0.03/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.32	% Computer   : n018.cluster.edu
0.12/0.32	% Model      : x86_64 x86_64
0.12/0.32	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.32	% Memory     : 8042.1875MB
0.12/0.32	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.32	% CPULimit   : 180
0.12/0.32	% DateTime   : Thu Aug 29 12:13:51 EDT 2019
0.12/0.33	% CPUTime    : 
0.17/0.50	% SZS status Theorem
0.17/0.50	
0.17/0.50	% SZS output start Proof
0.17/0.50	Take the following subset of the input axioms:
0.17/0.50	  fof(additive_associativity, axiom, ![A, B, C]: addition(addition(A, B), C)=addition(A, addition(B, C))).
0.17/0.50	  fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
0.17/0.50	  fof(distributivity2, axiom, ![A, B, C]: multiplication(addition(A, B), C)=addition(multiplication(A, C), multiplication(B, C))).
0.17/0.50	  fof(goals, conjecture, ![X0]: strong_iteration(one)=strong_iteration(star(X0))).
0.17/0.50	  fof(idempotence, axiom, ![A]: A=addition(A, A)).
0.17/0.50	  fof(infty_coinduction, axiom, ![A, B, C]: (leq(C, addition(multiplication(A, C), B)) => leq(C, multiplication(strong_iteration(A), B)))).
0.17/0.50	  fof(infty_unfold1, axiom, ![A]: strong_iteration(A)=addition(multiplication(A, strong_iteration(A)), one)).
0.17/0.50	  fof(multiplicative_left_identity, axiom, ![A]: A=multiplication(one, A)).
0.17/0.50	  fof(multiplicative_right_identity, axiom, ![A]: multiplication(A, one)=A).
0.17/0.50	  fof(order, axiom, ![A, B]: (addition(A, B)=B <=> leq(A, B))).
0.17/0.50	  fof(star_unfold2, axiom, ![A]: star(A)=addition(one, multiplication(star(A), A))).
0.17/0.50	
0.17/0.50	Now clausify the problem and encode Horn clauses using encoding 3 of
0.17/0.50	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.17/0.50	We repeatedly replace C & s=t => u=v by the two clauses:
0.17/0.50	  fresh(y, y, x1...xn) = u
0.17/0.50	  C => fresh(s, t, x1...xn) = v
0.17/0.50	where fresh is a fresh function symbol and x1..xn are the free
0.17/0.50	variables of u and v.
0.17/0.50	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.17/0.50	input problem has no model of domain size 1).
0.17/0.50	
0.17/0.50	The encoding turns the above axioms into the following unit equations and goals:
0.17/0.50	
0.17/0.50	Axiom 1 (infty_coinduction): fresh4(X, X, Y, Z, W) = true.
0.17/0.50	Axiom 2 (order): fresh5(X, X, Y, Z) = true.
0.17/0.50	Axiom 3 (order_1): fresh(X, X, Y, Z) = Z.
0.17/0.50	Axiom 4 (idempotence): X = addition(X, X).
0.17/0.50	Axiom 5 (star_unfold2): star(X) = addition(one, multiplication(star(X), X)).
0.17/0.50	Axiom 6 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
0.17/0.50	Axiom 7 (additive_associativity): addition(addition(X, Y), Z) = addition(X, addition(Y, Z)).
0.17/0.50	Axiom 8 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
0.17/0.50	Axiom 9 (order): fresh5(addition(X, Y), Y, X, Y) = leq(X, Y).
0.17/0.50	Axiom 10 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
0.17/0.50	Axiom 11 (multiplicative_left_identity): X = multiplication(one, X).
0.17/0.50	Axiom 12 (multiplicative_right_identity): multiplication(X, one) = X.
0.17/0.50	Axiom 13 (additive_commutativity): addition(X, Y) = addition(Y, X).
0.17/0.50	Axiom 14 (infty_coinduction): fresh4(leq(X, addition(multiplication(Y, X), Z)), true, Y, Z, X) = leq(X, multiplication(strong_iteration(Y), Z)).
0.17/0.50	
0.17/0.50	Lemma 15: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X).
0.17/0.50	Proof:
0.17/0.50	  addition(one, multiplication(X, strong_iteration(X)))
0.17/0.50	= { by axiom 13 (additive_commutativity) }
0.17/0.50	  addition(multiplication(X, strong_iteration(X)), one)
0.17/0.50	= { by axiom 6 (infty_unfold1) }
0.17/0.50	  strong_iteration(X)
0.17/0.50	
0.17/0.50	Lemma 16: addition(X, addition(X, Y)) = addition(X, Y).
0.17/0.50	Proof:
0.17/0.50	  addition(X, addition(X, Y))
0.17/0.50	= { by axiom 7 (additive_associativity) }
0.17/0.50	  addition(addition(X, X), Y)
0.17/0.50	= { by axiom 4 (idempotence) }
0.17/0.50	  addition(X, Y)
0.17/0.50	
0.17/0.50	Lemma 17: leq(X, multiplication(strong_iteration(addition(Y, one)), Z)) = true.
0.17/0.50	Proof:
0.17/0.50	  leq(X, multiplication(strong_iteration(addition(Y, one)), Z))
0.17/0.50	= { by axiom 14 (infty_coinduction) }
0.17/0.50	  fresh4(leq(X, addition(multiplication(addition(Y, one), X), Z)), true, addition(Y, one), Z, X)
0.17/0.50	= { by axiom 13 (additive_commutativity) }
0.17/0.50	  fresh4(leq(X, addition(multiplication(addition(one, Y), X), Z)), true, addition(Y, one), Z, X)
0.17/0.50	= { by axiom 10 (distributivity2) }
0.17/0.50	  fresh4(leq(X, addition(addition(multiplication(one, X), multiplication(Y, X)), Z)), true, addition(Y, one), Z, X)
0.17/0.50	= { by axiom 11 (multiplicative_left_identity) }
0.17/0.50	  fresh4(leq(X, addition(addition(X, multiplication(Y, X)), Z)), true, addition(Y, one), Z, X)
0.17/0.50	= { by axiom 7 (additive_associativity) }
0.17/0.50	  fresh4(leq(X, addition(X, addition(multiplication(Y, X), Z))), true, addition(Y, one), Z, X)
0.17/0.50	= { by axiom 13 (additive_commutativity) }
0.17/0.50	  fresh4(leq(X, addition(X, addition(Z, multiplication(Y, X)))), true, addition(Y, one), Z, X)
0.17/0.50	= { by axiom 9 (order) }
0.17/0.50	  fresh4(fresh5(addition(X, addition(X, addition(Z, multiplication(Y, X)))), addition(X, addition(Z, multiplication(Y, X))), X, addition(X, addition(Z, multiplication(Y, X)))), true, addition(Y, one), Z, X)
0.17/0.50	= { by lemma 16 }
0.17/0.50	  fresh4(fresh5(addition(X, addition(Z, multiplication(Y, X))), addition(X, addition(Z, multiplication(Y, X))), X, addition(X, addition(Z, multiplication(Y, X)))), true, addition(Y, one), Z, X)
0.17/0.50	= { by axiom 2 (order) }
0.17/0.50	  fresh4(true, true, addition(Y, one), Z, X)
0.17/0.50	= { by axiom 1 (infty_coinduction) }
0.17/0.50	  true
0.17/0.50	
0.17/0.50	Goal 1 (goals): strong_iteration(one) = strong_iteration(star(sK1_goals_X0)).
0.17/0.50	Proof:
0.17/0.50	  strong_iteration(one)
0.17/0.50	= { by axiom 3 (order_1) }
0.17/0.50	  fresh(true, true, strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)), strong_iteration(one))
0.17/0.50	= { by lemma 17 }
0.17/0.50	  fresh(leq(strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)), multiplication(strong_iteration(addition(one, one)), one)), true, strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)), strong_iteration(one))
0.17/0.50	= { by axiom 4 (idempotence) }
0.17/0.50	  fresh(leq(strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)), multiplication(strong_iteration(one), one)), true, strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)), strong_iteration(one))
0.17/0.50	= { by axiom 12 (multiplicative_right_identity) }
0.17/0.50	  fresh(leq(strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)), strong_iteration(one)), true, strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)), strong_iteration(one))
0.17/0.50	= { by axiom 8 (order_1) }
0.17/0.50	  addition(strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)), strong_iteration(one))
0.17/0.50	= { by axiom 13 (additive_commutativity) }
0.17/0.50	  addition(strong_iteration(one), strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)))
0.17/0.50	= { by axiom 6 (infty_unfold1) }
0.17/0.50	  addition(addition(multiplication(one, strong_iteration(one)), one), strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)))
0.17/0.50	= { by axiom 11 (multiplicative_left_identity) }
0.17/0.50	  addition(addition(strong_iteration(one), one), strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)))
0.17/0.50	= { by axiom 13 (additive_commutativity) }
0.17/0.50	  addition(addition(one, strong_iteration(one)), strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)))
0.17/0.50	= { by axiom 7 (additive_associativity) }
0.17/0.50	  addition(one, addition(strong_iteration(one), strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one))))
0.17/0.50	= { by axiom 8 (order_1) }
0.17/0.50	  addition(one, fresh(leq(strong_iteration(one), strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one))), true, strong_iteration(one), strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one))))
0.17/0.50	= { by axiom 12 (multiplicative_right_identity) }
0.17/0.50	  addition(one, fresh(leq(strong_iteration(one), multiplication(strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)), one)), true, strong_iteration(one), strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one))))
0.17/0.50	= { by lemma 17 }
0.17/0.50	  addition(one, fresh(true, true, strong_iteration(one), strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one))))
0.17/0.50	= { by axiom 3 (order_1) }
0.17/0.50	  addition(one, strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)))
0.17/0.50	= { by lemma 15 }
0.17/0.50	  addition(one, addition(one, multiplication(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one), strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one)))))
0.17/0.50	= { by lemma 16 }
0.17/0.50	  addition(one, multiplication(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one), strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one))))
0.17/0.50	= { by lemma 15 }
0.17/0.50	  strong_iteration(addition(multiplication(star(sK1_goals_X0), sK1_goals_X0), one))
0.17/0.50	= { by axiom 13 (additive_commutativity) }
0.17/0.50	  strong_iteration(addition(one, multiplication(star(sK1_goals_X0), sK1_goals_X0)))
0.17/0.50	= { by axiom 5 (star_unfold2) }
0.17/0.50	  strong_iteration(star(sK1_goals_X0))
0.17/0.50	% SZS output end Proof
0.17/0.50	
0.17/0.50	RESULT: Theorem (the conjecture is true).
0.17/0.51	EOF