0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.12	% 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   : n029.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 16:13:25 EDT 2021
0.12/0.33	% CPUTime    : 
0.18/0.44	% SZS status Theorem
0.18/0.44	
0.18/0.47	% SZS output start Proof
0.18/0.47	Take the following subset of the input axioms:
0.18/0.48	  fof(additive_associativity, axiom, ![A, C, B]: addition(addition(A, B), C)=addition(A, addition(B, C))).
0.18/0.48	  fof(additive_commutativity, axiom, ![A, B]: addition(B, A)=addition(A, B)).
0.18/0.48	  fof(additive_idempotence, axiom, ![A]: A=addition(A, A)).
0.18/0.48	  fof(goals, conjecture, ![X0, X1]: ((leq(one, addition(addition(addition(addition(multiplication(X1, X0), multiplication(c(X1), X0)), multiplication(X0, X1)), multiplication(c(X0), X1)), multiplication(c(X0), c(X1)))) & leq(addition(addition(addition(addition(multiplication(X1, X0), multiplication(c(X1), X0)), multiplication(X0, X1)), multiplication(c(X0), X1)), multiplication(c(X0), c(X1))), one)) <= (test(X0) & test(X1)))).
0.18/0.48	  fof(left_distributivity, axiom, ![A, C, B]: addition(multiplication(A, C), multiplication(B, C))=multiplication(addition(A, B), C)).
0.18/0.48	  fof(multiplicative_left_identity, axiom, ![A]: multiplication(one, A)=A).
0.18/0.48	  fof(multiplicative_right_identity, axiom, ![A]: multiplication(A, one)=A).
0.18/0.48	  fof(order, axiom, ![A, B]: (B=addition(A, B) <=> leq(A, B))).
0.18/0.48	  fof(right_distributivity, axiom, ![A, C, B]: addition(multiplication(A, B), multiplication(A, C))=multiplication(A, addition(B, C))).
0.18/0.48	  fof(test_1, axiom, ![X0]: (?[X1]: complement(X1, X0) <=> test(X0))).
0.18/0.48	  fof(test_2, axiom, ![X0, X1]: (complement(X1, X0) <=> (zero=multiplication(X0, X1) & (addition(X0, X1)=one & zero=multiplication(X1, X0))))).
0.18/0.48	  fof(test_3, axiom, ![X0, X1]: ((c(X0)=X1 <=> complement(X0, X1)) <= test(X0))).
0.18/0.48	
0.18/0.48	Now clausify the problem and encode Horn clauses using encoding 3 of
0.18/0.48	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.18/0.48	We repeatedly replace C & s=t => u=v by the two clauses:
0.18/0.48	  fresh(y, y, x1...xn) = u
0.18/0.48	  C => fresh(s, t, x1...xn) = v
0.18/0.48	where fresh is a fresh function symbol and x1..xn are the free
0.18/0.48	variables of u and v.
0.18/0.48	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.18/0.48	input problem has no model of domain size 1).
0.18/0.48	
0.18/0.48	The encoding turns the above axioms into the following unit equations and goals:
0.18/0.48	
0.18/0.48	Axiom 1 (goals): test(x1) = true.
0.18/0.48	Axiom 2 (goals_1): test(x0) = true.
0.18/0.48	Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
0.18/0.48	Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
0.18/0.48	Axiom 5 (additive_idempotence): X = addition(X, X).
0.18/0.48	Axiom 6 (additive_commutativity): addition(X, Y) = addition(Y, X).
0.18/0.48	Axiom 7 (test_1): fresh12(X, X, Y) = true.
0.18/0.48	Axiom 8 (additive_associativity): addition(addition(X, Y), Z) = addition(X, addition(Y, Z)).
0.18/0.48	Axiom 9 (test_1): fresh12(test(X), true, X) = complement(x1_2(X), X).
0.18/0.48	Axiom 10 (order): fresh11(X, X, Y, Z) = true.
0.18/0.48	Axiom 11 (test_2_3): fresh6(X, X, Y, Z) = one.
0.18/0.48	Axiom 12 (test_3): fresh5(X, X, Y, Z) = complement(Y, Z).
0.18/0.48	Axiom 13 (test_3): fresh4(X, X, Y, Z) = true.
0.18/0.48	Axiom 14 (test_3): fresh5(test(X), true, X, Y) = fresh4(c(X), Y, X, Y).
0.18/0.48	Axiom 15 (right_distributivity): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)).
0.18/0.48	Axiom 16 (left_distributivity): addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(X, Z), Y).
0.18/0.48	Axiom 17 (order): fresh11(X, addition(Y, X), Y, X) = leq(Y, X).
0.18/0.48	Axiom 18 (test_2_3): fresh6(complement(X, Y), true, Y, X) = addition(Y, X).
0.18/0.48	
0.18/0.48	Lemma 19: addition(x1, x1_2(x1)) = one.
0.18/0.48	Proof:
0.18/0.48	  addition(x1, x1_2(x1))
0.18/0.48	= { by axiom 18 (test_2_3) R->L }
0.18/0.48	  fresh6(complement(x1_2(x1), x1), true, x1, x1_2(x1))
0.18/0.48	= { by axiom 9 (test_1) R->L }
0.18/0.48	  fresh6(fresh12(test(x1), true, x1), true, x1, x1_2(x1))
0.18/0.48	= { by axiom 1 (goals) }
0.18/0.48	  fresh6(fresh12(true, true, x1), true, x1, x1_2(x1))
0.18/0.48	= { by axiom 7 (test_1) }
0.18/0.48	  fresh6(true, true, x1, x1_2(x1))
0.18/0.48	= { by axiom 11 (test_2_3) }
0.18/0.48	  one
0.18/0.48	
0.18/0.48	Lemma 20: addition(x1, one) = one.
0.18/0.48	Proof:
0.18/0.48	  addition(x1, one)
0.18/0.48	= { by lemma 19 R->L }
0.18/0.48	  addition(x1, addition(x1, x1_2(x1)))
0.18/0.48	= { by axiom 8 (additive_associativity) R->L }
0.18/0.48	  addition(addition(x1, x1), x1_2(x1))
0.18/0.48	= { by axiom 5 (additive_idempotence) R->L }
0.18/0.48	  addition(x1, x1_2(x1))
0.18/0.48	= { by lemma 19 }
0.18/0.48	  one
0.18/0.48	
0.18/0.48	Lemma 21: leq(X, X) = true.
0.18/0.48	Proof:
0.18/0.48	  leq(X, X)
0.18/0.48	= { by axiom 17 (order) R->L }
0.18/0.48	  fresh11(X, addition(X, X), X, X)
0.18/0.48	= { by axiom 5 (additive_idempotence) R->L }
0.18/0.48	  fresh11(X, X, X, X)
0.18/0.48	= { by axiom 10 (order) }
0.18/0.48	  true
0.18/0.48	
0.18/0.48	Lemma 22: fresh5(test(X), true, X, c(X)) = true.
0.18/0.48	Proof:
0.18/0.48	  fresh5(test(X), true, X, c(X))
0.18/0.48	= { by axiom 14 (test_3) }
0.18/0.48	  fresh4(c(X), c(X), X, c(X))
0.18/0.48	= { by axiom 13 (test_3) }
0.18/0.48	  true
0.18/0.48	
0.18/0.48	Lemma 23: addition(x1, c(x1)) = one.
0.18/0.48	Proof:
0.18/0.48	  addition(x1, c(x1))
0.18/0.48	= { by axiom 6 (additive_commutativity) R->L }
0.18/0.48	  addition(c(x1), x1)
0.18/0.48	= { by axiom 18 (test_2_3) R->L }
0.18/0.48	  fresh6(complement(x1, c(x1)), true, c(x1), x1)
0.18/0.48	= { by axiom 12 (test_3) R->L }
0.18/0.48	  fresh6(fresh5(true, true, x1, c(x1)), true, c(x1), x1)
0.18/0.48	= { by axiom 1 (goals) R->L }
0.18/0.48	  fresh6(fresh5(test(x1), true, x1, c(x1)), true, c(x1), x1)
0.18/0.48	= { by lemma 22 }
0.18/0.48	  fresh6(true, true, c(x1), x1)
0.18/0.48	= { by axiom 11 (test_2_3) }
0.18/0.48	  one
0.18/0.48	
0.18/0.48	Lemma 24: addition(x0, c(x0)) = one.
0.18/0.48	Proof:
0.18/0.48	  addition(x0, c(x0))
0.18/0.48	= { by axiom 6 (additive_commutativity) R->L }
0.18/0.48	  addition(c(x0), x0)
0.18/0.48	= { by axiom 18 (test_2_3) R->L }
0.18/0.48	  fresh6(complement(x0, c(x0)), true, c(x0), x0)
0.18/0.48	= { by axiom 12 (test_3) R->L }
0.18/0.48	  fresh6(fresh5(true, true, x0, c(x0)), true, c(x0), x0)
0.18/0.48	= { by axiom 2 (goals_1) R->L }
0.18/0.48	  fresh6(fresh5(test(x0), true, x0, c(x0)), true, c(x0), x0)
0.18/0.48	= { by lemma 22 }
0.18/0.48	  fresh6(true, true, c(x0), x0)
0.18/0.48	= { by axiom 11 (test_2_3) }
0.18/0.48	  one
0.18/0.48	
0.18/0.48	Lemma 25: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
0.18/0.48	Proof:
0.18/0.48	  addition(X, multiplication(X, Y))
0.18/0.48	= { by axiom 3 (multiplicative_right_identity) R->L }
0.18/0.48	  addition(multiplication(X, one), multiplication(X, Y))
0.18/0.48	= { by axiom 15 (right_distributivity) }
0.18/0.48	  multiplication(X, addition(one, Y))
0.18/0.48	= { by axiom 6 (additive_commutativity) }
0.18/0.49	  multiplication(X, addition(Y, one))
0.18/0.49	
0.18/0.49	Lemma 26: addition(Y, addition(X, Z)) = addition(X, addition(Y, Z)).
0.18/0.49	Proof:
0.18/0.49	  addition(Y, addition(X, Z))
0.18/0.49	= { by axiom 6 (additive_commutativity) R->L }
0.18/0.49	  addition(addition(X, Z), Y)
0.18/0.49	= { by axiom 8 (additive_associativity) }
0.18/0.49	  addition(X, addition(Z, Y))
0.18/0.49	= { by axiom 6 (additive_commutativity) }
0.18/0.49	  addition(X, addition(Y, Z))
0.18/0.49	
0.18/0.49	Goal 1 (goals_2): tuple(leq(one, addition(addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)), multiplication(c(x0), c(x1)))), leq(addition(addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)), multiplication(c(x0), c(x1))), one)) = tuple(true, true).
0.18/0.49	Proof:
0.18/0.49	  tuple(leq(one, addition(addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)), multiplication(c(x0), c(x1)))), leq(addition(addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)), multiplication(c(x0), c(x1))), one))
0.18/0.49	= { by axiom 6 (additive_commutativity) }
0.18/0.49	  tuple(leq(one, addition(multiplication(c(x0), c(x1)), addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)))), leq(addition(addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)), multiplication(c(x0), c(x1))), one))
0.18/0.49	= { by axiom 6 (additive_commutativity) }
0.18/0.49	  tuple(leq(one, addition(multiplication(c(x0), c(x1)), addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)))), leq(addition(multiplication(c(x0), c(x1)), addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1))), one))
0.18/0.49	= { by axiom 6 (additive_commutativity) }
0.18/0.49	  tuple(leq(one, addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1))))), leq(addition(multiplication(c(x0), c(x1)), addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1))), one))
0.18/0.49	= { by axiom 6 (additive_commutativity) }
0.18/0.49	  tuple(leq(one, addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1))))), leq(addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)))), one))
0.18/0.49	= { by axiom 6 (additive_commutativity) }
0.18/0.49	  tuple(leq(one, addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(multiplication(x0, x1), addition(multiplication(x1, x0), multiplication(c(x1), x0)))))), leq(addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)))), one))
0.18/0.49	= { by axiom 6 (additive_commutativity) }
0.18/0.49	  tuple(leq(one, addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(multiplication(x0, x1), addition(multiplication(x1, x0), multiplication(c(x1), x0)))))), leq(addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(multiplication(x0, x1), addition(multiplication(x1, x0), multiplication(c(x1), x0))))), one))
0.18/0.49	= { by axiom 16 (left_distributivity) }
0.18/0.49	  tuple(leq(one, addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(multiplication(x0, x1), multiplication(addition(x1, c(x1)), x0))))), leq(addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(multiplication(x0, x1), addition(multiplication(x1, x0), multiplication(c(x1), x0))))), one))
0.18/0.49	= { by axiom 16 (left_distributivity) }
0.18/0.49	  tuple(leq(one, addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(multiplication(x0, x1), multiplication(addition(x1, c(x1)), x0))))), leq(addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(multiplication(x0, x1), multiplication(addition(x1, c(x1)), x0)))), one))
0.18/0.49	= { by lemma 26 R->L }
0.18/0.49	  tuple(leq(one, addition(multiplication(c(x0), x1), addition(multiplication(c(x0), c(x1)), addition(multiplication(x0, x1), multiplication(addition(x1, c(x1)), x0))))), leq(addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(multiplication(x0, x1), multiplication(addition(x1, c(x1)), x0)))), one))
0.18/0.49	= { by lemma 26 R->L }
0.18/0.49	  tuple(leq(one, addition(multiplication(c(x0), x1), addition(multiplication(c(x0), c(x1)), addition(multiplication(x0, x1), multiplication(addition(x1, c(x1)), x0))))), leq(addition(multiplication(c(x0), x1), addition(multiplication(c(x0), c(x1)), addition(multiplication(x0, x1), multiplication(addition(x1, c(x1)), x0)))), one))
0.18/0.49	= { by lemma 26 R->L }
0.18/0.49	  tuple(leq(one, addition(multiplication(c(x0), x1), addition(multiplication(x0, x1), addition(multiplication(c(x0), c(x1)), multiplication(addition(x1, c(x1)), x0))))), leq(addition(multiplication(c(x0), x1), addition(multiplication(c(x0), c(x1)), addition(multiplication(x0, x1), multiplication(addition(x1, c(x1)), x0)))), one))
0.18/0.49	= { by lemma 26 R->L }
0.18/0.49	  tuple(leq(one, addition(multiplication(c(x0), x1), addition(multiplication(x0, x1), addition(multiplication(c(x0), c(x1)), multiplication(addition(x1, c(x1)), x0))))), leq(addition(multiplication(c(x0), x1), addition(multiplication(x0, x1), addition(multiplication(c(x0), c(x1)), multiplication(addition(x1, c(x1)), x0)))), one))
0.18/0.49	= { by lemma 26 R->L }
0.18/0.49	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), addition(multiplication(c(x0), c(x1)), multiplication(addition(x1, c(x1)), x0))))), leq(addition(multiplication(c(x0), x1), addition(multiplication(x0, x1), addition(multiplication(c(x0), c(x1)), multiplication(addition(x1, c(x1)), x0)))), one))
0.18/0.49	= { by lemma 26 R->L }
0.18/0.49	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), addition(multiplication(c(x0), c(x1)), multiplication(addition(x1, c(x1)), x0))))), leq(addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), addition(multiplication(c(x0), c(x1)), multiplication(addition(x1, c(x1)), x0)))), one))
0.18/0.49	= { by lemma 23 }
0.18/0.49	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), addition(multiplication(c(x0), c(x1)), multiplication(one, x0))))), leq(addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), addition(multiplication(c(x0), c(x1)), multiplication(addition(x1, c(x1)), x0)))), one))
0.18/0.49	= { by lemma 23 }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), addition(multiplication(c(x0), c(x1)), multiplication(one, x0))))), leq(addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), addition(multiplication(c(x0), c(x1)), multiplication(one, x0)))), one))
0.18/0.50	= { by axiom 6 (additive_commutativity) }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), addition(multiplication(one, x0), multiplication(c(x0), c(x1)))))), leq(addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), addition(multiplication(c(x0), c(x1)), multiplication(one, x0)))), one))
0.18/0.50	= { by axiom 6 (additive_commutativity) }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), addition(multiplication(one, x0), multiplication(c(x0), c(x1)))))), leq(addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), addition(multiplication(one, x0), multiplication(c(x0), c(x1))))), one))
0.18/0.50	= { by lemma 26 R->L }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(one, x0), addition(multiplication(c(x0), x1), multiplication(c(x0), c(x1)))))), leq(addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), addition(multiplication(one, x0), multiplication(c(x0), c(x1))))), one))
0.18/0.50	= { by lemma 26 R->L }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(one, x0), addition(multiplication(c(x0), x1), multiplication(c(x0), c(x1)))))), leq(addition(multiplication(x0, x1), addition(multiplication(one, x0), addition(multiplication(c(x0), x1), multiplication(c(x0), c(x1))))), one))
0.18/0.50	= { by axiom 15 (right_distributivity) }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(one, x0), multiplication(c(x0), addition(x1, c(x1)))))), leq(addition(multiplication(x0, x1), addition(multiplication(one, x0), addition(multiplication(c(x0), x1), multiplication(c(x0), c(x1))))), one))
0.18/0.50	= { by axiom 15 (right_distributivity) }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(one, x0), multiplication(c(x0), addition(x1, c(x1)))))), leq(addition(multiplication(x0, x1), addition(multiplication(one, x0), multiplication(c(x0), addition(x1, c(x1))))), one))
0.18/0.50	= { by lemma 23 }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(one, x0), multiplication(c(x0), one)))), leq(addition(multiplication(x0, x1), addition(multiplication(one, x0), multiplication(c(x0), addition(x1, c(x1))))), one))
0.18/0.50	= { by lemma 23 }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(one, x0), multiplication(c(x0), one)))), leq(addition(multiplication(x0, x1), addition(multiplication(one, x0), multiplication(c(x0), one))), one))
0.18/0.50	= { by axiom 3 (multiplicative_right_identity) }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(one, x0), c(x0)))), leq(addition(multiplication(x0, x1), addition(multiplication(one, x0), multiplication(c(x0), one))), one))
0.18/0.50	= { by axiom 3 (multiplicative_right_identity) }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(multiplication(one, x0), c(x0)))), leq(addition(multiplication(x0, x1), addition(multiplication(one, x0), c(x0))), one))
0.18/0.50	= { by axiom 6 (additive_commutativity) }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(c(x0), multiplication(one, x0)))), leq(addition(multiplication(x0, x1), addition(multiplication(one, x0), c(x0))), one))
0.18/0.50	= { by axiom 6 (additive_commutativity) }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(c(x0), multiplication(one, x0)))), leq(addition(multiplication(x0, x1), addition(c(x0), multiplication(one, x0))), one))
0.18/0.50	= { by axiom 4 (multiplicative_left_identity) }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(c(x0), x0))), leq(addition(multiplication(x0, x1), addition(c(x0), multiplication(one, x0))), one))
0.18/0.50	= { by axiom 4 (multiplicative_left_identity) }
0.18/0.50	  tuple(leq(one, addition(multiplication(x0, x1), addition(c(x0), x0))), leq(addition(multiplication(x0, x1), addition(c(x0), x0)), one))
0.18/0.50	= { by lemma 26 R->L }
0.18/0.50	  tuple(leq(one, addition(c(x0), addition(multiplication(x0, x1), x0))), leq(addition(multiplication(x0, x1), addition(c(x0), x0)), one))
0.18/0.50	= { by lemma 26 R->L }
0.18/0.50	  tuple(leq(one, addition(c(x0), addition(multiplication(x0, x1), x0))), leq(addition(c(x0), addition(multiplication(x0, x1), x0)), one))
0.18/0.50	= { by axiom 6 (additive_commutativity) }
0.18/0.50	  tuple(leq(one, addition(c(x0), addition(x0, multiplication(x0, x1)))), leq(addition(c(x0), addition(multiplication(x0, x1), x0)), one))
0.18/0.50	= { by axiom 6 (additive_commutativity) }
0.18/0.50	  tuple(leq(one, addition(c(x0), addition(x0, multiplication(x0, x1)))), leq(addition(c(x0), addition(x0, multiplication(x0, x1))), one))
0.18/0.50	= { by lemma 25 }
0.18/0.50	  tuple(leq(one, addition(c(x0), multiplication(x0, addition(x1, one)))), leq(addition(c(x0), addition(x0, multiplication(x0, x1))), one))
0.18/0.50	= { by lemma 25 }
0.18/0.50	  tuple(leq(one, addition(c(x0), multiplication(x0, addition(x1, one)))), leq(addition(c(x0), multiplication(x0, addition(x1, one))), one))
0.18/0.50	= { by lemma 20 }
0.18/0.50	  tuple(leq(one, addition(c(x0), multiplication(x0, one))), leq(addition(c(x0), multiplication(x0, addition(x1, one))), one))
0.18/0.50	= { by lemma 20 }
0.18/0.50	  tuple(leq(one, addition(c(x0), multiplication(x0, one))), leq(addition(c(x0), multiplication(x0, one)), one))
0.18/0.50	= { by axiom 3 (multiplicative_right_identity) }
0.18/0.50	  tuple(leq(one, addition(c(x0), x0)), leq(addition(c(x0), multiplication(x0, one)), one))
0.18/0.50	= { by axiom 3 (multiplicative_right_identity) }
0.18/0.50	  tuple(leq(one, addition(c(x0), x0)), leq(addition(c(x0), x0), one))
0.18/0.50	= { by axiom 6 (additive_commutativity) }
0.18/0.50	  tuple(leq(one, addition(x0, c(x0))), leq(addition(c(x0), x0), one))
0.18/0.50	= { by axiom 6 (additive_commutativity) }
0.18/0.50	  tuple(leq(one, addition(x0, c(x0))), leq(addition(x0, c(x0)), one))
0.18/0.50	= { by lemma 24 }
0.18/0.50	  tuple(leq(one, one), leq(addition(x0, c(x0)), one))
0.18/0.50	= { by lemma 24 }
0.18/0.50	  tuple(leq(one, one), leq(one, one))
0.18/0.50	= { by lemma 21 }
0.18/0.50	  tuple(true, leq(one, one))
0.18/0.50	= { by lemma 21 }
0.18/0.50	  tuple(true, true)
0.18/0.50	% SZS output end Proof
0.18/0.50	
0.18/0.50	RESULT: Theorem (the conjecture is true).
0.18/0.50	EOF