0.08/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.08/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
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   : 180
0.12/0.33	% DateTime   : Thu Aug 29 14:32:39 EDT 2019
0.12/0.33	% CPUTime    : 
3.65/3.86	% SZS status Theorem
3.65/3.86	
3.65/3.86	% SZS output start Proof
3.65/3.86	Take the following subset of the input axioms:
3.65/3.86	  fof(goals_13, conjecture, ![X17, X18]: '+'(X18, '==>'(X18, X17))='+'(X17, '==>'(X17, X18))).
3.65/3.86	  fof(sos_01, axiom, ![A, B, C]: '+'(A, '+'(B, C))='+'('+'(A, B), C)).
3.65/3.86	  fof(sos_02, axiom, ![A, B]: '+'(A, B)='+'(B, A)).
3.65/3.86	  fof(sos_03, axiom, ![A]: '+'(A, '0')=A).
3.65/3.86	  fof(sos_04, axiom, ![A]: '>='(A, A)).
3.65/3.86	  fof(sos_06, axiom, ![X3, X4]: (X4=X3 <= ('>='(X3, X4) & '>='(X4, X3)))).
3.65/3.86	  fof(sos_07, axiom, ![X5, X6, X7]: ('>='(X6, '==>'(X5, X7)) <=> '>='('+'(X5, X6), X7))).
3.65/3.86	  fof(sos_08, axiom, ![A]: '>='(A, '0')).
3.65/3.86	  fof(sos_09, axiom, ![X8, X9, X10]: ('>='('+'(X8, X10), '+'(X9, X10)) <= '>='(X8, X9))).
3.65/3.86	  fof(sos_11, axiom, ![X14, X15, X16]: ('>='(X14, X15) => '>='('==>'(X16, X14), '==>'(X16, X15)))).
3.65/3.86	  fof(sos_12, axiom, ![A, B, C]: '+'('+'(A, '==>'(A, B)), '==>'('+'(A, '==>'(A, B)), C))='+'(A, '==>'(A, '+'(B, '==>'(B, C))))).
3.65/3.86	
3.65/3.86	Now clausify the problem and encode Horn clauses using encoding 3 of
3.65/3.86	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
3.65/3.86	We repeatedly replace C & s=t => u=v by the two clauses:
3.65/3.86	  fresh(y, y, x1...xn) = u
3.65/3.86	  C => fresh(s, t, x1...xn) = v
3.65/3.86	where fresh is a fresh function symbol and x1..xn are the free
3.65/3.86	variables of u and v.
3.65/3.86	A predicate p(X) is encoded as p(X)=true (this is sound, because the
3.65/3.86	input problem has no model of domain size 1).
3.65/3.86	
3.65/3.86	The encoding turns the above axioms into the following unit equations and goals:
3.65/3.86	
3.65/3.86	Axiom 1 (sos_06): fresh(X, X, Y, Z) = Y.
3.65/3.86	Axiom 2 (sos_06): fresh2(X, X, Y, Z) = Z.
3.65/3.86	Axiom 3 (sos_07): fresh6(X, X, Y, Z, W) = true.
3.65/3.86	Axiom 4 (sos_07_1): fresh9(X, X, Y, Z, W) = true.
3.65/3.86	Axiom 5 (sos_09): fresh5(X, X, Y, Z, W) = true.
3.65/3.86	Axiom 6 (sos_11): fresh3(X, X, Y, Z, W) = true.
3.65/3.86	Axiom 7 (sos_02): X + Y = Y + X.
3.65/3.86	Axiom 8 (sos_07_1): fresh9((X + Y) >= Z, true, X, Y, Z) = Y >= (X ==> Z).
3.65/3.86	Axiom 9 (sos_07): fresh6(X >= (Y ==> Z), true, Y, X, Z) = (Y + X) >= Z.
3.65/3.86	Axiom 10 (sos_09): fresh5(X >= Y, true, X, Y, Z) = (X + Z) >= (Y + Z).
3.65/3.87	Axiom 11 (sos_06): fresh2(X >= Y, true, Y, X) = fresh(Y >= X, true, Y, X).
3.65/3.87	Axiom 12 (sos_04): X >= X = true.
3.65/3.87	Axiom 13 (sos_08): X >= 0 = true.
3.65/3.87	Axiom 14 (sos_01): X + (Y + Z) = (X + Y) + Z.
3.65/3.87	Axiom 15 (sos_11): fresh3(X >= Y, true, X, Y, Z) = (Z ==> X) >= (Z ==> Y).
3.65/3.87	Axiom 16 (sos_12): (X + (X ==> Y)) + ((X + (X ==> Y)) ==> Z) = X + (X ==> (Y + (Y ==> Z))).
3.65/3.88	Axiom 17 (sos_03): X + 0 = X.
3.65/3.88	
3.65/3.88	Lemma 18: (X + (X ==> Y)) >= Y = true.
3.65/3.88	Proof:
3.65/3.88	  (X + (X ==> Y)) >= Y
3.65/3.88	= { by axiom 9 (sos_07) }
3.65/3.88	  fresh6((X ==> Y) >= (X ==> Y), true, X, X ==> Y, Y)
3.65/3.88	= { by axiom 12 (sos_04) }
3.65/3.88	  fresh6(true, true, X, X ==> Y, Y)
3.65/3.88	= { by axiom 3 (sos_07) }
3.65/3.88	  true
3.65/3.88	
3.65/3.88	Lemma 19: X ==> (X + (X ==> Y)) = X ==> Y.
3.65/3.88	Proof:
3.65/3.88	  X ==> (X + (X ==> Y))
3.65/3.88	= { by axiom 1 (sos_06) }
3.65/3.88	  fresh(true, true, X ==> (X + (X ==> Y)), X ==> Y)
3.65/3.88	= { by axiom 6 (sos_11) }
3.65/3.88	  fresh(fresh3(true, true, X + (X ==> Y), Y, X), true, X ==> (X + (X ==> Y)), X ==> Y)
3.65/3.88	= { by lemma 18 }
3.65/3.88	  fresh(fresh3((X + (X ==> Y)) >= Y, true, X + (X ==> Y), Y, X), true, X ==> (X + (X ==> Y)), X ==> Y)
3.65/3.88	= { by axiom 15 (sos_11) }
3.65/3.88	  fresh((X ==> (X + (X ==> Y))) >= (X ==> Y), true, X ==> (X + (X ==> Y)), X ==> Y)
3.65/3.88	= { by axiom 7 (sos_02) }
3.65/3.88	  fresh((X ==> ((X ==> Y) + X)) >= (X ==> Y), true, X ==> (X + (X ==> Y)), X ==> Y)
3.65/3.88	= { by axiom 7 (sos_02) }
3.65/3.88	  fresh((X ==> ((X ==> Y) + X)) >= (X ==> Y), true, X ==> ((X ==> Y) + X), X ==> Y)
3.65/3.88	= { by axiom 11 (sos_06) }
3.65/3.88	  fresh2((X ==> Y) >= (X ==> ((X ==> Y) + X)), true, X ==> ((X ==> Y) + X), X ==> Y)
3.65/3.88	= { by axiom 7 (sos_02) }
3.65/3.88	  fresh2((X ==> Y) >= (X ==> (X + (X ==> Y))), true, X ==> ((X ==> Y) + X), X ==> Y)
3.65/3.88	= { by axiom 8 (sos_07_1) }
3.65/3.88	  fresh2(fresh9((X + (X ==> Y)) >= (X + (X ==> Y)), true, X, X ==> Y, X + (X ==> Y)), true, X ==> ((X ==> Y) + X), X ==> Y)
3.65/3.88	= { by axiom 12 (sos_04) }
3.65/3.88	  fresh2(fresh9(true, true, X, X ==> Y, X + (X ==> Y)), true, X ==> ((X ==> Y) + X), X ==> Y)
3.65/3.88	= { by axiom 4 (sos_07_1) }
3.65/3.88	  fresh2(true, true, X ==> ((X ==> Y) + X), X ==> Y)
3.65/3.88	= { by axiom 2 (sos_06) }
3.65/3.90	  X ==> Y
3.65/3.90	
3.65/3.90	Lemma 20: (X + (X ==> Y)) >= (Y + (Y ==> X)) = true.
3.65/3.90	Proof:
3.65/3.90	  (X + (X ==> Y)) >= (Y + (Y ==> X))
3.65/3.90	= { by lemma 19 }
3.65/3.90	  (X + (X ==> (X + (X ==> Y)))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 17 (sos_03) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + 0)))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 2 (sos_06) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + fresh2(true, true, (X + (X ==> Y)) ==> X, 0))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 4 (sos_07_1) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + fresh2(fresh9(true, true, X + (X ==> Y), 0, X), true, (X + (X ==> Y)) ==> X, 0))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 5 (sos_09) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + fresh2(fresh9(fresh5(true, true, X ==> Y, 0, X), true, X + (X ==> Y), 0, X), true, (X + (X ==> Y)) ==> X, 0))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 13 (sos_08) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + fresh2(fresh9(fresh5((X ==> Y) >= 0, true, X ==> Y, 0, X), true, X + (X ==> Y), 0, X), true, (X + (X ==> Y)) ==> X, 0))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 10 (sos_09) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + fresh2(fresh9(((X ==> Y) + X) >= (0 + X), true, X + (X ==> Y), 0, X), true, (X + (X ==> Y)) ==> X, 0))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 7 (sos_02) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + fresh2(fresh9((X + (X ==> Y)) >= (0 + X), true, X + (X ==> Y), 0, X), true, (X + (X ==> Y)) ==> X, 0))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 17 (sos_03) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + fresh2(fresh9(((X + (X ==> Y)) + 0) >= (0 + X), true, X + (X ==> Y), 0, X), true, (X + (X ==> Y)) ==> X, 0))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 7 (sos_02) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + fresh2(fresh9(((X + (X ==> Y)) + 0) >= (X + 0), true, X + (X ==> Y), 0, X), true, (X + (X ==> Y)) ==> X, 0))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 17 (sos_03) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + fresh2(fresh9(((X + (X ==> Y)) + 0) >= X, true, X + (X ==> Y), 0, X), true, (X + (X ==> Y)) ==> X, 0))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 8 (sos_07_1) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + fresh2(0 >= ((X + (X ==> Y)) ==> X), true, (X + (X ==> Y)) ==> X, 0))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 11 (sos_06) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + fresh(((X + (X ==> Y)) ==> X) >= 0, true, (X + (X ==> Y)) ==> X, 0))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 13 (sos_08) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + fresh(true, true, (X + (X ==> Y)) ==> X, 0))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 1 (sos_06) }
3.65/3.90	  (X + (X ==> (X + ((X ==> Y) + ((X + (X ==> Y)) ==> X))))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 14 (sos_01) }
3.65/3.90	  (X + (X ==> ((X + (X ==> Y)) + ((X + (X ==> Y)) ==> X)))) >= (Y + (Y ==> X))
3.65/3.90	= { by axiom 16 (sos_12) }
3.65/3.90	  (X + (X ==> (X + (X ==> (Y + (Y ==> X)))))) >= (Y + (Y ==> X))
3.65/3.90	= { by lemma 19 }
3.65/3.90	  (X + (X ==> (Y + (Y ==> X)))) >= (Y + (Y ==> X))
3.65/3.90	= { by lemma 18 }
3.65/3.90	  true
3.65/3.90	
3.65/3.90	Goal 1 (goals_13): sK1_goals_13_X18 + (sK1_goals_13_X18 ==> sK2_goals_13_X17) = sK2_goals_13_X17 + (sK2_goals_13_X17 ==> sK1_goals_13_X18).
3.65/3.90	Proof:
3.65/3.90	  sK1_goals_13_X18 + (sK1_goals_13_X18 ==> sK2_goals_13_X17)
3.65/3.90	= { by axiom 1 (sos_06) }
3.65/3.90	  fresh(true, true, sK1_goals_13_X18 + (sK1_goals_13_X18 ==> sK2_goals_13_X17), sK2_goals_13_X17 + (sK2_goals_13_X17 ==> sK1_goals_13_X18))
3.65/3.90	= { by lemma 20 }
3.65/3.90	  fresh((sK1_goals_13_X18 + (sK1_goals_13_X18 ==> sK2_goals_13_X17)) >= (sK2_goals_13_X17 + (sK2_goals_13_X17 ==> sK1_goals_13_X18)), true, sK1_goals_13_X18 + (sK1_goals_13_X18 ==> sK2_goals_13_X17), sK2_goals_13_X17 + (sK2_goals_13_X17 ==> sK1_goals_13_X18))
3.65/3.90	= { by axiom 11 (sos_06) }
3.65/3.90	  fresh2((sK2_goals_13_X17 + (sK2_goals_13_X17 ==> sK1_goals_13_X18)) >= (sK1_goals_13_X18 + (sK1_goals_13_X18 ==> sK2_goals_13_X17)), true, sK1_goals_13_X18 + (sK1_goals_13_X18 ==> sK2_goals_13_X17), sK2_goals_13_X17 + (sK2_goals_13_X17 ==> sK1_goals_13_X18))
3.65/3.90	= { by lemma 20 }
3.65/3.90	  fresh2(true, true, sK1_goals_13_X18 + (sK1_goals_13_X18 ==> sK2_goals_13_X17), sK2_goals_13_X17 + (sK2_goals_13_X17 ==> sK1_goals_13_X18))
3.65/3.90	= { by axiom 2 (sos_06) }
3.65/3.90	  sK2_goals_13_X17 + (sK2_goals_13_X17 ==> sK1_goals_13_X18)
3.65/3.90	% SZS output end Proof
3.65/3.90	
3.65/3.90	RESULT: Theorem (the conjecture is true).
3.65/3.91	EOF