| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/lnexp_fnd/ (Beweissystem der NASA Version 6.0.9©) Datei vom 28.9.2014 mit Größe 217 kB |
|
Quelle atanh_series.proof Sprache: unbekannt |
|
|
Spracherkennung für: .proof vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen] USED PATCH 873
----------------- Starting pvs-allegro6.2 -qq ... Allegro CL Enterprise Edition 6.2 [Linux (x86)] (Nov 3, 2004 23:30) Copyright (C) 1985-2002, Franz Inc., Berkeley, CA, USA. All Rights Reserved. This dynamic runtime copy of Allegro CL was built by: [TC8101] SRI International ;; Optimization settings: safety 1, space 1, speed 3, debug 1. ;; For a complete description of all compiler switches given the ;; current optimization settings evaluate (explain-compiler-settings). ;;--- ;; Current reader case mode: :case-sensitive-lower pvs(1): pvs(2): nil pvs(9): nil pvs(11): ; Loading /home/rwb/pvs-strategies Defining sqrt-rew. Defining sqrt-rew$. Defining sqrt-rew-off. Defining sqrt-rew-off$. Installing rewrite rule sq_abs_neg Installing rewrite rule sq_abs Installing rewrite rule sq_1 Installing rewrite rule sq_0 Installing rewrite rule sq_sqrt Installing rewrite rule sqrt_sq_neg Installing rewrite rule sqrt_sq Installing rewrite rule sqrt_square Installing rewrite rule sqrt_1 Installing rewrite rule sqrt_0 Installing rewrite rule sqrt_25 Installing rewrite rule sqrt_16 Installing rewrite rule sqrt_9 Installing rewrite rule sqrt_4 Installing rewrite rule factorial_0 Installing rewrite rule factorial_1 Installing rewrite rule ln_e Installing rewrite rule ln_1 Installing rewrite rule exp_1 Installing rewrite rule exp_0 Installing rewrite rule Riemann?_Rie Installing rewrite rule xis_lem Installing rewrite rule xis? Installing rewrite rule member Installing rewrite rule member Installing rewrite rule finseq_appl Installing rewrite rule not_one_element Installing rewrite rule sort_length Installing rewrite rule not_one_element Installing rewrite rule set2seq_length Installing rewrite rule set2part_length Installing rewrite rule not_one_element Installing rewrite rule not_one_element Installing rewrite rule not_one_element Installing rewrite rule not_one_element atanh_series : |------- {1} FORALL (n: nat, z: real_abs_lt1): abs(atanh(z) - atanh_series_n(z, n)) <= ((1 + z) ^ (2 * n + 3) + (1 - z) ^ (2 * n + 3)) / (2 * (2 * n + 3) * (1 - sq(z)) ^ (2 * n + 3)) Rerunning step: (skosimp*) Repeatedly Skolemizing and flattening, this simplifies to: atanh_series : |------- {1} abs(atanh(z!1) - atanh_series_n(z!1, n!1)) <= ((1 + z!1) ^ (2 * n!1 + 3) + (1 - z!1) ^ (2 * n!1 + 3)) / (2 * (2 * n!1 + 3) * (1 - sq(z!1)) ^ (2 * n!1 + 3)) Rerunning step: (lemma "atanh_taylors" ("z" "z!1" "n" "n!1")) Applying atanh_taylors where z gets z!1, n gets n!1, this simplifies to: atanh_series : {-1} EXISTS (c: between[real_abs_lt1](0, z!1)): atanh(z!1) = atanh_series_n(z!1, n!1) + nderiv[real_abs_lt1](2 * n!1 + 3, atanh)(c) * z!1 ^ (2 * n!1 + 3) / factorial(2 * n!1 + 3) |------- [1] abs(atanh(z!1) - atanh_series_n(z!1, n!1)) <= ((1 + z!1) ^ (2 * n!1 + 3) + (1 - z!1) ^ (2 * n!1 + 3)) / (2 * (2 * n!1 + 3) * (1 - sq(z!1)) ^ (2 * n!1 + 3)) Rerunning step: (skosimp*) Repeatedly Skolemizing and flattening, this simplifies to: atanh_series : {-1} atanh(z!1) = atanh_series_n(z!1, n!1) + nderiv[real_abs_lt1](2 * n!1 + 3, atanh)(c!1) * z!1 ^ (2 * n!1 + 3) / factorial(2 * n!1 + 3) |------- [1] abs(atanh(z!1) - atanh_series_n(z!1, n!1)) <= ((1 + z!1) ^ (2 * n!1 + 3) + (1 - z!1) ^ (2 * n!1 + 3)) / (2 * (2 * n!1 + 3) * (1 - sq(z!1)) ^ (2 * n!1 + 3)) Rerunning step: (replace -1 1) Replacing using formula -1, this simplifies to: atanh_series : [-1] atanh(z!1) = atanh_series_n(z!1, n!1) + nderiv[real_abs_lt1](2 * n!1 + 3, atanh)(c!1) * z!1 ^ (2 * n!1 + 3) / factorial(2 * n!1 + 3) |------- {1} abs(atanh_series_n(z!1, n!1) + nderiv[real_abs_lt1](2 * n!1 + 3, atanh)(c!1) * z!1 ^ (2 * n!1 + 3) / factorial(2 * n!1 + 3) - atanh_series_n(z!1, n!1)) <= ((1 + z!1) ^ (2 * n!1 + 3) + (1 - z!1) ^ (2 * n!1 + 3)) / (2 * (2 * n!1 + 3) * (1 - sq(z!1)) ^ (2 * n!1 + 3)) Rerunning step: (simplify 1) Simplifying with decision procedures, this simplifies to: atanh_series : [-1] atanh(z!1) = atanh_series_n(z!1, n!1) + nderiv[real_abs_lt1](2 * n!1 + 3, atanh)(c!1) * z!1 ^ (2 * n!1 + 3) / factorial(2 * n!1 + 3) |------- {1} abs((nderiv[real_abs_lt1](3 + 2 * n!1, atanh)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series : |------- [1] abs((nderiv[real_abs_lt1](3 + 2 * n!1, atanh)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (lemma "atanh_nderiv" ("n" "3+2*n!1")) Applying atanh_nderiv where n gets 3 + 2 * n!1, this simplifies to: atanh_series : {-1} nderiv[real_abs_lt1](3 + 2 * n!1, atanh) = IF 3 + 2 * n!1 = 0 THEN atanh ELSIF even?(3 + 2 * n!1) THEN deriv[real_abs_lt1](atanhND((3 + 2 * n!1) / 2 - 1)) ELSE atanhND((3 + 2 * n!1 - 1) / 2) ENDIF |------- [1] abs((nderiv[real_abs_lt1](3 + 2 * n!1, atanh)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (simplify -1) Simplifying with decision procedures, this simplifies to: atanh_series : {-1} nderiv[real_abs_lt1](3 + 2 * n!1, atanh) = IF even?(3 + 2 * n!1) THEN deriv[real_abs_lt1](atanhND((3 + 2 * n!1) / 2 - 1)) ELSE atanhND((2 + 2 * n!1) / 2) |------- [1] abs((nderiv[real_abs_lt1](3 + 2 * n!1, atanh)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (expand "even?" -1) Expanding the definition of even?, this simplifies to: atanh_series : {-1} nderiv[real_abs_lt1](3 + 2 * n!1, atanh) = atanhND((2 + 2 * n!1) / 2) |------- [1] abs((nderiv[real_abs_lt1](3 + 2 * n!1, atanh)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (lemma "div_cancel1" ("x" "1+n!1" "n0z" "2")) Applying div_cancel1 where x gets 1 + n!1, n0z gets 2, this simplifies to: atanh_series : {-1} 2 * ((1 + n!1) / 2) = 1 + n!1 [-2] nderiv[real_abs_lt1](3 + 2 * n!1, atanh) = atanhND((2 + 2 * n!1) / 2) |------- [1] abs((nderiv[real_abs_lt1](3 + 2 * n!1, atanh)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (replace -1 -2) Replacing using formula -1, this simplifies to: atanh_series : [-1] 2 * ((1 + n!1) / 2) = 1 + n!1 {-2} nderiv[real_abs_lt1](3 + 2 * n!1, atanh) = atanhND(1 + n!1) |------- [1] abs((nderiv[real_abs_lt1](3 + 2 * n!1, atanh)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (replace -2 1) Replacing using formula -2, this simplifies to: atanh_series : [-1] 2 * ((1 + n!1) / 2) = 1 + n!1 [-2] nderiv[real_abs_lt1](3 + 2 * n!1, atanh) = atanhND(1 + n!1) |------- {1} abs((atanhND(1 + n!1)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (hide -1 -2) Hiding formulas: -1, -2, this simplifies to: atanh_series : |------- [1] abs((atanhND(1 + n!1)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (case "1-sq(z!1)>0") Case splitting on 1 - sq(z!1) > 0, this yields 2 subgoals: atanh_series.1 : {-1} 1 - sq(z!1) > 0 |------- [1] abs((atanhND(1 + n!1)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (case "abs(z!1^(3+2*n!1))<1") Case splitting on abs(z!1 ^ (3 + 2 * n!1)) < 1, this yields 2 subgoals: atanh_series.1.1 : {-1} abs(z!1 ^ (3 + 2 * n!1)) < 1 [-2] 1 - sq(z!1) > 0 |------- [1] abs((atanhND(1 + n!1)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (rewrite "abs_div" 1) Found matching substitution: n0y: nonzero_real gets factorial(3 + 2 * n!1), x: real gets atanhND(1 + n!1)(c!1) * z!1 ^ (3 + 2 * n!1), Rewriting using abs_div, matching in 1, this simplifies to: atanh_series.1.1 : [-1] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-2] 1 - sq(z!1) > 0 |------- {1} abs(atanhND(1 + n!1)(c!1) * z!1 ^ (3 + 2 * n!1)) / abs(factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (rewrite "abs_mult" 1) Found matching substitution: y: real gets z!1 ^ (3 + 2 * n!1), x gets atanhND(1 + n!1)(c!1), Rewriting using abs_mult, matching in 1, this simplifies to: atanh_series.1.1 : [-1] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-2] 1 - sq(z!1) > 0 |------- {1} abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / abs(factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (expand "abs" 1 3) Expanding the definition of abs, this simplifies to: atanh_series.1.1 : [-1] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-2] 1 - sq(z!1) > 0 |------- {1} abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (lemma "expt_pos" ("px" "1-sq(z!1)" "i" "3+2*n!1")) Applying expt_pos where px gets 1 - sq(z!1), i gets 3 + 2 * n!1, this yields 2 subgoals: atanh_series.1.1.1 : {-1} (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- [1] abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (lemma "posreal_times_posreal_is_posreal" ("px" "6+4*n!1" "py" "(1 - sq(z!1)) ^ (3 + 2 * n!1)")) Applying posreal_times_posreal_is_posreal where px gets 6 + 4 * n!1, py gets (1 - sq(z!1)) ^ (3 + 2 * n!1), this yields 2 subgoals: atanh_series.1.1.1.1 : {-1} (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- [1] abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (case "atanhND(1 + n!1)(c!1)>0") Case splitting on atanhND(1 + n!1)(c!1) > 0, this yields 2 subgoals: atanh_series.1.1.1.1.1 : {-1} atanhND(1 + n!1)(c!1) > 0 [-2] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-5] 1 - sq(z!1) > 0 |------- [1] abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (case "abs(atanhND(1 + n!1)(c!1)) <= atanhND(1 + n!1)(z!1)") Case splitting on abs(atanhND(1 + n!1)(c!1)) <= atanhND(1 + n!1)(z!1), this yields 2 subgoals: atanh_series.1.1.1.1.1.1 : {-1} abs(atanhND(1 + n!1)(c!1)) <= atanhND(1 + n!1)(z!1) [-2] atanhND(1 + n!1)(c!1) > 0 [-3] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-5] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-6] 1 - sq(z!1) > 0 |------- [1] abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (case "atanhND(1 + n!1)(z!1) = factorial(2+2*n!1)*(((1 - z!1) ^ (3 + 2 * n!1) + ( Case splitting on atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))), this yields 3 subgoals: atanh_series.1.1.1.1.1.1.1 : {-1} atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) [-2] abs(atanhND(1 + n!1)(c!1)) <= atanhND(1 + n!1)(z!1) [-3] atanhND(1 + n!1)(c!1) > 0 [-4] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-5] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] 1 - sq(z!1) > 0 |------- [1] abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (rewrite "div_mult_pos_le1" 1) Found matching substitution: x gets ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)), py: posreal gets factorial(3 + 2 * n!1), z: real gets abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)), Rewriting using div_mult_pos_le1, matching in 1, this simplifies to: atanh_series.1.1.1.1.1.1.1 : [-1] atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) [-2] abs(atanhND(1 + n!1)(c!1)) <= atanhND(1 + n!1)(z!1) [-3] atanhND(1 + n!1)(c!1) > 0 [-4] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-5] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] 1 - sq(z!1) > 0 |------- {1} abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) <= (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1))) * factorial(3 + 2 * n!1) Rerunning step: (expand "abs" 1 1) Expanding the definition of abs, this simplifies to: atanh_series.1.1.1.1.1.1.1 : [-1] atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) [-2] abs(atanhND(1 + n!1)(c!1)) <= atanhND(1 + n!1)(z!1) [-3] atanhND(1 + n!1)(c!1) > 0 [-4] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-5] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] 1 - sq(z!1) > 0 |------- {1} IF atanhND(1 + n!1)(c!1) < 0 THEN -atanhND(1 + n!1)(c!1) ELSE atanhND(1 + n!1)(c!1) ENDIF * abs(z!1 ^ (3 + 2 * n!1)) <= (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1))) * factorial(3 + 2 * n!1) Rerunning step: (expand "abs" -2) Expanding the definition of abs, this simplifies to: atanh_series.1.1.1.1.1.1.1 : [-1] atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) {-2} IF atanhND(1 + n!1)(c!1) < 0 THEN -atanhND(1 + n!1)(c!1) ELSE atanhND(1 + n!1)(c!1) ENDIF <= atanhND(1 + n!1)(z!1) [-3] atanhND(1 + n!1)(c!1) > 0 [-4] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-5] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] 1 - sq(z!1) > 0 |------- [1] IF atanhND(1 + n!1)(c!1) < 0 THEN -atanhND(1 + n!1)(c!1) ELSE atanhND(1 + n!1)(c!1) ENDIF * abs(z!1 ^ (3 + 2 * n!1)) <= (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1))) * factorial(3 + 2 * n!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.1.1 : [-1] atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) {-2} atanhND(1 + n!1)(c!1) <= atanhND(1 + n!1)(z!1) [-3] atanhND(1 + n!1)(c!1) > 0 {-4} 6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * ((1 - sq(z!1)) ^ (3 + 2 * n!1) * n!1) > 0 [-5] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] 1 - sq(z!1) > 0 |------- {1} atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) <= (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1))) * factorial(3 + 2 * n!1) Rerunning step: (name-replace "K10" "1 - sq(z!1)") Using K10 to name and replace 1 - sq(z!1), this simplifies to: atanh_series.1.1.1.1.1.1.1 : {-1} atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * K10 ^ (3 + 2 * n!1))) [-2] atanhND(1 + n!1)(c!1) <= atanhND(1 + n!1)(z!1) [-3] atanhND(1 + n!1)(c!1) > 0 {-4} 6 * K10 ^ (3 + 2 * n!1) + 4 * (K10 ^ (3 + 2 * n!1) * n!1) > 0 {-5} K10 ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 {-7} K10 > 0 |------- {1} atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) <= (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * K10 ^ (3 + 2 * n!1) + 4 * ((K10 ^ (3 + 2 * n!1)) * n!1))) * factorial(3 + 2 * n!1) Rerunning step: (name-replace "K11" "K10^(3+2*n!1)") Using K11 to name and replace K10^(3+2*n!1), this simplifies to: atanh_series.1.1.1.1.1.1.1 : {-1} atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * K11)) [-2] atanhND(1 + n!1)(c!1) <= atanhND(1 + n!1)(z!1) [-3] atanhND(1 + n!1)(c!1) > 0 {-4} 6 * K11 + 4 * (K11 * n!1) > 0 {-5} K11 > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] K10 > 0 |------- {1} atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) <= (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * K11 + 4 * (K11 * n!1))) * factorial(3 + 2 * n!1) Rerunning step: (name-replace "K12" "(1-z!1)^(3+2*n!1)+(1+z!1)^(3+2*n!1)") Using K12 to name and replace (1-z!1)^(3+2*n!1)+(1+z!1)^(3+2*n!1), this simplifies to: atanh_series.1.1.1.1.1.1.1 : {-1} atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-2] atanhND(1 + n!1)(c!1) <= atanhND(1 + n!1)(z!1) [-3] atanhND(1 + n!1)(c!1) > 0 [-4] 6 * K11 + 4 * (K11 * n!1) > 0 [-5] K11 > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] K10 > 0 |------- {1} atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) <= (K12 / (6 * K11 + 4 * (K11 * n!1))) * factorial(3 + 2 * n!1) Rerunning step: (replace -1 -2) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.1 : [-1] atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (K12 / (2 * K11)) {-2} atanhND(1 + n!1)(c!1) <= factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-3] atanhND(1 + n!1)(c!1) > 0 [-4] 6 * K11 + 4 * (K11 * n!1) > 0 [-5] K11 > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] K10 > 0 |------- [1] atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) <= (K12 / (6 * K11 + 4 * (K11 * n!1))) * factorial(3 + 2 * n!1) Rerunning step: (lemma "lt_times_lt_pos1" ("px" "atanhND(1 + n!1)(c!1)" "y" "factorial(2 + 2 * n!1) * (K12 / (2 * K11))" "nnz" "abs(z!1 ^ (3 + 2 * n!1))" "w" "1")) Applying lt_times_lt_pos1 where px gets atanhND(1 + n!1)(c!1), y gets factorial(2 + 2 * n!1) * (K12 / (2 * K11)), nnz gets abs(z!1 ^ (3 + 2 * n!1)), w gets 1, this simplifies to: atanh_series.1.1.1.1.1.1.1 : {-1} atanhND(1 + n!1)(c!1) <= factorial(2 + 2 * n!1) * (K12 / (2 * K11)) AND abs(z!1 ^ (3 + 2 * n!1)) < 1 IMPLIES atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) < factorial(2 + 2 * n!1) * (K12 / (2 * K11)) * 1 [-2] atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-3] atanhND(1 + n!1)(c!1) <= factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-4] atanhND(1 + n!1)(c!1) > 0 [-5] 6 * K11 + 4 * (K11 * n!1) > 0 [-6] K11 > 0 [-7] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-8] K10 > 0 |------- [1] atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) <= (K12 / (6 * K11 + 4 * (K11 * n!1))) * factorial(3 + 2 * n!1) Rerunning step: (replace -3) Replacing using formula -3, this simplifies to: atanh_series.1.1.1.1.1.1.1 : {-1} abs(z!1 ^ (3 + 2 * n!1)) < 1 IMPLIES atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) < factorial(2 + 2 * n!1) * (K12 / (2 * K11)) * 1 [-2] atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-3] atanhND(1 + n!1)(c!1) <= factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-4] atanhND(1 + n!1)(c!1) > 0 [-5] 6 * K11 + 4 * (K11 * n!1) > 0 [-6] K11 > 0 [-7] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-8] K10 > 0 |------- [1] atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) <= (K12 / (6 * K11 + 4 * (K11 * n!1))) * factorial(3 + 2 * n!1) Rerunning step: (replace -7) Replacing using formula -7, this simplifies to: atanh_series.1.1.1.1.1.1.1 : {-1} atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) < factorial(2 + 2 * n!1) * (K12 / (2 * K11)) * 1 [-2] atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-3] atanhND(1 + n!1)(c!1) <= factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-4] atanhND(1 + n!1)(c!1) > 0 [-5] 6 * K11 + 4 * (K11 * n!1) > 0 [-6] K11 > 0 [-7] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-8] K10 > 0 |------- [1] atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) <= (K12 / (6 * K11 + 4 * (K11 * n!1))) * factorial(3 + 2 * n!1) Rerunning step: (expand "factorial" 1) Expanding the definition of factorial, this simplifies to: atanh_series.1.1.1.1.1.1.1 : [-1] atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) < factorial(2 + 2 * n!1) * (K12 / (2 * K11)) * 1 [-2] atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-3] atanhND(1 + n!1)(c!1) <= factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-4] atanhND(1 + n!1)(c!1) > 0 [-5] 6 * K11 + 4 * (K11 * n!1) > 0 [-6] K11 > 0 [-7] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-8] K10 > 0 |------- {1} atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) <= 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) Rerunning step: (case-replace "factorial(2 + 2 * n!1) * (K12 / (2 * K11)) * 1 = 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1)") Assuming and applying factorial(2 + 2 * n!1) * (K12 / (2 * K11)) * 1 = 3 * (factorial(2 + 2 * n!1) * (K12 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1), this yields 2 subgoals: atanh_series.1.1.1.1.1.1.1.1 : {-1} factorial(2 + 2 * n!1) * (K12 / (2 * K11)) * 1 = 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) {-2} atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) < 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) {-3} atanhND(1 + n!1)(z!1) = 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) {-4} atanhND(1 + n!1)(c!1) <= 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) [-5] atanhND(1 + n!1)(c!1) > 0 [-6] 6 * K11 + 4 * (K11 * n!1) > 0 [-7] K11 > 0 [-8] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-9] K10 > 0 |------- [1] atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) <= 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.1.1.1. atanh_series.1.1.1.1.1.1.1.2 : [-1] atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) < factorial(2 + 2 * n!1) * (K12 / (2 * K11)) * 1 [-2] atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-3] atanhND(1 + n!1)(c!1) <= factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-4] atanhND(1 + n!1)(c!1) > 0 [-5] 6 * K11 + 4 * (K11 * n!1) > 0 [-6] K11 > 0 [-7] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-8] K10 > 0 |------- {1} factorial(2 + 2 * n!1) * (K12 / (2 * K11)) * 1 = 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) [2] atanhND(1 + n!1)(c!1) * abs(z!1 ^ (3 + 2 * n!1)) <= 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) Rerunning step: (hide-all-but (-5 -6 1)) Keeping (-5 -6 1) and hiding *, this simplifies to: atanh_series.1.1.1.1.1.1.1.2 : [-1] 6 * K11 + 4 * (K11 * n!1) > 0 [-2] K11 > 0 |------- [1] factorial(2 + 2 * n!1) * (K12 / (2 * K11)) * 1 = 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) Rerunning step: (lemma "div_cancel1" ("x" "factorial(2 + 2 * n!1) * (K12 / (2 * K11))" "n0z" "3+2*n!1")) Applying div_cancel1 where x gets factorial(2 + 2 * n!1) * (K12 / (2 * K11)), n0z gets 3 + 2 * n!1, this simplifies to: atanh_series.1.1.1.1.1.1.1.2 : {-1} (3 + 2 * n!1) * (factorial(2 + 2 * n!1) * (K12 / (2 * K11)) / (3 + 2 * n!1)) = factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-2] 6 * K11 + 4 * (K11 * n!1) > 0 [-3] K11 > 0 |------- [1] factorial(2 + 2 * n!1) * (K12 / (2 * K11)) * 1 = 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) Rerunning step: (replace -1 1 rl) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.1.2 : [-1] (3 + 2 * n!1) * (factorial(2 + 2 * n!1) * (K12 / (2 * K11)) / (3 + 2 * n!1)) = factorial(2 + 2 * n!1) * (K12 / (2 * K11)) [-2] 6 * K11 + 4 * (K11 * n!1) > 0 [-3] K11 > 0 |------- {1} (3 + 2 * n!1) * (factorial(2 + 2 * n!1) * (K12 / (2 * K11)) / (3 + 2 * n!1)) = 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.1.1.2 : [-1] 6 * K11 + 4 * (K11 * n!1) > 0 [-2] K11 > 0 |------- [1] (3 + 2 * n!1) * (factorial(2 + 2 * n!1) * (K12 / (2 * K11)) / (3 + 2 * n!1)) = 3 * (factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((factorial(2 + 2 * n!1) * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) Rerunning step: (name-replace "K13" "factorial(2 + 2 * n!1)") Using K13 to name and replace factorial(2 + 2 * n!1), this simplifies to: atanh_series.1.1.1.1.1.1.1.2 : [-1] 6 * K11 + 4 * (K11 * n!1) > 0 [-2] K11 > 0 |------- {1} (3 + 2 * n!1) * (K13 * (K12 / (2 * K11)) / (3 + 2 * n!1)) = 3 * (K13 * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((K13 * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) Rerunning step: (case-replace "(K13 * (K12 / (2 * K11)) / (3 + 2 * n!1))=(K13 * (K12 / (6 * K11 + 4 * (K11 * n!1))))") Assuming and applying (K13 * (K12 / (2 * K11)) / (3 + 2 * n!1))=(K13 * (K12 / (6 * K11 + 4 * (K11 * n!1)))) this yields 2 subgoals: atanh_series.1.1.1.1.1.1.1.2.1 : {-1} (K13 * (K12 / (2 * K11)) / (3 + 2 * n!1)) = (K13 * (K12 / (6 * K11 + 4 * (K11 * n!1)))) [-2] 6 * K11 + 4 * (K11 * n!1) > 0 [-3] K11 > 0 |------- {1} (3 + 2 * n!1) * (K13 * (K12 / (6 * K11 + 4 * (K11 * n!1)))) = 3 * (K13 * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((K13 * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.1.1.2.1. atanh_series.1.1.1.1.1.1.1.2.2 : [-1] 6 * K11 + 4 * (K11 * n!1) > 0 [-2] K11 > 0 |------- {1} (K13 * (K12 / (2 * K11)) / (3 + 2 * n!1)) = (K13 * (K12 / (6 * K11 + 4 * (K11 * n!1)))) [2] (3 + 2 * n!1) * (K13 * (K12 / (2 * K11)) / (3 + 2 * n!1)) = 3 * (K13 * (K12 / (6 * K11 + 4 * (K11 * n!1)))) + 2 * ((K13 * (K12 / (6 * K11 + 4 * (K11 * n!1)))) * n!1) Rerunning step: (hide 2) Hiding formulas: 2, this simplifies to: atanh_series.1.1.1.1.1.1.1.2.2 : [-1] 6 * K11 + 4 * (K11 * n!1) > 0 [-2] K11 > 0 |------- [1] (K13 * (K12 / (2 * K11)) / (3 + 2 * n!1)) = (K13 * (K12 / (6 * K11 + 4 * (K11 * n!1)))) Rerunning step: (lemma "cross_mult" ("x" "K13 * (K12 / (2 * K11))" "n0x" "3+2*n!1" "y" "K13*K12" "n0y" "6 * K11 + 4 * (K11 * n!1)")) Applying cross_mult where x gets K13 * (K12 / (2 * K11)), n0x gets 3 + 2 * n!1, y gets K13 * K12, n0y gets 6 * K11 + 4 * (K11 * n!1), this simplifies to: atanh_series.1.1.1.1.1.1.1.2.2 : {-1} (K13 * (K12 / (2 * K11)) / (3 + 2 * n!1) = K13 * K12 / (6 * K11 + 4 * (K11 * n!1))) IFF (K13 * (K12 / (2 * K11)) * (6 * K11 + 4 * (K11 * n!1)) = K13 * K12 * (3 + 2 * n!1)) [-2] 6 * K11 + 4 * (K11 * n!1) > 0 [-3] K11 > 0 |------- [1] (K13 * (K12 / (2 * K11)) / (3 + 2 * n!1)) = (K13 * (K12 / (6 * K11 + 4 * (K11 * n!1)))) Rerunning step: (replace -1) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.1.2.2 : [-1] (K13 * (K12 / (2 * K11)) / (3 + 2 * n!1) = K13 * K12 / (6 * K11 + 4 * (K11 * n!1))) IFF (K13 * (K12 / (2 * K11)) * (6 * K11 + 4 * (K11 * n!1)) = K13 * K12 * (3 + 2 * n!1)) [-2] 6 * K11 + 4 * (K11 * n!1) > 0 [-3] K11 > 0 |------- {1} (K13 * (K12 / (2 * K11)) * (6 * K11 + 4 * (K11 * n!1)) = K13 * K12 * (3 + 2 * n!1)) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.1.1.2.2 : [-1] 6 * K11 + 4 * (K11 * n!1) > 0 [-2] K11 > 0 |------- [1] (K13 * (K12 / (2 * K11)) * (6 * K11 + 4 * (K11 * n!1)) = K13 * K12 * (3 + 2 * n!1)) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.1.1.2.2. This completes the proof of atanh_series.1.1.1.1.1.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.1.1. atanh_series.1.1.1.1.1.1.2 : [-1] abs(atanhND(1 + n!1)(c!1)) <= atanhND(1 + n!1)(z!1) [-2] atanhND(1 + n!1)(c!1) > 0 [-3] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-5] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-6] 1 - sq(z!1) > 0 |------- {1} atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) [2] abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (hide -1 -2 2) Hiding formulas: -1, -2, 2, this simplifies to: atanh_series.1.1.1.1.1.1.2 : [-1] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- [1] atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.1.2 : [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- [1] atanhND(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) Rerunning step: (expand "atanhND") Expanding the definition of atanhND, this simplifies to: atanh_series.1.1.1.1.1.1.2 : [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- {1} atanhN(1 + n!1)(z!1) / atanhD(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) Rerunning step: (expand "atanhN") Expanding the definition of atanhN, this simplifies to: atanh_series.1.1.1.1.1.1.2 : [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / atanhD(1 + n!1)(z!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) Rerunning step: (expand "atanhD") Expanding the definition of atanhD, this simplifies to: atanh_series.1.1.1.1.1.1.2 : [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) Rerunning step: (lemma "cross_mult" ("x" "polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1)" "n0x" "(1 - sq(z!1)) ^ (3 + 2 * n!1)" "y" "factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1))" "n0y" "2 * (1 - sq(z!1)) ^ (3 + 2 * n!1)")) Applying cross_mult where x gets polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1), n0x gets (1 - sq(z!1)) ^ (3 + 2 * n!1), y gets factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)), n0y gets 2 * (1 - sq(z!1)) ^ (3 + 2 * n!1), this simplifies to: atanh_series.1.1.1.1.1.1.2 : {-1} (polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1) = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) IFF (polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1)) = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1)) [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1) = factorial(2 + 2 * n!1) * (((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) Rerunning step: (replace -1 1) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.2 : [-1] (polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1) = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1))) IFF (polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1)) = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1)) [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- {1} (polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1)) = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1)) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.1.2 : [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- [1] (polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1)) = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1)) Rerunning step: (lemma "both_sides_times1" ("x" "polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2" "n0z" "(1 - sq(z!1)) ^ (3 + 2 * n!1)" "y" "factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1))")) Applying both_sides_times1 where x gets polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2, n0z gets (1 - sq(z!1)) ^ (3 + 2 * n!1), y gets factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)), this yields 2 subgoals: atanh_series.1.1.1.1.1.1.2.1 : {-1} (polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 * (1 - sq(z!1)) ^ (3 + 2 * n!1) = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1)) IFF polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- [1] (polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1)) = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1)) Rerunning step: (replace -1 1) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1 : [-1] (polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 * (1 - sq(z!1)) ^ (3 + 2 * n!1) = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1)) IFF polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1 : [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) Rerunning step: (hide -1 -2) Hiding formulas: -1, -2, this simplifies to: atanh_series.1.1.1.1.1.1.2.1 : [-1] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) Rerunning step: (lemma "power_polynomial" ("pn" "3+2*n!1")) Applying power_polynomial where pn gets 3 + 2 * n!1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1 : {-1} (LAMBDA (x: real): (1 + x) ^ (3 + 2 * n!1)) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-2] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) Rerunning step: (lemma "neg_power_polynomial" ("pn" "3+2*n!1")) Applying neg_power_polynomial where pn gets 3 + 2 * n!1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1 : {-1} (LAMBDA (x: real): (1 - x) ^ (3 + 2 * n!1)) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-2] (LAMBDA (x: real): (1 + x) ^ (3 + 2 * n!1)) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-3] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) Rerunning step: (case-replace "(1 - z!1) ^ (3 + 2 * n!1) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1)") Assuming and applying (1 - z!1) ^ (3 + 2 * n!1) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z this yields 2 subgoals: atanh_series.1.1.1.1.1.1.2.1.1 : {-1} (1 - z!1) ^ (3 + 2 * n!1) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) [-2] (LAMBDA (x: real): (1 - x) ^ (3 + 2 * n!1)) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-3] (LAMBDA (x: real): (1 + x) ^ (3 + 2 * n!1)) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-4] 1 - sq(z!1) > 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * (polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) + (1 + z!1) ^ (3 + 2 * n!1)) Rerunning step: (case-replace "(1 + z!1) ^ (3 + 2 * n!1) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1)") Assuming and applying (1 + z!1) ^ (3 + 2 * n!1) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1), this yields 2 subgoals: atanh_series.1.1.1.1.1.1.2.1.1.1 : {-1} (1 + z!1) ^ (3 + 2 * n!1) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) [-2] (1 - z!1) ^ (3 + 2 * n!1) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) [-3] (LAMBDA (x: real): (1 - x) ^ (3 + 2 * n!1)) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-4] (LAMBDA (x: real): (1 + x) ^ (3 + 2 * n!1)) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-5] 1 - sq(z!1) > 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * (polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) + polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1)) Rerunning step: (hide -1 -2 -3 -4) Hiding formulas: -1, -2, -3, -4, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * (polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) + polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1)) Rerunning step: (expand "polynomial") Expanding the definition of polynomial, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] 1 - sq(z!1) > 0 |------- {1} 2 * sigma(0, 2 + 2 * n!1, LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) = factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) + factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) Rerunning step: (lemma "sigma_scal[nat]" ("low" "0" "high" "2+2*n!1" "a" "2" "F" "LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)")) Free variables in expr i Applying sigma_scal[nat] where low gets 0, high gets 2 + 2 * n!1, a gets 2, F gets LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF), this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : {-1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = 2 * sigma(0, 2 + 2 * n!1, LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) [-2] 1 - sq(z!1) > 0 |------- [1] 2 * sigma(0, 2 + 2 * n!1, LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) = factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) + factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) Rerunning step: (replace -1 1 rl) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = 2 * sigma(0, 2 + 2 * n!1, LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) [-2] 1 - sq(z!1) > 0 |------- {1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) + factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) + factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) Rerunning step: (lemma "sigma_scal[nat]" ("low" "0" "high" "3+2*n!1" "a" "factorial(2 + 2 * n!1)" "F" "LAMBDA (i: nat):power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)")) Free variables in expr i Ignoring 1 repeated TCCs. Applying sigma_scal[nat] where low gets 0, high gets 3 + 2 * n!1, a gets factorial(2 + 2 * n!1), F gets LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF), this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : {-1} sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) [-2] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) + factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) Rerunning step: (replace -1 1 rl) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) [-2] 1 - sq(z!1) > 0 |------- {1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) + factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) + factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) Rerunning step: (lemma "sigma_scal[nat]" ("low" "0" "high" "3+2*n!1" "a" "factorial(2 + 2 * n!1)" "F" "LAMBDA (i: nat):neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)")) Free variables in expr i Ignoring 1 repeated TCCs. Applying sigma_scal[nat] where low gets 0, high gets 3 + 2 * n!1, a gets factorial(2 + 2 * n!1), F gets LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF), this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : {-1} sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) [-2] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) + factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) Rerunning step: (replace -1 1 rl) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = factorial(2 + 2 * n!1) * sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) [-2] 1 - sq(z!1) > 0 |------- {1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) + sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) + sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) Rerunning step: (lemma "sigma_sum" ("low" "0" "high" "3+2*n!1" "F" "LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF))(i)" "G" "LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF))(i)")) Free variables in expr i Free variables in expr i Applying sigma_sum where low gets 0, high gets 3 + 2 * n!1, F gets LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i), G gets LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i), this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : {-1} sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i)) + sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i)) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): (LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i)) (i_1) + (LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i)) (i_1)) [-2] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) + sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) Rerunning step: (replace -1 1) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i)) + sigma(0, 3 + 2 * n!1, LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i)) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): (LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i)) (i_1) + (LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i)) (i_1)) [-2] 1 - sq(z!1) > 0 |------- {1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): (LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i)) (i_1) + (LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i)) (i_1)) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (LAMBDA (i: nat): atanhF(1 + n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i_1)) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): (LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i)) (i_1) + (LAMBDA (i: nat): factorial(2 + 2 * n!1) * (LAMBDA (i: nat): neg_power_fs(3 + 2 * n!1)(i) * (IF i = 0 THEN 1 ELSE z!1 ^ i ENDIF)) (i)) (i_1)) Rerunning step: (simplify 1) Simplifying with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] 1 - sq(z!1) > 0 |------- {1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) Rerunning step: (lemma "sigma_last" ("low" "0" "high" "3+2*n!1" "F" "LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))")) Free variables in expr i_1 Free variables in expr i_1 Ignoring 1 repeated TCCs. Applying sigma_last where low gets 0, high gets 3 + 2 * n!1, F gets LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)), this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : {-1} 3 + 2 * n!1 > 0 IMPLIES sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 3 + 2 * n!1 - 1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + (LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) (3 + 2 * n!1) [-2] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : {-1} sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) [-2] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) Rerunning step: (replace -1 1) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] sigma(0, 3 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) [-2] 1 - sq(z!1) > 0 |------- {1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) Rerunning step: (expand "neg_power_fs" 1 2) Expanding the definition of neg_power_fs, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] 1 - sq(z!1) > 0 |------- {1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * ((C(3 + 2 * n!1, 3 + 2 * n!1) * (-1) ^ (3 + 2 * n!1)) * z!1 ^ (3 + 2 * n!1)) Rerunning step: (expand "power_fs" 1 2) Expanding the definition of power_fs, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1 : [-1] 1 - sq(z!1) > 0 |------- {1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * ((C(3 + 2 * n!1, 3 + 2 * n!1) * (-1) ^ (3 + 2 * n!1)) * z!1 ^ (3 + 2 * n!1)) Rerunning step: (case-replace "(-1) ^ (3 + 2 * n!1)=-1") Assuming and applying (-1) ^ (3 + 2 * n!1)=-1, this yields 2 subgoals: atanh_series.1.1.1.1.1.1.2.1.1.1.1 : {-1} (-1) ^ (3 + 2 * n!1) = -1 [-2] 1 - sq(z!1) > 0 |------- {1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * -1 * z!1 ^ (3 + 2 * n!1)) Rerunning step: (case-replace "sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * -1 * z!1 ^ (3 + 2 * n!1)) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)))") Assuming and applying sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * -1 * z!1 ^ (3 + 2 * n!1)) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))), this yields 2 subgoals: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1 : {-1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * -1 * z!1 ^ (3 + 2 * n!1)) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) [-2] (-1) ^ (3 + 2 * n!1) = -1 [-3] 1 - sq(z!1) > 0 |------- {1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) Rerunning step: (hide -1 -2) Hiding formulas: -1, -2, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1 : [-1] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) Rerunning step: (lemma "sigma_eq[nat]" ("low" "0" "high" "2+2*n!1" "F" "LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))" "G" "LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))")) Free variables in expr i_1 Free variables in expr i_1 Free variables in expr i_1 Ignoring 1 repeated TCCs. Applying sigma_eq[nat] where low gets 0, high gets 2 + 2 * n!1, F gets LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)), G gets LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)), this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1 : {-1} (FORALL (n: subrange(0, 2 + 2 * n!1)): (LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) (n) = (LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) (n)) IMPLIES sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) [-2] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) Rerunning step: (split -1) Splitting conjunctions, this yields 2 subgoals: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.1 : {-1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) [-2] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) which is trivially true. This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.1. atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2 : [-1] 1 - sq(z!1) > 0 |------- {1} FORALL (n: subrange(0, 2 + 2 * n!1)): (LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) (n) = (LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) (n) [2] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) Rerunning step: (hide 2) Hiding formulas: 2, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2 : [-1] 1 - sq(z!1) > 0 |------- [1] FORALL (n: subrange(0, 2 + 2 * n!1)): (LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) (n) = (LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) (n) Rerunning step: (skosimp*) Repeatedly Skolemizing and flattening, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2 : [-1] 1 - sq(z!1) > 0 |------- {1} 2 * (atanhF(1 + n!1)(n!2) * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) = factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(n!2) * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(n!2) * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) Rerunning step: (expand "atanhF") Expanding the definition of atanhF, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2 : [-1] 1 - sq(z!1) > 0 |------- {1} 2 * (IF odd?(n!2) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, n!2) ENDIF * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) = factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(n!2) * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(n!2) * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) Rerunning step: (expand "power_fs") Expanding the definition of power_fs, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2 : [-1] 1 - sq(z!1) > 0 |------- {1} 2 * (IF odd?(n!2) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, n!2) ENDIF * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) = factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, n!2) * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(n!2) * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) Rerunning step: (expand "neg_power_fs") Expanding the definition of neg_power_fs, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2 : [-1] 1 - sq(z!1) > 0 |------- {1} 2 * (IF odd?(n!2) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, n!2) ENDIF * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) = factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, n!2) * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, n!2) * (-1) ^ n!2 * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) Rerunning step: (name-replace "K400" "factorial(2 + 2 * n!1)") Using K400 to name and replace factorial(2 + 2 * n!1), this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2 : [-1] 1 - sq(z!1) > 0 |------- {1} 2 * (IF odd?(n!2) THEN 0 ELSE K400 * C(3 + 2 * n!1, n!2) ENDIF * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) = K400 * (C(3 + 2 * n!1, n!2) * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) + K400 * (C(3 + 2 * n!1, n!2) * (-1) ^ n!2 * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) Rerunning step: (name-replace "K401" "C(3 + 2 * n!1, n!2)") Using K401 to name and replace C(3 + 2 * n!1, n!2), this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2 : [-1] 1 - sq(z!1) > 0 |------- {1} 2 * (IF odd?(n!2) THEN 0 ELSE K400 * K401 ENDIF * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) = K400 * (K401 * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) + K400 * (K401 * (-1) ^ n!2 * (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)) Rerunning step: (name-replace "K402" "(IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF)") Using K402 to name and replace (IF n!2 = 0 THEN 1 ELSE z!1 ^ n!2 ENDIF), this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2 : [-1] 1 - sq(z!1) > 0 |------- {1} 2 * (IF odd?(n!2) THEN 0 ELSE K400 * K401 ENDIF * K402) = K400 * (K401 * K402) + K400 * (K401 * (-1) ^ n!2 * K402) Rerunning step: (case "even?(n!2)") Case splitting on even?(n!2), this yields 2 subgoals: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1 : {-1} even?(n!2) [-2] 1 - sq(z!1) > 0 |------- [1] 2 * (IF odd?(n!2) THEN 0 ELSE K400 * K401 ENDIF * K402) = K400 * (K401 * K402) + K400 * (K401 * (-1) ^ n!2 * K402) Rerunning step: (lemma "even_or_odd" ("x" "n!2")) Applying even_or_odd where x gets n!2, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1 : {-1} even?(n!2) IFF NOT odd?(n!2) [-2] even?(n!2) [-3] 1 - sq(z!1) > 0 |------- [1] 2 * (IF odd?(n!2) THEN 0 ELSE K400 * K401 ENDIF * K402) = K400 * (K401 * K402) + K400 * (K401 * (-1) ^ n!2 * K402) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1 : [-1] even?(n!2) [-2] 1 - sq(z!1) > 0 |------- {1} odd?(n!2) {2} 2 * (K400 * K401 * K402) = K400 * (K401 * K402) + K400 * ((-1) ^ n!2 * K401 * K402) Rerunning step: (expand "even?") Expanding the definition of even?, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1 : {-1} EXISTS j: n!2 = 2 * j [-2] 1 - sq(z!1) > 0 |------- [1] odd?(n!2) [2] 2 * (K400 * K401 * K402) = K400 * (K401 * K402) + K400 * ((-1) ^ n!2 * K401 * K402) Rerunning step: (skosimp*) Repeatedly Skolemizing and flattening, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1 : {-1} n!2 = 2 * j!1 [-2] 1 - sq(z!1) > 0 |------- [1] odd?(n!2) [2] 2 * (K400 * K401 * K402) = K400 * (K401 * K402) + K400 * ((-1) ^ n!2 * K401 * K402) Rerunning step: (replace -1 2) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1 : [-1] n!2 = 2 * j!1 [-2] 1 - sq(z!1) > 0 |------- [1] odd?(n!2) {2} 2 * (K400 * K401 * K402) = K400 * (K401 * K402) + K400 * ((-1) ^ (2 * j!1) * K401 * K402) Rerunning step: (lemma "expt_times" ("n0x" "-1" "i" "2" "j" "j!1")) Applying expt_times where n0x gets -1, i gets 2, j gets j!1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1 : {-1} (-1) ^ (2 * j!1) = ((-1) ^ 2) ^ j!1 [-2] n!2 = 2 * j!1 [-3] 1 - sq(z!1) > 0 |------- [1] odd?(n!2) [2] 2 * (K400 * K401 * K402) = K400 * (K401 * K402) + K400 * ((-1) ^ (2 * j!1) * K401 * K402) Rerunning step: (case-replace "(-1)^2=1") Assuming and applying (-1)^2=1, this yields 2 subgoals: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1.1 : {-1} (-1) ^ 2 = 1 {-2} (-1) ^ (2 * j!1) = 1 ^ j!1 [-3] n!2 = 2 * j!1 [-4] 1 - sq(z!1) > 0 |------- [1] odd?(n!2) [2] 2 * (K400 * K401 * K402) = K400 * (K401 * K402) + K400 * ((-1) ^ (2 * j!1) * K401 * K402) Rerunning step: (rewrite "expt_1i") Found matching substitution: i: int gets j!1, Rewriting using expt_1i, matching in *, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1.1 : [-1] (-1) ^ 2 = 1 {-2} (-1) ^ (2 * j!1) = 1 [-3] n!2 = 2 * j!1 [-4] 1 - sq(z!1) > 0 |------- [1] odd?(n!2) [2] 2 * (K400 * K401 * K402) = K400 * (K401 * K402) + K400 * ((-1) ^ (2 * j!1) * K401 * K402) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1.1. atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1.2 : [-1] (-1) ^ (2 * j!1) = ((-1) ^ 2) ^ j!1 [-2] n!2 = 2 * j!1 [-3] 1 - sq(z!1) > 0 |------- {1} (-1) ^ 2 = 1 [2] odd?(n!2) [3] 2 * (K400 * K401 * K402) = K400 * (K401 * K402) + K400 * ((-1) ^ (2 * j!1) * K401 * K402) Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1.2 : |------- [1] (-1) ^ 2 = 1 Rerunning step: (grind) expt rewrites expt((-1), 0) to 1 expt rewrites expt((-1), 1) to -1 expt rewrites expt((-1), 2) to (-1) * -1 ^ rewrites (-1) ^ 2 to (-1) * -1 Trying repeated skolemization, instantiation, and if-lifting, This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1.2. This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.1. atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2 : [-1] 1 - sq(z!1) > 0 |------- {1} even?(n!2) [2] 2 * (IF odd?(n!2) THEN 0 ELSE K400 * K401 ENDIF * K402) = K400 * (K401 * K402) + K400 * (K401 * (-1) ^ n!2 * K402) Rerunning step: (rewrite "even_or_odd" 1) Found matching substitution: x: int gets n!2, Rewriting using even_or_odd, matching in 1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2 : {-1} odd?(n!2) [-2] 1 - sq(z!1) > 0 |------- [1] 2 * (IF odd?(n!2) THEN 0 ELSE K400 * K401 ENDIF * K402) = K400 * (K401 * K402) + K400 * (K401 * (-1) ^ n!2 * K402) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2 : [-1] odd?(n!2) [-2] 1 - sq(z!1) > 0 |------- {1} 0 = K400 * (K401 * K402) + K400 * ((-1) ^ n!2 * K401 * K402) Rerunning step: (expand "odd?") Expanding the definition of odd?, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2 : {-1} EXISTS j: n!2 = 1 + 2 * j [-2] 1 - sq(z!1) > 0 |------- [1] 0 = K400 * (K401 * K402) + K400 * ((-1) ^ n!2 * K401 * K402) Rerunning step: (skosimp*) Repeatedly Skolemizing and flattening, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2 : {-1} n!2 = 1 + 2 * j!1 [-2] 1 - sq(z!1) > 0 |------- [1] 0 = K400 * (K401 * K402) + K400 * ((-1) ^ n!2 * K401 * K402) Rerunning step: (replace -1 1) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2 : [-1] n!2 = 1 + 2 * j!1 [-2] 1 - sq(z!1) > 0 |------- {1} 0 = K400 * (K401 * K402) + K400 * (((-1) ^ (1 + 2 * j!1)) * K401 * K402) Rerunning step: (lemma "expt_plus" ("n0x" "-1" "i" "1" "j" "2*j!1")) Applying expt_plus where n0x gets -1, i gets 1, j gets 2 * j!1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2 : {-1} (-1) ^ (1 + 2 * j!1) = (-1) ^ 1 * (-1) ^ (2 * j!1) [-2] n!2 = 1 + 2 * j!1 [-3] 1 - sq(z!1) > 0 |------- [1] 0 = K400 * (K401 * K402) + K400 * (((-1) ^ (1 + 2 * j!1)) * K401 * K402) Rerunning step: (rewrite "expt_x1") Found matching substitution: x: real gets -1, Rewriting using expt_x1, matching in *, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2 : {-1} (-1) ^ (1 + 2 * j!1) = -1 * (-1) ^ (2 * j!1) [-2] n!2 = 1 + 2 * j!1 [-3] 1 - sq(z!1) > 0 |------- [1] 0 = K400 * (K401 * K402) + K400 * (((-1) ^ (1 + 2 * j!1)) * K401 * K402) Rerunning step: (lemma "expt_times" ("n0x" "-1" "i" "2" "j" "j!1")) Applying expt_times where n0x gets -1, i gets 2, j gets j!1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2 : {-1} (-1) ^ (2 * j!1) = ((-1) ^ 2) ^ j!1 [-2] (-1) ^ (1 + 2 * j!1) = -1 * (-1) ^ (2 * j!1) [-3] n!2 = 1 + 2 * j!1 [-4] 1 - sq(z!1) > 0 |------- [1] 0 = K400 * (K401 * K402) + K400 * (((-1) ^ (1 + 2 * j!1)) * K401 * K402) Rerunning step: (case-replace "(-1)^2=1") Assuming and applying (-1)^2=1, this yields 2 subgoals: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2.1 : {-1} (-1) ^ 2 = 1 {-2} (-1) ^ (2 * j!1) = 1 ^ j!1 [-3] (-1) ^ (1 + 2 * j!1) = -1 * (-1) ^ (2 * j!1) [-4] n!2 = 1 + 2 * j!1 [-5] 1 - sq(z!1) > 0 |------- [1] 0 = K400 * (K401 * K402) + K400 * (((-1) ^ (1 + 2 * j!1)) * K401 * K402) Rerunning step: (rewrite "expt_1i") Found matching substitution: i: int gets j!1, Rewriting using expt_1i, matching in *, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2.1 : [-1] (-1) ^ 2 = 1 {-2} (-1) ^ (2 * j!1) = 1 [-3] (-1) ^ (1 + 2 * j!1) = -1 * (-1) ^ (2 * j!1) [-4] n!2 = 1 + 2 * j!1 [-5] 1 - sq(z!1) > 0 |------- [1] 0 = K400 * (K401 * K402) + K400 * (((-1) ^ (1 + 2 * j!1)) * K401 * K402) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2.1. atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2.2 : [-1] (-1) ^ (2 * j!1) = ((-1) ^ 2) ^ j!1 [-2] (-1) ^ (1 + 2 * j!1) = -1 * (-1) ^ (2 * j!1) [-3] n!2 = 1 + 2 * j!1 [-4] 1 - sq(z!1) > 0 |------- {1} (-1) ^ 2 = 1 [2] 0 = K400 * (K401 * K402) + K400 * (((-1) ^ (1 + 2 * j!1)) * K401 * K402) Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2.2 : |------- [1] (-1) ^ 2 = 1 Rerunning step: (grind) expt rewrites expt((-1), 0) to 1 expt rewrites expt((-1), 1) to -1 expt rewrites expt((-1), 2) to (-1) * -1 ^ rewrites (-1) ^ 2 to (-1) * -1 Trying repeated skolemization, instantiation, and if-lifting, This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2.2. This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2.2. This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.1.1. atanh_series.1.1.1.1.1.1.2.1.1.1.1.2 : [-1] (-1) ^ (3 + 2 * n!1) = -1 [-2] 1 - sq(z!1) > 0 |------- {1} sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * -1 * z!1 ^ (3 + 2 * n!1)) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) [2] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * -1 * z!1 ^ (3 + 2 * n!1)) Rerunning step: (hide 2) Hiding formulas: 2, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.1.2 : [-1] (-1) ^ (3 + 2 * n!1) = -1 [-2] 1 - sq(z!1) > 0 |------- [1] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * -1 * z!1 ^ (3 + 2 * n!1)) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) Rerunning step: (name-replace "K300" "sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)))") Using K300 to name and replace sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))), This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.1. atanh_series.1.1.1.1.1.1.2.1.1.1.2 : [-1] 1 - sq(z!1) > 0 |------- {1} (-1) ^ (3 + 2 * n!1) = -1 [2] sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): 2 * (atanhF(1 + n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) = sigma(0, 2 + 2 * n!1, LAMBDA (i_1: nat): factorial(2 + 2 * n!1) * (power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF)) + factorial(2 + 2 * n!1) * (neg_power_fs(3 + 2 * n!1)(i_1) * (IF i_1 = 0 THEN 1 ELSE z!1 ^ i_1 ENDIF))) + factorial(2 + 2 * n!1) * (C(3 + 2 * n!1, 3 + 2 * n!1) * z!1 ^ (3 + 2 * n!1)) + factorial(2 + 2 * n!1) * ((C(3 + 2 * n!1, 3 + 2 * n!1) * (-1) ^ (3 + 2 * n!1)) * z!1 ^ (3 + 2 * n!1)) Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.2 : |------- [1] (-1) ^ (3 + 2 * n!1) = -1 Rerunning step: (lemma "expt_plus" ("n0x" "-1" "i" "1" "j" "2+2*n!1")) Applying expt_plus where n0x gets -1, i gets 1, j gets 2 + 2 * n!1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.2 : {-1} (-1) ^ (1 + (2 + 2 * n!1)) = (-1) ^ 1 * (-1) ^ (2 + 2 * n!1) |------- [1] (-1) ^ (3 + 2 * n!1) = -1 Rerunning step: (rewrite "expt_x1") Found matching substitution: x: real gets -1, Rewriting using expt_x1, matching in *, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.2 : {-1} (-1) ^ (1 + (2 + 2 * n!1)) = -1 * (-1) ^ (2 + 2 * n!1) |------- [1] (-1) ^ (3 + 2 * n!1) = -1 Rerunning step: (lemma "expt_times") Applying expt_times this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.2 : {-1} FORALL (i, j: int, n0x: nzreal): n0x ^ (i * j) = (n0x ^ i) ^ j [-2] (-1) ^ (1 + (2 + 2 * n!1)) = -1 * (-1) ^ (2 + 2 * n!1) |------- [1] (-1) ^ (3 + 2 * n!1) = -1 Rerunning step: (inst - "2" "1+n!1" "-1") Instantiating the top quantifier in - with the terms: 2, 1+n!1, -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.2 : {-1} (-1) ^ (2 * (1 + n!1)) = ((-1) ^ 2) ^ (1 + n!1) [-2] (-1) ^ (1 + (2 + 2 * n!1)) = -1 * (-1) ^ (2 + 2 * n!1) |------- [1] (-1) ^ (3 + 2 * n!1) = -1 Rerunning step: (replace -1 -2) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.2 : [-1] (-1) ^ (2 * (1 + n!1)) = ((-1) ^ 2) ^ (1 + n!1) {-2} (-1) ^ (1 + (2 + 2 * n!1)) = -1 * ((-1) ^ 2) ^ (1 + n!1) |------- [1] (-1) ^ (3 + 2 * n!1) = -1 Rerunning step: (case-replace "(-1)^2=1") Assuming and applying (-1)^2=1, this yields 2 subgoals: atanh_series.1.1.1.1.1.1.2.1.1.1.2.1 : {-1} (-1) ^ 2 = 1 {-2} (-1) ^ (2 * (1 + n!1)) = 1 ^ (1 + n!1) {-3} (-1) ^ (1 + (2 + 2 * n!1)) = -1 * 1 ^ (1 + n!1) |------- [1] (-1) ^ (3 + 2 * n!1) = -1 Rerunning step: (rewrite "expt_1i" -2) Found matching substitution: i: int gets 1 + n!1, Rewriting using expt_1i, matching in -2, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.2.1 : [-1] (-1) ^ 2 = 1 {-2} (-1) ^ (2 * (1 + n!1)) = 1 {-3} (-1) ^ (1 + (2 + 2 * n!1)) = -1 * 1 |------- [1] (-1) ^ (3 + 2 * n!1) = -1 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.2.1. atanh_series.1.1.1.1.1.1.2.1.1.1.2.2 : [-1] (-1) ^ (2 * (1 + n!1)) = ((-1) ^ 2) ^ (1 + n!1) [-2] (-1) ^ (1 + (2 + 2 * n!1)) = -1 * ((-1) ^ 2) ^ (1 + n!1) |------- {1} (-1) ^ 2 = 1 [2] (-1) ^ (3 + 2 * n!1) = -1 Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.1.2.2 : |------- [1] (-1) ^ 2 = 1 Rerunning step: (grind) expt rewrites expt((-1), 0) to 1 expt rewrites expt((-1), 1) to -1 expt rewrites expt((-1), 2) to (-1) * -1 ^ rewrites (-1) ^ 2 to (-1) * -1 Trying repeated skolemization, instantiation, and if-lifting, This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.2.2. This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.1. atanh_series.1.1.1.1.1.1.2.1.1.2 : [-1] (1 - z!1) ^ (3 + 2 * n!1) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) [-2] (LAMBDA (x: real): (1 - x) ^ (3 + 2 * n!1)) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-3] (LAMBDA (x: real): (1 + x) ^ (3 + 2 * n!1)) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-4] 1 - sq(z!1) > 0 |------- {1} (1 + z!1) ^ (3 + 2 * n!1) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) [2] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * (polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) + (1 + z!1) ^ (3 + 2 * n!1)) Rerunning step: (replace -3 1 rl) Replacing using formula -3, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.2 : [-1] (1 - z!1) ^ (3 + 2 * n!1) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) [-2] (LAMBDA (x: real): (1 - x) ^ (3 + 2 * n!1)) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-3] (LAMBDA (x: real): (1 + x) ^ (3 + 2 * n!1)) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-4] 1 - sq(z!1) > 0 |------- {1} (1 + z!1) ^ (3 + 2 * n!1) = (LAMBDA (x: real): (1 + x) ^ (3 + 2 * n!1))(z!1) [2] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * (polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) + (1 + z!1) ^ (3 + 2 * n!1)) Rerunning step: (simplify 1) Simplifying with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.1.2 : [-1] (1 - z!1) ^ (3 + 2 * n!1) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) [-2] (LAMBDA (x: real): (1 - x) ^ (3 + 2 * n!1)) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-3] (LAMBDA (x: real): (1 + x) ^ (3 + 2 * n!1)) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-4] 1 - sq(z!1) > 0 |------- {1} TRUE [2] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * (polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) + (1 + z!1) ^ (3 + 2 * n!1)) which is trivially true. This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.1.2.1.1. atanh_series.1.1.1.1.1.1.2.1.2 : [-1] (LAMBDA (x: real): (1 - x) ^ (3 + 2 * n!1)) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-2] (LAMBDA (x: real): (1 + x) ^ (3 + 2 * n!1)) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-3] 1 - sq(z!1) > 0 |------- {1} (1 - z!1) ^ (3 + 2 * n!1) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1)(z!1) [2] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) Rerunning step: (replace -1 1 rl) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.2 : [-1] (LAMBDA (x: real): (1 - x) ^ (3 + 2 * n!1)) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-2] (LAMBDA (x: real): (1 + x) ^ (3 + 2 * n!1)) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-3] 1 - sq(z!1) > 0 |------- {1} (1 - z!1) ^ (3 + 2 * n!1) = (LAMBDA (x: real): (1 - x) ^ (3 + 2 * n!1))(z!1) [2] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) Rerunning step: (simplify 1) Simplifying with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.1.2.1.2 : [-1] (LAMBDA (x: real): (1 - x) ^ (3 + 2 * n!1)) = polynomial(neg_power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-2] (LAMBDA (x: real): (1 + x) ^ (3 + 2 * n!1)) = polynomial(power_fs(3 + 2 * n!1), 3 + 2 * n!1) [-3] 1 - sq(z!1) > 0 |------- {1} TRUE [2] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * 2 = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) which is trivially true. This completes the proof of atanh_series.1.1.1.1.1.1.2.1.2. This completes the proof of atanh_series.1.1.1.1.1.1.2.1. atanh_series.1.1.1.1.1.1.2.2 (TCC): [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- {1} (1 - sq(z!1)) ^ (3 + 2 * n!1) /= 0 [2] (polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1)) = factorial(2 + 2 * n!1) * ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1)) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.1.2.2. This completes the proof of atanh_series.1.1.1.1.1.1.2. atanh_series.1.1.1.1.1.1.3 (TCC): [-1] abs(atanhND(1 + n!1)(c!1)) <= atanhND(1 + n!1)(z!1) [-2] atanhND(1 + n!1)(c!1) > 0 [-3] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-5] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-6] 1 - sq(z!1) > 0 |------- {1} (2 * (1 - sq(z!1)) ^ (3 + 2 * n!1)) /= 0 [2] abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.1.3. This completes the proof of atanh_series.1.1.1.1.1.1. atanh_series.1.1.1.1.1.2 : [-1] atanhND(1 + n!1)(c!1) > 0 [-2] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-5] 1 - sq(z!1) > 0 |------- {1} abs(atanhND(1 + n!1)(c!1)) <= atanhND(1 + n!1)(z!1) [2] abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (hide 2) Hiding formulas: 2, this simplifies to: atanh_series.1.1.1.1.1.2 : [-1] atanhND(1 + n!1)(c!1) > 0 [-2] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-5] 1 - sq(z!1) > 0 |------- [1] abs(atanhND(1 + n!1)(c!1)) <= atanhND(1 + n!1)(z!1) Rerunning step: (expand "abs" 1) Expanding the definition of abs, this simplifies to: atanh_series.1.1.1.1.1.2 : [-1] atanhND(1 + n!1)(c!1) > 0 [-2] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-5] 1 - sq(z!1) > 0 |------- {1} IF atanhND(1 + n!1)(c!1) < 0 THEN -atanhND(1 + n!1)(c!1) ELSE atanhND(1 + n!1)(c!1) ENDIF <= atanhND(1 + n!1)(z!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.2 : [-1] atanhND(1 + n!1)(c!1) > 0 {-2} 6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * ((1 - sq(z!1)) ^ (3 + 2 * n!1) * n!1) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-5] 1 - sq(z!1) > 0 |------- {1} atanhND(1 + n!1)(c!1) <= atanhND(1 + n!1)(z!1) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.2 : [-1] 6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * ((1 - sq(z!1)) ^ (3 + 2 * n!1) * n!1) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- [1] atanhND(1 + n!1)(c!1) <= atanhND(1 + n!1)(z!1) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.2 : [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- [1] atanhND(1 + n!1)(c!1) <= atanhND(1 + n!1)(z!1) Rerunning step: (expand "atanhND") Expanding the definition of atanhND, this simplifies to: atanh_series.1.1.1.1.1.2 : [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- {1} atanhN(1 + n!1)(c!1) / atanhD(1 + n!1)(c!1) <= atanhN(1 + n!1)(z!1) / atanhD(1 + n!1)(z!1) Rerunning step: (expand "atanhN") Expanding the definition of atanhN, this simplifies to: atanh_series.1.1.1.1.1.2 : [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / atanhD(1 + n!1)(c!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / atanhD(1 + n!1)(z!1) Rerunning step: (expand "atanhD") Expanding the definition of atanhD, this simplifies to: atanh_series.1.1.1.1.1.2 : [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / (1 - sq(c!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1) Rerunning step: (case "1-sq(c!1)>0") Case splitting on 1 - sq(c!1) > 0, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1 : {-1} 1 - sq(c!1) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / (1 - sq(c!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1) Rerunning step: (lemma "expt_pos" ("px" "1-sq(c!1)" "i" "3+2*n!1")) Applying expt_pos where px gets 1 - sq(c!1), i gets 3 + 2 * n!1, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1 : {-1} (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-2] 1 - sq(c!1) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-5] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / (1 - sq(c!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1) Rerunning step: (rewrite "div_mult_pos_le1" 1) Found matching substitution: x gets polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1), py: posreal gets (1 - sq(c!1)) ^ (3 + 2 * n!1), z: real gets polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1), Rewriting using div_mult_pos_le1, matching in 1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1 : [-1] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-2] 1 - sq(c!1) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-5] 1 - sq(z!1) > 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) <= (polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (lemma "div_mult_pos_le2" ("z" "polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1)*(1 - sq(c!1)) ^ (3 + 2 * n!1)" "py" "(1 - sq(z!1)) ^ (3 + 2 * n!1)" "x" "polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1)")) Ignoring 1 repeated TCCs. Applying div_mult_pos_le2 where z gets polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1), py gets (1 - sq(z!1)) ^ (3 + 2 * n!1), x gets polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1), this simplifies to: atanh_series.1.1.1.1.1.2.1.1 : {-1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1) IFF polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1) [-2] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-3] 1 - sq(c!1) > 0 [-4] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-5] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-6] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) <= (polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (replace -1 1) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1 : [-1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1) IFF polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1) [-2] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-3] 1 - sq(c!1) > 0 [-4] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-5] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-6] 1 - sq(z!1) > 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1 : [-1] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-2] 1 - sq(c!1) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-5] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (lemma "even_polynomial" ("a" "atanhF(1 + n!1)" "n" "1 + n!1")) Applying even_polynomial where a gets atanhF(1 + n!1), n gets 1 + n!1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1 : {-1} FORALL (x: real): even_fs?(atanhF(1 + n!1)) IMPLIES polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(x) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (x ^ 2) [-2] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-3] 1 - sq(c!1) > 0 [-4] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-5] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-6] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (case "even_fs?(atanhF(1 + n!1))") Case splitting on even_fs?(atanhF(1 + n!1)), this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1 : {-1} even_fs?(atanhF(1 + n!1)) [-2] FORALL (x: real): even_fs?(atanhF(1 + n!1)) IMPLIES polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(x) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (x ^ 2) [-3] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-4] 1 - sq(c!1) > 0 [-5] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (replace -1 -2) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1 : [-1] even_fs?(atanhF(1 + n!1)) {-2} FORALL (x: real): polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(x) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(x ^ 2) [-3] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-4] 1 - sq(c!1) > 0 [-5] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (inst-cp - "c!1") Instantiating (with copying) the top quantifier in - with the terms: c!1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1 : [-1] even_fs?(atanhF(1 + n!1)) [-2] FORALL (x: real): polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(x) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(x ^ 2) {-3} polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(c!1) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) [-4] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-5] 1 - sq(c!1) > 0 [-6] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-7] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-8] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (inst - "z!1") Instantiating the top quantifier in - with the terms: z!1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1 : [-1] even_fs?(atanhF(1 + n!1)) {-2} polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(z!1) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(z!1 ^ 2) [-3] polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(c!1) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) [-4] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-5] 1 - sq(c!1) > 0 [-6] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-7] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-8] 1 - sq(z!1) > 0 |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (replace -2) Replacing using formula -2, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1 : [-1] even_fs?(atanhF(1 + n!1)) [-2] polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(z!1) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(z!1 ^ 2) [-3] polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(c!1) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) [-4] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-5] 1 - sq(c!1) > 0 [-6] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-7] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-8] 1 - sq(z!1) > 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (z!1 ^ 2) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (replace -3) Replacing using formula -3, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1 : [-1] even_fs?(atanhF(1 + n!1)) [-2] polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(z!1) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(z!1 ^ 2) [-3] polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(c!1) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) [-4] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-5] 1 - sq(c!1) > 0 [-6] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-7] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-8] 1 - sq(z!1) > 0 |------- {1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (z!1 ^ 2) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (hide -1 -2 -3) Hiding formulas: -1, -2, -3, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1 : [-1] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-2] 1 - sq(c!1) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-5] 1 - sq(z!1) > 0 |------- [1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (z!1 ^ 2) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (case "FORALL (x:real): x^2 = sq(x)") Case splitting on FORALL (x: real): x ^ 2 = sq(x), this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1 : {-1} FORALL (x: real): x ^ 2 = sq(x) [-2] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-3] 1 - sq(c!1) > 0 [-4] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-5] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-6] 1 - sq(z!1) > 0 |------- [1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (z!1 ^ 2) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (inst-cp - "c!1") Instantiating (with copying) the top quantifier in - with the terms: c!1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1 : [-1] FORALL (x: real): x ^ 2 = sq(x) {-2} c!1 ^ 2 = sq(c!1) [-3] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-4] 1 - sq(c!1) > 0 [-5] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] 1 - sq(z!1) > 0 |------- [1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (z!1 ^ 2) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (inst - "z!1") Instantiating the top quantifier in - with the terms: z!1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1 : {-1} z!1 ^ 2 = sq(z!1) [-2] c!1 ^ 2 = sq(c!1) [-3] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-4] 1 - sq(c!1) > 0 [-5] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] 1 - sq(z!1) > 0 |------- [1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (z!1 ^ 2) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (replace -1) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1 : [-1] z!1 ^ 2 = sq(z!1) [-2] c!1 ^ 2 = sq(c!1) [-3] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-4] 1 - sq(c!1) > 0 [-5] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] 1 - sq(z!1) > 0 |------- {1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (replace -2) Replacing using formula -2, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1 : [-1] z!1 ^ 2 = sq(z!1) [-2] c!1 ^ 2 = sq(c!1) [-3] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-4] 1 - sq(c!1) > 0 [-5] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] 1 - sq(z!1) > 0 |------- {1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (case "sq(c!1)<=sq(z!1)") Case splitting on sq(c!1) <= sq(z!1), this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1 : {-1} sq(c!1) <= sq(z!1) [-2] z!1 ^ 2 = sq(z!1) [-3] c!1 ^ 2 = sq(c!1) [-4] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-5] 1 - sq(c!1) > 0 [-6] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-7] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-8] 1 - sq(z!1) > 0 |------- [1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (lemma "both_sides_expt_pos_le" ("px" "1-sq(z!1)" "py" "1-sq(c!1)" "pm" "3+2*n!1")) Ignoring 2 repeated TCCs. Applying both_sides_expt_pos_le where px gets 1 - sq(z!1), py gets 1 - sq(c!1), pm gets 3 + 2 * n!1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1 : {-1} (1 - sq(z!1)) ^ (3 + 2 * n!1) <= (1 - sq(c!1)) ^ (3 + 2 * n!1) IFF 1 - sq(z!1) <= 1 - sq(c!1) [-2] sq(c!1) <= sq(z!1) [-3] z!1 ^ 2 = sq(z!1) [-4] c!1 ^ 2 = sq(c!1) [-5] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-6] 1 - sq(c!1) > 0 [-7] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-8] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-9] 1 - sq(z!1) > 0 |------- [1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1 : {-1} (1 - sq(z!1)) ^ (3 + 2 * n!1) <= (1 - sq(c!1)) ^ (3 + 2 * n!1) [-2] sq(c!1) <= sq(z!1) [-3] z!1 ^ 2 = sq(z!1) [-4] c!1 ^ 2 = sq(c!1) [-5] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-6] 1 - sq(c!1) > 0 [-7] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-8] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-9] 1 - sq(z!1) > 0 |------- [1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (case "FORALL (n:nat): polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n) Case splitting on FORALL (n: nat): polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(c!1)) > 0, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1 : {-1} FORALL (n: nat): polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(c!1)) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) <= (1 - sq(c!1)) ^ (3 + 2 * n!1) [-3] sq(c!1) <= sq(z!1) [-4] z!1 ^ 2 = sq(z!1) [-5] c!1 ^ 2 = sq(c!1) [-6] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-7] 1 - sq(c!1) > 0 [-8] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-9] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-10] 1 - sq(z!1) > 0 |------- [1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (inst - "1+n!1") Instantiating the top quantifier in - with the terms: 1+n!1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1 : {-1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) <= (1 - sq(c!1)) ^ (3 + 2 * n!1) [-3] sq(c!1) <= sq(z!1) [-4] z!1 ^ 2 = sq(z!1) [-5] c!1 ^ 2 = sq(c!1) [-6] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-7] 1 - sq(c!1) > 0 [-8] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-9] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-10] 1 - sq(z!1) > 0 |------- [1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (case "FORALL (n:nat): polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(z!1))") Case splitting on FORALL (n: nat): polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(z!1)), this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.1 : {-1} FORALL (n: nat): polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(z!1)) [-2] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) <= (1 - sq(c!1)) ^ (3 + 2 * n!1) [-4] sq(c!1) <= sq(z!1) [-5] z!1 ^ 2 = sq(z!1) [-6] c!1 ^ 2 = sq(c!1) [-7] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-8] 1 - sq(c!1) > 0 [-9] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-10] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-11] 1 - sq(z!1) > 0 |------- [1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (inst - "1+n!1") Instantiating the top quantifier in - with the terms: 1+n!1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.1 : {-1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(z!1)) [-2] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) <= (1 - sq(c!1)) ^ (3 + 2 * n!1) [-4] sq(c!1) <= sq(z!1) [-5] z!1 ^ 2 = sq(z!1) [-6] c!1 ^ 2 = sq(c!1) [-7] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-8] 1 - sq(c!1) > 0 [-9] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-10] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-11] 1 - sq(z!1) > 0 |------- [1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (lemma "le_times_le_pos" ("nnx" "polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1))" "nnz" "(1 - sq(z!1)) ^ (3 + 2 * n!1)" "y" "polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(z!1))" "w" "(1 - sq(c!1)) ^ (3 + 2 * n!1)")) Applying le_times_le_pos where nnx gets polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(c!1)), nnz gets (1 - sq(z!1)) ^ (3 + 2 * n!1), y gets polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)), w gets (1 - sq(c!1)) ^ (3 + 2 * n!1), this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.1.1 : {-1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) AND (1 - sq(z!1)) ^ (3 + 2 * n!1) <= (1 - sq(c!1)) ^ (3 + 2 * n!1) IMPLIES polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) [-2] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(z!1)) [-3] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) > 0 [-4] (1 - sq(z!1)) ^ (3 + 2 * n!1) <= (1 - sq(c!1)) ^ (3 + 2 * n!1) [-5] sq(c!1) <= sq(z!1) [-6] z!1 ^ 2 = sq(z!1) [-7] c!1 ^ 2 = sq(c!1) [-8] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-9] 1 - sq(c!1) > 0 [-10] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-11] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-12] 1 - sq(z!1) > 0 |------- [1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.1.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.1.2 (TCC): [-1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(z!1)) [-2] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) <= (1 - sq(c!1)) ^ (3 + 2 * n!1) [-4] sq(c!1) <= sq(z!1) [-5] z!1 ^ 2 = sq(z!1) [-6] c!1 ^ 2 = sq(c!1) [-7] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-8] 1 - sq(c!1) > 0 [-9] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-10] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-11] 1 - sq(z!1) > 0 |------- {1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) >= 0 [2] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2 : [-1] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) <= (1 - sq(c!1)) ^ (3 + 2 * n!1) [-3] sq(c!1) <= sq(z!1) [-4] z!1 ^ 2 = sq(z!1) [-5] c!1 ^ 2 = sq(c!1) [-6] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-7] 1 - sq(c!1) > 0 [-8] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-9] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-10] 1 - sq(z!1) > 0 |------- {1} FORALL (n: nat): polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(z!1)) [2] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (hide-all-but (1 -3)) Keeping (1 -3) and hiding *, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2 : [-1] sq(c!1) <= sq(z!1) |------- [1] FORALL (n: nat): polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(z!1)) Rerunning step: (induct "n") Inducting on n on formula 1, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.1 : [-1] sq(c!1) <= sq(z!1) |------- {1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 0)(sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 0)(sq(z!1)) Rerunning step: (rewrite "polynomial_n0") Found matching substitution: a: sequence[real] gets LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), Rewriting using polynomial_n0, matching in *, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.1 : [-1] sq(c!1) <= sq(z!1) |------- {1} (LAMBDA (x: real): atanhF(1 + n!1)(0))(sq(c!1)) <= (LAMBDA (x: real): atanhF(1 + n!1)(0))(sq(z!1)) Rerunning step: (simplify 1) Simplifying with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.1 : [-1] sq(c!1) <= sq(z!1) |------- {1} atanhF(1 + n!1)(0) <= atanhF(1 + n!1)(0) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2 : [-1] sq(c!1) <= sq(z!1) |------- {1} FORALL j: polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), j)(sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), j)(sq(z!1)) IMPLIES polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), j + 1) (sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), j + 1) (sq(z!1)) Rerunning step: (skosimp*) Repeatedly Skolemizing and flattening, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2 : {-1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), j!1)(sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), j!1)(sq(z!1)) [-2] sq(c!1) <= sq(z!1) |------- {1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), j!1 + 1)(sq(c!1)) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), j!1 + 1)(sq(z!1)) Rerunning step: (expand "polynomial") Expanding the definition of polynomial, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2 : {-1} sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(c!1) ^ i_1 ENDIF)) <= sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(z!1) ^ i_1 ENDIF)) [-2] sq(c!1) <= sq(z!1) |------- {1} sigma(0, 1 + j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(c!1) ^ i_1 ENDIF)) <= sigma(0, 1 + j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(z!1) ^ i_1 ENDIF)) Rerunning step: (expand "sigma" 1) Expanding the definition of sigma, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2 : [-1] sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(c!1) ^ i_1 ENDIF)) <= sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(z!1) ^ i_1 ENDIF)) [-2] sq(c!1) <= sq(z!1) |------- {1} sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(c!1) ^ i_1 ENDIF)) + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(z!1) ^ i_1 ENDIF)) + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (name-replace "K100" "sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(c!1) ^ i_1 ENDIF))") Using K100 to name and replace sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(c!1) ^ i_1 ENDIF)), this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2 : {-1} K100 <= sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(z!1) ^ i_1 ENDIF)) [-2] sq(c!1) <= sq(z!1) |------- {1} K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(z!1) ^ i_1 ENDIF)) + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (name-replace "K101" "sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(z!1) ^ i_1 ENDIF))") Using K101 to name and replace sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(z!1) ^ i_1 ENDIF)), this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2 : {-1} K100 <= K101 [-2] sq(c!1) <= sq(z!1) |------- {1} K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (case-replace "sq(c!1)=0") Assuming and applying sq(c!1)=0, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1 : {-1} sq(c!1) = 0 [-2] K100 <= K101 {-3} 0 <= sq(z!1) |------- {1} K100 + atanhF(1 + n!1)(2 + 2 * j!1) * 0 ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (expand "^" 1 1) Expanding the definition of ^, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1 : [-1] sq(c!1) = 0 [-2] K100 <= K101 [-3] 0 <= sq(z!1) |------- {1} K100 + atanhF(1 + n!1)(2 + 2 * j!1) * expt(0, 1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (expand "expt" 1 1) Expanding the definition of expt, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1 : [-1] sq(c!1) = 0 [-2] K100 <= K101 [-3] 0 <= sq(z!1) |------- {1} K100 <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (lemma "expt_pos" ("px" "sq(z!1)" "i" "1+j!1")) Applying expt_pos where px gets sq(z!1), i gets 1 + j!1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1 : {-1} sq(z!1) ^ (1 + j!1) > 0 [-2] sq(c!1) = 0 [-3] K100 <= K101 [-4] 0 <= sq(z!1) |------- [1] K100 <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (expand "atanhF") Expanding the definition of atanhF, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1 : [-1] sq(z!1) ^ (1 + j!1) > 0 [-2] sq(c!1) = 0 [-3] K100 <= K101 [-4] 0 <= sq(z!1) |------- {1} K100 <= K101 + IF 2 + 2 * j!1 > 2 + 2 * n!1 OR odd?(2 + 2 * j!1) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF * sq(z!1) ^ (1 + j!1) Rerunning step: (expand "odd?") Expanding the definition of odd?, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1 : [-1] sq(z!1) ^ (1 + j!1) > 0 [-2] sq(c!1) = 0 [-3] K100 <= K101 [-4] 0 <= sq(z!1) |------- {1} K100 <= K101 + IF 2 + 2 * j!1 > 2 + 2 * n!1 THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF * sq(z!1) ^ (1 + j!1) Rerunning step: (case "j!1>n!1") Case splitting on j!1 > n!1, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1.1 : {-1} j!1 > n!1 [-2] sq(z!1) ^ (1 + j!1) > 0 [-3] sq(c!1) = 0 [-4] K100 <= K101 [-5] 0 <= sq(z!1) |------- [1] K100 <= K101 + IF 2 + 2 * j!1 > 2 + 2 * n!1 THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF * sq(z!1) ^ (1 + j!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1.2 : [-1] sq(z!1) ^ (1 + j!1) > 0 [-2] sq(c!1) = 0 [-3] K100 <= K101 [-4] 0 <= sq(z!1) |------- {1} j!1 > n!1 [2] K100 <= K101 + IF 2 + 2 * j!1 > 2 + 2 * n!1 THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF * sq(z!1) ^ (1 + j!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1.2 : [-1] sq(z!1) ^ (1 + j!1) > 0 [-2] sq(c!1) = 0 [-3] K100 <= K101 [-4] 0 <= sq(z!1) |------- [1] j!1 > n!1 {2} K100 <= K101 + factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (lemma "posreal_times_posreal_is_posreal" ("px" "factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1)" "py" "sq(z!1) ^ (1 + j!1)")) Applying posreal_times_posreal_is_posreal where px gets factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1), py gets sq(z!1) ^ (1 + j!1), this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1.2 : {-1} factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) > 0 [-2] sq(z!1) ^ (1 + j!1) > 0 [-3] sq(c!1) = 0 [-4] K100 <= K101 [-5] 0 <= sq(z!1) |------- [1] j!1 > n!1 [2] K100 <= K101 + factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2 : [-1] K100 <= K101 [-2] sq(c!1) <= sq(z!1) |------- {1} sq(c!1) = 0 [2] K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (lemma "both_sides_expt_pos_le" ("px" "sq(c!1)" "py" "sq(z!1)" "pm" "1+j!1")) Applying both_sides_expt_pos_le where px gets sq(c!1), py gets sq(z!1), pm gets 1 + j!1, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1 : {-1} sq(c!1) ^ (1 + j!1) <= sq(z!1) ^ (1 + j!1) IFF sq(c!1) <= sq(z!1) [-2] K100 <= K101 [-3] sq(c!1) <= sq(z!1) |------- [1] sq(c!1) = 0 [2] K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (case "0 <= atanhF(1 + n!1)(2 + 2 * j!1)") Case splitting on 0 <= atanhF(1 + n!1)(2 + 2 * j!1), this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.1 : {-1} 0 <= atanhF(1 + n!1)(2 + 2 * j!1) [-2] sq(c!1) ^ (1 + j!1) <= sq(z!1) ^ (1 + j!1) IFF sq(c!1) <= sq(z!1) [-3] K100 <= K101 [-4] sq(c!1) <= sq(z!1) |------- [1] sq(c!1) = 0 [2] K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (expand "<=" -1) Expanding the definition of <=, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.1 : {-1} 0 < atanhF(1 + n!1)(2 + 2 * j!1) OR 0 = atanhF(1 + n!1)(2 + 2 * j!1) [-2] sq(c!1) ^ (1 + j!1) <= sq(z!1) ^ (1 + j!1) IFF sq(c!1) <= sq(z!1) [-3] K100 <= K101 [-4] sq(c!1) <= sq(z!1) |------- [1] sq(c!1) = 0 [2] K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (split -1) Splitting conjunctions, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.1.1 : {-1} 0 < atanhF(1 + n!1)(2 + 2 * j!1) [-2] sq(c!1) ^ (1 + j!1) <= sq(z!1) ^ (1 + j!1) IFF sq(c!1) <= sq(z!1) [-3] K100 <= K101 [-4] sq(c!1) <= sq(z!1) |------- [1] sq(c!1) = 0 [2] K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (lemma "both_sides_times_pos_le1" ("x" "sq(c!1) ^ (1 + j!1)" "y" "sq(z!1) ^ (1 + j!1)" "pz" "atanhF(1 + n!1)(2 + 2 * j!1)")) Applying both_sides_times_pos_le1 where x gets sq(c!1) ^ (1 + j!1), y gets sq(z!1) ^ (1 + j!1), pz gets atanhF(1 + n!1)(2 + 2 * j!1), this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.1.1.1 : {-1} sq(c!1) ^ (1 + j!1) * atanhF(1 + n!1)(2 + 2 * j!1) <= sq(z!1) ^ (1 + j!1) * atanhF(1 + n!1)(2 + 2 * j!1) IFF sq(c!1) ^ (1 + j!1) <= sq(z!1) ^ (1 + j!1) [-2] 0 < atanhF(1 + n!1)(2 + 2 * j!1) [-3] sq(c!1) ^ (1 + j!1) <= sq(z!1) ^ (1 + j!1) IFF sq(c!1) <= sq(z!1) [-4] K100 <= K101 [-5] sq(c!1) <= sq(z!1) |------- [1] sq(c!1) = 0 [2] K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.1.1.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.1.1.2 (TCC): [-1] 0 < atanhF(1 + n!1)(2 + 2 * j!1) [-2] sq(c!1) ^ (1 + j!1) <= sq(z!1) ^ (1 + j!1) IFF sq(c!1) <= sq(z!1) [-3] K100 <= K101 [-4] sq(c!1) <= sq(z!1) |------- {1} atanhF(1 + n!1)(2 + 2 * j!1) >= 0 AND atanhF(1 + n!1)(2 + 2 * j!1) > 0 [2] sq(c!1) = 0 [3] K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.1.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.1.2 : {-1} 0 = atanhF(1 + n!1)(2 + 2 * j!1) [-2] sq(c!1) ^ (1 + j!1) <= sq(z!1) ^ (1 + j!1) IFF sq(c!1) <= sq(z!1) [-3] K100 <= K101 [-4] sq(c!1) <= sq(z!1) |------- [1] sq(c!1) = 0 [2] K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (replace -1 * rl) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.1.2 : [-1] 0 = atanhF(1 + n!1)(2 + 2 * j!1) [-2] sq(c!1) ^ (1 + j!1) <= sq(z!1) ^ (1 + j!1) IFF sq(c!1) <= sq(z!1) [-3] K100 <= K101 [-4] sq(c!1) <= sq(z!1) |------- [1] sq(c!1) = 0 {2} K100 + 0 * sq(c!1) ^ (1 + j!1) <= K101 + 0 * sq(z!1) ^ (1 + j!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.2 : [-1] sq(c!1) ^ (1 + j!1) <= sq(z!1) ^ (1 + j!1) IFF sq(c!1) <= sq(z!1) [-2] K100 <= K101 [-3] sq(c!1) <= sq(z!1) |------- {1} 0 <= atanhF(1 + n!1)(2 + 2 * j!1) [2] sq(c!1) = 0 [3] K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.2 : |------- [1] 0 <= atanhF(1 + n!1)(2 + 2 * j!1) Rerunning step: (expand "atanhF") Expanding the definition of atanhF, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.2 : |------- {1} 0 <= IF 2 + 2 * j!1 > 2 + 2 * n!1 OR odd?(2 + 2 * j!1) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF Rerunning step: (expand "odd?") Expanding the definition of odd?, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.2 : |------- {1} 0 <= IF 2 + 2 * j!1 > 2 + 2 * n!1 THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF Rerunning step: (case "j!1>n!1") Case splitting on j!1 > n!1, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.2.1 : {-1} j!1 > n!1 |------- [1] 0 <= IF 2 + 2 * j!1 > 2 + 2 * n!1 THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.2.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.2.2 : |------- {1} j!1 > n!1 [2] 0 <= IF 2 + 2 * j!1 > 2 + 2 * n!1 THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.2.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.2 (TCC): [-1] K100 <= K101 [-2] sq(c!1) <= sq(z!1) |------- {1} sq(c!1) > 0 [2] sq(c!1) = 0 [3] K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (typepred "sq(c!1)") Adding type constraints for sq(c!1), this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.2 : {-1} sq(c!1) >= 0 [-2] K100 <= K101 [-3] sq(c!1) <= sq(z!1) |------- [1] sq(c!1) > 0 [2] sq(c!1) = 0 [3] K100 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) <= K101 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(z!1) ^ (1 + j!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.2 : [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) <= (1 - sq(c!1)) ^ (3 + 2 * n!1) [-2] sq(c!1) <= sq(z!1) [-3] z!1 ^ 2 = sq(z!1) [-4] c!1 ^ 2 = sq(c!1) [-5] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-6] 1 - sq(c!1) > 0 [-7] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-8] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-9] 1 - sq(z!1) > 0 |------- {1} FORALL (n: nat): polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(c!1)) > 0 [2] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2 : |------- [1] FORALL (n: nat): polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), n)(sq(c!1)) > 0 Rerunning step: (induct "n") Inducting on n on formula 1, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.1 : |------- {1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 0)(sq(c!1)) > 0 Rerunning step: (rewrite "polynomial_n0") Found matching substitution: a: sequence[real] gets LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), Rewriting using polynomial_n0, matching in *, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.1 : |------- {1} (LAMBDA (x: real): atanhF(1 + n!1)(0))(sq(c!1)) > 0 Rerunning step: (simplify) Simplifying with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.1 : |------- {1} atanhF(1 + n!1)(0) > 0 Rerunning step: (expand "atanhF") Expanding the definition of atanhF, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.1 : |------- {1} IF odd?(0) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 0) ENDIF > 0 Rerunning step: (expand "odd?") Expanding the definition of odd?, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.1 : |------- {1} factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 0) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2 : |------- {1} FORALL j: polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), j)(sq(c!1)) > 0 IMPLIES polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), j + 1) (sq(c!1)) > 0 Rerunning step: (skosimp*) Repeatedly Skolemizing and flattening, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2 : {-1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), j!1)(sq(c!1)) > 0 |------- {1} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), j!1 + 1)(sq(c!1)) > 0 Rerunning step: (expand "polynomial") Expanding the definition of polynomial, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2 : {-1} sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(c!1) ^ i_1 ENDIF)) > 0 |------- {1} sigma(0, 1 + j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(c!1) ^ i_1 ENDIF)) > 0 Rerunning step: (expand "sigma" 1) Expanding the definition of sigma, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2 : [-1] sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(c!1) ^ i_1 ENDIF)) > 0 |------- {1} sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(c!1) ^ i_1 ENDIF)) + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (name-replace "K103" "sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(c!1) ^ i_1 ENDIF))") Using K103 to name and replace sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE sq(c!1) ^ i_1 ENDIF)), this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2 : {-1} K103 > 0 |------- {1} K103 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (typepred "sq(c!1)") Adding type constraints for sq(c!1), this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2 : {-1} sq(c!1) >= 0 [-2] K103 > 0 |------- [1] K103 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (expand ">=" -1) Expanding the definition of >=, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2 : {-1} 0 <= sq(c!1) [-2] K103 > 0 |------- [1] K103 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (expand "<=" -1) Expanding the definition of <=, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2 : {-1} 0 < sq(c!1) OR 0 = sq(c!1) [-2] K103 > 0 |------- [1] K103 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (split -1) Splitting conjunctions, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1 : {-1} 0 < sq(c!1) [-2] K103 > 0 |------- [1] K103 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (lemma "expt_pos" ("px" "sq(c!1)" "i" "1+j!1")) Applying expt_pos where px gets sq(c!1), i gets 1 + j!1, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1.1 : {-1} sq(c!1) ^ (1 + j!1) > 0 [-2] 0 < sq(c!1) [-3] K103 > 0 |------- [1] K103 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (expand "atanhF") Expanding the definition of atanhF, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1.1 : [-1] sq(c!1) ^ (1 + j!1) > 0 [-2] 0 < sq(c!1) [-3] K103 > 0 |------- {1} K103 + IF 2 + 2 * j!1 > 2 + 2 * n!1 OR odd?(2 + 2 * j!1) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (expand "odd?") Expanding the definition of odd?, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1.1 : [-1] sq(c!1) ^ (1 + j!1) > 0 [-2] 0 < sq(c!1) [-3] K103 > 0 |------- {1} K103 + IF 2 + 2 * j!1 > 2 + 2 * n!1 THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (case "j!1>n!1") Case splitting on j!1 > n!1, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1.1.1 : {-1} j!1 > n!1 [-2] sq(c!1) ^ (1 + j!1) > 0 [-3] 0 < sq(c!1) [-4] K103 > 0 |------- [1] K103 + IF 2 + 2 * j!1 > 2 + 2 * n!1 THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1.1.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1.1.2 : [-1] sq(c!1) ^ (1 + j!1) > 0 [-2] 0 < sq(c!1) [-3] K103 > 0 |------- {1} j!1 > n!1 [2] K103 + IF 2 + 2 * j!1 > 2 + 2 * n!1 THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1.1.2 : [-1] sq(c!1) ^ (1 + j!1) > 0 [-2] 0 < sq(c!1) [-3] K103 > 0 |------- [1] j!1 > n!1 {2} K103 + factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (lemma "posreal_times_posreal_is_posreal" ("px" "factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1)" "py" "sq(c!1) ^ (1 + j!1)")) Applying posreal_times_posreal_is_posreal where px gets factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1), py gets sq(c!1) ^ (1 + j!1), this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1.1.2 : {-1} factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) > 0 [-2] sq(c!1) ^ (1 + j!1) > 0 [-3] 0 < sq(c!1) [-4] K103 > 0 |------- [1] j!1 > n!1 [2] K103 + factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1.2 (TCC): [-1] 0 < sq(c!1) [-2] K103 > 0 |------- {1} sq(c!1) > 0 [2] K103 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.1. atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.2 : {-1} 0 = sq(c!1) [-2] K103 > 0 |------- [1] K103 + atanhF(1 + n!1)(2 + 2 * j!1) * sq(c!1) ^ (1 + j!1) > 0 Rerunning step: (replace -1 * rl) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.2 : [-1] 0 = sq(c!1) [-2] K103 > 0 |------- {1} K103 + atanhF(1 + n!1)(2 + 2 * j!1) * 0 ^ (1 + j!1) > 0 Rerunning step: (expand "^" 1) Expanding the definition of ^, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.2 : [-1] 0 = sq(c!1) [-2] K103 > 0 |------- {1} K103 + atanhF(1 + n!1)(2 + 2 * j!1) * expt(0, 1 + j!1) > 0 Rerunning step: (expand "expt" 1) Expanding the definition of expt, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.2 : [-1] 0 = sq(c!1) [-2] K103 > 0 |------- {1} K103 > 0 which is trivially true. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.2.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.1. atanh_series.1.1.1.1.1.2.1.1.1.1.2 : [-1] z!1 ^ 2 = sq(z!1) [-2] c!1 ^ 2 = sq(c!1) [-3] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-4] 1 - sq(c!1) > 0 [-5] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-6] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-7] 1 - sq(z!1) > 0 |------- {1} sq(c!1) <= sq(z!1) [2] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(sq(c!1)) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (sq(z!1)) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.2 : |------- [1] sq(c!1) <= sq(z!1) Rerunning step: (typepred "c!1") Adding type constraints for c!1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.2 : {-1} -1 < c!1 {-2} c!1 < 1 {-3} (0 < z!1 IMPLIES 0 < c!1 AND c!1 < z!1) {-4} (z!1 < 0 IMPLIES z!1 < c!1 AND c!1 < 0) {-5} (0 = z!1 IMPLIES c!1 = 0) |------- [1] sq(c!1) <= sq(z!1) Rerunning step: (lemma "trichotomy" ("x" "z!1")) Applying trichotomy where x gets z!1, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.2 : {-1} z!1 > 0 OR z!1 = 0 OR 0 > z!1 [-2] -1 < c!1 [-3] c!1 < 1 [-4] (0 < z!1 IMPLIES 0 < c!1 AND c!1 < z!1) [-5] (z!1 < 0 IMPLIES z!1 < c!1 AND c!1 < 0) [-6] (0 = z!1 IMPLIES c!1 = 0) |------- [1] sq(c!1) <= sq(z!1) Rerunning step: (split -1) Splitting conjunctions, this yields 3 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.2.1 : {-1} z!1 > 0 [-2] -1 < c!1 [-3] c!1 < 1 [-4] (0 < z!1 IMPLIES 0 < c!1 AND c!1 < z!1) [-5] (z!1 < 0 IMPLIES z!1 < c!1 AND c!1 < 0) [-6] (0 = z!1 IMPLIES c!1 = 0) |------- [1] sq(c!1) <= sq(z!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.2.1 : [-1] z!1 > 0 [-2] -1 < c!1 [-3] c!1 < 1 {-4} 0 < c!1 AND c!1 < z!1 {-5} (FALSE IMPLIES z!1 < c!1 AND c!1 < 0) {-6} (FALSE IMPLIES c!1 = 0) |------- [1] sq(c!1) <= sq(z!1) Rerunning step: (flatten -4) Applying disjunctive simplification to flatten sequent, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.2.1 : [-1] z!1 > 0 [-2] -1 < c!1 [-3] c!1 < 1 {-4} 0 < c!1 {-5} c!1 < z!1 [-6] (FALSE IMPLIES z!1 < c!1 AND c!1 < 0) [-7] (FALSE IMPLIES c!1 = 0) |------- [1] sq(c!1) <= sq(z!1) Rerunning step: (lemma "sq_lt" ("nna" "c!1" "nnb" "z!1")) Applying sq_lt where nna gets c!1, nnb gets z!1, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.2.1.1 : {-1} sq(c!1) < sq(z!1) IFF c!1 < z!1 [-2] z!1 > 0 [-3] -1 < c!1 [-4] c!1 < 1 [-5] 0 < c!1 [-6] c!1 < z!1 [-7] (FALSE IMPLIES z!1 < c!1 AND c!1 < 0) [-8] (FALSE IMPLIES c!1 = 0) |------- [1] sq(c!1) <= sq(z!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.2.1.1. atanh_series.1.1.1.1.1.2.1.1.1.1.2.1.2 (TCC): [-1] z!1 > 0 [-2] -1 < c!1 [-3] c!1 < 1 [-4] 0 < c!1 [-5] c!1 < z!1 [-6] (FALSE IMPLIES z!1 < c!1 AND c!1 < 0) [-7] (FALSE IMPLIES c!1 = 0) |------- {1} c!1 >= 0 [2] sq(c!1) <= sq(z!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.2.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.2.1. atanh_series.1.1.1.1.1.2.1.1.1.1.2.2 : {-1} z!1 = 0 [-2] -1 < c!1 [-3] c!1 < 1 [-4] (0 < z!1 IMPLIES 0 < c!1 AND c!1 < z!1) [-5] (z!1 < 0 IMPLIES z!1 < c!1 AND c!1 < 0) [-6] (0 = z!1 IMPLIES c!1 = 0) |------- [1] sq(c!1) <= sq(z!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.2.2. atanh_series.1.1.1.1.1.2.1.1.1.1.2.3 : {-1} 0 > z!1 [-2] -1 < c!1 [-3] c!1 < 1 [-4] (0 < z!1 IMPLIES 0 < c!1 AND c!1 < z!1) [-5] (z!1 < 0 IMPLIES z!1 < c!1 AND c!1 < 0) [-6] (0 = z!1 IMPLIES c!1 = 0) |------- [1] sq(c!1) <= sq(z!1) Rerunning step: (lemma "sq_lt" ("nna" "-c!1" "nnb" "-z!1")) Applying sq_lt where nna gets -c!1, nnb gets -z!1, this yields 3 subgoals: atanh_series.1.1.1.1.1.2.1.1.1.1.2.3.1 : {-1} sq(-c!1) < sq(-z!1) IFF -c!1 < -z!1 [-2] 0 > z!1 [-3] -1 < c!1 [-4] c!1 < 1 [-5] (0 < z!1 IMPLIES 0 < c!1 AND c!1 < z!1) [-6] (z!1 < 0 IMPLIES z!1 < c!1 AND c!1 < 0) [-7] (0 = z!1 IMPLIES c!1 = 0) |------- [1] sq(c!1) <= sq(z!1) Rerunning step: (rewrite "sq_neg") Found matching substitution: a: real gets c!1, Rewriting using sq_neg, matching in *, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.2.3.1 : {-1} sq(c!1) < sq(-z!1) IFF -c!1 < -z!1 [-2] 0 > z!1 [-3] -1 < c!1 [-4] c!1 < 1 [-5] (0 < z!1 IMPLIES 0 < c!1 AND c!1 < z!1) [-6] (z!1 < 0 IMPLIES z!1 < c!1 AND c!1 < 0) [-7] (0 = z!1 IMPLIES c!1 = 0) |------- [1] sq(c!1) <= sq(z!1) Rerunning step: (rewrite "sq_neg") Found matching substitution: a: real gets z!1, Rewriting using sq_neg, matching in *, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.1.2.3.1 : {-1} sq(c!1) < sq(z!1) IFF -c!1 < -z!1 [-2] 0 > z!1 [-3] -1 < c!1 [-4] c!1 < 1 [-5] (0 < z!1 IMPLIES 0 < c!1 AND c!1 < z!1) [-6] (z!1 < 0 IMPLIES z!1 < c!1 AND c!1 < 0) [-7] (0 = z!1 IMPLIES c!1 = 0) |------- [1] sq(c!1) <= sq(z!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.2.3.1. atanh_series.1.1.1.1.1.2.1.1.1.1.2.3.2 (TCC): [-1] 0 > z!1 [-2] -1 < c!1 [-3] c!1 < 1 [-4] (0 < z!1 IMPLIES 0 < c!1 AND c!1 < z!1) [-5] (z!1 < 0 IMPLIES z!1 < c!1 AND c!1 < 0) [-6] (0 = z!1 IMPLIES c!1 = 0) |------- {1} -z!1 >= 0 [2] sq(c!1) <= sq(z!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.2.3.2. atanh_series.1.1.1.1.1.2.1.1.1.1.2.3.3 (TCC): [-1] 0 > z!1 [-2] -1 < c!1 [-3] c!1 < 1 [-4] (0 < z!1 IMPLIES 0 < c!1 AND c!1 < z!1) [-5] (z!1 < 0 IMPLIES z!1 < c!1 AND c!1 < 0) [-6] (0 = z!1 IMPLIES c!1 = 0) |------- {1} -c!1 >= 0 [2] sq(c!1) <= sq(z!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.2.3.3. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.2.3. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.1. atanh_series.1.1.1.1.1.2.1.1.1.2 : [-1] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-2] 1 - sq(c!1) > 0 [-3] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-4] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-5] 1 - sq(z!1) > 0 |------- {1} FORALL (x: real): x ^ 2 = sq(x) [2] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (z!1 ^ 2) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.1.2 : |------- [1] FORALL (x: real): x ^ 2 = sq(x) Rerunning step: (grind) expt rewrites expt(x, 0) to 1 expt rewrites expt(x, 1) to x expt rewrites expt(x, 2) to x * x ^ rewrites x ^ 2 to x * x sq rewrites sq(x) to x * x Trying repeated skolemization, instantiation, and if-lifting, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1.1. atanh_series.1.1.1.1.1.2.1.1.2 : [-1] FORALL (x: real): even_fs?(atanhF(1 + n!1)) IMPLIES polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(x) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (x ^ 2) [-2] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-3] 1 - sq(c!1) > 0 [-4] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-5] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-6] 1 - sq(z!1) > 0 |------- {1} even_fs?(atanhF(1 + n!1)) [2] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.2 : |------- [1] even_fs?(atanhF(1 + n!1)) Rerunning step: (expand "even_fs?") Expanding the definition of even_fs?, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.2 : |------- {1} FORALL (i: nat): odd?(i) => atanhF(1 + n!1)(i) = 0 Rerunning step: (expand "atanhF") Expanding the definition of atanhF, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.2 : |------- {1} FORALL (i: nat): odd?(i) => IF i > 2 + 2 * n!1 OR odd?(i) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, i) ENDIF = 0 Rerunning step: (skosimp*) Repeatedly Skolemizing and flattening, this simplifies to: atanh_series.1.1.1.1.1.2.1.1.2 : {-1} odd?(i!1) |------- {1} IF i!1 > 2 + 2 * n!1 OR odd?(i!1) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, i!1) ENDIF = 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1.1. atanh_series.1.1.1.1.1.2.1.2 (TCC): [-1] 1 - sq(c!1) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- {1} 1 - sq(c!1) >= 0 AND 1 - sq(c!1) > 0 [2] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / (1 - sq(c!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.1. atanh_series.1.1.1.1.1.2.2 : [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- {1} 1 - sq(c!1) > 0 [2] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / (1 - sq(c!1)) ^ (3 + 2 * n!1) <= polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(z!1) / (1 - sq(z!1)) ^ (3 + 2 * n!1) Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.1.1.1.1.2.2 : |------- [1] 1 - sq(c!1) > 0 Rerunning step: (case "c!1>=0") Case splitting on c!1 >= 0, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.2.1 : {-1} c!1 >= 0 |------- [1] 1 - sq(c!1) > 0 Rerunning step: (lemma "sq_gt" ("nna" "1" "nnb" "c!1")) Applying sq_gt where nna gets 1, nnb gets c!1, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.2.1.1 : {-1} sq(1) > sq(c!1) IFF 1 > c!1 [-2] c!1 >= 0 |------- [1] 1 - sq(c!1) > 0 Rerunning step: (rewrite "sq_1") Found matching substitution: Rewriting using sq_1, matching in *, this simplifies to: atanh_series.1.1.1.1.1.2.2.1.1 : {-1} 1 > sq(c!1) IFF 1 > c!1 [-2] c!1 >= 0 |------- [1] 1 - sq(c!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.2.1.1. atanh_series.1.1.1.1.1.2.2.1.2 (TCC): [-1] c!1 >= 0 |------- {1} c!1 >= 0 [2] 1 - sq(c!1) > 0 which is trivially true. This completes the proof of atanh_series.1.1.1.1.1.2.2.1.2. This completes the proof of atanh_series.1.1.1.1.1.2.2.1. atanh_series.1.1.1.1.1.2.2.2 : |------- {1} c!1 >= 0 [2] 1 - sq(c!1) > 0 Rerunning step: (lemma "sq_gt" ("nna" "1" "nnb" "-c!1")) Applying sq_gt where nna gets 1, nnb gets -c!1, this yields 2 subgoals: atanh_series.1.1.1.1.1.2.2.2.1 : {-1} sq(1) > sq(-c!1) IFF 1 > -c!1 |------- [1] c!1 >= 0 [2] 1 - sq(c!1) > 0 Rerunning step: (rewrite "sq_1") Found matching substitution: Rewriting using sq_1, matching in *, this simplifies to: atanh_series.1.1.1.1.1.2.2.2.1 : {-1} 1 > sq(-c!1) IFF 1 > -c!1 |------- [1] c!1 >= 0 [2] 1 - sq(c!1) > 0 Rerunning step: (rewrite "sq_neg") Found matching substitution: a: real gets c!1, Rewriting using sq_neg, matching in *, this simplifies to: atanh_series.1.1.1.1.1.2.2.2.1 : {-1} 1 > sq(c!1) IFF 1 > -c!1 |------- [1] c!1 >= 0 [2] 1 - sq(c!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.2.2.1. atanh_series.1.1.1.1.1.2.2.2.2 (TCC): |------- {1} -c!1 >= 0 [2] c!1 >= 0 [3] 1 - sq(c!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.1.2.2.2.2. This completes the proof of atanh_series.1.1.1.1.1.2.2.2. This completes the proof of atanh_series.1.1.1.1.1.2.2. This completes the proof of atanh_series.1.1.1.1.1.2. This completes the proof of atanh_series.1.1.1.1.1. atanh_series.1.1.1.1.2 : [-1] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- {1} atanhND(1 + n!1)(c!1) > 0 [2] abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (hide 2) Hiding formulas: 2, this simplifies to: atanh_series.1.1.1.1.2 : [-1] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- [1] atanhND(1 + n!1)(c!1) > 0 Rerunning step: (expand "atanhND") Expanding the definition of atanhND, this simplifies to: atanh_series.1.1.1.1.2 : [-1] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- {1} atanhN(1 + n!1)(c!1) / atanhD(1 + n!1)(c!1) > 0 Rerunning step: (expand "atanhN") Expanding the definition of atanhN, this simplifies to: atanh_series.1.1.1.1.2 : [-1] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / atanhD(1 + n!1)(c!1) > 0 Rerunning step: (expand "atanhD") Expanding the definition of atanhD, this simplifies to: atanh_series.1.1.1.1.2 : [-1] (6 + 4 * n!1) * (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-3] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-4] 1 - sq(z!1) > 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (hide -1 -2 -3 -4) Hiding formulas: -1, -2, -3, -4, this simplifies to: atanh_series.1.1.1.1.2 : |------- [1] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (case-replace "c!1=0") Assuming and applying c!1=0, this yields 2 subgoals: atanh_series.1.1.1.1.2.1 : {-1} c!1 = 0 |------- {1} polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(0) / (1 - sq(0)) ^ (3 + 2 * n!1) > 0 Rerunning step: (rewrite "polynomial_x0") Found matching substitution: n: nat gets 2 + 2 * n!1, a: sequence[real] gets atanhF(1 + n!1), Rewriting using polynomial_x0, matching in *, this simplifies to: atanh_series.1.1.1.1.2.1 : [-1] c!1 = 0 |------- {1} atanhF(1 + n!1)(0) / (1 - sq(0)) ^ (3 + 2 * n!1) > 0 Rerunning step: (expand "atanhF") Expanding the definition of atanhF, this simplifies to: atanh_series.1.1.1.1.2.1 : [-1] c!1 = 0 |------- {1} IF odd?(0) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 0) ENDIF / (1 - sq(0)) ^ (3 + 2 * n!1) > 0 Rerunning step: (expand "odd?") Expanding the definition of odd?, this simplifies to: atanh_series.1.1.1.1.2.1 : [-1] c!1 = 0 |------- {1} factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 0) / (1 - sq(0)) ^ (3 + 2 * n!1) > 0 Rerunning step: (rewrite "C_0") Found matching substitution: n: nat gets 3 + 2 * n!1, Rewriting using C_0, matching in *, this simplifies to: atanh_series.1.1.1.1.2.1 : [-1] c!1 = 0 |------- {1} factorial(2 + 2 * n!1) * 1 / (1 - sq(0)) ^ (3 + 2 * n!1) > 0 Rerunning step: (assert) sq_0 rewrites sq(0) to 0 Simplifying, rewriting, and recording with decision procedures, this simplifies to: atanh_series.1.1.1.1.2.1 : [-1] c!1 = 0 |------- {1} factorial(2 + 2 * n!1) / 1 ^ (3 + 2 * n!1) > 0 Rerunning step: (cross-mult) Multiplying both sides of selected formulas by LHS/RHS divisor(s), this simplifies to: atanh_series.1.1.1.1.2.1 : [-1] c!1 = 0 |------- {1} factorial(2 + 2 * n!1) > 0 * 1 ^ (3 + 2 * n!1) Rerunning step: (ground) Applying propositional simplification and decision procedures, This completes the proof of atanh_series.1.1.1.1.2.1. atanh_series.1.1.1.1.2.2 : |------- {1} c!1 = 0 [2] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (lemma "even_polynomial" ("a" "atanhF(1 + n!1)" "n" "1 + n!1" "x" "c!1")) Applying even_polynomial where a gets atanhF(1 + n!1), n gets 1 + n!1, x gets c!1, this simplifies to: atanh_series.1.1.1.1.2.2 : {-1} even_fs?(atanhF(1 + n!1)) IMPLIES polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(c!1) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1) (c!1 ^ 2) |------- [1] c!1 = 0 [2] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (split -1) Splitting conjunctions, this yields 2 subgoals: atanh_series.1.1.1.1.2.2.1 : {-1} polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(c!1) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) |------- [1] c!1 = 0 [2] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (replace -1 2) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.2.2.1 : [-1] polynomial(atanhF(1 + n!1), 2 * (1 + n!1))(c!1) = polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) |------- [1] c!1 = 0 {2} polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (hide -1) Hiding formulas: -1, this simplifies to: atanh_series.1.1.1.1.2.2.1 : |------- [1] c!1 = 0 [2] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (case "c!1^2>0") Case splitting on c!1 ^ 2 > 0, this yields 2 subgoals: atanh_series.1.1.1.1.2.2.1.1 : {-1} c!1 ^ 2 > 0 |------- [1] c!1 = 0 [2] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (expand "polynomial") Expanding the definition of polynomial, this simplifies to: atanh_series.1.1.1.1.2.2.1.1 : [-1] c!1 ^ 2 > 0 |------- [1] c!1 = 0 {2} sigma(0, 1 + n!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE c!1 ^ 2 ^ i_1 ENDIF)) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (name-replace "KC" "c!1^2") Using KC to name and replace c!1^2, this simplifies to: atanh_series.1.1.1.1.2.2.1.1 : {-1} KC > 0 |------- [1] c!1 = 0 {2} sigma(0, 1 + n!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (case "FORALL (n:nat): sigma(0, n, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) > 0") Free variables in expr i_1 Case splitting on FORALL (n: nat): sigma(0, n, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) > 0, this yields 2 subgoals: atanh_series.1.1.1.1.2.2.1.1.1 : {-1} FORALL (n: nat): sigma(0, n, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) > 0 [-2] KC > 0 |------- [1] c!1 = 0 [2] sigma(0, 1 + n!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (inst - "1+n!1") Instantiating the top quantifier in - with the terms: 1+n!1, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.1 : {-1} sigma(0, 1 + n!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) > 0 [-2] KC > 0 |------- [1] c!1 = 0 [2] sigma(0, 1 + n!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (name-replace "SS" "sigma(0, 1 + n!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF))") Using SS to name and replace sigma(0, 1 + n!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)), this simplifies to: atanh_series.1.1.1.1.2.2.1.1.1 : {-1} SS > 0 [-2] KC > 0 |------- [1] c!1 = 0 {2} SS / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (case "(1 - sq(c!1)) ^ (3 + 2 * n!1) > 0") Case splitting on (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0, this yields 2 subgoals: atanh_series.1.1.1.1.2.2.1.1.1.1 : {-1} (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-2] SS > 0 [-3] KC > 0 |------- [1] c!1 = 0 [2] SS / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (cross-mult 2) Multiplying both sides of selected formulas by LHS/RHS divisor(s), this simplifies to: atanh_series.1.1.1.1.2.2.1.1.1.1 : [-1] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [-2] SS > 0 [-3] KC > 0 |------- [1] c!1 = 0 {2} SS > 0 * (1 - sq(c!1)) ^ (3 + 2 * n!1) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.2.2.1.1.1.1. atanh_series.1.1.1.1.2.2.1.1.1.2 : [-1] SS > 0 [-2] KC > 0 |------- {1} (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [2] c!1 = 0 [3] SS / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (typepred "c!1") Adding type constraints for c!1, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.1.2 : {-1} -1 < c!1 {-2} c!1 < 1 {-3} (0 < z!1 IMPLIES 0 < c!1 AND c!1 < z!1) {-4} (z!1 < 0 IMPLIES z!1 < c!1 AND c!1 < 0) {-5} (0 = z!1 IMPLIES c!1 = 0) [-6] SS > 0 [-7] KC > 0 |------- [1] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 [2] c!1 = 0 [3] SS / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (hide-all-but (-1 -2 1)) Keeping (-1 -2 1) and hiding *, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.1.2 : [-1] -1 < c!1 [-2] c!1 < 1 |------- [1] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (lemma "expt_pos_aux") Applying expt_pos_aux this simplifies to: atanh_series.1.1.1.1.2.2.1.1.1.2 : {-1} FORALL (n: nat, px: posreal): expt(px, n) > 0 [-2] -1 < c!1 [-3] c!1 < 1 |------- [1] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (inst -1 "3+2*n!1" "1 - sq(c!1)") Instantiating the top quantifier in -1 with the terms: 3+2*n!1, 1 - sq(c!1), this yields 2 subgoals: atanh_series.1.1.1.1.2.2.1.1.1.2.1 : {-1} expt(1 - sq(c!1), 3 + 2 * n!1) > 0 [-2] -1 < c!1 [-3] c!1 < 1 |------- [1] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (expand "^") Expanding the definition of ^, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.1.2.1 : [-1] expt(1 - sq(c!1), 3 + 2 * n!1) > 0 [-2] -1 < c!1 [-3] c!1 < 1 |------- {1} expt((1 - sq(c!1)), (3 + 2 * n!1)) > 0 which is trivially true. This completes the proof of atanh_series.1.1.1.1.2.2.1.1.1.2.1. atanh_series.1.1.1.1.2.2.1.1.1.2.2 (TCC): [-1] -1 < c!1 [-2] c!1 < 1 |------- {1} 1 - sq(c!1) >= 0 AND 1 - sq(c!1) > 0 [2] (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (hide 2) Hiding formulas: 2, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.1.2.2 : [-1] -1 < c!1 [-2] c!1 < 1 |------- [1] 1 - sq(c!1) >= 0 AND 1 - sq(c!1) > 0 Rerunning step: (lemma "neg_pos_sq_lt") Applying neg_pos_sq_lt this simplifies to: atanh_series.1.1.1.1.2.2.1.1.1.2.2 : {-1} FORALL (a, b: real): (-b < a AND a < b) IMPLIES sq(a) < sq(b) [-2] -1 < c!1 [-3] c!1 < 1 |------- [1] 1 - sq(c!1) >= 0 AND 1 - sq(c!1) > 0 Rerunning step: (inst?) Found substitution: a gets c!1, b: real gets 1, Using template: -b < a Instantiating quantified variables, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.1.2.2 : {-1} (-1 < c!1 AND c!1 < 1) IMPLIES sq(c!1) < sq(1) [-2] -1 < c!1 [-3] c!1 < 1 |------- [1] 1 - sq(c!1) >= 0 AND 1 - sq(c!1) > 0 Rerunning step: (rewrite "sq_1") Found matching substitution: Rewriting using sq_1, matching in *, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.1.2.2 : {-1} (-1 < c!1 AND c!1 < 1) IMPLIES sq(c!1) < 1 [-2] -1 < c!1 [-3] c!1 < 1 |------- [1] 1 - sq(c!1) >= 0 AND 1 - sq(c!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.2.2.1.1.1.2.2. This completes the proof of atanh_series.1.1.1.1.2.2.1.1.1.2. This completes the proof of atanh_series.1.1.1.1.2.2.1.1.1. atanh_series.1.1.1.1.2.2.1.1.2 : [-1] KC > 0 |------- {1} FORALL (n: nat): sigma(0, n, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) > 0 [2] c!1 = 0 [3] sigma(0, 1 + n!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (hide 2 3) Hiding formulas: 2, 3, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.2 : [-1] KC > 0 |------- [1] FORALL (n: nat): sigma(0, n, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) > 0 Rerunning step: (induct "n") Inducting on n on formula 1, this yields 2 subgoals: atanh_series.1.1.1.1.2.2.1.1.2.1 : [-1] KC > 0 |------- {1} sigma(0, 0, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) > 0 Rerunning step: (expand "sigma" 1) Expanding the definition of sigma, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.2.1 : [-1] KC > 0 |------- {1} atanhF(1 + n!1)(0) > 0 Rerunning step: (expand "atanhF" 1) Expanding the definition of atanhF, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.2.1 : [-1] KC > 0 |------- {1} IF odd?(0) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 0) ENDIF > 0 Rerunning step: (expand "odd?" 1) Expanding the definition of odd?, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.2.1 : [-1] KC > 0 |------- {1} factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 0) > 0 Rerunning step: (rewrite "C_0") Found matching substitution: n: nat gets 3 + 2 * n!1, Rewriting using C_0, matching in *, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.2.1 : [-1] KC > 0 |------- {1} factorial(2 + 2 * n!1) * 1 > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.2.2.1.1.2.1. atanh_series.1.1.1.1.2.2.1.1.2.2 : [-1] KC > 0 |------- {1} FORALL j: sigma(0, j, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) > 0 IMPLIES sigma(0, j + 1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) > 0 Rerunning step: (skosimp*) Repeatedly Skolemizing and flattening, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.2.2 : {-1} sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) > 0 [-2] KC > 0 |------- {1} sigma(0, j!1 + 1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) > 0 Rerunning step: (expand "sigma" 1) Expanding the definition of sigma, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.2.2 : [-1] sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) > 0 [-2] KC > 0 |------- {1} sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)) + atanhF(1 + n!1)(2 + 2 * j!1) * KC ^ (1 + j!1) > 0 Rerunning step: (name-replace "K20" "sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF))") Using K20 to name and replace sigma(0, j!1, LAMBDA (i_1: nat): atanhF(1 + n!1)(2 * i_1) * (IF i_1 = 0 THEN 1 ELSE KC ^ i_1 ENDIF)), this simplifies to: atanh_series.1.1.1.1.2.2.1.1.2.2 : {-1} K20 > 0 [-2] KC > 0 |------- {1} K20 + atanhF(1 + n!1)(2 + 2 * j!1) * KC ^ (1 + j!1) > 0 Rerunning step: (lemma "expt_pos" ("px" "KC" "i" "1+j!1")) Applying expt_pos where px gets KC, i gets 1 + j!1, this yields 2 subgoals: atanh_series.1.1.1.1.2.2.1.1.2.2.1 : {-1} KC ^ (1 + j!1) > 0 [-2] K20 > 0 [-3] KC > 0 |------- [1] K20 + atanhF(1 + n!1)(2 + 2 * j!1) * KC ^ (1 + j!1) > 0 Rerunning step: (case "atanhF(1 + n!1)(2 + 2 * j!1) >=0") Case splitting on atanhF(1 + n!1)(2 + 2 * j!1) >= 0, this yields 2 subgoals: atanh_series.1.1.1.1.2.2.1.1.2.2.1.1 : {-1} atanhF(1 + n!1)(2 + 2 * j!1) >= 0 [-2] KC ^ (1 + j!1) > 0 [-3] K20 > 0 [-4] KC > 0 |------- [1] K20 + atanhF(1 + n!1)(2 + 2 * j!1) * KC ^ (1 + j!1) > 0 Rerunning step: (lemma "both_sides_times_pos_ge1" ("x" "atanhF(1 + n!1)(2 + 2 * j!1)" "y" "0" "pz" "KC ^ (1 + j!1)")) Applying both_sides_times_pos_ge1 where x gets atanhF(1 + n!1)(2 + 2 * j!1), y gets 0, pz gets KC ^ (1 + j!1), this yields 2 subgoals: atanh_series.1.1.1.1.2.2.1.1.2.2.1.1.1 : {-1} atanhF(1 + n!1)(2 + 2 * j!1) * KC ^ (1 + j!1) >= 0 * KC ^ (1 + j!1) IFF atanhF(1 + n!1)(2 + 2 * j!1) >= 0 [-2] atanhF(1 + n!1)(2 + 2 * j!1) >= 0 [-3] KC ^ (1 + j!1) > 0 [-4] K20 > 0 [-5] KC > 0 |------- [1] K20 + atanhF(1 + n!1)(2 + 2 * j!1) * KC ^ (1 + j!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.2.2.1.1.2.2.1.1.1. atanh_series.1.1.1.1.2.2.1.1.2.2.1.1.2 (TCC): [-1] atanhF(1 + n!1)(2 + 2 * j!1) >= 0 [-2] KC ^ (1 + j!1) > 0 [-3] K20 > 0 [-4] KC > 0 |------- {1} KC ^ (1 + j!1) >= 0 AND KC ^ (1 + j!1) > 0 [2] K20 + atanhF(1 + n!1)(2 + 2 * j!1) * KC ^ (1 + j!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.2.2.1.1.2.2.1.1.2. This completes the proof of atanh_series.1.1.1.1.2.2.1.1.2.2.1.1. atanh_series.1.1.1.1.2.2.1.1.2.2.1.2 : [-1] KC ^ (1 + j!1) > 0 [-2] K20 > 0 [-3] KC > 0 |------- {1} atanhF(1 + n!1)(2 + 2 * j!1) >= 0 [2] K20 + atanhF(1 + n!1)(2 + 2 * j!1) * KC ^ (1 + j!1) > 0 Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.2.2.1.2 : |------- [1] atanhF(1 + n!1)(2 + 2 * j!1) >= 0 Rerunning step: (expand "atanhF") Expanding the definition of atanhF, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.2.2.1.2 : |------- {1} IF 2 + 2 * j!1 > 2 + 2 * n!1 OR odd?(2 + 2 * j!1) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF >= 0 Rerunning step: (expand "odd?") Expanding the definition of odd?, this simplifies to: atanh_series.1.1.1.1.2.2.1.1.2.2.1.2 : |------- {1} IF 2 + 2 * j!1 > 2 + 2 * n!1 THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF >= 0 Rerunning step: (case "j!1>n!1") Case splitting on j!1 > n!1, this yields 2 subgoals: atanh_series.1.1.1.1.2.2.1.1.2.2.1.2.1 : {-1} j!1 > n!1 |------- [1] IF 2 + 2 * j!1 > 2 + 2 * n!1 THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF >= 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.2.2.1.1.2.2.1.2.1. atanh_series.1.1.1.1.2.2.1.1.2.2.1.2.2 : |------- {1} j!1 > n!1 [2] IF 2 + 2 * j!1 > 2 + 2 * n!1 THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, 2 + 2 * j!1) ENDIF >= 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.2.2.1.1.2.2.1.2.2. This completes the proof of atanh_series.1.1.1.1.2.2.1.1.2.2.1.2. This completes the proof of atanh_series.1.1.1.1.2.2.1.1.2.2.1. atanh_series.1.1.1.1.2.2.1.1.2.2.2 (TCC): [-1] K20 > 0 [-2] KC > 0 |------- {1} KC >= 0 AND KC > 0 [2] K20 + atanhF(1 + n!1)(2 + 2 * j!1) * KC ^ (1 + j!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.2.2.1.1.2.2.2. This completes the proof of atanh_series.1.1.1.1.2.2.1.1.2.2. This completes the proof of atanh_series.1.1.1.1.2.2.1.1.2. This completes the proof of atanh_series.1.1.1.1.2.2.1.1. atanh_series.1.1.1.1.2.2.1.2 : |------- {1} c!1 ^ 2 > 0 [2] c!1 = 0 [3] polynomial(LAMBDA (i: nat): atanhF(1 + n!1)(2 * i), 1 + n!1)(c!1 ^ 2) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (hide 3) Hiding formulas: 3, this simplifies to: atanh_series.1.1.1.1.2.2.1.2 : |------- [1] c!1 ^ 2 > 0 [2] c!1 = 0 Rerunning step: (expand "^") Expanding the definition of ^, this simplifies to: atanh_series.1.1.1.1.2.2.1.2 : |------- {1} expt(c!1, 2) > 0 [2] c!1 = 0 Rerunning step: (expand "expt") Expanding the definition of expt, this simplifies to: atanh_series.1.1.1.1.2.2.1.2 : |------- {1} c!1 * expt(c!1, 1) > 0 [2] c!1 = 0 Rerunning step: (expand "expt") Expanding the definition of expt, this simplifies to: atanh_series.1.1.1.1.2.2.1.2 : |------- {1} c!1 * (c!1 * expt(c!1, 0)) > 0 [2] c!1 = 0 Rerunning step: (expand "expt") Expanding the definition of expt, this simplifies to: atanh_series.1.1.1.1.2.2.1.2 : |------- {1} c!1 * c!1 > 0 [2] c!1 = 0 Rerunning step: (rewrite "sq_rew") Found matching substitution: a: real gets c!1, Rewriting using sq_rew, matching in *, this simplifies to: atanh_series.1.1.1.1.2.2.1.2 : |------- {1} sq(c!1) > 0 [2] c!1 = 0 Rerunning step: (lemma "sq_nz_pos" ("nz" "c!1")) Applying sq_nz_pos where nz gets c!1, this yields 2 subgoals: atanh_series.1.1.1.1.2.2.1.2.1 : {-1} sq(c!1) > 0 |------- [1] sq(c!1) > 0 [2] c!1 = 0 which is trivially true. This completes the proof of atanh_series.1.1.1.1.2.2.1.2.1. atanh_series.1.1.1.1.2.2.1.2.2 (TCC): |------- {1} c!1 /= 0 [2] sq(c!1) > 0 [3] c!1 = 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.1.2.2.1.2.2. This completes the proof of atanh_series.1.1.1.1.2.2.1.2. This completes the proof of atanh_series.1.1.1.1.2.2.1. atanh_series.1.1.1.1.2.2.2 : |------- {1} even_fs?(atanhF(1 + n!1)) [2] c!1 = 0 [3] polynomial(atanhF(1 + n!1), 2 + 2 * n!1)(c!1) / (1 - sq(c!1)) ^ (3 + 2 * n!1) > 0 Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.1.1.1.2.2.2 : |------- [1] even_fs?(atanhF(1 + n!1)) Rerunning step: (expand "even_fs?") Expanding the definition of even_fs?, this simplifies to: atanh_series.1.1.1.1.2.2.2 : |------- {1} FORALL (i: nat): odd?(i) => atanhF(1 + n!1)(i) = 0 Rerunning step: (expand "atanhF") Expanding the definition of atanhF, this simplifies to: atanh_series.1.1.1.1.2.2.2 : |------- {1} FORALL (i: nat): odd?(i) => IF i > 2 + 2 * n!1 OR odd?(i) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, i) ENDIF = 0 Rerunning step: (skosimp*) Repeatedly Skolemizing and flattening, this simplifies to: atanh_series.1.1.1.1.2.2.2 : {-1} odd?(i!1) |------- {1} IF i!1 > 2 + 2 * n!1 OR odd?(i!1) THEN 0 ELSE factorial(2 + 2 * n!1) * C(3 + 2 * n!1, i!1) ENDIF = 0 Rerunning step: (replace -1) Replacing using formula -1, this simplifies to: atanh_series.1.1.1.1.2.2.2 : [-1] odd?(i!1) |------- {1} TRUE which is trivially true. This completes the proof of atanh_series.1.1.1.1.2.2.2. This completes the proof of atanh_series.1.1.1.1.2.2. This completes the proof of atanh_series.1.1.1.1.2. This completes the proof of atanh_series.1.1.1.1. atanh_series.1.1.1.2 (TCC): [-1] (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [-2] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-3] 1 - sq(z!1) > 0 |------- {1} (1 - sq(z!1)) ^ (3 + 2 * n!1) >= 0 AND (1 - sq(z!1)) ^ (3 + 2 * n!1) > 0 [2] abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.1.2. This completes the proof of atanh_series.1.1.1. atanh_series.1.1.2 (TCC): [-1] abs(z!1 ^ (3 + 2 * n!1)) < 1 [-2] 1 - sq(z!1) > 0 |------- {1} 1 - sq(z!1) >= 0 AND 1 - sq(z!1) > 0 [2] abs(atanhND(1 + n!1)(c!1)) * abs(z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.1.2. This completes the proof of atanh_series.1.1. atanh_series.1.2 : [-1] 1 - sq(z!1) > 0 |------- {1} abs(z!1 ^ (3 + 2 * n!1)) < 1 [2] abs((atanhND(1 + n!1)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (hide -1 2) Hiding formulas: -1, 2, this simplifies to: atanh_series.1.2 : |------- [1] abs(z!1 ^ (3 + 2 * n!1)) < 1 Rerunning step: (case "FORALL (pn:posnat): abs(z!1^pn)<1") Free variables in expr pn Case splitting on FORALL (pn: posnat): abs(z!1 ^ pn) < 1, this yields 2 subgoals: atanh_series.1.2.1 : {-1} FORALL (pn: posnat): abs(z!1 ^ pn) < 1 |------- [1] abs(z!1 ^ (3 + 2 * n!1)) < 1 Rerunning step: (inst - "3+2*n!1") Instantiating the top quantifier in - with the terms: 3+2*n!1, This completes the proof of atanh_series.1.2.1. atanh_series.1.2.2 : |------- {1} FORALL (pn: posnat): abs(z!1 ^ pn) < 1 [2] abs(z!1 ^ (3 + 2 * n!1)) < 1 Rerunning step: (hide 2) Hiding formulas: 2, this simplifies to: atanh_series.1.2.2 : |------- [1] FORALL (pn: posnat): abs(z!1 ^ pn) < 1 Rerunning step: (induct "pn") Inducting on pn on formula 1, this yields 3 subgoals: atanh_series.1.2.2.1 : |------- {1} pn!1 > 0 {2} abs(z!1 ^ pn!1) < 1 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.2.2.1. atanh_series.1.2.2.2 : |------- {1} 0 > 0 IMPLIES abs(z!1 ^ 0) < 1 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.2.2.2. atanh_series.1.2.2.3 : |------- {1} FORALL j: (j > 0 IMPLIES abs(z!1 ^ j) < 1) IMPLIES j + 1 > 0 IMPLIES abs(z!1 ^ (j + 1)) < 1 Rerunning step: (skosimp*) Repeatedly Skolemizing and flattening, this simplifies to: atanh_series.1.2.2.3 : {-1} j!1 > 0 IMPLIES abs(z!1 ^ j!1) < 1 {-2} j!1 + 1 > 0 |------- {1} abs(z!1 ^ (j!1 + 1)) < 1 Rerunning step: (case-replace "j!1=0") Assuming and applying j!1=0, this yields 2 subgoals: atanh_series.1.2.2.3.1 : {-1} j!1 = 0 {-2} 0 > 0 IMPLIES abs(z!1 ^ 0) < 1 {-3} 0 + 1 > 0 |------- {1} abs(z!1 ^ (0 + 1)) < 1 Rerunning step: (rewrite "expt_x1") Found matching substitution: x: real gets z!1, Rewriting using expt_x1, matching in *, this simplifies to: atanh_series.1.2.2.3.1 : [-1] j!1 = 0 [-2] 0 > 0 IMPLIES abs(z!1 ^ 0) < 1 [-3] 0 + 1 > 0 |------- {1} abs(z!1) < 1 Rerunning step: (expand "abs") Expanding the definition of abs, this simplifies to: atanh_series.1.2.2.3.1 : [-1] j!1 = 0 {-2} FALSE IMPLIES IF z!1 ^ 0 < 0 THEN -z!1 ^ 0 ELSE z!1 ^ 0 ENDIF < 1 [-3] 0 + 1 > 0 |------- {1} IF z!1 < 0 THEN -z!1 ELSE z!1 ENDIF < 1 Rerunning step: (case "z!1<0") Case splitting on z!1 < 0, this yields 2 subgoals: atanh_series.1.2.2.3.1.1 : {-1} z!1 < 0 [-2] j!1 = 0 [-3] FALSE IMPLIES IF z!1 ^ 0 < 0 THEN -z!1 ^ 0 ELSE z!1 ^ 0 ENDIF < 1 [-4] 0 + 1 > 0 |------- [1] IF z!1 < 0 THEN -z!1 ELSE z!1 ENDIF < 1 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.2.2.3.1.1. atanh_series.1.2.2.3.1.2 : [-1] j!1 = 0 [-2] FALSE IMPLIES IF z!1 ^ 0 < 0 THEN -z!1 ^ 0 ELSE z!1 ^ 0 ENDIF < 1 [-3] 0 + 1 > 0 |------- {1} z!1 < 0 [2] IF z!1 < 0 THEN -z!1 ELSE z!1 ENDIF < 1 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.2.2.3.1.2. This completes the proof of atanh_series.1.2.2.3.1. atanh_series.1.2.2.3.2 : [-1] j!1 > 0 IMPLIES abs(z!1 ^ j!1) < 1 [-2] j!1 + 1 > 0 |------- {1} j!1 = 0 [2] abs(z!1 ^ (j!1 + 1)) < 1 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, this simplifies to: atanh_series.1.2.2.3.2 : {-1} abs(z!1 ^ j!1) < 1 {-2} 1 + j!1 > 0 |------- [1] j!1 = 0 {2} abs(z!1 ^ (1 + j!1)) < 1 Rerunning step: (case-replace "z!1=0") Assuming and applying z!1=0, this yields 2 subgoals: atanh_series.1.2.2.3.2.1 : {-1} z!1 = 0 {-2} abs(0 ^ j!1) < 1 [-3] 1 + j!1 > 0 |------- [1] j!1 = 0 {2} abs(0 ^ (1 + j!1)) < 1 Rerunning step: (expand "^" 2) Expanding the definition of ^, this simplifies to: atanh_series.1.2.2.3.2.1 : [-1] z!1 = 0 [-2] abs(0 ^ j!1) < 1 [-3] 1 + j!1 > 0 |------- [1] j!1 = 0 {2} abs(expt(0, 1 + j!1)) < 1 Rerunning step: (expand "expt" 2) Expanding the definition of expt, this simplifies to: atanh_series.1.2.2.3.2.1 : [-1] z!1 = 0 [-2] abs(0 ^ j!1) < 1 [-3] 1 + j!1 > 0 |------- [1] j!1 = 0 {2} abs(0 * expt(0, j!1)) < 1 Rerunning step: (expand "expt" 2) Expanding the definition of expt, this simplifies to: atanh_series.1.2.2.3.2.1 : [-1] z!1 = 0 [-2] abs(0 ^ j!1) < 1 [-3] 1 + j!1 > 0 |------- [1] j!1 = 0 {2} abs(0) < 1 Rerunning step: (expand "abs" 2) Expanding the definition of abs, this simplifies to: atanh_series.1.2.2.3.2.1 : [-1] z!1 = 0 [-2] abs(0 ^ j!1) < 1 [-3] 1 + j!1 > 0 |------- [1] j!1 = 0 {2} 0 < 1 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.2.2.3.2.1. atanh_series.1.2.2.3.2.2 : [-1] abs(z!1 ^ j!1) < 1 [-2] 1 + j!1 > 0 |------- {1} z!1 = 0 [2] j!1 = 0 [3] abs(z!1 ^ (1 + j!1)) < 1 Rerunning step: (lemma "expt_plus" ("n0x" "z!1" "i" "1" "j" "j!1")) Applying expt_plus where n0x gets z!1, i gets 1, j gets j!1, this yields 2 subgoals: atanh_series.1.2.2.3.2.2.1 : {-1} z!1 ^ (1 + j!1) = z!1 ^ 1 * z!1 ^ j!1 [-2] abs(z!1 ^ j!1) < 1 [-3] 1 + j!1 > 0 |------- [1] z!1 = 0 [2] j!1 = 0 [3] abs(z!1 ^ (1 + j!1)) < 1 Rerunning step: (replace -1 3) Replacing using formula -1, this simplifies to: atanh_series.1.2.2.3.2.2.1 : [-1] z!1 ^ (1 + j!1) = z!1 ^ 1 * z!1 ^ j!1 [-2] abs(z!1 ^ j!1) < 1 [-3] 1 + j!1 > 0 |------- [1] z!1 = 0 [2] j!1 = 0 {3} abs(z!1 ^ 1 * z!1 ^ j!1) < 1 Rerunning step: (rewrite "expt_x1" 3) Found matching substitution: x: real gets z!1, Rewriting using expt_x1, matching in 3, this simplifies to: atanh_series.1.2.2.3.2.2.1 : {-1} z!1 ^ (1 + j!1) = z!1 * z!1 ^ j!1 [-2] abs(z!1 ^ j!1) < 1 [-3] 1 + j!1 > 0 |------- [1] z!1 = 0 [2] j!1 = 0 {3} abs(z!1 * z!1 ^ j!1) < 1 Rerunning step: (rewrite "abs_mult" 3) Found matching substitution: y: real gets z!1 ^ j!1, x gets z!1, Rewriting using abs_mult, matching in 3, this simplifies to: atanh_series.1.2.2.3.2.2.1 : [-1] z!1 ^ (1 + j!1) = z!1 * z!1 ^ j!1 [-2] abs(z!1 ^ j!1) < 1 [-3] 1 + j!1 > 0 |------- [1] z!1 = 0 [2] j!1 = 0 {3} abs(z!1) * abs(z!1 ^ j!1) < 1 Rerunning step: (lemma "lt_times_lt_pos1" ("px" "abs(z!1)" "y" "1" "nnz" "abs(z!1^j!1)" "w" "1")) Applying lt_times_lt_pos1 where px gets abs(z!1), y gets 1, nnz gets abs(z!1 ^ j!1), w gets 1, this yields 2 subgoals: atanh_series.1.2.2.3.2.2.1.1 : {-1} abs(z!1) <= 1 AND abs(z!1 ^ j!1) < 1 IMPLIES abs(z!1) * abs(z!1 ^ j!1) < 1 * 1 [-2] z!1 ^ (1 + j!1) = z!1 * z!1 ^ j!1 [-3] abs(z!1 ^ j!1) < 1 [-4] 1 + j!1 > 0 |------- [1] z!1 = 0 [2] j!1 = 0 [3] abs(z!1) * abs(z!1 ^ j!1) < 1 Rerunning step: (split -1) Splitting conjunctions, this yields 3 subgoals: atanh_series.1.2.2.3.2.2.1.1.1 : {-1} abs(z!1) * abs(z!1 ^ j!1) < 1 * 1 [-2] z!1 ^ (1 + j!1) = z!1 * z!1 ^ j!1 [-3] abs(z!1 ^ j!1) < 1 [-4] 1 + j!1 > 0 |------- [1] z!1 = 0 [2] j!1 = 0 [3] abs(z!1) * abs(z!1 ^ j!1) < 1 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.2.2.3.2.2.1.1.1. atanh_series.1.2.2.3.2.2.1.1.2 : [-1] z!1 ^ (1 + j!1) = z!1 * z!1 ^ j!1 [-2] abs(z!1 ^ j!1) < 1 [-3] 1 + j!1 > 0 |------- {1} abs(z!1) <= 1 [2] z!1 = 0 [3] j!1 = 0 [4] abs(z!1) * abs(z!1 ^ j!1) < 1 Rerunning step: (hide-all-but 1) Keeping 1 and hiding *, this simplifies to: atanh_series.1.2.2.3.2.2.1.1.2 : |------- [1] abs(z!1) <= 1 Rerunning step: (expand "abs") Expanding the definition of abs, this simplifies to: atanh_series.1.2.2.3.2.2.1.1.2 : |------- {1} IF z!1 < 0 THEN -z!1 ELSE z!1 ENDIF <= 1 Rerunning step: (case "z!1<0") Case splitting on z!1 < 0, this yields 2 subgoals: atanh_series.1.2.2.3.2.2.1.1.2.1 : {-1} z!1 < 0 |------- [1] IF z!1 < 0 THEN -z!1 ELSE z!1 ENDIF <= 1 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.2.2.3.2.2.1.1.2.1. atanh_series.1.2.2.3.2.2.1.1.2.2 : |------- {1} z!1 < 0 [2] IF z!1 < 0 THEN -z!1 ELSE z!1 ENDIF <= 1 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.2.2.3.2.2.1.1.2.2. This completes the proof of atanh_series.1.2.2.3.2.2.1.1.2. atanh_series.1.2.2.3.2.2.1.1.3 : [-1] z!1 ^ (1 + j!1) = z!1 * z!1 ^ j!1 [-2] abs(z!1 ^ j!1) < 1 [-3] 1 + j!1 > 0 |------- {1} abs(z!1 ^ j!1) < 1 [2] z!1 = 0 [3] j!1 = 0 [4] abs(z!1) * abs(z!1 ^ j!1) < 1 which is trivially true. This completes the proof of atanh_series.1.2.2.3.2.2.1.1.3. This completes the proof of atanh_series.1.2.2.3.2.2.1.1. atanh_series.1.2.2.3.2.2.1.2 (TCC): [-1] z!1 ^ (1 + j!1) = z!1 * z!1 ^ j!1 [-2] abs(z!1 ^ j!1) < 1 [-3] 1 + j!1 > 0 |------- {1} abs(z!1) > 0 [2] z!1 = 0 [3] j!1 = 0 [4] abs(z!1) * abs(z!1 ^ j!1) < 1 Rerunning step: (hide-all-but (1 2)) Keeping (1 2) and hiding *, this simplifies to: atanh_series.1.2.2.3.2.2.1.2 : |------- [1] abs(z!1) > 0 [2] z!1 = 0 Rerunning step: (expand "abs") Expanding the definition of abs, this simplifies to: atanh_series.1.2.2.3.2.2.1.2 : |------- {1} IF z!1 < 0 THEN -z!1 ELSE z!1 ENDIF > 0 [2] z!1 = 0 Rerunning step: (lemma "trichotomy" ("x" "z!1")) Applying trichotomy where x gets z!1, this simplifies to: atanh_series.1.2.2.3.2.2.1.2 : {-1} z!1 > 0 OR z!1 = 0 OR 0 > z!1 |------- [1] IF z!1 < 0 THEN -z!1 ELSE z!1 ENDIF > 0 [2] z!1 = 0 Rerunning step: (split -1) Splitting conjunctions, this yields 3 subgoals: atanh_series.1.2.2.3.2.2.1.2.1 : {-1} z!1 > 0 |------- [1] IF z!1 < 0 THEN -z!1 ELSE z!1 ENDIF > 0 [2] z!1 = 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.2.2.3.2.2.1.2.1. atanh_series.1.2.2.3.2.2.1.2.2 : {-1} z!1 = 0 |------- [1] IF z!1 < 0 THEN -z!1 ELSE z!1 ENDIF > 0 [2] z!1 = 0 which is trivially true. This completes the proof of atanh_series.1.2.2.3.2.2.1.2.2. atanh_series.1.2.2.3.2.2.1.2.3 : {-1} 0 > z!1 |------- [1] IF z!1 < 0 THEN -z!1 ELSE z!1 ENDIF > 0 [2] z!1 = 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.2.2.3.2.2.1.2.3. This completes the proof of atanh_series.1.2.2.3.2.2.1.2. This completes the proof of atanh_series.1.2.2.3.2.2.1. atanh_series.1.2.2.3.2.2.2 (TCC): [-1] abs(z!1 ^ j!1) < 1 [-2] 1 + j!1 > 0 |------- {1} z!1 /= 0 [2] z!1 = 0 [3] j!1 = 0 [4] abs(z!1 ^ (1 + j!1)) < 1 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.1.2.2.3.2.2.2. This completes the proof of atanh_series.1.2.2.3.2.2. This completes the proof of atanh_series.1.2.2.3.2. This completes the proof of atanh_series.1.2.2.3. This completes the proof of atanh_series.1.2.2. This completes the proof of atanh_series.1.2. This completes the proof of atanh_series.1. atanh_series.2 : |------- {1} 1 - sq(z!1) > 0 [2] abs((atanhND(1 + n!1)(c!1) * z!1 ^ (3 + 2 * n!1)) / factorial(3 + 2 * n!1)) <= ((1 - z!1) ^ (3 + 2 * n!1) + (1 + z!1) ^ (3 + 2 * n!1)) / (6 * (1 - sq(z!1)) ^ (3 + 2 * n!1) + 4 * (((1 - sq(z!1)) ^ (3 + 2 * n!1)) * n!1)) Rerunning step: (hide 2) Hiding formulas: 2, this simplifies to: atanh_series.2 : |------- [1] 1 - sq(z!1) > 0 Rerunning step: (case "z!1>=0") Case splitting on z!1 >= 0, this yields 2 subgoals: atanh_series.2.1 : {-1} z!1 >= 0 |------- [1] 1 - sq(z!1) > 0 Rerunning step: (lemma "sq_gt" ("nna" "1" "nnb" "z!1")) Applying sq_gt where nna gets 1, nnb gets z!1, this yields 2 subgoals: atanh_series.2.1.1 : {-1} sq(1) > sq(z!1) IFF 1 > z!1 [-2] z!1 >= 0 |------- [1] 1 - sq(z!1) > 0 Rerunning step: (rewrite "sq_1") Found matching substitution: Rewriting using sq_1, matching in *, this simplifies to: atanh_series.2.1.1 : {-1} 1 > sq(z!1) IFF 1 > z!1 [-2] z!1 >= 0 |------- [1] 1 - sq(z!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.2.1.1. atanh_series.2.1.2 (TCC): [-1] z!1 >= 0 |------- {1} z!1 >= 0 [2] 1 - sq(z!1) > 0 which is trivially true. This completes the proof of atanh_series.2.1.2. This completes the proof of atanh_series.2.1. atanh_series.2.2 : |------- {1} z!1 >= 0 [2] 1 - sq(z!1) > 0 Rerunning step: (lemma "sq_gt" ("nna" "1" "nnb" "-z!1")) Applying sq_gt where nna gets 1, nnb gets -z!1, this yields 2 subgoals: atanh_series.2.2.1 : {-1} sq(1) > sq(-z!1) IFF 1 > -z!1 |------- [1] z!1 >= 0 [2] 1 - sq(z!1) > 0 Rerunning step: (rewrite "sq_1") Found matching substitution: Rewriting using sq_1, matching in *, this simplifies to: atanh_series.2.2.1 : {-1} 1 > sq(-z!1) IFF 1 > -z!1 |------- [1] z!1 >= 0 [2] 1 - sq(z!1) > 0 Rerunning step: (rewrite "sq_neg") Found matching substitution: a: real gets z!1, Rewriting using sq_neg, matching in *, this simplifies to: atanh_series.2.2.1 : {-1} 1 > sq(z!1) IFF 1 > -z!1 |------- [1] z!1 >= 0 [2] 1 - sq(z!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.2.2.1. atanh_series.2.2.2 (TCC): |------- {1} -z!1 >= 0 [2] z!1 >= 0 [3] 1 - sq(z!1) > 0 Rerunning step: (assert) Simplifying, rewriting, and recording with decision procedures, This completes the proof of atanh_series.2.2.2. This completes the proof of atanh_series.2.2. This completes the proof of atanh_series.2. Q.E.D. Run time = 18.02 secs. Real time = 36.13 secs. nil pvs(26): [Dauer der Verarbeitung: 0.232 Sekunden, vorverarbeitet 2026-04-26] |
2026-05-26