0.00/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.12	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.13/0.33	% Computer   : n003.cluster.edu
0.13/0.33	% Model      : x86_64 x86_64
0.13/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.33	% Memory     : 8042.1875MB
0.13/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.33	% CPULimit   : 1200
0.13/0.33	% WCLimit    : 120
0.13/0.33	% DateTime   : Tue Jul 13 16:14:54 EDT 2021
0.13/0.34	% CPUTime    : 
0.19/0.49	% SZS status Theorem
0.19/0.49	
0.19/0.50	% SZS output start Proof
0.19/0.50	Take the following subset of the input axioms:
0.19/0.50	  fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
0.19/0.50	  fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
0.19/0.50	  fof(distributivity2, axiom, ![A, B, C]: multiplication(addition(A, B), C)=addition(multiplication(A, C), multiplication(B, C))).
0.19/0.50	  fof(goals, conjecture, ![X0]: (leq(strong_iteration(strong_iteration(X0)), strong_iteration(one)) & leq(strong_iteration(one), strong_iteration(strong_iteration(X0))))).
0.19/0.50	  fof(idempotence, axiom, ![A]: addition(A, A)=A).
0.19/0.50	  fof(infty_coinduction, axiom, ![A, B, C]: (leq(C, addition(multiplication(A, C), B)) => leq(C, multiplication(strong_iteration(A), B)))).
0.19/0.50	  fof(infty_unfold1, axiom, ![A]: strong_iteration(A)=addition(multiplication(A, strong_iteration(A)), one)).
0.19/0.50	  fof(multiplicative_left_identity, axiom, ![A]: multiplication(one, A)=A).
0.19/0.50	  fof(multiplicative_right_identity, axiom, ![A]: A=multiplication(A, one)).
0.19/0.50	  fof(order, axiom, ![A, B]: (B=addition(A, B) <=> leq(A, B))).
0.19/0.50	
0.19/0.50	Now clausify the problem and encode Horn clauses using encoding 3 of
0.19/0.50	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.19/0.50	We repeatedly replace C & s=t => u=v by the two clauses:
0.19/0.50	  fresh(y, y, x1...xn) = u
0.19/0.50	  C => fresh(s, t, x1...xn) = v
0.19/0.50	where fresh is a fresh function symbol and x1..xn are the free
0.19/0.50	variables of u and v.
0.19/0.50	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.19/0.50	input problem has no model of domain size 1).
0.19/0.50	
0.19/0.50	The encoding turns the above axioms into the following unit equations and goals:
0.19/0.50	
0.19/0.50	Axiom 1 (idempotence): addition(X, X) = X.
0.19/0.50	Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
0.19/0.50	Axiom 3 (multiplicative_right_identity): X = multiplication(X, one).
0.19/0.50	Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
0.19/0.50	Axiom 5 (order): fresh5(X, X, Y, Z) = true.
0.19/0.50	Axiom 6 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
0.19/0.50	Axiom 7 (infty_coinduction): fresh4(X, X, Y, Z, W) = true.
0.19/0.50	Axiom 8 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
0.19/0.50	Axiom 9 (order): fresh5(X, addition(Y, X), Y, X) = leq(Y, X).
0.19/0.50	Axiom 10 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
0.19/0.50	Axiom 11 (infty_coinduction): fresh4(leq(X, addition(multiplication(Y, X), Z)), true, Y, Z, X) = leq(X, multiplication(strong_iteration(Y), Z)).
0.19/0.50	
0.19/0.50	Lemma 12: leq(X, multiplication(strong_iteration(addition(Y, one)), Z)) = true.
0.19/0.50	Proof:
0.19/0.50	  leq(X, multiplication(strong_iteration(addition(Y, one)), Z))
0.19/0.50	= { by axiom 11 (infty_coinduction) R->L }
0.19/0.50	  fresh4(leq(X, addition(multiplication(addition(Y, one), X), Z)), true, addition(Y, one), Z, X)
0.19/0.50	= { by axiom 2 (additive_commutativity) R->L }
0.19/0.50	  fresh4(leq(X, addition(multiplication(addition(one, Y), X), Z)), true, addition(Y, one), Z, X)
0.19/0.50	= { by axiom 10 (distributivity2) }
0.19/0.50	  fresh4(leq(X, addition(addition(multiplication(one, X), multiplication(Y, X)), Z)), true, addition(Y, one), Z, X)
0.19/0.50	= { by axiom 4 (multiplicative_left_identity) }
0.19/0.50	  fresh4(leq(X, addition(addition(X, multiplication(Y, X)), Z)), true, addition(Y, one), Z, X)
0.19/0.50	= { by axiom 6 (additive_associativity) R->L }
0.19/0.50	  fresh4(leq(X, addition(X, addition(multiplication(Y, X), Z))), true, addition(Y, one), Z, X)
0.19/0.50	= { by axiom 9 (order) R->L }
0.19/0.50	  fresh4(fresh5(addition(X, addition(multiplication(Y, X), Z)), addition(X, addition(X, addition(multiplication(Y, X), Z))), X, addition(X, addition(multiplication(Y, X), Z))), true, addition(Y, one), Z, X)
0.19/0.50	= { by axiom 6 (additive_associativity) }
0.19/0.50	  fresh4(fresh5(addition(X, addition(multiplication(Y, X), Z)), addition(addition(X, X), addition(multiplication(Y, X), Z)), X, addition(X, addition(multiplication(Y, X), Z))), true, addition(Y, one), Z, X)
0.19/0.50	= { by axiom 1 (idempotence) }
0.19/0.50	  fresh4(fresh5(addition(X, addition(multiplication(Y, X), Z)), addition(X, addition(multiplication(Y, X), Z)), X, addition(X, addition(multiplication(Y, X), Z))), true, addition(Y, one), Z, X)
0.19/0.50	= { by axiom 5 (order) }
0.19/0.50	  fresh4(true, true, addition(Y, one), Z, X)
0.19/0.50	= { by axiom 7 (infty_coinduction) }
0.19/0.50	  true
0.19/0.50	
0.19/0.50	Goal 1 (goals): tuple(leq(strong_iteration(strong_iteration(x0_2)), strong_iteration(one)), leq(strong_iteration(one), strong_iteration(strong_iteration(x0)))) = tuple(true, true).
0.19/0.50	Proof:
0.19/0.50	  tuple(leq(strong_iteration(strong_iteration(x0_2)), strong_iteration(one)), leq(strong_iteration(one), strong_iteration(strong_iteration(x0))))
0.19/0.50	= { by axiom 3 (multiplicative_right_identity) }
0.19/0.50	  tuple(leq(strong_iteration(strong_iteration(x0_2)), multiplication(strong_iteration(one), one)), leq(strong_iteration(one), strong_iteration(strong_iteration(x0))))
0.19/0.50	= { by axiom 1 (idempotence) R->L }
0.19/0.50	  tuple(leq(strong_iteration(strong_iteration(x0_2)), multiplication(strong_iteration(addition(one, one)), one)), leq(strong_iteration(one), strong_iteration(strong_iteration(x0))))
0.19/0.50	= { by lemma 12 }
0.19/0.50	  tuple(true, leq(strong_iteration(one), strong_iteration(strong_iteration(x0))))
0.19/0.50	= { by axiom 8 (infty_unfold1) }
0.19/0.50	  tuple(true, leq(strong_iteration(one), strong_iteration(addition(multiplication(x0, strong_iteration(x0)), one))))
0.19/0.50	= { by axiom 3 (multiplicative_right_identity) }
0.19/0.50	  tuple(true, leq(strong_iteration(one), multiplication(strong_iteration(addition(multiplication(x0, strong_iteration(x0)), one)), one)))
0.19/0.50	= { by lemma 12 }
0.19/0.50	  tuple(true, true)
0.19/0.50	% SZS output end Proof
0.19/0.50	
0.19/0.50	RESULT: Theorem (the conjecture is true).
0.19/0.50	EOF