TSTP Solution File: KLE120+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE120+1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 05:35:57 EDT 2023
% Result : Theorem 11.45s 1.84s
% Output : Proof 11.94s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : KLE120+1 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.13 % 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 : n016.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 : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Tue Aug 29 12:52:31 EDT 2023
% 0.13/0.34 % CPUTime :
% 11.45/1.84 Command-line arguments: --flatten
% 11.45/1.84
% 11.45/1.84 % SZS status Theorem
% 11.45/1.84
% 11.94/1.88 % SZS output start Proof
% 11.94/1.88 Take the following subset of the input axioms:
% 11.94/1.88 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 11.94/1.88 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 11.94/1.88 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 11.94/1.88 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 11.94/1.88 fof(backward_diamond, axiom, ![X0, X1]: backward_diamond(X0, X1)=codomain(multiplication(codomain(X1), X0))).
% 11.94/1.88 fof(codomain1, axiom, ![X0_2]: multiplication(X0_2, coantidomain(X0_2))=zero).
% 11.94/1.88 fof(codomain2, axiom, ![X0_2, X1_2]: addition(coantidomain(multiplication(X0_2, X1_2)), coantidomain(multiplication(coantidomain(coantidomain(X0_2)), X1_2)))=coantidomain(multiplication(coantidomain(coantidomain(X0_2)), X1_2))).
% 11.94/1.88 fof(codomain3, axiom, ![X0_2]: addition(coantidomain(coantidomain(X0_2)), coantidomain(X0_2))=one).
% 11.94/1.88 fof(codomain4, axiom, ![X0_2]: codomain(X0_2)=coantidomain(coantidomain(X0_2))).
% 11.94/1.88 fof(complement, axiom, ![X0_2]: c(X0_2)=antidomain(domain(X0_2))).
% 11.94/1.88 fof(domain1, axiom, ![X0_2]: multiplication(antidomain(X0_2), X0_2)=zero).
% 11.94/1.88 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 11.94/1.88 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 11.94/1.88 fof(domain_difference, axiom, ![X0_2, X1_2]: domain_difference(X0_2, X1_2)=multiplication(domain(X0_2), antidomain(X1_2))).
% 11.94/1.88 fof(goals, conjecture, ![X0_2]: addition(backward_diamond(one, domain(X0_2)), domain(X0_2))=domain(X0_2)).
% 11.94/1.88 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 11.94/1.88 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 11.94/1.88 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 11.94/1.88 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 11.94/1.88 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 11.94/1.88 fof(right_annihilation, axiom, ![A3]: multiplication(A3, zero)=zero).
% 11.94/1.88 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 11.94/1.88
% 11.94/1.88 Now clausify the problem and encode Horn clauses using encoding 3 of
% 11.94/1.88 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 11.94/1.88 We repeatedly replace C & s=t => u=v by the two clauses:
% 11.94/1.88 fresh(y, y, x1...xn) = u
% 11.94/1.88 C => fresh(s, t, x1...xn) = v
% 11.94/1.88 where fresh is a fresh function symbol and x1..xn are the free
% 11.94/1.88 variables of u and v.
% 11.94/1.88 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 11.94/1.88 input problem has no model of domain size 1).
% 11.94/1.88
% 11.94/1.88 The encoding turns the above axioms into the following unit equations and goals:
% 11.94/1.88
% 11.94/1.88 Axiom 1 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 11.94/1.88 Axiom 2 (complement): c(X) = antidomain(domain(X)).
% 11.94/1.88 Axiom 3 (domain4): domain(X) = antidomain(antidomain(X)).
% 11.94/1.88 Axiom 4 (additive_idempotence): addition(X, X) = X.
% 11.94/1.88 Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 11.94/1.88 Axiom 6 (additive_identity): addition(X, zero) = X.
% 11.94/1.88 Axiom 7 (right_annihilation): multiplication(X, zero) = zero.
% 11.94/1.88 Axiom 8 (multiplicative_right_identity): multiplication(X, one) = X.
% 11.94/1.88 Axiom 9 (multiplicative_left_identity): multiplication(one, X) = X.
% 11.94/1.88 Axiom 10 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 11.94/1.88 Axiom 11 (domain1): multiplication(antidomain(X), X) = zero.
% 11.94/1.88 Axiom 12 (order): fresh(X, X, Y, Z) = true.
% 11.94/1.88 Axiom 13 (order_1): fresh2(X, X, Y, Z) = Z.
% 11.94/1.88 Axiom 14 (backward_diamond): backward_diamond(X, Y) = codomain(multiplication(codomain(Y), X)).
% 11.94/1.88 Axiom 15 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 11.94/1.88 Axiom 16 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 11.94/1.88 Axiom 17 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 11.94/1.88 Axiom 18 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 11.94/1.88 Axiom 19 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 11.94/1.88 Axiom 20 (order): fresh(addition(X, Y), Y, X, Y) = leq(X, Y).
% 11.94/1.88 Axiom 21 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
% 11.94/1.88 Axiom 22 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 11.94/1.88 Axiom 23 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 11.94/1.88 Axiom 24 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
% 11.94/1.88
% 11.94/1.88 Lemma 25: addition(domain(X), antidomain(X)) = one.
% 11.94/1.88 Proof:
% 11.94/1.88 addition(domain(X), antidomain(X))
% 11.94/1.88 = { by axiom 3 (domain4) }
% 11.94/1.88 addition(antidomain(antidomain(X)), antidomain(X))
% 11.94/1.88 = { by axiom 19 (domain3) }
% 11.94/1.88 one
% 11.94/1.88
% 11.94/1.88 Lemma 26: multiplication(domain(X), X) = X.
% 11.94/1.88 Proof:
% 11.94/1.88 multiplication(domain(X), X)
% 11.94/1.88 = { by axiom 6 (additive_identity) R->L }
% 11.94/1.88 addition(multiplication(domain(X), X), zero)
% 11.94/1.88 = { by axiom 11 (domain1) R->L }
% 11.94/1.88 addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
% 11.94/1.88 = { by axiom 23 (left_distributivity) R->L }
% 11.94/1.88 multiplication(addition(domain(X), antidomain(X)), X)
% 11.94/1.88 = { by lemma 25 }
% 11.94/1.88 multiplication(one, X)
% 11.94/1.88 = { by axiom 9 (multiplicative_left_identity) }
% 11.94/1.88 X
% 11.94/1.88
% 11.94/1.88 Lemma 27: domain(antidomain(X)) = c(X).
% 11.94/1.88 Proof:
% 11.94/1.88 domain(antidomain(X))
% 11.94/1.88 = { by axiom 3 (domain4) }
% 11.94/1.88 antidomain(antidomain(antidomain(X)))
% 11.94/1.88 = { by axiom 3 (domain4) R->L }
% 11.94/1.88 antidomain(domain(X))
% 11.94/1.88 = { by axiom 2 (complement) R->L }
% 11.94/1.88 c(X)
% 11.94/1.88
% 11.94/1.88 Lemma 28: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 11.94/1.88 Proof:
% 11.94/1.88 multiplication(antidomain(X), addition(X, Y))
% 11.94/1.88 = { by axiom 5 (additive_commutativity) R->L }
% 11.94/1.88 multiplication(antidomain(X), addition(Y, X))
% 11.94/1.88 = { by axiom 22 (right_distributivity) }
% 11.94/1.88 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 11.94/1.88 = { by axiom 11 (domain1) }
% 11.94/1.88 addition(multiplication(antidomain(X), Y), zero)
% 11.94/1.88 = { by axiom 6 (additive_identity) }
% 11.94/1.88 multiplication(antidomain(X), Y)
% 11.94/1.88
% 11.94/1.88 Lemma 29: antidomain(X) = c(X).
% 11.94/1.88 Proof:
% 11.94/1.88 antidomain(X)
% 11.94/1.88 = { by lemma 26 R->L }
% 11.94/1.88 multiplication(domain(antidomain(X)), antidomain(X))
% 11.94/1.88 = { by lemma 27 }
% 11.94/1.88 multiplication(c(X), antidomain(X))
% 11.94/1.88 = { by axiom 2 (complement) }
% 11.94/1.88 multiplication(antidomain(domain(X)), antidomain(X))
% 11.94/1.88 = { by lemma 28 R->L }
% 11.94/1.88 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 11.94/1.88 = { by lemma 25 }
% 11.94/1.88 multiplication(antidomain(domain(X)), one)
% 11.94/1.88 = { by axiom 8 (multiplicative_right_identity) }
% 11.94/1.88 antidomain(domain(X))
% 11.94/1.88 = { by axiom 2 (complement) R->L }
% 11.94/1.88 c(X)
% 11.94/1.88
% 11.94/1.88 Lemma 30: coantidomain(one) = zero.
% 11.94/1.88 Proof:
% 11.94/1.88 coantidomain(one)
% 11.94/1.88 = { by axiom 9 (multiplicative_left_identity) R->L }
% 11.94/1.88 multiplication(one, coantidomain(one))
% 11.94/1.88 = { by axiom 10 (codomain1) }
% 11.94/1.88 zero
% 11.94/1.88
% 11.94/1.88 Lemma 31: antidomain(c(X)) = c(antidomain(X)).
% 11.94/1.88 Proof:
% 11.94/1.88 antidomain(c(X))
% 11.94/1.88 = { by lemma 27 R->L }
% 11.94/1.88 antidomain(domain(antidomain(X)))
% 11.94/1.88 = { by axiom 2 (complement) R->L }
% 11.94/1.88 c(antidomain(X))
% 11.94/1.88
% 11.94/1.88 Lemma 32: coantidomain(codomain(X)) = codomain(coantidomain(X)).
% 11.94/1.88 Proof:
% 11.94/1.88 coantidomain(codomain(X))
% 11.94/1.88 = { by axiom 1 (codomain4) }
% 11.94/1.88 coantidomain(coantidomain(coantidomain(X)))
% 11.94/1.88 = { by axiom 1 (codomain4) R->L }
% 11.94/1.88 codomain(coantidomain(X))
% 11.94/1.88
% 11.94/1.88 Lemma 33: addition(codomain(X), coantidomain(X)) = one.
% 11.94/1.88 Proof:
% 11.94/1.88 addition(codomain(X), coantidomain(X))
% 11.94/1.88 = { by axiom 1 (codomain4) }
% 11.94/1.88 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 11.94/1.88 = { by axiom 18 (codomain3) }
% 11.94/1.88 one
% 11.94/1.88
% 11.94/1.88 Lemma 34: multiplication(X, codomain(X)) = X.
% 11.94/1.88 Proof:
% 11.94/1.88 multiplication(X, codomain(X))
% 11.94/1.88 = { by axiom 6 (additive_identity) R->L }
% 11.94/1.88 addition(multiplication(X, codomain(X)), zero)
% 11.94/1.88 = { by axiom 10 (codomain1) R->L }
% 11.94/1.88 addition(multiplication(X, codomain(X)), multiplication(X, coantidomain(X)))
% 11.94/1.88 = { by axiom 22 (right_distributivity) R->L }
% 11.94/1.88 multiplication(X, addition(codomain(X), coantidomain(X)))
% 11.94/1.88 = { by lemma 33 }
% 11.94/1.88 multiplication(X, one)
% 11.94/1.88 = { by axiom 8 (multiplicative_right_identity) }
% 11.94/1.88 X
% 11.94/1.88
% 11.94/1.88 Lemma 35: codomain(coantidomain(X)) = coantidomain(X).
% 11.94/1.88 Proof:
% 11.94/1.88 codomain(coantidomain(X))
% 11.94/1.88 = { by lemma 32 R->L }
% 11.94/1.88 coantidomain(codomain(X))
% 11.94/1.88 = { by axiom 9 (multiplicative_left_identity) R->L }
% 11.94/1.88 multiplication(one, coantidomain(codomain(X)))
% 11.94/1.88 = { by lemma 33 R->L }
% 11.94/1.88 multiplication(addition(codomain(X), coantidomain(X)), coantidomain(codomain(X)))
% 11.94/1.88 = { by axiom 5 (additive_commutativity) R->L }
% 11.94/1.88 multiplication(addition(coantidomain(X), codomain(X)), coantidomain(codomain(X)))
% 11.94/1.88 = { by axiom 23 (left_distributivity) }
% 11.94/1.88 addition(multiplication(coantidomain(X), coantidomain(codomain(X))), multiplication(codomain(X), coantidomain(codomain(X))))
% 11.94/1.88 = { by axiom 10 (codomain1) }
% 11.94/1.89 addition(multiplication(coantidomain(X), coantidomain(codomain(X))), zero)
% 11.94/1.89 = { by axiom 6 (additive_identity) }
% 11.94/1.89 multiplication(coantidomain(X), coantidomain(codomain(X)))
% 11.94/1.89 = { by lemma 32 }
% 11.94/1.89 multiplication(coantidomain(X), codomain(coantidomain(X)))
% 11.94/1.89 = { by lemma 34 }
% 11.94/1.89 coantidomain(X)
% 11.94/1.89
% 11.94/1.89 Lemma 36: codomain(codomain(X)) = codomain(X).
% 11.94/1.89 Proof:
% 11.94/1.89 codomain(codomain(X))
% 11.94/1.89 = { by axiom 1 (codomain4) }
% 11.94/1.89 codomain(coantidomain(coantidomain(X)))
% 11.94/1.89 = { by lemma 35 }
% 11.94/1.89 coantidomain(coantidomain(X))
% 11.94/1.89 = { by axiom 1 (codomain4) R->L }
% 11.94/1.89 codomain(X)
% 11.94/1.89
% 11.94/1.89 Lemma 37: c(domain(X)) = c(X).
% 11.94/1.89 Proof:
% 11.94/1.89 c(domain(X))
% 11.94/1.89 = { by lemma 29 R->L }
% 11.94/1.89 antidomain(domain(X))
% 11.94/1.89 = { by axiom 2 (complement) R->L }
% 11.94/1.89 c(X)
% 11.94/1.89
% 11.94/1.89 Lemma 38: backward_diamond(one, X) = codomain(codomain(X)).
% 11.94/1.89 Proof:
% 11.94/1.89 backward_diamond(one, X)
% 11.94/1.89 = { by axiom 14 (backward_diamond) }
% 11.94/1.89 codomain(multiplication(codomain(X), one))
% 11.94/1.89 = { by axiom 8 (multiplicative_right_identity) }
% 11.94/1.89 codomain(codomain(X))
% 11.94/1.89
% 11.94/1.89 Lemma 39: backward_diamond(one, X) = codomain(X).
% 11.94/1.89 Proof:
% 11.94/1.89 backward_diamond(one, X)
% 11.94/1.89 = { by lemma 38 }
% 11.94/1.89 codomain(codomain(X))
% 11.94/1.89 = { by lemma 36 }
% 11.94/1.89 codomain(X)
% 11.94/1.89
% 11.94/1.89 Lemma 40: addition(X, addition(X, Y)) = addition(X, Y).
% 11.94/1.89 Proof:
% 11.94/1.89 addition(X, addition(X, Y))
% 11.94/1.89 = { by axiom 15 (additive_associativity) }
% 11.94/1.89 addition(addition(X, X), Y)
% 11.94/1.89 = { by axiom 4 (additive_idempotence) }
% 11.94/1.89 addition(X, Y)
% 11.94/1.89
% 11.94/1.89 Lemma 41: addition(c(X), domain(X)) = one.
% 11.94/1.89 Proof:
% 11.94/1.89 addition(c(X), domain(X))
% 11.94/1.89 = { by axiom 3 (domain4) }
% 11.94/1.89 addition(c(X), antidomain(antidomain(X)))
% 11.94/1.89 = { by lemma 27 R->L }
% 11.94/1.89 addition(domain(antidomain(X)), antidomain(antidomain(X)))
% 11.94/1.89 = { by lemma 25 }
% 11.94/1.89 one
% 11.94/1.89
% 11.94/1.89 Lemma 42: addition(backward_diamond(one, X), one) = one.
% 11.94/1.89 Proof:
% 11.94/1.89 addition(backward_diamond(one, X), one)
% 11.94/1.89 = { by axiom 5 (additive_commutativity) R->L }
% 11.94/1.89 addition(one, backward_diamond(one, X))
% 11.94/1.89 = { by lemma 38 }
% 11.94/1.89 addition(one, codomain(codomain(X)))
% 11.94/1.89 = { by axiom 5 (additive_commutativity) R->L }
% 11.94/1.89 addition(codomain(codomain(X)), one)
% 11.94/1.89 = { by lemma 33 R->L }
% 11.94/1.89 addition(codomain(codomain(X)), addition(codomain(codomain(X)), coantidomain(codomain(X))))
% 11.94/1.89 = { by lemma 40 }
% 11.94/1.89 addition(codomain(codomain(X)), coantidomain(codomain(X)))
% 11.94/1.89 = { by lemma 33 }
% 11.94/1.89 one
% 11.94/1.89
% 11.94/1.89 Lemma 43: multiplication(domain(X), domain(Y)) = domain_difference(X, antidomain(Y)).
% 11.94/1.89 Proof:
% 11.94/1.89 multiplication(domain(X), domain(Y))
% 11.94/1.89 = { by axiom 3 (domain4) }
% 11.94/1.89 multiplication(domain(X), antidomain(antidomain(Y)))
% 11.94/1.89 = { by axiom 16 (domain_difference) R->L }
% 11.94/1.89 domain_difference(X, antidomain(Y))
% 11.94/1.89
% 11.94/1.89 Lemma 44: multiplication(addition(X, one), Y) = addition(Y, multiplication(X, Y)).
% 11.94/1.89 Proof:
% 11.94/1.89 multiplication(addition(X, one), Y)
% 11.94/1.89 = { by axiom 5 (additive_commutativity) R->L }
% 11.94/1.89 multiplication(addition(one, X), Y)
% 11.94/1.89 = { by axiom 23 (left_distributivity) }
% 11.94/1.89 addition(multiplication(one, Y), multiplication(X, Y))
% 11.94/1.89 = { by axiom 9 (multiplicative_left_identity) }
% 11.94/1.89 addition(Y, multiplication(X, Y))
% 11.94/1.89
% 11.94/1.89 Lemma 45: addition(domain(X), c(domain(X))) = one.
% 11.94/1.89 Proof:
% 11.94/1.89 addition(domain(X), c(domain(X)))
% 11.94/1.89 = { by axiom 5 (additive_commutativity) R->L }
% 11.94/1.89 addition(c(domain(X)), domain(X))
% 11.94/1.89 = { by lemma 37 }
% 11.94/1.89 addition(c(X), domain(X))
% 11.94/1.89 = { by lemma 41 }
% 11.94/1.89 one
% 11.94/1.89
% 11.94/1.89 Lemma 46: backward_diamond(c(domain(X)), domain(X)) = zero.
% 11.94/1.89 Proof:
% 11.94/1.89 backward_diamond(c(domain(X)), domain(X))
% 11.94/1.89 = { by axiom 14 (backward_diamond) }
% 11.94/1.89 codomain(multiplication(codomain(domain(X)), c(domain(X))))
% 11.94/1.89 = { by lemma 36 R->L }
% 11.94/1.89 codomain(codomain(multiplication(codomain(domain(X)), c(domain(X)))))
% 11.94/1.89 = { by axiom 14 (backward_diamond) R->L }
% 11.94/1.89 codomain(backward_diamond(c(domain(X)), domain(X)))
% 11.94/1.89 = { by axiom 1 (codomain4) }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), domain(X))))
% 11.94/1.89 = { by lemma 26 R->L }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(domain(domain(X)), domain(X)))))
% 11.94/1.89 = { by axiom 3 (domain4) }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(domain(antidomain(antidomain(X))), domain(X)))))
% 11.94/1.89 = { by lemma 27 }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(c(antidomain(X)), domain(X)))))
% 11.94/1.89 = { by lemma 31 R->L }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(antidomain(c(X)), domain(X)))))
% 11.94/1.89 = { by axiom 8 (multiplicative_right_identity) R->L }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(multiplication(antidomain(c(X)), one), domain(X)))))
% 11.94/1.89 = { by lemma 41 R->L }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(multiplication(antidomain(c(X)), addition(c(X), domain(X))), domain(X)))))
% 11.94/1.89 = { by lemma 28 }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(multiplication(antidomain(c(X)), domain(X)), domain(X)))))
% 11.94/1.89 = { by lemma 31 }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(multiplication(c(antidomain(X)), domain(X)), domain(X)))))
% 11.94/1.89 = { by lemma 27 R->L }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(multiplication(domain(antidomain(antidomain(X))), domain(X)), domain(X)))))
% 11.94/1.89 = { by lemma 43 }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(domain_difference(antidomain(antidomain(X)), antidomain(X)), domain(X)))))
% 11.94/1.89 = { by axiom 16 (domain_difference) }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(multiplication(domain(antidomain(antidomain(X))), antidomain(antidomain(X))), domain(X)))))
% 11.94/1.89 = { by lemma 26 }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(antidomain(antidomain(X)), domain(X)))))
% 11.94/1.89 = { by axiom 3 (domain4) R->L }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(domain(X)), multiplication(domain(X), domain(X)))))
% 11.94/1.89 = { by lemma 37 }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(c(X), multiplication(domain(X), domain(X)))))
% 11.94/1.89 = { by lemma 27 R->L }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(domain(antidomain(X)), multiplication(domain(X), domain(X)))))
% 11.94/1.89 = { by lemma 43 }
% 11.94/1.89 coantidomain(coantidomain(backward_diamond(domain(antidomain(X)), domain_difference(X, antidomain(X)))))
% 11.94/1.89 = { by axiom 14 (backward_diamond) }
% 11.94/1.89 coantidomain(coantidomain(codomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by lemma 32 }
% 11.94/1.89 coantidomain(codomain(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by lemma 35 }
% 11.94/1.89 coantidomain(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))))
% 11.94/1.89 = { by axiom 1 (codomain4) }
% 11.94/1.89 coantidomain(coantidomain(multiplication(coantidomain(coantidomain(domain_difference(X, antidomain(X)))), domain(antidomain(X)))))
% 11.94/1.89 = { by axiom 24 (codomain2) R->L }
% 11.94/1.89 coantidomain(addition(coantidomain(multiplication(domain_difference(X, antidomain(X)), domain(antidomain(X)))), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(X, antidomain(X)))), domain(antidomain(X))))))
% 11.94/1.89 = { by axiom 1 (codomain4) R->L }
% 11.94/1.89 coantidomain(addition(coantidomain(multiplication(domain_difference(X, antidomain(X)), domain(antidomain(X)))), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by axiom 16 (domain_difference) }
% 11.94/1.89 coantidomain(addition(coantidomain(multiplication(multiplication(domain(X), antidomain(antidomain(X))), domain(antidomain(X)))), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by axiom 17 (multiplicative_associativity) R->L }
% 11.94/1.89 coantidomain(addition(coantidomain(multiplication(domain(X), multiplication(antidomain(antidomain(X)), domain(antidomain(X))))), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by lemma 29 }
% 11.94/1.89 coantidomain(addition(coantidomain(multiplication(domain(X), multiplication(c(antidomain(X)), domain(antidomain(X))))), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by axiom 2 (complement) }
% 11.94/1.89 coantidomain(addition(coantidomain(multiplication(domain(X), multiplication(antidomain(domain(antidomain(X))), domain(antidomain(X))))), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by axiom 11 (domain1) }
% 11.94/1.89 coantidomain(addition(coantidomain(multiplication(domain(X), zero)), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by axiom 7 (right_annihilation) }
% 11.94/1.89 coantidomain(addition(coantidomain(zero), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by lemma 30 R->L }
% 11.94/1.89 coantidomain(addition(coantidomain(coantidomain(one)), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by axiom 1 (codomain4) R->L }
% 11.94/1.89 coantidomain(addition(codomain(one), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by axiom 6 (additive_identity) R->L }
% 11.94/1.89 coantidomain(addition(addition(codomain(one), zero), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by lemma 30 R->L }
% 11.94/1.89 coantidomain(addition(addition(codomain(one), coantidomain(one)), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by lemma 33 }
% 11.94/1.89 coantidomain(addition(one, coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))))
% 11.94/1.89 = { by axiom 5 (additive_commutativity) R->L }
% 11.94/1.89 coantidomain(addition(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), one))
% 11.94/1.89 = { by axiom 21 (order_1) R->L }
% 11.94/1.90 coantidomain(fresh2(leq(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), one), true, coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), one))
% 11.94/1.90 = { by lemma 33 R->L }
% 11.94/1.90 coantidomain(fresh2(leq(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), addition(codomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))))), true, coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), one))
% 11.94/1.90 = { by axiom 5 (additive_commutativity) R->L }
% 11.94/1.90 coantidomain(fresh2(leq(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), addition(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), codomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))))), true, coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), one))
% 11.94/1.90 = { by axiom 20 (order) R->L }
% 11.94/1.90 coantidomain(fresh2(fresh(addition(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), addition(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), codomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))))), addition(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), codomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), addition(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), codomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))))), true, coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), one))
% 11.94/1.90 = { by lemma 40 }
% 11.94/1.90 coantidomain(fresh2(fresh(addition(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), codomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))), addition(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), codomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X))))), coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), addition(coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), codomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))))), true, coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), one))
% 11.94/1.90 = { by axiom 12 (order) }
% 11.94/1.90 coantidomain(fresh2(true, true, coantidomain(multiplication(codomain(domain_difference(X, antidomain(X))), domain(antidomain(X)))), one))
% 11.94/1.90 = { by axiom 13 (order_1) }
% 11.94/1.90 coantidomain(one)
% 11.94/1.90 = { by lemma 30 }
% 11.94/1.90 zero
% 11.94/1.90
% 11.94/1.90 Lemma 47: addition(backward_diamond(one, domain(X)), domain(X)) = backward_diamond(one, domain(X)).
% 11.94/1.90 Proof:
% 11.94/1.90 addition(backward_diamond(one, domain(X)), domain(X))
% 11.94/1.90 = { by axiom 5 (additive_commutativity) }
% 11.94/1.90 addition(domain(X), backward_diamond(one, domain(X)))
% 11.94/1.90 = { by lemma 39 }
% 11.94/1.90 addition(domain(X), codomain(domain(X)))
% 11.94/1.90 = { by axiom 5 (additive_commutativity) R->L }
% 11.94/1.90 addition(codomain(domain(X)), domain(X))
% 11.94/1.90 = { by lemma 34 R->L }
% 11.94/1.90 addition(codomain(domain(X)), multiplication(domain(X), codomain(domain(X))))
% 11.94/1.90 = { by lemma 44 R->L }
% 11.94/1.90 multiplication(addition(domain(X), one), codomain(domain(X)))
% 11.94/1.90 = { by lemma 25 R->L }
% 11.94/1.90 multiplication(addition(domain(X), addition(domain(X), antidomain(X))), codomain(domain(X)))
% 11.94/1.90 = { by lemma 40 }
% 11.94/1.90 multiplication(addition(domain(X), antidomain(X)), codomain(domain(X)))
% 11.94/1.90 = { by lemma 25 }
% 11.94/1.90 multiplication(one, codomain(domain(X)))
% 11.94/1.90 = { by axiom 9 (multiplicative_left_identity) }
% 11.94/1.90 codomain(domain(X))
% 11.94/1.90 = { by lemma 39 R->L }
% 11.94/1.90 backward_diamond(one, domain(X))
% 11.94/1.90
% 11.94/1.90 Lemma 48: addition(domain(X), addition(Y, c(domain(X)))) = addition(Y, one).
% 11.94/1.90 Proof:
% 11.94/1.90 addition(domain(X), addition(Y, c(domain(X))))
% 11.94/1.90 = { by axiom 5 (additive_commutativity) R->L }
% 11.94/1.90 addition(domain(X), addition(c(domain(X)), Y))
% 11.94/1.90 = { by lemma 37 }
% 11.94/1.90 addition(domain(X), addition(c(X), Y))
% 11.94/1.90 = { by lemma 29 R->L }
% 11.94/1.90 addition(domain(X), addition(antidomain(X), Y))
% 11.94/1.90 = { by axiom 15 (additive_associativity) }
% 11.94/1.90 addition(addition(domain(X), antidomain(X)), Y)
% 11.94/1.90 = { by lemma 25 }
% 11.94/1.90 addition(one, Y)
% 11.94/1.90 = { by axiom 5 (additive_commutativity) }
% 11.94/1.90 addition(Y, one)
% 11.94/1.90
% 11.94/1.90 Lemma 49: codomain(multiplication(addition(backward_diamond(one, domain(X)), domain(X)), Y)) = backward_diamond(Y, domain(X)).
% 11.94/1.90 Proof:
% 11.94/1.90 codomain(multiplication(addition(backward_diamond(one, domain(X)), domain(X)), Y))
% 11.94/1.90 = { by lemma 47 }
% 11.94/1.90 codomain(multiplication(backward_diamond(one, domain(X)), Y))
% 11.94/1.90 = { by lemma 39 }
% 11.94/1.90 codomain(multiplication(codomain(domain(X)), Y))
% 11.94/1.90 = { by axiom 14 (backward_diamond) R->L }
% 11.94/1.90 backward_diamond(Y, domain(X))
% 11.94/1.90
% 11.94/1.90 Goal 1 (goals): addition(backward_diamond(one, domain(x0)), domain(x0)) = domain(x0).
% 11.94/1.90 Proof:
% 11.94/1.90 addition(backward_diamond(one, domain(x0)), domain(x0))
% 11.94/1.90 = { by axiom 8 (multiplicative_right_identity) R->L }
% 11.94/1.90 multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), one)
% 11.94/1.90 = { by lemma 45 R->L }
% 11.94/1.90 multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), addition(domain(x0), c(domain(x0))))
% 11.94/1.90 = { by axiom 22 (right_distributivity) }
% 11.94/1.90 addition(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))
% 11.94/1.90 = { by axiom 6 (additive_identity) R->L }
% 11.94/1.90 addition(addition(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)), zero), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))
% 11.94/1.90 = { by axiom 5 (additive_commutativity) }
% 11.94/1.90 addition(addition(zero, multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))
% 11.94/1.90 = { by axiom 11 (domain1) R->L }
% 11.94/1.90 addition(addition(multiplication(antidomain(domain(x0)), domain(x0)), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))
% 11.94/1.90 = { by axiom 23 (left_distributivity) R->L }
% 11.94/1.90 addition(multiplication(addition(antidomain(domain(x0)), addition(backward_diamond(one, domain(x0)), domain(x0))), domain(x0)), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))
% 11.94/1.90 = { by lemma 29 }
% 11.94/1.90 addition(multiplication(addition(c(domain(x0)), addition(backward_diamond(one, domain(x0)), domain(x0))), domain(x0)), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))
% 11.94/1.90 = { by axiom 5 (additive_commutativity) R->L }
% 11.94/1.90 addition(multiplication(addition(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))), domain(x0)), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))
% 11.94/1.90 = { by axiom 5 (additive_commutativity) }
% 11.94/1.90 addition(multiplication(addition(addition(domain(x0), backward_diamond(one, domain(x0))), c(domain(x0))), domain(x0)), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))
% 11.94/1.90 = { by axiom 15 (additive_associativity) R->L }
% 11.94/1.90 addition(multiplication(addition(domain(x0), addition(backward_diamond(one, domain(x0)), c(domain(x0)))), domain(x0)), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))
% 11.94/1.90 = { by lemma 48 }
% 11.94/1.90 addition(multiplication(addition(backward_diamond(one, domain(x0)), one), domain(x0)), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))
% 11.94/1.90 = { by lemma 42 }
% 11.94/1.90 addition(multiplication(one, domain(x0)), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))
% 11.94/1.90 = { by axiom 9 (multiplicative_left_identity) }
% 11.94/1.90 addition(domain(x0), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))
% 11.94/1.90 = { by axiom 6 (additive_identity) R->L }
% 11.94/1.90 addition(domain(x0), addition(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))), zero))
% 11.94/1.90 = { by lemma 46 R->L }
% 11.94/1.90 addition(domain(x0), addition(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))), backward_diamond(c(domain(x0)), domain(x0))))
% 11.94/1.90 = { by lemma 49 R->L }
% 11.94/1.90 addition(domain(x0), addition(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))), codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))))
% 11.94/1.90 = { by axiom 5 (additive_commutativity) R->L }
% 11.94/1.90 addition(domain(x0), addition(codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0)))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0)))))
% 11.94/1.90 = { by lemma 34 R->L }
% 11.94/1.90 addition(domain(x0), addition(codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0)))), multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))), codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0)))))))
% 11.94/1.90 = { by lemma 44 R->L }
% 11.94/1.90 addition(domain(x0), multiplication(addition(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))), one), codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))))
% 11.94/1.90 = { by lemma 48 R->L }
% 11.94/1.90 addition(domain(x0), multiplication(addition(domain(x0), addition(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))), c(domain(x0)))), codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))))
% 11.94/1.90 = { by axiom 5 (additive_commutativity) }
% 11.94/1.90 addition(domain(x0), multiplication(addition(domain(x0), addition(c(domain(x0)), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))), codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))))
% 11.94/1.90 = { by lemma 47 }
% 11.94/1.90 addition(domain(x0), multiplication(addition(domain(x0), addition(c(domain(x0)), multiplication(backward_diamond(one, domain(x0)), c(domain(x0))))), codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))))
% 11.94/1.90 = { by lemma 44 R->L }
% 11.94/1.90 addition(domain(x0), multiplication(addition(domain(x0), multiplication(addition(backward_diamond(one, domain(x0)), one), c(domain(x0)))), codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))))
% 11.94/1.90 = { by lemma 42 }
% 11.94/1.90 addition(domain(x0), multiplication(addition(domain(x0), multiplication(one, c(domain(x0)))), codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))))
% 11.94/1.90 = { by axiom 9 (multiplicative_left_identity) }
% 11.94/1.90 addition(domain(x0), multiplication(addition(domain(x0), c(domain(x0))), codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))))
% 11.94/1.90 = { by lemma 45 }
% 11.94/1.90 addition(domain(x0), multiplication(one, codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0))))))
% 11.94/1.90 = { by axiom 9 (multiplicative_left_identity) }
% 11.94/1.90 addition(domain(x0), codomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), c(domain(x0)))))
% 11.94/1.90 = { by lemma 49 }
% 11.94/1.90 addition(domain(x0), backward_diamond(c(domain(x0)), domain(x0)))
% 11.94/1.90 = { by lemma 46 }
% 11.94/1.90 addition(domain(x0), zero)
% 11.94/1.90 = { by axiom 6 (additive_identity) }
% 11.94/1.90 domain(x0)
% 11.94/1.90 % SZS output end Proof
% 11.94/1.90
% 11.94/1.90 RESULT: Theorem (the conjecture is true).
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