0.07/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.12	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.12/0.33	% Computer   : n007.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   : 1200
0.12/0.33	% WCLimit    : 120
0.12/0.33	% DateTime   : Tue Jul 13 16:11:19 EDT 2021
0.12/0.33	% CPUTime    : 
0.19/0.43	% SZS status Theorem
0.19/0.43	
0.19/0.44	% SZS output start Proof
0.19/0.44	Take the following subset of the input axioms:
0.19/0.44	  fof(additive_commutativity, axiom, ![A, B]: addition(B, A)=addition(A, B)).
0.19/0.44	  fof(domain1, axiom, ![X0]: multiplication(domain(X0), X0)=addition(X0, multiplication(domain(X0), X0))).
0.19/0.44	  fof(domain3, axiom, ![X0]: addition(domain(X0), one)=one).
0.19/0.44	  fof(goals, conjecture, ![X0, X1]: (addition(domain(X0), domain(X1))=domain(X1) => addition(X0, multiplication(domain(X1), X0))=multiplication(domain(X1), X0))).
0.19/0.44	  fof(left_distributivity, axiom, ![A, C, B]: addition(multiplication(A, C), multiplication(B, C))=multiplication(addition(A, B), C)).
0.19/0.44	  fof(multiplicative_left_identity, axiom, ![A]: multiplication(one, A)=A).
0.19/0.44	
0.19/0.44	Now clausify the problem and encode Horn clauses using encoding 3 of
0.19/0.44	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.19/0.44	We repeatedly replace C & s=t => u=v by the two clauses:
0.19/0.44	  fresh(y, y, x1...xn) = u
0.19/0.44	  C => fresh(s, t, x1...xn) = v
0.19/0.44	where fresh is a fresh function symbol and x1..xn are the free
0.19/0.44	variables of u and v.
0.19/0.44	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.19/0.44	input problem has no model of domain size 1).
0.19/0.44	
0.19/0.44	The encoding turns the above axioms into the following unit equations and goals:
0.19/0.44	
0.19/0.44	Axiom 1 (multiplicative_left_identity): multiplication(one, X) = X.
0.19/0.44	Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
0.19/0.44	Axiom 3 (domain3): addition(domain(X), one) = one.
0.19/0.44	Axiom 4 (goals): addition(domain(x0), domain(x1)) = domain(x1).
0.19/0.44	Axiom 5 (domain1): multiplication(domain(X), X) = addition(X, multiplication(domain(X), X)).
0.19/0.44	Axiom 6 (left_distributivity): addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(X, Z), Y).
0.19/0.44	
0.19/0.44	Lemma 7: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
0.19/0.44	Proof:
0.19/0.44	  addition(X, multiplication(Y, X))
0.19/0.44	= { by axiom 1 (multiplicative_left_identity) R->L }
0.19/0.44	  addition(multiplication(one, X), multiplication(Y, X))
0.19/0.44	= { by axiom 6 (left_distributivity) }
0.19/0.44	  multiplication(addition(one, Y), X)
0.19/0.44	= { by axiom 2 (additive_commutativity) }
0.19/0.44	  multiplication(addition(Y, one), X)
0.19/0.44	
0.19/0.44	Lemma 8: addition(X, multiplication(domain(Y), X)) = X.
0.19/0.44	Proof:
0.19/0.44	  addition(X, multiplication(domain(Y), X))
0.19/0.44	= { by lemma 7 }
0.19/0.44	  multiplication(addition(domain(Y), one), X)
0.19/0.44	= { by axiom 3 (domain3) }
0.19/0.44	  multiplication(one, X)
0.19/0.44	= { by axiom 1 (multiplicative_left_identity) }
0.19/0.44	  X
0.19/0.44	
0.19/0.44	Goal 1 (goals_1): addition(x0, multiplication(domain(x1), x0)) = multiplication(domain(x1), x0).
0.19/0.44	Proof:
0.19/0.44	  addition(x0, multiplication(domain(x1), x0))
0.19/0.44	= { by axiom 2 (additive_commutativity) }
0.19/0.44	  addition(multiplication(domain(x1), x0), x0)
0.19/0.44	= { by axiom 1 (multiplicative_left_identity) R->L }
0.19/0.44	  addition(multiplication(domain(x1), x0), multiplication(one, x0))
0.19/0.44	= { by axiom 3 (domain3) R->L }
0.19/0.44	  addition(multiplication(domain(x1), x0), multiplication(addition(domain(x0), one), x0))
0.19/0.44	= { by lemma 7 R->L }
0.19/0.44	  addition(multiplication(domain(x1), x0), addition(x0, multiplication(domain(x0), x0)))
0.19/0.44	= { by axiom 5 (domain1) R->L }
0.19/0.44	  addition(multiplication(domain(x1), x0), multiplication(domain(x0), x0))
0.19/0.44	= { by axiom 6 (left_distributivity) }
0.19/0.44	  multiplication(addition(domain(x1), domain(x0)), x0)
0.19/0.44	= { by lemma 8 R->L }
0.19/0.44	  multiplication(addition(domain(x1), domain(x0)), addition(x0, multiplication(domain(X), x0)))
0.19/0.44	= { by axiom 2 (additive_commutativity) R->L }
0.19/0.44	  multiplication(addition(domain(x0), domain(x1)), addition(x0, multiplication(domain(X), x0)))
0.19/0.44	= { by axiom 4 (goals) }
0.19/0.44	  multiplication(domain(x1), addition(x0, multiplication(domain(X), x0)))
0.19/0.44	= { by lemma 8 }
0.19/0.44	  multiplication(domain(x1), x0)
0.19/0.44	% SZS output end Proof
0.19/0.44	
0.19/0.44	RESULT: Theorem (the conjecture is true).
0.19/0.44	EOF