0.00/0.10	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.10/0.10	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.10/0.31	% Computer   : n021.cluster.edu
0.10/0.31	% Model      : x86_64 x86_64
0.10/0.31	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.10/0.31	% Memory     : 8042.1875MB
0.10/0.31	% OS         : Linux 3.10.0-693.el7.x86_64
0.10/0.31	% CPULimit   : 1200
0.10/0.31	% WCLimit    : 120
0.10/0.31	% DateTime   : Tue Jul 13 16:18:11 EDT 2021
0.10/0.31	% CPUTime    : 
0.16/0.47	% SZS status Theorem
0.16/0.47	
0.16/0.48	% SZS output start Proof
0.16/0.48	Take the following subset of the input axioms:
0.16/0.49	  fof(a, conjecture, leq(multiplication(a, a), star(a))).
0.16/0.49	  fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
0.16/0.49	  fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
0.16/0.49	  fof(additive_idempotence, axiom, ![A]: addition(A, A)=A).
0.16/0.49	  fof(left_distributivity, axiom, ![A, B, C]: addition(multiplication(A, C), multiplication(B, C))=multiplication(addition(A, B), C)).
0.16/0.49	  fof(multiplicative_left_identity, axiom, ![A]: multiplication(one, A)=A).
0.16/0.49	  fof(multiplicative_right_identity, axiom, ![A]: A=multiplication(A, one)).
0.16/0.49	  fof(order, axiom, ![A, B]: (B=addition(A, B) <=> leq(A, B))).
0.16/0.49	  fof(right_distributivity, axiom, ![A, B, C]: addition(multiplication(A, B), multiplication(A, C))=multiplication(A, addition(B, C))).
0.16/0.49	  fof(star_unfold_left, axiom, ![A]: leq(addition(one, multiplication(star(A), A)), star(A))).
0.16/0.49	  fof(star_unfold_right, axiom, ![A]: leq(addition(one, multiplication(A, star(A))), star(A))).
0.16/0.49	
0.16/0.49	Now clausify the problem and encode Horn clauses using encoding 3 of
0.16/0.49	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.16/0.49	We repeatedly replace C & s=t => u=v by the two clauses:
0.16/0.49	  fresh(y, y, x1...xn) = u
0.16/0.49	  C => fresh(s, t, x1...xn) = v
0.16/0.49	where fresh is a fresh function symbol and x1..xn are the free
0.16/0.49	variables of u and v.
0.16/0.49	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.16/0.49	input problem has no model of domain size 1).
0.16/0.49	
0.16/0.49	The encoding turns the above axioms into the following unit equations and goals:
0.16/0.49	
0.16/0.49	Axiom 1 (additive_idempotence): addition(X, X) = X.
0.16/0.49	Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
0.16/0.49	Axiom 3 (multiplicative_right_identity): X = multiplication(X, one).
0.16/0.49	Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
0.16/0.49	Axiom 5 (order_1): fresh(X, X, Y, Z) = Z.
0.16/0.49	Axiom 6 (order): fresh3(X, X, Y, Z) = true.
0.16/0.49	Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
0.16/0.49	Axiom 8 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
0.16/0.49	Axiom 9 (order): fresh3(X, addition(Y, X), Y, X) = leq(Y, X).
0.16/0.49	Axiom 10 (right_distributivity): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)).
0.16/0.49	Axiom 11 (left_distributivity): addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(X, Z), Y).
0.16/0.49	Axiom 12 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
0.16/0.49	Axiom 13 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = true.
0.16/0.49	
0.16/0.49	Lemma 14: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
0.16/0.49	Proof:
0.16/0.49	  multiplication(addition(one, Y), X)
0.16/0.49	= { by axiom 11 (left_distributivity) R->L }
0.16/0.49	  addition(multiplication(one, X), multiplication(Y, X))
0.16/0.49	= { by axiom 4 (multiplicative_left_identity) }
0.16/0.49	  addition(X, multiplication(Y, X))
0.16/0.49	
0.16/0.49	Lemma 15: addition(one, multiplication(addition(X, one), star(X))) = star(X).
0.16/0.49	Proof:
0.16/0.49	  addition(one, multiplication(addition(X, one), star(X)))
0.16/0.49	= { by axiom 2 (additive_commutativity) R->L }
0.16/0.49	  addition(one, multiplication(addition(one, X), star(X)))
0.16/0.49	= { by lemma 14 }
0.16/0.49	  addition(one, addition(star(X), multiplication(X, star(X))))
0.16/0.49	= { by axiom 2 (additive_commutativity) R->L }
0.16/0.49	  addition(one, addition(multiplication(X, star(X)), star(X)))
0.16/0.49	= { by axiom 7 (additive_associativity) }
0.16/0.49	  addition(addition(one, multiplication(X, star(X))), star(X))
0.16/0.49	= { by axiom 8 (order_1) R->L }
0.16/0.49	  fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
0.16/0.49	= { by axiom 12 (star_unfold_right) }
0.16/0.49	  fresh(true, true, addition(one, multiplication(X, star(X))), star(X))
0.16/0.49	= { by axiom 5 (order_1) }
0.16/0.49	  star(X)
0.16/0.49	
0.16/0.49	Lemma 16: addition(one, star(X)) = star(X).
0.16/0.49	Proof:
0.16/0.49	  addition(one, star(X))
0.16/0.49	= { by lemma 15 R->L }
0.16/0.49	  addition(one, addition(one, multiplication(addition(X, one), star(X))))
0.16/0.49	= { by axiom 7 (additive_associativity) }
0.16/0.49	  addition(addition(one, one), multiplication(addition(X, one), star(X)))
0.16/0.49	= { by axiom 1 (additive_idempotence) }
0.16/0.49	  addition(one, multiplication(addition(X, one), star(X)))
0.16/0.49	= { by lemma 15 }
0.16/0.49	  star(X)
0.16/0.49	
0.16/0.49	Lemma 17: multiplication(X, addition(Y, one)) = addition(X, multiplication(X, Y)).
0.16/0.49	Proof:
0.16/0.49	  multiplication(X, addition(Y, one))
0.16/0.49	= { by axiom 2 (additive_commutativity) R->L }
0.16/0.49	  multiplication(X, addition(one, Y))
0.16/0.49	= { by axiom 10 (right_distributivity) R->L }
0.16/0.49	  addition(multiplication(X, one), multiplication(X, Y))
0.16/0.49	= { by axiom 3 (multiplicative_right_identity) R->L }
0.16/0.49	  addition(X, multiplication(X, Y))
0.16/0.49	
0.16/0.49	Lemma 18: multiplication(star(X), addition(X, one)) = star(X).
0.16/0.49	Proof:
0.16/0.49	  multiplication(star(X), addition(X, one))
0.16/0.49	= { by lemma 17 }
0.16/0.49	  addition(star(X), multiplication(star(X), X))
0.16/0.49	= { by lemma 16 R->L }
0.16/0.49	  addition(addition(one, star(X)), multiplication(star(X), X))
0.16/0.49	= { by axiom 7 (additive_associativity) R->L }
0.16/0.49	  addition(one, addition(star(X), multiplication(star(X), X)))
0.16/0.49	= { by axiom 2 (additive_commutativity) }
0.16/0.49	  addition(one, addition(multiplication(star(X), X), star(X)))
0.16/0.49	= { by axiom 7 (additive_associativity) }
0.16/0.49	  addition(addition(one, multiplication(star(X), X)), star(X))
0.16/0.49	= { by axiom 8 (order_1) R->L }
0.16/0.49	  fresh(leq(addition(one, multiplication(star(X), X)), star(X)), true, addition(one, multiplication(star(X), X)), star(X))
0.16/0.49	= { by axiom 13 (star_unfold_left) }
0.16/0.49	  fresh(true, true, addition(one, multiplication(star(X), X)), star(X))
0.16/0.49	= { by axiom 5 (order_1) }
0.16/0.49	  star(X)
0.16/0.49	
0.16/0.49	Lemma 19: addition(star(X), multiplication(star(X), X)) = star(X).
0.16/0.49	Proof:
0.16/0.49	  addition(star(X), multiplication(star(X), X))
0.16/0.49	= { by lemma 17 R->L }
0.16/0.49	  multiplication(star(X), addition(X, one))
0.16/0.49	= { by lemma 18 }
0.16/0.49	  star(X)
0.16/0.49	
0.16/0.49	Goal 1 (a): leq(multiplication(a, a), star(a)) = true.
0.16/0.49	Proof:
0.16/0.49	  leq(multiplication(a, a), star(a))
0.16/0.49	= { by axiom 9 (order) R->L }
0.16/0.49	  fresh3(star(a), addition(multiplication(a, a), star(a)), multiplication(a, a), star(a))
0.16/0.49	= { by axiom 2 (additive_commutativity) R->L }
0.16/0.49	  fresh3(star(a), addition(star(a), multiplication(a, a)), multiplication(a, a), star(a))
0.16/0.49	= { by lemma 19 R->L }
0.16/0.49	  fresh3(star(a), addition(addition(star(a), multiplication(star(a), a)), multiplication(a, a)), multiplication(a, a), star(a))
0.16/0.49	= { by axiom 7 (additive_associativity) R->L }
0.16/0.49	  fresh3(star(a), addition(star(a), addition(multiplication(star(a), a), multiplication(a, a))), multiplication(a, a), star(a))
0.16/0.49	= { by axiom 2 (additive_commutativity) }
0.16/0.49	  fresh3(star(a), addition(star(a), addition(multiplication(a, a), multiplication(star(a), a))), multiplication(a, a), star(a))
0.16/0.49	= { by axiom 11 (left_distributivity) }
0.16/0.49	  fresh3(star(a), addition(star(a), multiplication(addition(a, star(a)), a)), multiplication(a, a), star(a))
0.16/0.49	= { by lemma 16 R->L }
0.16/0.49	  fresh3(star(a), addition(star(a), multiplication(addition(a, addition(one, star(a))), a)), multiplication(a, a), star(a))
0.16/0.49	= { by axiom 7 (additive_associativity) }
0.16/0.49	  fresh3(star(a), addition(star(a), multiplication(addition(addition(a, one), star(a)), a)), multiplication(a, a), star(a))
0.16/0.49	= { by lemma 18 R->L }
0.16/0.49	  fresh3(star(a), addition(star(a), multiplication(addition(addition(a, one), multiplication(star(a), addition(a, one))), a)), multiplication(a, a), star(a))
0.16/0.49	= { by lemma 14 R->L }
0.16/0.49	  fresh3(star(a), addition(star(a), multiplication(multiplication(addition(one, star(a)), addition(a, one)), a)), multiplication(a, a), star(a))
0.16/0.49	= { by lemma 16 }
0.16/0.49	  fresh3(star(a), addition(star(a), multiplication(multiplication(star(a), addition(a, one)), a)), multiplication(a, a), star(a))
0.16/0.49	= { by lemma 18 }
0.16/0.49	  fresh3(star(a), addition(star(a), multiplication(star(a), a)), multiplication(a, a), star(a))
0.16/0.49	= { by lemma 19 }
0.16/0.49	  fresh3(star(a), star(a), multiplication(a, a), star(a))
0.16/0.49	= { by axiom 6 (order) }
0.16/0.49	  true
0.16/0.49	% SZS output end Proof
0.16/0.49	
0.16/0.49	RESULT: Theorem (the conjecture is true).
0.16/0.50	EOF