0.10/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.10/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.33	% Computer   : n024.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   : 180
0.13/0.33	% DateTime   : Thu Aug 29 11:32:50 EDT 2019
0.13/0.34	% CPUTime    : 
6.76/6.94	% SZS status Unsatisfiable
6.76/6.94	
6.76/6.94	% SZS output start Proof
6.76/6.94	Take the following subset of the input axioms:
6.97/7.17	  fof(a_times_b_is_c, negated_conjecture, product(a, b, c)=true).
6.97/7.17	  fof(associativity1, axiom, ![X, Y, Z, V, W, U]: ifeq(product(U, Z, W), true, ifeq(product(Y, Z, V), true, ifeq(product(X, Y, U), true, product(X, V, W), true), true), true)=true).
6.97/7.17	  fof(associativity2, axiom, ![X, Y, Z, V, W, U]: ifeq(product(Y, Z, V), true, ifeq(product(X, V, W), true, ifeq(product(X, Y, U), true, product(U, Z, W), true), true), true)=true).
6.97/7.17	  fof(c_times_inverse_a_is_d, negated_conjecture, true=product(c, inverse(a), d)).
6.97/7.17	  fof(d_times_inverse_b_is_h, negated_conjecture, product(d, inverse(b), h)=true).
6.97/7.17	  fof(h_times_b_is_j, negated_conjecture, product(h, b, j)=true).
6.97/7.17	  fof(ifeq_axiom, axiom, ![A, B, C]: ifeq2(A, A, B, C)=B).
6.97/7.17	  fof(ifeq_axiom_001, axiom, ![A, B, C]: ifeq(A, A, B, C)=B).
6.97/7.17	  fof(j_times_inverse_h_is_k, negated_conjecture, true=product(j, inverse(h), k)).
6.97/7.17	  fof(left_identity, axiom, ![X]: true=product(identity, X, X)).
6.97/7.17	  fof(left_inverse, axiom, ![X]: true=product(inverse(X), X, identity)).
6.97/7.17	  fof(prove_k_times_inverse_b_is_e, negated_conjecture, product(k, inverse(b), identity)!=true).
6.97/7.17	  fof(right_identity, axiom, ![X]: true=product(X, identity, X)).
6.97/7.17	  fof(right_inverse, axiom, ![X]: true=product(X, inverse(X), identity)).
6.97/7.17	  fof(total_function1, axiom, ![X, Y]: product(X, Y, multiply(X, Y))=true).
6.97/7.17	  fof(total_function2, axiom, ![X, Y, Z, W]: W=ifeq2(product(X, Y, W), true, ifeq2(product(X, Y, Z), true, Z, W), W)).
6.97/7.17	  fof(x_cubed_is_identity_2, hypothesis, ![X, Y]: ifeq(product(X, X, Y), true, product(Y, X, identity), true)=true).
6.97/7.17	
6.97/7.17	Now clausify the problem and encode Horn clauses using encoding 3 of
6.97/7.17	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
6.97/7.17	We repeatedly replace C & s=t => u=v by the two clauses:
6.97/7.17	  fresh(y, y, x1...xn) = u
6.97/7.17	  C => fresh(s, t, x1...xn) = v
6.97/7.17	where fresh is a fresh function symbol and x1..xn are the free
6.97/7.17	variables of u and v.
6.97/7.17	A predicate p(X) is encoded as p(X)=true (this is sound, because the
6.97/7.17	input problem has no model of domain size 1).
6.97/7.17	
6.97/7.17	The encoding turns the above axioms into the following unit equations and goals:
6.97/7.17	
6.97/7.17	Axiom 1 (associativity1): ifeq(product(X, Y, Z), true, ifeq(product(W, Y, V), true, ifeq(product(U, W, X), true, product(U, V, Z), true), true), true) = true.
6.97/7.17	Axiom 2 (x_cubed_is_identity_2): ifeq(product(X, X, Y), true, product(Y, X, identity), true) = true.
6.97/7.17	Axiom 3 (right_inverse): true = product(X, inverse(X), identity).
6.97/7.17	Axiom 4 (left_identity): true = product(identity, X, X).
6.97/7.17	Axiom 5 (h_times_b_is_j): product(h, b, j) = true.
6.97/7.17	Axiom 6 (d_times_inverse_b_is_h): product(d, inverse(b), h) = true.
6.97/7.17	Axiom 7 (total_function2): X = ifeq2(product(Y, Z, X), true, ifeq2(product(Y, Z, W), true, W, X), X).
6.97/7.17	Axiom 8 (j_times_inverse_h_is_k): true = product(j, inverse(h), k).
6.97/7.17	Axiom 9 (c_times_inverse_a_is_d): true = product(c, inverse(a), d).
6.97/7.17	Axiom 10 (ifeq_axiom_001): ifeq(X, X, Y, Z) = Y.
6.97/7.17	Axiom 11 (associativity2): ifeq(product(X, Y, Z), true, ifeq(product(W, Z, V), true, ifeq(product(W, X, U), true, product(U, Y, V), true), true), true) = true.
6.97/7.17	Axiom 12 (left_inverse): true = product(inverse(X), X, identity).
6.97/7.17	Axiom 13 (a_times_b_is_c): product(a, b, c) = true.
6.97/7.17	Axiom 14 (total_function1): product(X, Y, multiply(X, Y)) = true.
6.97/7.17	Axiom 15 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
6.97/7.17	Axiom 16 (right_identity): true = product(X, identity, X).
6.97/7.17	
6.97/7.17	Lemma 17: ifeq2(product(identity, X, Y), true, Y, X) = X.
6.97/7.17	Proof:
6.97/7.17	  ifeq2(product(identity, X, Y), true, Y, X)
6.97/7.17	= { by axiom 15 (ifeq_axiom) }
6.97/7.17	  ifeq2(true, true, ifeq2(product(identity, X, Y), true, Y, X), X)
6.97/7.17	= { by axiom 4 (left_identity) }
6.97/7.17	  ifeq2(product(identity, X, X), true, ifeq2(product(identity, X, Y), true, Y, X), X)
6.97/7.17	= { by axiom 7 (total_function2) }
6.97/7.17	  X
6.97/7.17	
6.97/7.17	Lemma 18: ifeq(product(X, Y, Z), true, product(Z, inverse(Y), X), true) = true.
6.97/7.17	Proof:
6.97/7.17	  ifeq(product(X, Y, Z), true, product(Z, inverse(Y), X), true)
6.97/7.17	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.17	  ifeq(true, true, ifeq(product(X, Y, Z), true, product(Z, inverse(Y), X), true), true)
6.97/7.17	= { by axiom 3 (right_inverse) }
6.97/7.17	  ifeq(product(Y, inverse(Y), identity), true, ifeq(product(X, Y, Z), true, product(Z, inverse(Y), X), true), true)
6.97/7.17	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.17	  ifeq(product(Y, inverse(Y), identity), true, ifeq(true, true, ifeq(product(X, Y, Z), true, product(Z, inverse(Y), X), true), true), true)
6.97/7.17	= { by axiom 16 (right_identity) }
6.97/7.17	  ifeq(product(Y, inverse(Y), identity), true, ifeq(product(X, identity, X), true, ifeq(product(X, Y, Z), true, product(Z, inverse(Y), X), true), true), true)
6.97/7.17	= { by axiom 11 (associativity2) }
6.97/7.18	  true
6.97/7.18	
6.97/7.18	Lemma 19: inverse(inverse(X)) = X.
6.97/7.18	Proof:
6.97/7.18	  inverse(inverse(X))
6.97/7.18	= { by lemma 17 }
6.97/7.18	  ifeq2(product(identity, inverse(inverse(X)), X), true, X, inverse(inverse(X)))
6.97/7.18	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.18	  ifeq2(ifeq(true, true, product(identity, inverse(inverse(X)), X), true), true, X, inverse(inverse(X)))
6.97/7.18	= { by axiom 3 (right_inverse) }
6.97/7.18	  ifeq2(ifeq(product(X, inverse(X), identity), true, product(identity, inverse(inverse(X)), X), true), true, X, inverse(inverse(X)))
6.97/7.18	= { by lemma 18 }
6.97/7.18	  ifeq2(true, true, X, inverse(inverse(X)))
6.97/7.18	= { by axiom 15 (ifeq_axiom) }
6.97/7.18	  X
6.97/7.18	
6.97/7.18	Lemma 20: product(multiply(X, Y), inverse(Y), X) = true.
6.97/7.18	Proof:
6.97/7.18	  product(multiply(X, Y), inverse(Y), X)
6.97/7.18	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.18	  ifeq(true, true, product(multiply(X, Y), inverse(Y), X), true)
6.97/7.18	= { by axiom 14 (total_function1) }
6.97/7.18	  ifeq(product(X, Y, multiply(X, Y)), true, product(multiply(X, Y), inverse(Y), X), true)
6.97/7.18	= { by lemma 18 }
6.97/7.18	  true
6.97/7.18	
6.97/7.18	Lemma 21: ifeq2(product(Y, Z, X), true, multiply(Y, Z), X) = X.
6.97/7.18	Proof:
6.97/7.18	  ifeq2(product(Y, Z, X), true, multiply(Y, Z), X)
6.97/7.18	= { by axiom 15 (ifeq_axiom) }
6.97/7.18	  ifeq2(product(Y, Z, X), true, ifeq2(true, true, multiply(Y, Z), X), X)
6.97/7.18	= { by axiom 14 (total_function1) }
6.97/7.18	  ifeq2(product(Y, Z, X), true, ifeq2(product(Y, Z, multiply(Y, Z)), true, multiply(Y, Z), X), X)
6.97/7.18	= { by axiom 7 (total_function2) }
6.97/7.18	  X
6.97/7.18	
6.97/7.18	Lemma 22: multiply(multiply(X, Y), inverse(Y)) = X.
6.97/7.18	Proof:
6.97/7.18	  multiply(multiply(X, Y), inverse(Y))
6.97/7.18	= { by axiom 15 (ifeq_axiom) }
6.97/7.18	  ifeq2(true, true, multiply(multiply(X, Y), inverse(Y)), X)
6.97/7.18	= { by lemma 20 }
6.97/7.18	  ifeq2(product(multiply(X, Y), inverse(Y), X), true, multiply(multiply(X, Y), inverse(Y)), X)
6.97/7.18	= { by lemma 21 }
6.97/7.18	  X
6.97/7.18	
6.97/7.18	Lemma 23: multiply(identity, X) = X.
6.97/7.18	Proof:
6.97/7.18	  multiply(identity, X)
6.97/7.18	= { by axiom 15 (ifeq_axiom) }
6.97/7.18	  ifeq2(true, true, multiply(identity, X), X)
6.97/7.18	= { by axiom 14 (total_function1) }
6.97/7.18	  ifeq2(product(identity, X, multiply(identity, X)), true, multiply(identity, X), X)
6.97/7.18	= { by lemma 17 }
6.97/7.18	  X
6.97/7.18	
6.97/7.18	Lemma 24: multiply(X, multiply(inverse(X), Y)) = Y.
6.97/7.18	Proof:
6.97/7.18	  multiply(X, multiply(inverse(X), Y))
6.97/7.18	= { by lemma 22 }
6.97/7.18	  multiply(multiply(multiply(X, multiply(inverse(X), Y)), inverse(Y)), inverse(inverse(Y)))
6.97/7.18	= { by axiom 15 (ifeq_axiom) }
6.97/7.18	  multiply(ifeq2(true, true, multiply(multiply(X, multiply(inverse(X), Y)), inverse(Y)), identity), inverse(inverse(Y)))
6.97/7.18	= { by axiom 11 (associativity2) }
6.97/7.18	  multiply(ifeq2(ifeq(product(multiply(inverse(X), Y), inverse(Y), inverse(X)), true, ifeq(product(X, inverse(X), identity), true, ifeq(product(X, multiply(inverse(X), Y), multiply(X, multiply(inverse(X), Y))), true, product(multiply(X, multiply(inverse(X), Y)), inverse(Y), identity), true), true), true), true, multiply(multiply(X, multiply(inverse(X), Y)), inverse(Y)), identity), inverse(inverse(Y)))
6.97/7.18	= { by lemma 20 }
6.97/7.18	  multiply(ifeq2(ifeq(true, true, ifeq(product(X, inverse(X), identity), true, ifeq(product(X, multiply(inverse(X), Y), multiply(X, multiply(inverse(X), Y))), true, product(multiply(X, multiply(inverse(X), Y)), inverse(Y), identity), true), true), true), true, multiply(multiply(X, multiply(inverse(X), Y)), inverse(Y)), identity), inverse(inverse(Y)))
6.97/7.18	= { by axiom 3 (right_inverse) }
6.97/7.18	  multiply(ifeq2(ifeq(true, true, ifeq(true, true, ifeq(product(X, multiply(inverse(X), Y), multiply(X, multiply(inverse(X), Y))), true, product(multiply(X, multiply(inverse(X), Y)), inverse(Y), identity), true), true), true), true, multiply(multiply(X, multiply(inverse(X), Y)), inverse(Y)), identity), inverse(inverse(Y)))
6.97/7.18	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.18	  multiply(ifeq2(ifeq(true, true, ifeq(product(X, multiply(inverse(X), Y), multiply(X, multiply(inverse(X), Y))), true, product(multiply(X, multiply(inverse(X), Y)), inverse(Y), identity), true), true), true, multiply(multiply(X, multiply(inverse(X), Y)), inverse(Y)), identity), inverse(inverse(Y)))
6.97/7.18	= { by axiom 14 (total_function1) }
6.97/7.18	  multiply(ifeq2(ifeq(true, true, ifeq(true, true, product(multiply(X, multiply(inverse(X), Y)), inverse(Y), identity), true), true), true, multiply(multiply(X, multiply(inverse(X), Y)), inverse(Y)), identity), inverse(inverse(Y)))
6.97/7.18	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.18	  multiply(ifeq2(ifeq(true, true, product(multiply(X, multiply(inverse(X), Y)), inverse(Y), identity), true), true, multiply(multiply(X, multiply(inverse(X), Y)), inverse(Y)), identity), inverse(inverse(Y)))
6.97/7.18	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.18	  multiply(ifeq2(product(multiply(X, multiply(inverse(X), Y)), inverse(Y), identity), true, multiply(multiply(X, multiply(inverse(X), Y)), inverse(Y)), identity), inverse(inverse(Y)))
6.97/7.18	= { by lemma 21 }
6.97/7.18	  multiply(identity, inverse(inverse(Y)))
6.97/7.18	= { by lemma 23 }
6.97/7.18	  inverse(inverse(Y))
6.97/7.18	= { by lemma 19 }
6.97/7.18	  Y
6.97/7.18	
6.97/7.18	Lemma 25: multiply(inverse(Y), multiply(Y, X)) = X.
6.97/7.18	Proof:
6.97/7.18	  multiply(inverse(Y), multiply(Y, X))
6.97/7.18	= { by lemma 19 }
6.97/7.18	  multiply(inverse(Y), multiply(inverse(inverse(Y)), X))
6.97/7.18	= { by lemma 24 }
6.97/7.18	  X
6.97/7.18	
6.97/7.18	Lemma 26: multiply(inverse(X), inverse(Y)) = inverse(multiply(Y, X)).
6.97/7.18	Proof:
6.97/7.18	  multiply(inverse(X), inverse(Y))
6.97/7.18	= { by lemma 22 }
6.97/7.18	  multiply(inverse(X), multiply(multiply(inverse(Y), multiply(inverse(inverse(Y)), X)), inverse(multiply(inverse(inverse(Y)), X))))
6.97/7.18	= { by lemma 24 }
6.97/7.18	  multiply(inverse(X), multiply(X, inverse(multiply(inverse(inverse(Y)), X))))
6.97/7.18	= { by lemma 25 }
6.97/7.18	  inverse(multiply(inverse(inverse(Y)), X))
6.97/7.18	= { by lemma 19 }
6.97/7.18	  inverse(multiply(Y, X))
6.97/7.18	
6.97/7.18	Lemma 27: product(j, a, c) = true.
6.97/7.18	Proof:
6.97/7.18	  product(j, a, c)
6.97/7.18	= { by axiom 15 (ifeq_axiom) }
6.97/7.18	  product(ifeq2(true, true, j, d), a, c)
6.97/7.18	= { by lemma 18 }
6.97/7.18	  product(ifeq2(ifeq(product(d, inverse(b), h), true, product(h, inverse(inverse(b)), d), true), true, j, d), a, c)
6.97/7.18	= { by axiom 6 (d_times_inverse_b_is_h) }
6.97/7.18	  product(ifeq2(ifeq(true, true, product(h, inverse(inverse(b)), d), true), true, j, d), a, c)
6.97/7.18	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.18	  product(ifeq2(product(h, inverse(inverse(b)), d), true, j, d), a, c)
6.97/7.18	= { by lemma 19 }
6.97/7.18	  product(ifeq2(product(h, b, d), true, j, d), a, c)
6.97/7.18	= { by axiom 15 (ifeq_axiom) }
6.97/7.18	  product(ifeq2(product(h, b, d), true, ifeq2(true, true, j, d), d), a, c)
6.97/7.18	= { by axiom 5 (h_times_b_is_j) }
6.97/7.18	  product(ifeq2(product(h, b, d), true, ifeq2(product(h, b, j), true, j, d), d), a, c)
6.97/7.18	= { by axiom 7 (total_function2) }
6.97/7.18	  product(d, a, c)
6.97/7.18	= { by lemma 19 }
6.97/7.18	  product(d, inverse(inverse(a)), c)
6.97/7.18	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.18	  ifeq(true, true, product(d, inverse(inverse(a)), c), true)
6.97/7.18	= { by axiom 9 (c_times_inverse_a_is_d) }
6.97/7.18	  ifeq(product(c, inverse(a), d), true, product(d, inverse(inverse(a)), c), true)
6.97/7.18	= { by lemma 18 }
6.97/7.18	  true
6.97/7.18	
6.97/7.18	Lemma 28: ifeq(product(X, Y, Z), true, product(X, multiply(Y, W), multiply(Z, W)), true) = true.
6.97/7.18	Proof:
6.97/7.18	  ifeq(product(X, Y, Z), true, product(X, multiply(Y, W), multiply(Z, W)), true)
6.97/7.18	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.18	  ifeq(true, true, ifeq(product(X, Y, Z), true, product(X, multiply(Y, W), multiply(Z, W)), true), true)
6.97/7.18	= { by axiom 14 (total_function1) }
6.97/7.18	  ifeq(product(Y, W, multiply(Y, W)), true, ifeq(product(X, Y, Z), true, product(X, multiply(Y, W), multiply(Z, W)), true), true)
6.97/7.18	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.18	  ifeq(true, true, ifeq(product(Y, W, multiply(Y, W)), true, ifeq(product(X, Y, Z), true, product(X, multiply(Y, W), multiply(Z, W)), true), true), true)
6.97/7.18	= { by axiom 14 (total_function1) }
6.97/7.18	  ifeq(product(Z, W, multiply(Z, W)), true, ifeq(product(Y, W, multiply(Y, W)), true, ifeq(product(X, Y, Z), true, product(X, multiply(Y, W), multiply(Z, W)), true), true), true)
6.97/7.18	= { by axiom 1 (associativity1) }
6.97/7.18	  true
6.97/7.18	
6.97/7.18	Lemma 29: multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
6.97/7.18	Proof:
6.97/7.18	  multiply(multiply(X, Y), Z)
6.97/7.18	= { by lemma 21 }
6.97/7.18	  ifeq2(product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
6.97/7.18	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.18	  ifeq2(ifeq(true, true, product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true), true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
6.97/7.18	= { by axiom 14 (total_function1) }
6.97/7.18	  ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true), true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
6.97/7.18	= { by lemma 28 }
6.97/7.18	  ifeq2(true, true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
6.97/7.18	= { by axiom 15 (ifeq_axiom) }
6.97/7.18	  multiply(X, multiply(Y, Z))
6.97/7.18	
6.97/7.18	Lemma 30: multiply(X, X) = inverse(X).
6.97/7.18	Proof:
6.97/7.18	  multiply(X, X)
6.97/7.18	= { by lemma 22 }
6.97/7.18	  multiply(multiply(multiply(X, X), X), inverse(X))
6.97/7.18	= { by axiom 7 (total_function2) }
6.97/7.18	  multiply(ifeq2(product(multiply(X, X), X, multiply(multiply(X, X), X)), true, ifeq2(product(multiply(X, X), X, identity), true, identity, multiply(multiply(X, X), X)), multiply(multiply(X, X), X)), inverse(X))
6.97/7.18	= { by axiom 14 (total_function1) }
6.97/7.18	  multiply(ifeq2(true, true, ifeq2(product(multiply(X, X), X, identity), true, identity, multiply(multiply(X, X), X)), multiply(multiply(X, X), X)), inverse(X))
6.97/7.18	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.18	  multiply(ifeq2(true, true, ifeq2(ifeq(true, true, product(multiply(X, X), X, identity), true), true, identity, multiply(multiply(X, X), X)), multiply(multiply(X, X), X)), inverse(X))
6.97/7.18	= { by axiom 14 (total_function1) }
6.97/7.18	  multiply(ifeq2(true, true, ifeq2(ifeq(product(X, X, multiply(X, X)), true, product(multiply(X, X), X, identity), true), true, identity, multiply(multiply(X, X), X)), multiply(multiply(X, X), X)), inverse(X))
6.97/7.18	= { by axiom 2 (x_cubed_is_identity_2) }
6.97/7.18	  multiply(ifeq2(true, true, ifeq2(true, true, identity, multiply(multiply(X, X), X)), multiply(multiply(X, X), X)), inverse(X))
6.97/7.18	= { by axiom 15 (ifeq_axiom) }
6.97/7.18	  multiply(ifeq2(true, true, identity, multiply(multiply(X, X), X)), inverse(X))
6.97/7.18	= { by axiom 15 (ifeq_axiom) }
6.97/7.18	  multiply(identity, inverse(X))
6.97/7.18	= { by lemma 23 }
6.97/7.18	  inverse(X)
6.97/7.18	
6.97/7.18	Lemma 31: ifeq(product(X, b, Y), true, ifeq(product(Z, a, X), true, product(Z, c, Y), true), true) = true.
6.97/7.18	Proof:
6.97/7.18	  ifeq(product(X, b, Y), true, ifeq(product(Z, a, X), true, product(Z, c, Y), true), true)
6.97/7.18	= { by axiom 10 (ifeq_axiom_001) }
6.97/7.18	  ifeq(product(X, b, Y), true, ifeq(true, true, ifeq(product(Z, a, X), true, product(Z, c, Y), true), true), true)
6.97/7.18	= { by axiom 13 (a_times_b_is_c) }
6.97/7.18	  ifeq(product(X, b, Y), true, ifeq(product(a, b, c), true, ifeq(product(Z, a, X), true, product(Z, c, Y), true), true), true)
6.97/7.18	= { by axiom 1 (associativity1) }
7.26/7.42	  true
7.26/7.42	
7.26/7.42	Goal 1 (prove_k_times_inverse_b_is_e): product(k, inverse(b), identity) = true.
7.26/7.42	Proof:
7.26/7.42	  product(k, inverse(b), identity)
7.26/7.42	= { by lemma 25 }
7.26/7.42	  product(multiply(inverse(a), multiply(a, k)), inverse(b), identity)
7.26/7.42	= { by lemma 21 }
7.26/7.42	  product(multiply(inverse(a), multiply(a, ifeq2(product(c, inverse(multiply(h, a)), k), true, multiply(c, inverse(multiply(h, a))), k))), inverse(b), identity)
7.26/7.42	= { by lemma 26 }
7.26/7.42	  product(multiply(inverse(a), multiply(a, ifeq2(product(c, multiply(inverse(a), inverse(h)), k), true, multiply(c, inverse(multiply(h, a))), k))), inverse(b), identity)
7.26/7.42	= { by axiom 10 (ifeq_axiom_001) }
7.26/7.42	  product(multiply(inverse(a), multiply(a, ifeq2(ifeq(true, true, product(c, multiply(inverse(a), inverse(h)), k), true), true, multiply(c, inverse(multiply(h, a))), k))), inverse(b), identity)
7.26/7.42	= { by lemma 18 }
7.26/7.42	  product(multiply(inverse(a), multiply(a, ifeq2(ifeq(ifeq(product(j, a, c), true, product(c, inverse(a), j), true), true, product(c, multiply(inverse(a), inverse(h)), k), true), true, multiply(c, inverse(multiply(h, a))), k))), inverse(b), identity)
7.26/7.42	= { by lemma 27 }
7.26/7.42	  product(multiply(inverse(a), multiply(a, ifeq2(ifeq(ifeq(true, true, product(c, inverse(a), j), true), true, product(c, multiply(inverse(a), inverse(h)), k), true), true, multiply(c, inverse(multiply(h, a))), k))), inverse(b), identity)
7.26/7.42	= { by axiom 10 (ifeq_axiom_001) }
7.26/7.42	  product(multiply(inverse(a), multiply(a, ifeq2(ifeq(product(c, inverse(a), j), true, product(c, multiply(inverse(a), inverse(h)), k), true), true, multiply(c, inverse(multiply(h, a))), k))), inverse(b), identity)
7.26/7.42	= { by axiom 10 (ifeq_axiom_001) }
7.26/7.42	  product(multiply(inverse(a), multiply(a, ifeq2(ifeq(true, true, ifeq(product(c, inverse(a), j), true, product(c, multiply(inverse(a), inverse(h)), k), true), true), true, multiply(c, inverse(multiply(h, a))), k))), inverse(b), identity)
7.26/7.42	= { by axiom 14 (total_function1) }
7.26/7.42	  product(multiply(inverse(a), multiply(a, ifeq2(ifeq(product(inverse(a), inverse(h), multiply(inverse(a), inverse(h))), true, ifeq(product(c, inverse(a), j), true, product(c, multiply(inverse(a), inverse(h)), k), true), true), true, multiply(c, inverse(multiply(h, a))), k))), inverse(b), identity)
7.26/7.42	= { by axiom 10 (ifeq_axiom_001) }
7.26/7.42	  product(multiply(inverse(a), multiply(a, ifeq2(ifeq(true, true, ifeq(product(inverse(a), inverse(h), multiply(inverse(a), inverse(h))), true, ifeq(product(c, inverse(a), j), true, product(c, multiply(inverse(a), inverse(h)), k), true), true), true), true, multiply(c, inverse(multiply(h, a))), k))), inverse(b), identity)
7.26/7.42	= { by axiom 8 (j_times_inverse_h_is_k) }
7.26/7.42	  product(multiply(inverse(a), multiply(a, ifeq2(ifeq(product(j, inverse(h), k), true, ifeq(product(inverse(a), inverse(h), multiply(inverse(a), inverse(h))), true, ifeq(product(c, inverse(a), j), true, product(c, multiply(inverse(a), inverse(h)), k), true), true), true), true, multiply(c, inverse(multiply(h, a))), k))), inverse(b), identity)
7.26/7.42	= { by axiom 1 (associativity1) }
7.26/7.42	  product(multiply(inverse(a), multiply(a, ifeq2(true, true, multiply(c, inverse(multiply(h, a))), k))), inverse(b), identity)
7.26/7.42	= { by axiom 15 (ifeq_axiom) }
7.26/7.42	  product(multiply(inverse(a), multiply(a, multiply(c, inverse(multiply(h, a))))), inverse(b), identity)
7.26/7.42	= { by lemma 29 }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), inverse(multiply(h, a)))), inverse(b), identity)
7.26/7.42	= { by lemma 26 }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(inverse(a), inverse(h)))), inverse(b), identity)
7.26/7.42	= { by lemma 22 }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(multiply(multiply(inverse(a), inverse(h)), j), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by lemma 21 }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(product(multiply(multiply(inverse(a), inverse(h)), h), b, multiply(multiply(inverse(a), inverse(h)), j)), true, multiply(multiply(multiply(inverse(a), inverse(h)), h), b), multiply(multiply(inverse(a), inverse(h)), j)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 10 (ifeq_axiom_001) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(ifeq(true, true, product(multiply(multiply(inverse(a), inverse(h)), h), b, multiply(multiply(inverse(a), inverse(h)), j)), true), true, multiply(multiply(multiply(inverse(a), inverse(h)), h), b), multiply(multiply(inverse(a), inverse(h)), j)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 14 (total_function1) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(ifeq(product(multiply(inverse(a), inverse(h)), h, multiply(multiply(inverse(a), inverse(h)), h)), true, product(multiply(multiply(inverse(a), inverse(h)), h), b, multiply(multiply(inverse(a), inverse(h)), j)), true), true, multiply(multiply(multiply(inverse(a), inverse(h)), h), b), multiply(multiply(inverse(a), inverse(h)), j)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 10 (ifeq_axiom_001) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(ifeq(true, true, ifeq(product(multiply(inverse(a), inverse(h)), h, multiply(multiply(inverse(a), inverse(h)), h)), true, product(multiply(multiply(inverse(a), inverse(h)), h), b, multiply(multiply(inverse(a), inverse(h)), j)), true), true), true, multiply(multiply(multiply(inverse(a), inverse(h)), h), b), multiply(multiply(inverse(a), inverse(h)), j)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 14 (total_function1) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(ifeq(product(multiply(inverse(a), inverse(h)), j, multiply(multiply(inverse(a), inverse(h)), j)), true, ifeq(product(multiply(inverse(a), inverse(h)), h, multiply(multiply(inverse(a), inverse(h)), h)), true, product(multiply(multiply(inverse(a), inverse(h)), h), b, multiply(multiply(inverse(a), inverse(h)), j)), true), true), true, multiply(multiply(multiply(inverse(a), inverse(h)), h), b), multiply(multiply(inverse(a), inverse(h)), j)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 10 (ifeq_axiom_001) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(ifeq(true, true, ifeq(product(multiply(inverse(a), inverse(h)), j, multiply(multiply(inverse(a), inverse(h)), j)), true, ifeq(product(multiply(inverse(a), inverse(h)), h, multiply(multiply(inverse(a), inverse(h)), h)), true, product(multiply(multiply(inverse(a), inverse(h)), h), b, multiply(multiply(inverse(a), inverse(h)), j)), true), true), true), true, multiply(multiply(multiply(inverse(a), inverse(h)), h), b), multiply(multiply(inverse(a), inverse(h)), j)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 5 (h_times_b_is_j) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(ifeq(product(h, b, j), true, ifeq(product(multiply(inverse(a), inverse(h)), j, multiply(multiply(inverse(a), inverse(h)), j)), true, ifeq(product(multiply(inverse(a), inverse(h)), h, multiply(multiply(inverse(a), inverse(h)), h)), true, product(multiply(multiply(inverse(a), inverse(h)), h), b, multiply(multiply(inverse(a), inverse(h)), j)), true), true), true), true, multiply(multiply(multiply(inverse(a), inverse(h)), h), b), multiply(multiply(inverse(a), inverse(h)), j)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 11 (associativity2) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(true, true, multiply(multiply(multiply(inverse(a), inverse(h)), h), b), multiply(multiply(inverse(a), inverse(h)), j)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 15 (ifeq_axiom) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(multiply(multiply(multiply(inverse(a), inverse(h)), h), b), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by lemma 19 }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(multiply(multiply(multiply(inverse(a), inverse(h)), inverse(inverse(h))), b), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by lemma 22 }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(multiply(inverse(a), b), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by lemma 30 }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(multiply(multiply(a, a), b), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by lemma 21 }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(product(a, c, multiply(multiply(a, a), b)), true, multiply(a, c), multiply(multiply(a, a), b)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 10 (ifeq_axiom_001) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(ifeq(true, true, product(a, c, multiply(multiply(a, a), b)), true), true, multiply(a, c), multiply(multiply(a, a), b)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 14 (total_function1) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(ifeq(product(a, a, multiply(a, a)), true, product(a, c, multiply(multiply(a, a), b)), true), true, multiply(a, c), multiply(multiply(a, a), b)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 10 (ifeq_axiom_001) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(ifeq(true, true, ifeq(product(a, a, multiply(a, a)), true, product(a, c, multiply(multiply(a, a), b)), true), true), true, multiply(a, c), multiply(multiply(a, a), b)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 14 (total_function1) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(ifeq(product(multiply(a, a), b, multiply(multiply(a, a), b)), true, ifeq(product(a, a, multiply(a, a)), true, product(a, c, multiply(multiply(a, a), b)), true), true), true, multiply(a, c), multiply(multiply(a, a), b)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by lemma 31 }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(ifeq2(true, true, multiply(a, c), multiply(multiply(a, a), b)), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by axiom 15 (ifeq_axiom) }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(a, c), multiply(multiply(a, c), inverse(j)))), inverse(b), identity)
7.26/7.42	= { by lemma 29 }
7.26/7.42	  product(multiply(inverse(a), multiply(multiply(multiply(a, c), multiply(a, c)), inverse(j))), inverse(b), identity)
7.26/7.42	= { by lemma 30 }
7.26/7.42	  product(multiply(inverse(a), multiply(inverse(multiply(a, c)), inverse(j))), inverse(b), identity)
7.26/7.42	= { by lemma 26 }
7.26/7.42	  product(multiply(inverse(a), inverse(multiply(j, multiply(a, c)))), inverse(b), identity)
7.26/7.42	= { by axiom 15 (ifeq_axiom) }
7.26/7.42	  product(multiply(inverse(a), inverse(ifeq2(true, true, multiply(j, multiply(a, c)), multiply(c, c)))), inverse(b), identity)
7.26/7.42	= { by lemma 28 }
7.26/7.42	  product(multiply(inverse(a), inverse(ifeq2(ifeq(product(j, a, c), true, product(j, multiply(a, c), multiply(c, c)), true), true, multiply(j, multiply(a, c)), multiply(c, c)))), inverse(b), identity)
7.26/7.42	= { by lemma 27 }
7.26/7.42	  product(multiply(inverse(a), inverse(ifeq2(ifeq(true, true, product(j, multiply(a, c), multiply(c, c)), true), true, multiply(j, multiply(a, c)), multiply(c, c)))), inverse(b), identity)
7.26/7.42	= { by axiom 10 (ifeq_axiom_001) }
7.26/7.42	  product(multiply(inverse(a), inverse(ifeq2(product(j, multiply(a, c), multiply(c, c)), true, multiply(j, multiply(a, c)), multiply(c, c)))), inverse(b), identity)
7.26/7.42	= { by lemma 21 }
7.26/7.42	  product(multiply(inverse(a), inverse(multiply(c, c))), inverse(b), identity)
7.26/7.42	= { by lemma 30 }
7.26/7.42	  product(multiply(inverse(a), inverse(inverse(c))), inverse(b), identity)
7.26/7.42	= { by lemma 19 }
7.26/7.42	  product(multiply(inverse(a), c), inverse(b), identity)
7.26/7.42	= { by axiom 15 (ifeq_axiom) }
7.26/7.42	  product(ifeq2(true, true, multiply(inverse(a), c), b), inverse(b), identity)
7.26/7.42	= { by lemma 31 }
7.26/7.42	  product(ifeq2(ifeq(product(identity, b, b), true, ifeq(product(inverse(a), a, identity), true, product(inverse(a), c, b), true), true), true, multiply(inverse(a), c), b), inverse(b), identity)
7.26/7.42	= { by axiom 4 (left_identity) }
7.26/7.42	  product(ifeq2(ifeq(true, true, ifeq(product(inverse(a), a, identity), true, product(inverse(a), c, b), true), true), true, multiply(inverse(a), c), b), inverse(b), identity)
7.26/7.42	= { by axiom 10 (ifeq_axiom_001) }
7.26/7.42	  product(ifeq2(ifeq(product(inverse(a), a, identity), true, product(inverse(a), c, b), true), true, multiply(inverse(a), c), b), inverse(b), identity)
7.26/7.42	= { by axiom 12 (left_inverse) }
7.26/7.42	  product(ifeq2(ifeq(true, true, product(inverse(a), c, b), true), true, multiply(inverse(a), c), b), inverse(b), identity)
7.26/7.42	= { by axiom 10 (ifeq_axiom_001) }
7.26/7.42	  product(ifeq2(product(inverse(a), c, b), true, multiply(inverse(a), c), b), inverse(b), identity)
7.26/7.42	= { by lemma 21 }
7.26/7.42	  product(b, inverse(b), identity)
7.26/7.42	= { by axiom 3 (right_inverse) }
7.26/7.42	  true
7.26/7.42	% SZS output end Proof
7.26/7.42	
7.26/7.42	RESULT: Unsatisfiable (the axioms are contradictory).
7.26/7.42	EOF