","%n x. sin(x + 1/2*real (n)*pi)")]
|
|
560 |
Maclaurin2_objl 1);
|
|
561 |
by (Step_tac 1);
|
|
562 |
by (Asm_full_simp_tac 1);
|
|
563 |
by (Simp_tac 1);
|
|
564 |
by (dtac ssubst 1 THEN assume_tac 2);
|
|
565 |
by (res_inst_tac [("x","t")] exI 1);
|
|
566 |
by (rtac conjI 1 THEN rtac conjI 2);
|
|
567 |
by (assume_tac 1 THEN assume_tac 1);
|
|
568 |
by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);
|
|
569 |
by (rtac sumr_fun_eq 1);
|
|
570 |
by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));
|
|
571 |
by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex] delsimps [fact_Suc,realpow_Suc]));
|
|
572 |
qed "Maclaurin_sin_expansion4";
|
|
573 |
|
|
574 |
(*-----------------------------------------------------------------------------*)
|
|
575 |
(* Maclaurin expansion for cos *)
|
|
576 |
(*-----------------------------------------------------------------------------*)
|
|
577 |
|
|
578 |
Goal "sumr 0 (Suc n) \
|
|
579 |
\ (%m. (if even m \
|
|
580 |
\ then (- 1) ^ (m div 2)/(real (fact m)) \
|
|
581 |
\ else 0) * \
|
|
582 |
\ 0 ^ m) = 1";
|
|
583 |
by (induct_tac "n" 1);
|
|
584 |
by Auto_tac;
|
|
585 |
qed "sumr_cos_zero_one";
|
|
586 |
Addsimps [sumr_cos_zero_one];
|
|
587 |
|
|
588 |
Goal "EX t. abs t <= abs x & \
|
|
589 |
\ cos x = \
|
|
590 |
\ (sumr 0 n (%m. (if even m \
|
|
591 |
\ then (- 1) ^ (m div 2)/(real (fact m)) \
|
|
592 |
\ else 0) * \
|
|
593 |
\ x ^ m)) \
|
|
594 |
\ + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)";
|
|
595 |
by (cut_inst_tac [("f","cos"),("n","n"),("x","x"),
|
|
596 |
("diff","%n x. cos(x + 1/2*real (n)*pi)")]
|
|
597 |
Maclaurin_all_lt_objl 1);
|
|
598 |
by (Step_tac 1);
|
|
599 |
by (Simp_tac 1);
|
|
600 |
by (Simp_tac 1);
|
|
601 |
by (case_tac "n" 1);
|
|
602 |
by (Asm_full_simp_tac 1);
|
|
603 |
by (asm_full_simp_tac (simpset() delsimps [sumr_Suc]) 1);
|
|
604 |
by (rtac ccontr 1);
|
|
605 |
by (Asm_full_simp_tac 1);
|
|
606 |
by (dres_inst_tac [("x","x")] spec 1 THEN Asm_full_simp_tac 1);
|
|
607 |
by (dtac ssubst 1 THEN assume_tac 2);
|
|
608 |
by (res_inst_tac [("x","t")] exI 1);
|
|
609 |
by (rtac conjI 1);
|
|
610 |
by (arith_tac 1);
|
|
611 |
by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);
|
|
612 |
by (rtac sumr_fun_eq 1);
|
|
613 |
by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));
|
|
614 |
by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex,left_distrib,cos_add] delsimps
|
|
615 |
[fact_Suc,realpow_Suc]));
|
|
616 |
by (auto_tac (claset(),simpset() addsimps [real_mult_commute]));
|
|
617 |
qed "Maclaurin_cos_expansion";
|
|
618 |
|
|
619 |
Goal "[| 0 < x; 0 < n |] ==> \
|
|
620 |
\ EX t. 0 < t & t < x & \
|
|
621 |
\ cos x = \
|
|
622 |
\ (sumr 0 n (%m. (if even m \
|
|
623 |
\ then (- 1) ^ (m div 2)/(real (fact m)) \
|
|
624 |
\ else 0) * \
|
|
625 |
\ x ^ m)) \
|
|
626 |
\ + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)";
|
|
627 |
by (cut_inst_tac [("f","cos"),("n","n"),("h","x"),
|
|
628 |
("diff","%n x. cos(x + 1/2*real (n)*pi)")]
|
|
629 |
Maclaurin_objl 1);
|
|
630 |
by (Step_tac 1);
|
|
631 |
by (Asm_full_simp_tac 1);
|
|
632 |
by (Simp_tac 1);
|
|
633 |
by (dtac ssubst 1 THEN assume_tac 2);
|
|
634 |
by (res_inst_tac [("x","t")] exI 1);
|
|
635 |
by (rtac conjI 1 THEN rtac conjI 2);
|
|
636 |
by (assume_tac 1 THEN assume_tac 1);
|
|
637 |
by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);
|
|
638 |
by (rtac sumr_fun_eq 1);
|
|
639 |
by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));
|
|
640 |
by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex,left_distrib,cos_add] delsimps [fact_Suc,realpow_Suc]));
|
|
641 |
by (auto_tac (claset(),simpset() addsimps [real_mult_commute]));
|
|
642 |
qed "Maclaurin_cos_expansion2";
|
|
643 |
|
|
644 |
Goal "[| x < 0; 0 < n |] ==> \
|
|
645 |
\ EX t. x < t & t < 0 & \
|
|
646 |
\ cos x = \
|
|
647 |
\ (sumr 0 n (%m. (if even m \
|
|
648 |
\ then (- 1) ^ (m div 2)/(real (fact m)) \
|
|
649 |
\ else 0) * \
|
|
650 |
\ x ^ m)) \
|
|
651 |
\ + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)";
|
|
652 |
by (cut_inst_tac [("f","cos"),("n","n"),("h","x"),
|
|
653 |
("diff","%n x. cos(x + 1/2*real (n)*pi)")]
|
|
654 |
Maclaurin_minus_objl 1);
|
|
655 |
by (Step_tac 1);
|
|
656 |
by (Asm_full_simp_tac 1);
|
|
657 |
by (Simp_tac 1);
|
|
658 |
by (dtac ssubst 1 THEN assume_tac 2);
|
|
659 |
by (res_inst_tac [("x","t")] exI 1);
|
|
660 |
by (rtac conjI 1 THEN rtac conjI 2);
|
|
661 |
by (assume_tac 1 THEN assume_tac 1);
|
|
662 |
by (rtac (CLAIM "[|x = y; x' = y'|] ==> x + x' = y + (y'::real)") 1);
|
|
663 |
by (rtac sumr_fun_eq 1);
|
|
664 |
by (auto_tac (claset(),simpset() addsimps [odd_Suc_mult_two_ex]));
|
|
665 |
by (auto_tac (claset(),simpset() addsimps [even_mult_two_ex,left_distrib,cos_add] delsimps [fact_Suc,realpow_Suc]));
|
|
666 |
by (auto_tac (claset(),simpset() addsimps [real_mult_commute]));
|
|
667 |
qed "Maclaurin_minus_cos_expansion";
|
|
668 |
|
|
669 |
(* ------------------------------------------------------------------------- *)
|
|
670 |
(* Version for ln(1 +/- x). Where is it?? *)
|
|
671 |
(* ------------------------------------------------------------------------- *)
|
|
672 |
|