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SSL atanh_series.proof
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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) + (1 + z!1) ^ (3 + 2 * n!1)) / (2 * (1 - sq(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 / (6 * K11 + 4 * (K11 * n!1)))) +
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!1),
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)(sq(c!1)) > 0")
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)(sq(c!1))
<=
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):
[Verzeichnis aufwärts0.256unsichere VerbindungÜbersetzung europäischer Sprachen durch Browser2026-04-26]
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2026-05-26
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