TSTP Solution File: SEU130+2 by CSE---1.6
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : CSE---1.6
% Problem : SEU130+2 : TPTP v8.1.2. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : java -jar /export/starexec/sandbox/solver/bin/mcs_scs.jar %s %d
% Computer : n020.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 : Thu Aug 31 16:17:39 EDT 2023
% Result : Theorem 0.20s 0.65s
% Output : CNFRefutation 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SEU130+2 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.13 % Command : java -jar /export/starexec/sandbox/solver/bin/mcs_scs.jar %s %d
% 0.14/0.34 % Computer : n020.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Wed Aug 23 14:13:00 EDT 2023
% 0.14/0.34 % CPUTime :
% 0.20/0.57 start to proof:theBenchmark
% 0.20/0.63 %-------------------------------------------
% 0.20/0.63 % File :CSE---1.6
% 0.20/0.63 % Problem :theBenchmark
% 0.20/0.63 % Transform :cnf
% 0.20/0.63 % Format :tptp:raw
% 0.20/0.63 % Command :java -jar mcs_scs.jar %d %s
% 0.20/0.63
% 0.20/0.63 % Result :Theorem 0.010000s
% 0.20/0.64 % Output :CNFRefutation 0.010000s
% 0.20/0.64 %-------------------------------------------
% 0.20/0.64 %------------------------------------------------------------------------------
% 0.20/0.64 % File : SEU130+2 : TPTP v8.1.2. Released v3.3.0.
% 0.20/0.64 % Domain : Set theory
% 0.20/0.64 % Problem : MPTP chainy problem t28_xboole_1
% 0.20/0.64 % Version : [Urb07] axioms : Especial.
% 0.20/0.64 % English :
% 0.20/0.64
% 0.20/0.64 % Refs : [Ban01] Bancerek et al. (2001), On the Characterizations of Co
% 0.20/0.64 % : [Urb07] Urban (2006), Email to G. Sutcliffe
% 0.20/0.64 % Source : [Urb07]
% 0.20/0.64 % Names : chainy-t28_xboole_1 [Urb07]
% 0.20/0.64
% 0.20/0.64 % Status : Theorem
% 0.20/0.64 % Rating : 0.06 v8.1.0, 0.08 v7.5.0, 0.09 v7.4.0, 0.00 v7.0.0, 0.03 v6.4.0, 0.08 v6.3.0, 0.12 v6.2.0, 0.16 v6.1.0, 0.20 v6.0.0, 0.17 v5.5.0, 0.19 v5.4.0, 0.21 v5.3.0, 0.26 v5.2.0, 0.00 v5.0.0, 0.12 v4.1.0, 0.13 v4.0.1, 0.17 v4.0.0, 0.21 v3.7.0, 0.20 v3.5.0, 0.21 v3.4.0, 0.16 v3.3.0
% 0.20/0.64 % Syntax : Number of formulae : 38 ( 16 unt; 0 def)
% 0.20/0.64 % Number of atoms : 76 ( 16 equ)
% 0.20/0.64 % Maximal formula atoms : 6 ( 2 avg)
% 0.20/0.64 % Number of connectives : 56 ( 18 ~; 1 |; 16 &)
% 0.20/0.64 % ( 8 <=>; 13 =>; 0 <=; 0 <~>)
% 0.20/0.64 % Maximal formula depth : 9 ( 4 avg)
% 0.20/0.64 % Maximal term depth : 2 ( 1 avg)
% 0.20/0.64 % Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% 0.20/0.64 % Number of functors : 3 ( 3 usr; 1 con; 0-2 aty)
% 0.20/0.64 % Number of variables : 74 ( 70 !; 4 ?)
% 0.20/0.64 % SPC : FOF_THM_RFO_SEQ
% 0.20/0.64
% 0.20/0.64 % Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% 0.20/0.64 % library, www.mizar.org
% 0.20/0.64 %------------------------------------------------------------------------------
% 0.20/0.64 fof(antisymmetry_r2_hidden,axiom,
% 0.20/0.64 ! [A,B] :
% 0.20/0.64 ( in(A,B)
% 0.20/0.64 => ~ in(B,A) ) ).
% 0.20/0.64
% 0.20/0.64 fof(commutativity_k2_xboole_0,axiom,
% 0.20/0.64 ! [A,B] : set_union2(A,B) = set_union2(B,A) ).
% 0.20/0.64
% 0.20/0.64 fof(commutativity_k3_xboole_0,axiom,
% 0.20/0.64 ! [A,B] : set_intersection2(A,B) = set_intersection2(B,A) ).
% 0.20/0.64
% 0.20/0.64 fof(d10_xboole_0,axiom,
% 0.20/0.64 ! [A,B] :
% 0.20/0.64 ( A = B
% 0.20/0.64 <=> ( subset(A,B)
% 0.20/0.64 & subset(B,A) ) ) ).
% 0.20/0.64
% 0.20/0.64 fof(d1_xboole_0,axiom,
% 0.20/0.64 ! [A] :
% 0.20/0.64 ( A = empty_set
% 0.20/0.64 <=> ! [B] : ~ in(B,A) ) ).
% 0.20/0.64
% 0.20/0.64 fof(d2_xboole_0,axiom,
% 0.20/0.64 ! [A,B,C] :
% 0.20/0.64 ( C = set_union2(A,B)
% 0.20/0.64 <=> ! [D] :
% 0.20/0.64 ( in(D,C)
% 0.20/0.64 <=> ( in(D,A)
% 0.20/0.64 | in(D,B) ) ) ) ).
% 0.20/0.64
% 0.20/0.64 fof(d3_tarski,axiom,
% 0.20/0.64 ! [A,B] :
% 0.20/0.64 ( subset(A,B)
% 0.20/0.64 <=> ! [C] :
% 0.20/0.64 ( in(C,A)
% 0.20/0.64 => in(C,B) ) ) ).
% 0.20/0.64
% 0.20/0.64 fof(d3_xboole_0,axiom,
% 0.20/0.64 ! [A,B,C] :
% 0.20/0.64 ( C = set_intersection2(A,B)
% 0.20/0.64 <=> ! [D] :
% 0.20/0.64 ( in(D,C)
% 0.20/0.64 <=> ( in(D,A)
% 0.20/0.64 & in(D,B) ) ) ) ).
% 0.20/0.64
% 0.20/0.64 fof(d7_xboole_0,axiom,
% 0.20/0.64 ! [A,B] :
% 0.20/0.64 ( disjoint(A,B)
% 0.20/0.64 <=> set_intersection2(A,B) = empty_set ) ).
% 0.20/0.64
% 0.20/0.64 fof(dt_k1_xboole_0,axiom,
% 0.20/0.64 $true ).
% 0.20/0.64
% 0.20/0.64 fof(dt_k2_xboole_0,axiom,
% 0.20/0.64 $true ).
% 0.20/0.64
% 0.20/0.64 fof(dt_k3_xboole_0,axiom,
% 0.20/0.64 $true ).
% 0.20/0.64
% 0.20/0.64 fof(fc1_xboole_0,axiom,
% 0.20/0.64 empty(empty_set) ).
% 0.20/0.64
% 0.20/0.64 fof(fc2_xboole_0,axiom,
% 0.20/0.64 ! [A,B] :
% 0.20/0.64 ( ~ empty(A)
% 0.20/0.64 => ~ empty(set_union2(A,B)) ) ).
% 0.20/0.64
% 0.20/0.64 fof(fc3_xboole_0,axiom,
% 0.20/0.64 ! [A,B] :
% 0.20/0.64 ( ~ empty(A)
% 0.20/0.64 => ~ empty(set_union2(B,A)) ) ).
% 0.20/0.64
% 0.20/0.64 fof(idempotence_k2_xboole_0,axiom,
% 0.20/0.64 ! [A,B] : set_union2(A,A) = A ).
% 0.20/0.64
% 0.20/0.64 fof(idempotence_k3_xboole_0,axiom,
% 0.20/0.64 ! [A,B] : set_intersection2(A,A) = A ).
% 0.20/0.64
% 0.20/0.64 fof(rc1_xboole_0,axiom,
% 0.20/0.64 ? [A] : empty(A) ).
% 0.20/0.64
% 0.20/0.64 fof(rc2_xboole_0,axiom,
% 0.20/0.64 ? [A] : ~ empty(A) ).
% 0.20/0.64
% 0.20/0.64 fof(reflexivity_r1_tarski,axiom,
% 0.20/0.64 ! [A,B] : subset(A,A) ).
% 0.20/0.64
% 0.20/0.64 fof(symmetry_r1_xboole_0,axiom,
% 0.20/0.64 ! [A,B] :
% 0.20/0.64 ( disjoint(A,B)
% 0.20/0.64 => disjoint(B,A) ) ).
% 0.20/0.64
% 0.20/0.64 fof(t12_xboole_1,lemma,
% 0.20/0.64 ! [A,B] :
% 0.20/0.64 ( subset(A,B)
% 0.20/0.64 => set_union2(A,B) = B ) ).
% 0.20/0.64
% 0.20/0.64 fof(t17_xboole_1,lemma,
% 0.20/0.64 ! [A,B] : subset(set_intersection2(A,B),A) ).
% 0.20/0.64
% 0.20/0.64 fof(t19_xboole_1,lemma,
% 0.20/0.64 ! [A,B,C] :
% 0.20/0.64 ( ( subset(A,B)
% 0.20/0.64 & subset(A,C) )
% 0.20/0.64 => subset(A,set_intersection2(B,C)) ) ).
% 0.20/0.64
% 0.20/0.64 fof(t1_boole,axiom,
% 0.20/0.64 ! [A] : set_union2(A,empty_set) = A ).
% 0.20/0.64
% 0.20/0.64 fof(t1_xboole_1,lemma,
% 0.20/0.64 ! [A,B,C] :
% 0.20/0.64 ( ( subset(A,B)
% 0.20/0.64 & subset(B,C) )
% 0.20/0.64 => subset(A,C) ) ).
% 0.20/0.64
% 0.20/0.64 fof(t26_xboole_1,lemma,
% 0.20/0.64 ! [A,B,C] :
% 0.20/0.64 ( subset(A,B)
% 0.20/0.64 => subset(set_intersection2(A,C),set_intersection2(B,C)) ) ).
% 0.20/0.64
% 0.20/0.64 fof(t28_xboole_1,conjecture,
% 0.20/0.64 ! [A,B] :
% 0.20/0.64 ( subset(A,B)
% 0.20/0.64 => set_intersection2(A,B) = A ) ).
% 0.20/0.64
% 0.20/0.64 fof(t2_boole,axiom,
% 0.20/0.64 ! [A] : set_intersection2(A,empty_set) = empty_set ).
% 0.20/0.64
% 0.20/0.64 fof(t2_xboole_1,lemma,
% 0.20/0.64 ! [A] : subset(empty_set,A) ).
% 0.20/0.64
% 0.20/0.64 fof(t3_xboole_0,lemma,
% 0.20/0.64 ! [A,B] :
% 0.20/0.64 ( ~ ( ~ disjoint(A,B)
% 0.20/0.64 & ! [C] :
% 0.20/0.64 ~ ( in(C,A)
% 0.20/0.64 & in(C,B) ) )
% 0.20/0.64 & ~ ( ? [C] :
% 0.20/0.64 ( in(C,A)
% 0.20/0.64 & in(C,B) )
% 0.20/0.64 & disjoint(A,B) ) ) ).
% 0.20/0.64
% 0.20/0.64 fof(t3_xboole_1,lemma,
% 0.20/0.64 ! [A] :
% 0.20/0.64 ( subset(A,empty_set)
% 0.20/0.64 => A = empty_set ) ).
% 0.20/0.64
% 0.20/0.64 fof(t4_xboole_0,lemma,
% 0.20/0.64 ! [A,B] :
% 0.20/0.64 ( ~ ( ~ disjoint(A,B)
% 0.20/0.64 & ! [C] : ~ in(C,set_intersection2(A,B)) )
% 0.20/0.64 & ~ ( ? [C] : in(C,set_intersection2(A,B))
% 0.20/0.64 & disjoint(A,B) ) ) ).
% 0.20/0.64
% 0.20/0.64 fof(t6_boole,axiom,
% 0.20/0.64 ! [A] :
% 0.20/0.64 ( empty(A)
% 0.20/0.64 => A = empty_set ) ).
% 0.20/0.64
% 0.20/0.64 fof(t7_boole,axiom,
% 0.20/0.65 ! [A,B] :
% 0.20/0.65 ~ ( in(A,B)
% 0.20/0.65 & empty(B) ) ).
% 0.20/0.65
% 0.20/0.65 fof(t7_xboole_1,lemma,
% 0.20/0.65 ! [A,B] : subset(A,set_union2(A,B)) ).
% 0.20/0.65
% 0.20/0.65 fof(t8_boole,axiom,
% 0.20/0.65 ! [A,B] :
% 0.20/0.65 ~ ( empty(A)
% 0.20/0.65 & A != B
% 0.20/0.65 & empty(B) ) ).
% 0.20/0.65
% 0.20/0.65 fof(t8_xboole_1,lemma,
% 0.20/0.65 ! [A,B,C] :
% 0.20/0.65 ( ( subset(A,B)
% 0.20/0.65 & subset(C,B) )
% 0.20/0.65 => subset(set_union2(A,C),B) ) ).
% 0.20/0.65
% 0.20/0.65 %------------------------------------------------------------------------------
% 0.20/0.65 %-------------------------------------------
% 0.20/0.65 % Proof found
% 0.20/0.65 % SZS status Theorem for theBenchmark
% 0.20/0.65 % SZS output start Proof
% 0.20/0.65 %ClaNum:82(EqnAxiom:27)
% 0.20/0.65 %VarNum:269(SingletonVarNum:114)
% 0.20/0.65 %MaxLitNum:4
% 0.20/0.65 %MaxfuncDepth:1
% 0.20/0.65 %SharedTerms:11
% 0.20/0.65 %goalClause: 30 42
% 0.20/0.65 %singleGoalClaCount:2
% 0.20/0.65 [28]P1(a1)
% 0.20/0.65 [29]P1(a2)
% 0.20/0.65 [30]P3(a8,a10)
% 0.20/0.65 [41]~P1(a9)
% 0.20/0.65 [42]~E(f11(a8,a10),a8)
% 0.20/0.65 [32]P3(a1,x321)
% 0.20/0.65 [34]P3(x341,x341)
% 0.20/0.65 [31]E(f11(x311,a1),a1)
% 0.20/0.65 [33]E(f13(x331,a1),x331)
% 0.20/0.65 [35]E(f13(x351,x351),x351)
% 0.20/0.65 [36]E(f11(x361,x361),x361)
% 0.20/0.65 [37]E(f13(x371,x372),f13(x372,x371))
% 0.20/0.65 [38]E(f11(x381,x382),f11(x382,x381))
% 0.20/0.65 [39]P3(x391,f13(x391,x392))
% 0.20/0.65 [40]P3(f11(x401,x402),x401)
% 0.20/0.65 [43]~P1(x431)+E(x431,a1)
% 0.20/0.65 [47]~P3(x471,a1)+E(x471,a1)
% 0.20/0.65 [48]P4(f3(x481),x481)+E(x481,a1)
% 0.20/0.65 [46]~E(x461,x462)+P3(x461,x462)
% 0.20/0.65 [49]~P4(x492,x491)+~E(x491,a1)
% 0.20/0.65 [50]~P1(x501)+~P4(x502,x501)
% 0.20/0.65 [53]~P2(x532,x531)+P2(x531,x532)
% 0.20/0.65 [55]~P4(x552,x551)+~P4(x551,x552)
% 0.20/0.65 [51]~P2(x511,x512)+E(f11(x511,x512),a1)
% 0.20/0.65 [52]P2(x521,x522)+~E(f11(x521,x522),a1)
% 0.20/0.65 [54]~P3(x541,x542)+E(f13(x541,x542),x542)
% 0.20/0.65 [57]P1(x571)+~P1(f13(x572,x571))
% 0.20/0.65 [58]P1(x581)+~P1(f13(x581,x582))
% 0.20/0.65 [59]P3(x591,x592)+P4(f5(x591,x592),x591)
% 0.20/0.65 [60]P2(x601,x602)+P4(f12(x601,x602),x602)
% 0.20/0.65 [61]P2(x611,x612)+P4(f12(x611,x612),x611)
% 0.20/0.65 [70]P3(x701,x702)+~P4(f5(x701,x702),x702)
% 0.20/0.65 [71]P2(x711,x712)+P4(f4(x711,x712),f11(x711,x712))
% 0.20/0.65 [75]~P2(x751,x752)+~P4(x753,f11(x751,x752))
% 0.20/0.65 [76]~P3(x761,x763)+P3(f11(x761,x762),f11(x763,x762))
% 0.20/0.65 [44]~P1(x442)+~P1(x441)+E(x441,x442)
% 0.20/0.65 [56]~P3(x562,x561)+~P3(x561,x562)+E(x561,x562)
% 0.20/0.65 [62]~P3(x623,x622)+P4(x621,x622)+~P4(x621,x623)
% 0.20/0.65 [63]~P3(x631,x633)+P3(x631,x632)+~P3(x633,x632)
% 0.20/0.65 [68]~P2(x683,x682)+~P4(x681,x682)+~P4(x681,x683)
% 0.20/0.65 [73]~P3(x731,x733)+~P3(x731,x732)+P3(x731,f11(x732,x733))
% 0.20/0.65 [74]~P3(x742,x743)+~P3(x741,x743)+P3(f13(x741,x742),x743)
% 0.20/0.65 [77]P4(f7(x772,x773,x771),x771)+P4(f7(x772,x773,x771),x773)+E(x771,f11(x772,x773))
% 0.20/0.65 [78]P4(f7(x782,x783,x781),x781)+P4(f7(x782,x783,x781),x782)+E(x781,f11(x782,x783))
% 0.20/0.65 [80]~P4(f6(x802,x803,x801),x801)+~P4(f6(x802,x803,x801),x803)+E(x801,f13(x802,x803))
% 0.20/0.65 [81]~P4(f6(x812,x813,x811),x811)+~P4(f6(x812,x813,x811),x812)+E(x811,f13(x812,x813))
% 0.20/0.65 [64]~P4(x641,x644)+P4(x641,x642)+~E(x642,f13(x643,x644))
% 0.20/0.65 [65]~P4(x651,x653)+P4(x651,x652)+~E(x652,f13(x653,x654))
% 0.20/0.65 [66]~P4(x661,x663)+P4(x661,x662)+~E(x663,f11(x664,x662))
% 0.20/0.65 [67]~P4(x671,x673)+P4(x671,x672)+~E(x673,f11(x672,x674))
% 0.20/0.65 [79]P4(f6(x792,x793,x791),x791)+P4(f6(x792,x793,x791),x793)+P4(f6(x792,x793,x791),x792)+E(x791,f13(x792,x793))
% 0.20/0.65 [82]~P4(f7(x822,x823,x821),x821)+~P4(f7(x822,x823,x821),x823)+~P4(f7(x822,x823,x821),x822)+E(x821,f11(x822,x823))
% 0.20/0.65 [69]~P4(x691,x694)+P4(x691,x692)+P4(x691,x693)+~E(x694,f13(x693,x692))
% 0.20/0.65 [72]~P4(x721,x724)+~P4(x721,x723)+P4(x721,x722)+~E(x722,f11(x723,x724))
% 0.20/0.65 %EqnAxiom
% 0.20/0.65 [1]E(x11,x11)
% 0.20/0.65 [2]E(x22,x21)+~E(x21,x22)
% 0.20/0.65 [3]E(x31,x33)+~E(x31,x32)+~E(x32,x33)
% 0.20/0.65 [4]~E(x41,x42)+E(f11(x41,x43),f11(x42,x43))
% 0.20/0.65 [5]~E(x51,x52)+E(f11(x53,x51),f11(x53,x52))
% 0.20/0.65 [6]~E(x61,x62)+E(f13(x61,x63),f13(x62,x63))
% 0.20/0.65 [7]~E(x71,x72)+E(f13(x73,x71),f13(x73,x72))
% 0.20/0.65 [8]~E(x81,x82)+E(f7(x81,x83,x84),f7(x82,x83,x84))
% 0.20/0.65 [9]~E(x91,x92)+E(f7(x93,x91,x94),f7(x93,x92,x94))
% 0.20/0.65 [10]~E(x101,x102)+E(f7(x103,x104,x101),f7(x103,x104,x102))
% 0.20/0.65 [11]~E(x111,x112)+E(f6(x111,x113,x114),f6(x112,x113,x114))
% 0.20/0.65 [12]~E(x121,x122)+E(f6(x123,x121,x124),f6(x123,x122,x124))
% 0.20/0.65 [13]~E(x131,x132)+E(f6(x133,x134,x131),f6(x133,x134,x132))
% 0.20/0.65 [14]~E(x141,x142)+E(f5(x141,x143),f5(x142,x143))
% 0.20/0.65 [15]~E(x151,x152)+E(f5(x153,x151),f5(x153,x152))
% 0.20/0.65 [16]~E(x161,x162)+E(f12(x161,x163),f12(x162,x163))
% 0.20/0.65 [17]~E(x171,x172)+E(f12(x173,x171),f12(x173,x172))
% 0.20/0.65 [18]~E(x181,x182)+E(f4(x181,x183),f4(x182,x183))
% 0.20/0.65 [19]~E(x191,x192)+E(f4(x193,x191),f4(x193,x192))
% 0.20/0.65 [20]~E(x201,x202)+E(f3(x201),f3(x202))
% 0.20/0.65 [21]~P1(x211)+P1(x212)+~E(x211,x212)
% 0.20/0.65 [22]P4(x222,x223)+~E(x221,x222)+~P4(x221,x223)
% 0.20/0.65 [23]P4(x233,x232)+~E(x231,x232)+~P4(x233,x231)
% 0.20/0.65 [24]P3(x242,x243)+~E(x241,x242)+~P3(x241,x243)
% 0.20/0.65 [25]P3(x253,x252)+~E(x251,x252)+~P3(x253,x251)
% 0.20/0.65 [26]P2(x262,x263)+~E(x261,x262)+~P2(x261,x263)
% 0.20/0.65 [27]P2(x273,x272)+~E(x271,x272)+~P2(x273,x271)
% 0.20/0.65
% 0.20/0.65 %-------------------------------------------
% 0.20/0.65 cnf(87,plain,
% 0.20/0.65 (E(f13(x871,x871),x871)),
% 0.20/0.65 inference(rename_variables,[],[35])).
% 0.20/0.65 cnf(97,plain,
% 0.20/0.65 (P3(x971,x971)),
% 0.20/0.65 inference(rename_variables,[],[34])).
% 0.20/0.65 cnf(100,plain,
% 0.20/0.65 (E(f13(x1001,x1001),x1001)),
% 0.20/0.65 inference(rename_variables,[],[35])).
% 0.20/0.65 cnf(102,plain,
% 0.20/0.65 (E(f13(x1021,x1021),x1021)),
% 0.20/0.65 inference(rename_variables,[],[35])).
% 0.20/0.65 cnf(110,plain,
% 0.20/0.65 (E(f13(x1101,x1101),x1101)),
% 0.20/0.65 inference(rename_variables,[],[35])).
% 0.20/0.65 cnf(115,plain,
% 0.20/0.65 ($false),
% 0.20/0.65 inference(scs_inference,[],[30,34,97,32,28,41,42,35,87,100,102,110,39,40,31,2,50,49,61,60,52,26,25,24,21,3,63,56,67,66,73]),
% 0.20/0.65 ['proof']).
% 0.20/0.65 % SZS output end Proof
% 0.20/0.65 % Total time :0.010000s
%------------------------------------------------------------------------------