TSTP Solution File: GRP127-4.004 by Otter---3.3
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Otter---3.3
% Problem : GRP127-4.004 : TPTP v8.1.0. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp:raw
% Command : otter-tptp-script %s
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Jul 27 12:56:12 EDT 2022
% Result : Unsatisfiable 2.04s 2.25s
% Output : Refutation 2.04s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 18
% Syntax : Number of clauses : 55 ( 16 unt; 34 nHn; 54 RR)
% Number of literals : 129 ( 0 equ; 23 neg)
% Maximal clause size : 6 ( 2 avg)
% Maximal term depth : 1 ( 1 avg)
% Number of predicates : 4 ( 3 usr; 1 prp; 0-3 aty)
% Number of functors : 4 ( 4 usr; 4 con; 0-0 aty)
% Number of variables : 27 ( 0 sgn)
% Comments :
%------------------------------------------------------------------------------
cnf(1,axiom,
( ~ group_element(A)
| ~ group_element(B)
| product(e_1,A,B)
| product(e_2,A,B)
| product(e_3,A,B)
| product(e_4,A,B) ),
file('GRP127-4.004.p',unknown),
[] ).
cnf(2,axiom,
( ~ group_element(A)
| ~ group_element(B)
| product(A,e_1,B)
| product(A,e_2,B)
| product(A,e_3,B)
| product(A,e_4,B) ),
file('GRP127-4.004.p',unknown),
[] ).
cnf(3,axiom,
( product(A,B,C)
| ~ product(D,A,B)
| ~ product(C,A,D) ),
file('GRP127-4.004.p',unknown),
[] ).
cnf(5,axiom,
~ e_qualish(e_1,e_2),
file('GRP127-4.004.p',unknown),
[] ).
cnf(6,axiom,
~ e_qualish(e_1,e_3),
file('GRP127-4.004.p',unknown),
[] ).
cnf(7,axiom,
~ e_qualish(e_1,e_4),
file('GRP127-4.004.p',unknown),
[] ).
cnf(8,axiom,
~ e_qualish(e_2,e_1),
file('GRP127-4.004.p',unknown),
[] ).
cnf(9,axiom,
~ e_qualish(e_2,e_3),
file('GRP127-4.004.p',unknown),
[] ).
cnf(10,axiom,
~ e_qualish(e_2,e_4),
file('GRP127-4.004.p',unknown),
[] ).
cnf(15,axiom,
~ e_qualish(e_4,e_2),
file('GRP127-4.004.p',unknown),
[] ).
cnf(17,axiom,
( ~ group_element(A)
| ~ group_element(B)
| product(A,B,e_1)
| product(A,B,e_2)
| product(A,B,e_3)
| product(A,B,e_4) ),
file('GRP127-4.004.p',unknown),
[] ).
cnf(18,axiom,
( ~ product(A,B,C)
| ~ product(A,B,D)
| e_qualish(C,D) ),
file('GRP127-4.004.p',unknown),
[] ).
cnf(19,axiom,
( ~ product(A,B,C)
| ~ product(A,D,C)
| e_qualish(B,D) ),
file('GRP127-4.004.p',unknown),
[] ).
cnf(20,axiom,
( ~ product(A,B,C)
| ~ product(D,B,C)
| e_qualish(A,D) ),
file('GRP127-4.004.p',unknown),
[] ).
cnf(21,axiom,
( ~ product(A,B,C)
| ~ product(C,A,D)
| product(D,A,B) ),
file('GRP127-4.004.p',unknown),
[] ).
cnf(29,axiom,
group_element(e_1),
file('GRP127-4.004.p',unknown),
[] ).
cnf(30,axiom,
group_element(e_2),
file('GRP127-4.004.p',unknown),
[] ).
cnf(33,axiom,
product(A,A,A),
file('GRP127-4.004.p',unknown),
[] ).
cnf(34,plain,
( product(e_1,e_2,e_1)
| product(e_1,e_2,e_2)
| product(e_1,e_2,e_3)
| product(e_1,e_2,e_4) ),
inference(hyper,[status(thm)],[30,17,29]),
[iquote('hyper,30,17,29')] ).
cnf(35,plain,
( product(e_2,e_1,e_1)
| product(e_2,e_1,e_2)
| product(e_2,e_1,e_3)
| product(e_2,e_1,e_4) ),
inference(hyper,[status(thm)],[30,17,29]),
[iquote('hyper,30,17,29')] ).
cnf(36,plain,
( product(e_1,e_1,e_2)
| product(e_1,e_2,e_2)
| product(e_1,e_3,e_2)
| product(e_1,e_4,e_2) ),
inference(hyper,[status(thm)],[30,2,29]),
[iquote('hyper,30,2,29')] ).
cnf(37,plain,
( product(e_2,e_1,e_1)
| product(e_2,e_2,e_1)
| product(e_2,e_3,e_1)
| product(e_2,e_4,e_1) ),
inference(hyper,[status(thm)],[30,2,29]),
[iquote('hyper,30,2,29')] ).
cnf(38,plain,
( product(e_1,e_1,e_2)
| product(e_2,e_1,e_2)
| product(e_3,e_1,e_2)
| product(e_4,e_1,e_2) ),
inference(hyper,[status(thm)],[30,1,29]),
[iquote('hyper,30,1,29')] ).
cnf(73,plain,
( product(e_1,e_2,e_2)
| product(e_1,e_2,e_3)
| product(e_1,e_2,e_4) ),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[34,19,33]),5]),
[iquote('hyper,34,19,33,unit_del,5')] ).
cnf(75,plain,
( product(e_1,e_2,e_3)
| product(e_1,e_2,e_4) ),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[73,20,33]),8]),
[iquote('hyper,73,20,33,unit_del,8')] ).
cnf(78,plain,
( product(e_2,e_1,e_2)
| product(e_2,e_1,e_3)
| product(e_2,e_1,e_4) ),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[35,20,33]),5]),
[iquote('hyper,35,20,33,unit_del,5')] ).
cnf(83,plain,
( product(e_1,e_2,e_2)
| product(e_1,e_3,e_2)
| product(e_1,e_4,e_2) ),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[36,18,33]),5]),
[iquote('hyper,36,18,33,unit_del,5')] ).
cnf(87,plain,
( product(e_2,e_1,e_3)
| product(e_2,e_1,e_4) ),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[78,19,33]),8]),
[iquote('hyper,78,19,33,unit_del,8')] ).
cnf(90,plain,
( product(e_1,e_3,e_2)
| product(e_1,e_4,e_2) ),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[83,20,33]),8]),
[iquote('hyper,83,20,33,unit_del,8')] ).
cnf(98,plain,
( product(e_1,e_4,e_2)
| product(e_3,e_1,e_3)
| product(e_2,e_1,e_4) ),
inference(hyper,[status(thm)],[90,21,87]),
[iquote('hyper,90,21,87')] ).
cnf(100,plain,
( product(e_1,e_3,e_2)
| product(e_4,e_1,e_4)
| product(e_2,e_1,e_3) ),
inference(hyper,[status(thm)],[90,21,87]),
[iquote('hyper,90,21,87')] ).
cnf(107,plain,
( product(e_2,e_2,e_1)
| product(e_2,e_3,e_1)
| product(e_2,e_4,e_1) ),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[37,20,33]),5]),
[iquote('hyper,37,20,33,unit_del,5')] ).
cnf(132,plain,
( product(e_1,e_4,e_2)
| product(e_2,e_1,e_4) ),
inference(factor_simp,[status(thm)],[inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[98,20,87]),9])]),
[iquote('hyper,98,20,87,unit_del,9,factor_simp')] ).
cnf(133,plain,
( product(e_2,e_1,e_4)
| product(e_3,e_1,e_4) ),
inference(factor_simp,[status(thm)],[inference(hyper,[status(thm)],[132,21,87])]),
[iquote('hyper,132,21,87,factor_simp')] ).
cnf(134,plain,
( product(e_1,e_4,e_2)
| product(e_4,e_1,e_3) ),
inference(factor_simp,[status(thm)],[inference(hyper,[status(thm)],[132,21,90])]),
[iquote('hyper,132,21,90,factor_simp')] ).
cnf(146,plain,
( product(e_2,e_1,e_2)
| product(e_3,e_1,e_2)
| product(e_4,e_1,e_2) ),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[38,18,33]),5]),
[iquote('hyper,38,18,33,unit_del,5')] ).
cnf(174,plain,
( product(e_1,e_3,e_2)
| product(e_2,e_1,e_3) ),
inference(factor_simp,[status(thm)],[inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[100,20,87]),10])]),
[iquote('hyper,100,20,87,unit_del,10,factor_simp')] ).
cnf(176,plain,
( product(e_2,e_1,e_3)
| product(e_4,e_1,e_3) ),
inference(factor_simp,[status(thm)],[inference(hyper,[status(thm)],[174,21,87])]),
[iquote('hyper,174,21,87,factor_simp')] ).
cnf(179,plain,
( product(e_4,e_1,e_3)
| product(e_3,e_1,e_4) ),
inference(factor_simp,[status(thm)],[inference(hyper,[status(thm)],[176,21,134])]),
[iquote('hyper,176,21,134,factor_simp')] ).
cnf(222,plain,
( product(e_2,e_3,e_1)
| product(e_2,e_4,e_1) ),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[107,18,33]),8]),
[iquote('hyper,107,18,33,unit_del,8')] ).
cnf(247,plain,
( product(e_2,e_4,e_1)
| product(e_4,e_2,e_3)
| product(e_1,e_2,e_3) ),
inference(hyper,[status(thm)],[222,21,75]),
[iquote('hyper,222,21,75')] ).
cnf(308,plain,
( product(e_3,e_1,e_2)
| product(e_4,e_1,e_2) ),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[146,19,33]),8]),
[iquote('hyper,146,19,33,unit_del,8')] ).
cnf(328,plain,
( product(e_4,e_1,e_2)
| product(e_4,e_1,e_3) ),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[308,18,179]),15]),
[iquote('hyper,308,18,179,unit_del,15')] ).
cnf(330,plain,
( product(e_4,e_1,e_2)
| product(e_2,e_1,e_4) ),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[308,18,133]),15]),
[iquote('hyper,308,18,133,unit_del,15')] ).
cnf(357,plain,
( product(e_4,e_1,e_3)
| product(e_1,e_2,e_3) ),
inference(factor_simp,[status(thm)],[inference(hyper,[status(thm)],[328,3,179])]),
[iquote('hyper,328,3,179,factor_simp')] ).
cnf(385,plain,
( product(e_4,e_1,e_2)
| product(e_1,e_4,e_3) ),
inference(factor_simp,[status(thm)],[inference(hyper,[status(thm)],[330,3,308])]),
[iquote('hyper,330,3,308,factor_simp')] ).
cnf(954,plain,
( product(e_2,e_4,e_1)
| product(e_1,e_2,e_3) ),
inference(factor_simp,[status(thm)],[inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[247,19,357]),5])]),
[iquote('hyper,247,19,357,unit_del,5,factor_simp')] ).
cnf(965,plain,
( product(e_1,e_2,e_3)
| product(e_4,e_2,e_4) ),
inference(factor_simp,[status(thm)],[inference(hyper,[status(thm)],[954,21,75])]),
[iquote('hyper,954,21,75,factor_simp')] ).
cnf(975,plain,
( product(e_2,e_4,e_1)
| product(e_3,e_2,e_3) ),
inference(factor_simp,[status(thm)],[inference(hyper,[status(thm)],[954,21,222])]),
[iquote('hyper,954,21,222,factor_simp')] ).
cnf(1039,plain,
product(e_1,e_2,e_3),
inference(factor_simp,[status(thm)],[inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[965,20,75]),7])]),
[iquote('hyper,965,20,75,unit_del,7,factor_simp')] ).
cnf(1050,plain,
product(e_4,e_1,e_2),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[1039,19,385]),15]),
[iquote('hyper,1039,19,385,unit_del,15')] ).
cnf(1089,plain,
product(e_2,e_4,e_1),
inference(unit_del,[status(thm)],[inference(hyper,[status(thm)],[975,20,1039]),6]),
[iquote('hyper,975,20,1039,unit_del,6')] ).
cnf(1100,plain,
product(e_1,e_4,e_1),
inference(hyper,[status(thm)],[1089,21,1050]),
[iquote('hyper,1089,21,1050')] ).
cnf(1106,plain,
e_qualish(e_2,e_1),
inference(hyper,[status(thm)],[1100,20,1089]),
[iquote('hyper,1100,20,1089')] ).
cnf(1107,plain,
$false,
inference(binary,[status(thm)],[1106,8]),
[iquote('binary,1106.1,8.1')] ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : GRP127-4.004 : TPTP v8.1.0. Bugfixed v1.2.1.
% 0.10/0.12 % Command : otter-tptp-script %s
% 0.12/0.33 % Computer : n006.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Wed Jul 27 05:30:31 EDT 2022
% 0.12/0.33 % CPUTime :
% 2.04/2.25 ----- Otter 3.3f, August 2004 -----
% 2.04/2.25 The process was started by sandbox on n006.cluster.edu,
% 2.04/2.25 Wed Jul 27 05:30:32 2022
% 2.04/2.25 The command was "./otter". The process ID is 28006.
% 2.04/2.25
% 2.04/2.25 set(prolog_style_variables).
% 2.04/2.25 set(auto).
% 2.04/2.25 dependent: set(auto1).
% 2.04/2.25 dependent: set(process_input).
% 2.04/2.25 dependent: clear(print_kept).
% 2.04/2.25 dependent: clear(print_new_demod).
% 2.04/2.25 dependent: clear(print_back_demod).
% 2.04/2.25 dependent: clear(print_back_sub).
% 2.04/2.25 dependent: set(control_memory).
% 2.04/2.25 dependent: assign(max_mem, 12000).
% 2.04/2.25 dependent: assign(pick_given_ratio, 4).
% 2.04/2.25 dependent: assign(stats_level, 1).
% 2.04/2.25 dependent: assign(max_seconds, 10800).
% 2.04/2.25 clear(print_given).
% 2.04/2.25
% 2.04/2.25 list(usable).
% 2.04/2.25 0 [] -group_element(X)| -group_element(Y)|product(e_1,X,Y)|product(e_2,X,Y)|product(e_3,X,Y)|product(e_4,X,Y).
% 2.04/2.25 0 [] -group_element(X)| -group_element(Y)|product(X,e_1,Y)|product(X,e_2,Y)|product(X,e_3,Y)|product(X,e_4,Y).
% 2.04/2.25 0 [] product(Y,X,Z1)| -product(Z2,Y,X)| -product(Z1,Y,Z2).
% 2.04/2.25 0 [] product(Z1,Y,Z2)| -product(Z2,Y,X)| -product(Y,X,Z1).
% 2.04/2.25 0 [] group_element(e_1).
% 2.04/2.25 0 [] group_element(e_2).
% 2.04/2.25 0 [] group_element(e_3).
% 2.04/2.25 0 [] group_element(e_4).
% 2.04/2.25 0 [] -e_qualish(e_1,e_2).
% 2.04/2.25 0 [] -e_qualish(e_1,e_3).
% 2.04/2.25 0 [] -e_qualish(e_1,e_4).
% 2.04/2.25 0 [] -e_qualish(e_2,e_1).
% 2.04/2.25 0 [] -e_qualish(e_2,e_3).
% 2.04/2.25 0 [] -e_qualish(e_2,e_4).
% 2.04/2.25 0 [] -e_qualish(e_3,e_1).
% 2.04/2.25 0 [] -e_qualish(e_3,e_2).
% 2.04/2.25 0 [] -e_qualish(e_3,e_4).
% 2.04/2.25 0 [] -e_qualish(e_4,e_1).
% 2.04/2.25 0 [] -e_qualish(e_4,e_2).
% 2.04/2.25 0 [] -e_qualish(e_4,e_3).
% 2.04/2.25 0 [] -group_element(X)| -group_element(Y)|product(X,Y,e_1)|product(X,Y,e_2)|product(X,Y,e_3)|product(X,Y,e_4).
% 2.04/2.25 0 [] -product(X,Y,W)| -product(X,Y,Z)|e_qualish(W,Z).
% 2.04/2.25 0 [] -product(X,W,Y)| -product(X,Z,Y)|e_qualish(W,Z).
% 2.04/2.25 0 [] -product(W,Y,X)| -product(Z,Y,X)|e_qualish(W,Z).
% 2.04/2.25 0 [] product(X,X,X).
% 2.04/2.25 0 [] -product(Y,X,Z1)| -product(Z1,Y,Z2)|product(Z2,Y,X).
% 2.04/2.25 end_of_list.
% 2.04/2.25
% 2.04/2.25 SCAN INPUT: prop=0, horn=0, equality=0, symmetry=0, max_lits=6.
% 2.04/2.25
% 2.04/2.25 This is a non-Horn set without equality. The strategy will
% 2.04/2.25 be ordered hyper_res, unit deletion, and factoring, with
% 2.04/2.25 satellites in sos and with nuclei in usable.
% 2.04/2.25
% 2.04/2.25 dependent: set(hyper_res).
% 2.04/2.25 dependent: set(factor).
% 2.04/2.25 dependent: set(unit_deletion).
% 2.04/2.25
% 2.04/2.25 ------------> process usable:
% 2.04/2.25 ** KEPT (pick-wt=20): 1 [] -group_element(A)| -group_element(B)|product(e_1,A,B)|product(e_2,A,B)|product(e_3,A,B)|product(e_4,A,B).
% 2.04/2.25 ** KEPT (pick-wt=20): 2 [] -group_element(A)| -group_element(B)|product(A,e_1,B)|product(A,e_2,B)|product(A,e_3,B)|product(A,e_4,B).
% 2.04/2.25 ** KEPT (pick-wt=12): 3 [] product(A,B,C)| -product(D,A,B)| -product(C,A,D).
% 2.04/2.25 ** KEPT (pick-wt=12): 4 [] product(A,B,C)| -product(C,B,D)| -product(B,D,A).
% 2.04/2.25 ** KEPT (pick-wt=3): 5 [] -e_qualish(e_1,e_2).
% 2.04/2.25 ** KEPT (pick-wt=3): 6 [] -e_qualish(e_1,e_3).
% 2.04/2.25 ** KEPT (pick-wt=3): 7 [] -e_qualish(e_1,e_4).
% 2.04/2.25 ** KEPT (pick-wt=3): 8 [] -e_qualish(e_2,e_1).
% 2.04/2.25 ** KEPT (pick-wt=3): 9 [] -e_qualish(e_2,e_3).
% 2.04/2.25 ** KEPT (pick-wt=3): 10 [] -e_qualish(e_2,e_4).
% 2.04/2.25 ** KEPT (pick-wt=3): 11 [] -e_qualish(e_3,e_1).
% 2.04/2.25 ** KEPT (pick-wt=3): 12 [] -e_qualish(e_3,e_2).
% 2.04/2.25 ** KEPT (pick-wt=3): 13 [] -e_qualish(e_3,e_4).
% 2.04/2.25 ** KEPT (pick-wt=3): 14 [] -e_qualish(e_4,e_1).
% 2.04/2.25 ** KEPT (pick-wt=3): 15 [] -e_qualish(e_4,e_2).
% 2.04/2.25 ** KEPT (pick-wt=3): 16 [] -e_qualish(e_4,e_3).
% 2.04/2.25 ** KEPT (pick-wt=20): 17 [] -group_element(A)| -group_element(B)|product(A,B,e_1)|product(A,B,e_2)|product(A,B,e_3)|product(A,B,e_4).
% 2.04/2.25 ** KEPT (pick-wt=11): 18 [] -product(A,B,C)| -product(A,B,D)|e_qualish(C,D).
% 2.04/2.25 ** KEPT (pick-wt=11): 19 [] -product(A,B,C)| -product(A,D,C)|e_qualish(B,D).
% 2.04/2.25 ** KEPT (pick-wt=11): 20 [] -product(A,B,C)| -product(D,B,C)|e_qualish(A,D).
% 2.04/2.25 ** KEPT (pick-wt=12): 21 [] -product(A,B,C)| -product(C,A,D)|product(D,A,B).
% 2.04/2.25
% 2.04/2.25 ------------> process sos:
% 2.04/2.25 ** KEPT (pick-wt=2): 29 [] group_element(e_1).
% 2.04/2.25 ** KEPT (pick-wt=2): 30 [] group_element(e_2).
% 2.04/2.25 ** KEPT (pick-wt=2): 31 [] group_element(e_3).
% 2.04/2.25 ** KEPT (pick-wt=2): 32 [] group_element(e_4).
% 2.04/2.25 ** KEPT (pick-wt=4): 33 [] product(A,A,A).
% 2.04/2.25
% 2.04/2.25 ======= end of input processing =======
% 2.04/2.25
% 2.04/2.25 =========== start of search ===========
% 2.04/2.25
% 2.04/2.25 -------- PROOF --------
% 2.04/2.25
% 2.04/2.25 ----> UNIT CONFLICT at 0.15 sec ----> 1107 [binary,1106.1,8.1] $F.
% 2.04/2.25
% 2.04/2.25 Length of proof is 36. Level of proof is 15.
% 2.04/2.25
% 2.04/2.25 ---------------- PROOF ----------------
% 2.04/2.25 % SZS status Unsatisfiable
% 2.04/2.25 % SZS output start Refutation
% See solution above
% 2.04/2.25 ------------ end of proof -------------
% 2.04/2.25
% 2.04/2.25
% 2.04/2.25 Search stopped by max_proofs option.
% 2.04/2.25
% 2.04/2.25
% 2.04/2.25 Search stopped by max_proofs option.
% 2.04/2.25
% 2.04/2.25 ============ end of search ============
% 2.04/2.25
% 2.04/2.25 -------------- statistics -------------
% 2.04/2.25 clauses given 83
% 2.04/2.25 clauses generated 8146
% 2.04/2.25 clauses kept 1106
% 2.04/2.25 clauses forward subsumed 7064
% 2.04/2.25 clauses back subsumed 959
% 2.04/2.25 Kbytes malloced 976
% 2.04/2.25
% 2.04/2.25 ----------- times (seconds) -----------
% 2.04/2.25 user CPU time 0.15 (0 hr, 0 min, 0 sec)
% 2.04/2.25 system CPU time 0.00 (0 hr, 0 min, 0 sec)
% 2.04/2.25 wall-clock time 1 (0 hr, 0 min, 1 sec)
% 2.04/2.25
% 2.04/2.25 That finishes the proof of the theorem.
% 2.04/2.25
% 2.04/2.25 Process 28006 finished Wed Jul 27 05:30:33 2022
% 2.04/2.25 Otter interrupted
% 2.04/2.25 PROOF FOUND
%------------------------------------------------------------------------------