0.00/0.04	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.04	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.02/0.23	% Computer   : n159.star.cs.uiowa.edu
0.02/0.23	% Model      : x86_64 x86_64
0.02/0.23	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.02/0.23	% Memory     : 32218.625MB
0.02/0.23	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.02/0.23	% CPULimit   : 300
0.02/0.23	% DateTime   : Sat Jul 14 05:06:55 CDT 2018
0.02/0.24	% CPUTime    : 
0.07/0.59	% SZS status Theorem
0.07/0.59	
0.07/0.59	% SZS output start Proof
0.07/0.59	Take the following subset of the input axioms:
0.07/0.59	  fof(a, conjecture, leq(multiplication(a, a), star(a))).
0.07/0.59	  fof(additive_associativity, axiom,
0.07/0.59	      ![A, B, C]:
0.07/0.59	        addition(addition(A, B), C)=addition(A, addition(B, C))).
0.07/0.59	  fof(additive_commutativity, axiom,
0.07/0.59	      ![A, B]: addition(A, B)=addition(B, A)).
0.07/0.59	  fof(additive_idempotence, axiom, ![A]: A=addition(A, A)).
0.07/0.59	  fof(left_distributivity, axiom,
0.07/0.59	      ![A, B, C]:
0.07/0.59	        addition(multiplication(A, C),
0.07/0.59	                 multiplication(B, C))=multiplication(addition(A, B), C)).
0.07/0.59	  fof(multiplicative_left_identity, axiom,
0.07/0.59	      ![A]: A=multiplication(one, A)).
0.07/0.59	  fof(multiplicative_right_identity, axiom,
0.07/0.59	      ![A]: multiplication(A, one)=A).
0.41/0.59	  fof(order, axiom, ![A, B]: (addition(A, B)=B <=> leq(A, B))).
0.41/0.59	  fof(right_distributivity, axiom,
0.41/0.59	      ![A, B, C]:
0.41/0.59	        addition(multiplication(A, B),
0.41/0.59	                 multiplication(A, C))=multiplication(A, addition(B, C))).
0.41/0.59	  fof(star_unfold_left, axiom,
0.41/0.59	      ![A]: leq(addition(one, multiplication(star(A), A)), star(A))).
0.41/0.59	
0.41/0.59	Now clausify the problem and encode Horn clauses using encoding 3 of
0.41/0.59	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.41/0.59	We repeatedly replace C & s=t => u=v by the two clauses:
0.41/0.59	  $$fresh(y, y, x1...xn) = u
0.41/0.59	  C => $$fresh(s, t, x1...xn) = v
0.41/0.59	where $$fresh is a fresh function symbol and x1..xn are the free
0.41/0.59	variables of u and v.
0.41/0.59	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
0.41/0.59	input problem has no model of domain size 1).
0.41/0.59	
0.41/0.59	The encoding turns the above axioms into the following unit equations and goals:
0.41/0.59	
0.41/0.59	Axiom 1 (order): $$fresh4(X, X, Y, Z) = $$true.
0.41/0.59	Axiom 2 (order_1): $$fresh(X, X, Y, Z) = Z.
0.41/0.59	Axiom 5 (order_1): $$fresh(leq(X, Y), $$true, X, Y) = addition(X, Y).
0.41/0.59	Axiom 6 (order): $$fresh4(addition(X, Y), Y, X, Y) = leq(X, Y).
0.41/0.59	Axiom 8 (additive_idempotence): X = addition(X, X).
0.41/0.59	Axiom 10 (multiplicative_left_identity): X = multiplication(one, X).
0.41/0.59	Axiom 11 (right_distributivity): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)).
0.41/0.59	Axiom 13 (left_distributivity): addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(X, Z), Y).
0.41/0.59	Axiom 14 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = $$true.
0.41/0.59	Axiom 16 (multiplicative_right_identity): multiplication(X, one) = X.
0.41/0.59	Axiom 20 (additive_associativity): addition(addition(X, Y), Z) = addition(X, addition(Y, Z)).
0.41/0.59	Axiom 21 (additive_commutativity): addition(X, Y) = addition(Y, X).
0.41/0.59	
0.41/0.59	Lemma 22: addition(X, addition(X, Y)) = addition(X, Y).
0.41/0.59	Proof:
0.41/0.59	  addition(X, addition(X, Y))
0.41/0.59	= { by axiom 20 (additive_associativity) }
0.41/0.59	  addition(addition(X, X), Y)
0.41/0.59	= { by axiom 8 (additive_idempotence) }
0.41/0.59	  addition(X, Y)
0.41/0.59	
0.41/0.59	Lemma 23: addition(Y, multiplication(X, Y)) = multiplication(addition(X, one), Y).
0.41/0.59	Proof:
0.41/0.59	  addition(Y, multiplication(X, Y))
0.41/0.59	= { by axiom 10 (multiplicative_left_identity) }
0.41/0.59	  addition(multiplication(one, Y), multiplication(X, Y))
0.41/0.59	= { by axiom 13 (left_distributivity) }
0.41/0.59	  multiplication(addition(one, X), Y)
0.41/0.59	= { by axiom 21 (additive_commutativity) }
0.41/0.59	  multiplication(addition(X, one), Y)
0.41/0.59	
0.41/0.59	Lemma 24: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
0.41/0.59	Proof:
0.41/0.59	  addition(X, multiplication(X, Y))
0.41/0.59	= { by axiom 16 (multiplicative_right_identity) }
0.41/0.59	  addition(multiplication(X, one), multiplication(X, Y))
0.41/0.59	= { by axiom 11 (right_distributivity) }
0.41/0.59	  multiplication(X, addition(one, Y))
0.41/0.59	= { by axiom 21 (additive_commutativity) }
0.41/0.59	  multiplication(X, addition(Y, one))
0.41/0.59	
0.41/0.59	Lemma 25: addition(one, multiplication(star(X), addition(X, one))) = star(X).
0.41/0.59	Proof:
0.41/0.59	  addition(one, multiplication(star(X), addition(X, one)))
0.41/0.59	= { by lemma 24 }
0.41/0.59	  addition(one, addition(star(X), multiplication(star(X), X)))
0.41/0.59	= { by axiom 21 (additive_commutativity) }
0.41/0.59	  addition(one, addition(multiplication(star(X), X), star(X)))
0.41/0.59	= { by axiom 20 (additive_associativity) }
0.41/0.59	  addition(addition(one, multiplication(star(X), X)), star(X))
0.41/0.59	= { by axiom 5 (order_1) }
0.41/0.59	  $$fresh(leq(addition(one, multiplication(star(X), X)), star(X)), $$true, addition(one, multiplication(star(X), X)), star(X))
0.41/0.59	= { by axiom 14 (star_unfold_left) }
0.41/0.59	  $$fresh($$true, $$true, addition(one, multiplication(star(X), X)), star(X))
0.41/0.59	= { by axiom 2 (order_1) }
0.41/0.59	  star(X)
0.41/0.59	
0.41/0.59	Lemma 26: addition(one, star(X)) = star(X).
0.41/0.59	Proof:
0.41/0.59	  addition(one, star(X))
0.41/0.59	= { by lemma 25 }
0.41/0.59	  addition(one, addition(one, multiplication(star(X), addition(X, one))))
0.41/0.59	= { by lemma 22 }
0.41/0.59	  addition(one, multiplication(star(X), addition(X, one)))
0.41/0.59	= { by lemma 25 }
0.41/0.59	  star(X)
0.41/0.59	
0.41/0.59	Lemma 27: addition(X, star(X)) = star(X).
0.41/0.59	Proof:
0.41/0.59	  addition(X, star(X))
0.41/0.59	= { by lemma 26 }
0.41/0.59	  addition(X, addition(one, star(X)))
0.41/0.59	= { by axiom 20 (additive_associativity) }
0.41/0.59	  addition(addition(X, one), star(X))
0.41/0.59	= { by lemma 25 }
0.41/0.59	  addition(addition(X, one), addition(one, multiplication(star(X), addition(X, one))))
0.41/0.59	= { by axiom 21 (additive_commutativity) }
0.41/0.59	  addition(addition(X, one), addition(multiplication(star(X), addition(X, one)), one))
0.41/0.59	= { by axiom 20 (additive_associativity) }
0.41/0.60	  addition(addition(addition(X, one), multiplication(star(X), addition(X, one))), one)
0.41/0.60	= { by axiom 21 (additive_commutativity) }
0.41/0.60	  addition(one, addition(addition(X, one), multiplication(star(X), addition(X, one))))
0.41/0.60	= { by lemma 23 }
0.41/0.60	  addition(one, multiplication(addition(star(X), one), addition(X, one)))
0.41/0.60	= { by axiom 21 (additive_commutativity) }
0.41/0.60	  addition(one, multiplication(addition(one, star(X)), addition(X, one)))
0.41/0.60	= { by lemma 26 }
0.41/0.60	  addition(one, multiplication(star(X), addition(X, one)))
0.41/0.60	= { by lemma 25 }
0.41/0.60	  star(X)
0.41/0.60	
0.41/0.60	Lemma 28: multiplication(star(X), addition(X, one)) = star(X).
0.41/0.60	Proof:
0.41/0.60	  multiplication(star(X), addition(X, one))
0.41/0.60	= { by lemma 26 }
0.41/0.60	  multiplication(addition(one, star(X)), addition(X, one))
0.41/0.60	= { by axiom 21 (additive_commutativity) }
0.41/0.60	  multiplication(addition(star(X), one), addition(X, one))
0.41/0.60	= { by lemma 23 }
0.41/0.60	  addition(addition(X, one), multiplication(star(X), addition(X, one)))
0.41/0.60	= { by axiom 20 (additive_associativity) }
0.41/0.60	  addition(X, addition(one, multiplication(star(X), addition(X, one))))
0.41/0.60	= { by lemma 25 }
0.41/0.60	  addition(X, star(X))
0.41/0.60	= { by lemma 27 }
0.41/0.60	  star(X)
0.41/0.60	
0.41/0.60	Goal 1 (a): leq(multiplication(a, a), star(a)) = $$true.
0.41/0.60	Proof:
0.41/0.60	  leq(multiplication(a, a), star(a))
0.41/0.60	= { by lemma 28 }
0.41/0.60	  leq(multiplication(a, a), multiplication(star(a), addition(a, one)))
0.41/0.60	= { by lemma 24 }
0.41/0.60	  leq(multiplication(a, a), addition(star(a), multiplication(star(a), a)))
0.41/0.60	= { by lemma 27 }
0.41/0.60	  leq(multiplication(a, a), addition(star(a), multiplication(addition(a, star(a)), a)))
0.41/0.60	= { by axiom 21 (additive_commutativity) }
0.41/0.60	  leq(multiplication(a, a), addition(multiplication(addition(a, star(a)), a), star(a)))
0.41/0.60	= { by axiom 13 (left_distributivity) }
0.41/0.60	  leq(multiplication(a, a), addition(addition(multiplication(a, a), multiplication(star(a), a)), star(a)))
0.41/0.60	= { by axiom 20 (additive_associativity) }
0.41/0.60	  leq(multiplication(a, a), addition(multiplication(a, a), addition(multiplication(star(a), a), star(a))))
0.41/0.60	= { by axiom 21 (additive_commutativity) }
0.41/0.60	  leq(multiplication(a, a), addition(multiplication(a, a), addition(star(a), multiplication(star(a), a))))
0.41/0.60	= { by lemma 24 }
0.41/0.60	  leq(multiplication(a, a), addition(multiplication(a, a), multiplication(star(a), addition(a, one))))
0.41/0.60	= { by lemma 28 }
0.41/0.60	  leq(multiplication(a, a), addition(multiplication(a, a), star(a)))
0.41/0.60	= { by axiom 6 (order) }
0.41/0.60	  $$fresh4(addition(multiplication(a, a), addition(multiplication(a, a), star(a))), addition(multiplication(a, a), star(a)), multiplication(a, a), addition(multiplication(a, a), star(a)))
0.41/0.60	= { by lemma 22 }
0.41/0.60	  $$fresh4(addition(multiplication(a, a), star(a)), addition(multiplication(a, a), star(a)), multiplication(a, a), addition(multiplication(a, a), star(a)))
0.41/0.60	= { by axiom 1 (order) }
0.41/0.60	  $$true
0.41/0.60	% SZS output end Proof
0.41/0.60	
0.41/0.60	RESULT: Theorem (the conjecture is true).
0.41/0.60	EOF