TSTP Solution File: HWV003-2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HWV003-2 : TPTP v8.1.2. Released v2.7.0.
% 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 : n028.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 02:31:30 EDT 2023
% Result : Unsatisfiable 11.49s 1.81s
% Output : Proof 13.80s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.10 % Problem : HWV003-2 : TPTP v8.1.2. Released v2.7.0.
% 0.10/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.10/0.31 % Computer : n028.cluster.edu
% 0.10/0.31 % Model : x86_64 x86_64
% 0.10/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.31 % Memory : 8042.1875MB
% 0.10/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.31 % CPULimit : 300
% 0.10/0.31 % WCLimit : 300
% 0.10/0.31 % DateTime : Tue Aug 29 16:38:11 EDT 2023
% 0.10/0.32 % CPUTime :
% 11.49/1.81 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 11.49/1.81
% 11.49/1.81 % SZS status Unsatisfiable
% 11.49/1.81
% 12.94/2.05 % SZS output start Proof
% 12.94/2.05 Take the following subset of the input axioms:
% 12.94/2.05 fof(and_absorption, axiom, ![X, Y]: and(X, or(X, Y))=X).
% 12.94/2.05 fof(and_associative, axiom, ![Z, X2, Y2]: and(and(X2, Y2), Z)=and(X2, and(Y2, Z))).
% 12.94/2.05 fof(and_boundedness, axiom, ![X2]: and(X2, ll0)=ll0).
% 12.94/2.05 fof(and_commutative, axiom, ![X2, Y2]: and(X2, Y2)=and(Y2, X2)).
% 12.94/2.05 fof(and_complement, axiom, ![X2]: and(X2, not(X2))=ll0).
% 12.94/2.05 fof(and_demorgan, axiom, ![X2, Y2]: not(and(X2, Y2))=or(not(X2), not(Y2))).
% 12.94/2.05 fof(and_distributive, axiom, ![X2, Y2, Z2]: or(X2, and(Y2, Z2))=and(or(X2, Y2), or(X2, Z2))).
% 12.94/2.05 fof(and_identity, axiom, ![X2]: and(X2, ll1)=X2).
% 12.94/2.05 fof(carry, negated_conjecture, not(and(t1, t5))=carry).
% 12.94/2.05 fof(carry_def, negated_conjecture, or(and(cin, or(a, b)), and(not(cin), and(a, b)))=carry_def).
% 12.94/2.05 fof(ll0_inverse, axiom, not(ll0)=ll1).
% 12.94/2.05 fof(ll1_inverse, axiom, not(ll1)=ll0).
% 12.94/2.05 fof(not_involution, axiom, ![X2]: not(not(X2))=X2).
% 12.94/2.05 fof(or_absorption, axiom, ![X2, Y2]: or(X2, and(X2, Y2))=X2).
% 12.94/2.05 fof(or_commutative, axiom, ![X2, Y2]: or(X2, Y2)=or(Y2, X2)).
% 12.94/2.05 fof(or_complement, axiom, ![X2]: or(X2, not(X2))=ll1).
% 12.94/2.05 fof(or_demorgan, axiom, ![X2, Y2]: not(or(X2, Y2))=and(not(X2), not(Y2))).
% 12.94/2.05 fof(or_distributive, axiom, ![X2, Y2, Z2]: and(X2, or(Y2, Z2))=or(and(X2, Y2), and(X2, Z2))).
% 12.94/2.05 fof(or_identity, axiom, ![X2]: or(X2, ll0)=X2).
% 12.94/2.05 fof(prove_circuit, negated_conjecture, sum!=sum_def | carry!=carry_def).
% 12.94/2.05 fof(sum, negated_conjecture, not(and(t6, t7))=sum).
% 12.94/2.05 fof(sum_def, negated_conjecture, xor(xor(a, b), cin)=sum_def).
% 12.94/2.05 fof(t1, negated_conjecture, not(and(a, b))=t1).
% 12.94/2.05 fof(t2, negated_conjecture, not(and(a, t1))=t2).
% 12.94/2.05 fof(t3, negated_conjecture, not(and(b, t1))=t3).
% 12.94/2.05 fof(t4, negated_conjecture, not(and(t2, t3))=t4).
% 12.94/2.05 fof(t5, negated_conjecture, not(and(t4, cin))=t5).
% 12.94/2.05 fof(t6, negated_conjecture, not(and(t4, t5))=t6).
% 12.94/2.05 fof(t7, negated_conjecture, not(and(t5, cin))=t7).
% 12.94/2.05 fof(xor_associative, axiom, ![X2, Y2, Z2]: xor(X2, xor(Y2, Z2))=xor(xor(X2, Y2), Z2)).
% 12.94/2.05 fof(xor_commutative, axiom, ![X2, Y2]: xor(X2, Y2)=xor(Y2, X2)).
% 12.94/2.05 fof(xor_definition, axiom, ![X2, Y2]: or(and(not(X2), Y2), and(X2, not(Y2)))=xor(X2, Y2)).
% 12.94/2.05
% 12.94/2.05 Now clausify the problem and encode Horn clauses using encoding 3 of
% 12.94/2.05 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 12.94/2.05 We repeatedly replace C & s=t => u=v by the two clauses:
% 12.94/2.05 fresh(y, y, x1...xn) = u
% 12.94/2.05 C => fresh(s, t, x1...xn) = v
% 12.94/2.05 where fresh is a fresh function symbol and x1..xn are the free
% 12.94/2.05 variables of u and v.
% 12.94/2.05 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 12.94/2.05 input problem has no model of domain size 1).
% 12.94/2.05
% 12.94/2.05 The encoding turns the above axioms into the following unit equations and goals:
% 12.94/2.05
% 12.94/2.05 Axiom 1 (ll0_inverse): not(ll0) = ll1.
% 12.94/2.05 Axiom 2 (ll1_inverse): not(ll1) = ll0.
% 12.94/2.05 Axiom 3 (and_commutative): and(X, Y) = and(Y, X).
% 12.94/2.05 Axiom 4 (and_boundedness): and(X, ll0) = ll0.
% 12.94/2.05 Axiom 5 (and_identity): and(X, ll1) = X.
% 12.94/2.05 Axiom 6 (not_involution): not(not(X)) = X.
% 12.94/2.05 Axiom 7 (or_commutative): or(X, Y) = or(Y, X).
% 12.94/2.05 Axiom 8 (or_identity): or(X, ll0) = X.
% 12.94/2.05 Axiom 9 (xor_commutative): xor(X, Y) = xor(Y, X).
% 12.94/2.05 Axiom 10 (and_complement): and(X, not(X)) = ll0.
% 12.94/2.05 Axiom 11 (t1): not(and(a, b)) = t1.
% 12.94/2.05 Axiom 12 (t2): not(and(a, t1)) = t2.
% 12.94/2.05 Axiom 13 (t3): not(and(b, t1)) = t3.
% 12.94/2.05 Axiom 14 (carry): not(and(t1, t5)) = carry.
% 12.94/2.05 Axiom 15 (t7): not(and(t5, cin)) = t7.
% 12.94/2.05 Axiom 16 (t5): not(and(t4, cin)) = t5.
% 12.94/2.05 Axiom 17 (t6): not(and(t4, t5)) = t6.
% 12.94/2.05 Axiom 18 (t4): not(and(t2, t3)) = t4.
% 12.94/2.05 Axiom 19 (sum): not(and(t6, t7)) = sum.
% 12.94/2.06 Axiom 20 (or_complement): or(X, not(X)) = ll1.
% 12.94/2.06 Axiom 21 (and_absorption): and(X, or(X, Y)) = X.
% 12.94/2.06 Axiom 22 (or_demorgan): not(or(X, Y)) = and(not(X), not(Y)).
% 12.94/2.06 Axiom 23 (and_associative): and(and(X, Y), Z) = and(X, and(Y, Z)).
% 12.94/2.06 Axiom 24 (or_absorption): or(X, and(X, Y)) = X.
% 12.94/2.06 Axiom 25 (and_demorgan): not(and(X, Y)) = or(not(X), not(Y)).
% 12.94/2.06 Axiom 26 (xor_associative): xor(X, xor(Y, Z)) = xor(xor(X, Y), Z).
% 12.94/2.06 Axiom 27 (sum_def): xor(xor(a, b), cin) = sum_def.
% 12.94/2.06 Axiom 28 (and_distributive): or(X, and(Y, Z)) = and(or(X, Y), or(X, Z)).
% 12.94/2.06 Axiom 29 (or_distributive): and(X, or(Y, Z)) = or(and(X, Y), and(X, Z)).
% 12.94/2.06 Axiom 30 (xor_definition): or(and(not(X), Y), and(X, not(Y))) = xor(X, Y).
% 12.94/2.06 Axiom 31 (carry_def): or(and(cin, or(a, b)), and(not(cin), and(a, b))) = carry_def.
% 12.94/2.06
% 13.61/2.06 Lemma 32: or(ll0, X) = X.
% 13.61/2.06 Proof:
% 13.61/2.06 or(ll0, X)
% 13.61/2.06 = { by axiom 7 (or_commutative) R->L }
% 13.61/2.06 or(X, ll0)
% 13.61/2.06 = { by axiom 8 (or_identity) }
% 13.61/2.06 X
% 13.61/2.06
% 13.61/2.06 Lemma 33: or(X, and(Y, not(X))) = or(X, Y).
% 13.61/2.06 Proof:
% 13.61/2.06 or(X, and(Y, not(X)))
% 13.61/2.06 = { by axiom 28 (and_distributive) }
% 13.61/2.06 and(or(X, Y), or(X, not(X)))
% 13.61/2.06 = { by axiom 20 (or_complement) }
% 13.61/2.06 and(or(X, Y), ll1)
% 13.61/2.06 = { by axiom 5 (and_identity) }
% 13.61/2.06 or(X, Y)
% 13.61/2.06
% 13.61/2.06 Lemma 34: and(b, t1) = not(t3).
% 13.61/2.06 Proof:
% 13.61/2.06 and(b, t1)
% 13.61/2.06 = { by axiom 6 (not_involution) R->L }
% 13.61/2.06 not(not(and(b, t1)))
% 13.61/2.06 = { by axiom 13 (t3) }
% 13.61/2.06 not(t3)
% 13.61/2.06
% 13.61/2.06 Lemma 35: and(a, not(t3)) = ll0.
% 13.61/2.06 Proof:
% 13.61/2.06 and(a, not(t3))
% 13.61/2.06 = { by lemma 34 R->L }
% 13.61/2.06 and(a, and(b, t1))
% 13.61/2.06 = { by axiom 23 (and_associative) R->L }
% 13.61/2.06 and(and(a, b), t1)
% 13.61/2.06 = { by axiom 11 (t1) R->L }
% 13.61/2.06 and(and(a, b), not(and(a, b)))
% 13.61/2.06 = { by axiom 10 (and_complement) }
% 13.61/2.06 ll0
% 13.61/2.06
% 13.61/2.06 Lemma 36: or(a, t3) = t3.
% 13.61/2.06 Proof:
% 13.61/2.06 or(a, t3)
% 13.61/2.06 = { by axiom 7 (or_commutative) R->L }
% 13.61/2.06 or(t3, a)
% 13.61/2.06 = { by lemma 33 R->L }
% 13.61/2.06 or(t3, and(a, not(t3)))
% 13.61/2.06 = { by lemma 35 }
% 13.61/2.06 or(t3, ll0)
% 13.61/2.06 = { by axiom 8 (or_identity) }
% 13.61/2.06 t3
% 13.61/2.06
% 13.61/2.06 Lemma 37: and(a, b) = not(t1).
% 13.61/2.06 Proof:
% 13.61/2.06 and(a, b)
% 13.61/2.06 = { by axiom 6 (not_involution) R->L }
% 13.61/2.06 not(not(and(a, b)))
% 13.61/2.06 = { by axiom 11 (t1) }
% 13.61/2.06 not(t1)
% 13.61/2.06
% 13.61/2.06 Lemma 38: and(a, t1) = not(t2).
% 13.61/2.06 Proof:
% 13.61/2.06 and(a, t1)
% 13.61/2.06 = { by axiom 6 (not_involution) R->L }
% 13.61/2.06 not(not(and(a, t1)))
% 13.61/2.06 = { by axiom 12 (t2) }
% 13.61/2.06 not(t2)
% 13.61/2.06
% 13.61/2.06 Lemma 39: not(and(cin, t4)) = t5.
% 13.61/2.06 Proof:
% 13.61/2.06 not(and(cin, t4))
% 13.61/2.06 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.06 not(and(t4, cin))
% 13.61/2.06 = { by axiom 16 (t5) }
% 13.61/2.06 t5
% 13.61/2.06
% 13.61/2.06 Lemma 40: and(cin, t4) = not(t5).
% 13.61/2.06 Proof:
% 13.61/2.06 and(cin, t4)
% 13.61/2.06 = { by axiom 6 (not_involution) R->L }
% 13.61/2.06 not(not(and(cin, t4)))
% 13.61/2.06 = { by lemma 39 }
% 13.61/2.06 not(t5)
% 13.61/2.06
% 13.61/2.06 Lemma 41: and(cin, t5) = not(t7).
% 13.61/2.06 Proof:
% 13.61/2.06 and(cin, t5)
% 13.61/2.06 = { by axiom 6 (not_involution) R->L }
% 13.61/2.06 not(not(and(cin, t5)))
% 13.61/2.06 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.06 not(not(and(t5, cin)))
% 13.61/2.06 = { by axiom 15 (t7) }
% 13.61/2.06 not(t7)
% 13.61/2.06
% 13.61/2.06 Lemma 42: not(or(X, not(Y))) = and(Y, not(X)).
% 13.61/2.06 Proof:
% 13.61/2.06 not(or(X, not(Y)))
% 13.61/2.06 = { by axiom 7 (or_commutative) R->L }
% 13.61/2.06 not(or(not(Y), X))
% 13.61/2.06 = { by axiom 22 (or_demorgan) }
% 13.61/2.06 and(not(not(Y)), not(X))
% 13.61/2.06 = { by axiom 6 (not_involution) }
% 13.61/2.06 and(Y, not(X))
% 13.61/2.06
% 13.61/2.06 Lemma 43: or(X, and(Y, X)) = X.
% 13.61/2.06 Proof:
% 13.61/2.06 or(X, and(Y, X))
% 13.61/2.06 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.06 or(X, and(X, Y))
% 13.61/2.06 = { by axiom 24 (or_absorption) }
% 13.61/2.06 X
% 13.61/2.06
% 13.61/2.06 Lemma 44: and(t2, not(t1)) = not(t1).
% 13.61/2.06 Proof:
% 13.61/2.06 and(t2, not(t1))
% 13.61/2.06 = { by lemma 42 R->L }
% 13.61/2.06 not(or(t1, not(t2)))
% 13.61/2.06 = { by lemma 38 R->L }
% 13.61/2.06 not(or(t1, and(a, t1)))
% 13.61/2.06 = { by lemma 43 }
% 13.61/2.06 not(t1)
% 13.61/2.06
% 13.61/2.06 Lemma 45: and(t2, t3) = not(t4).
% 13.61/2.06 Proof:
% 13.61/2.06 and(t2, t3)
% 13.61/2.06 = { by axiom 6 (not_involution) R->L }
% 13.61/2.06 not(not(and(t2, t3)))
% 13.61/2.06 = { by axiom 18 (t4) }
% 13.61/2.06 not(t4)
% 13.61/2.06
% 13.61/2.06 Lemma 46: and(t2, and(X, t3)) = and(X, not(t4)).
% 13.61/2.06 Proof:
% 13.61/2.06 and(t2, and(X, t3))
% 13.61/2.06 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.06 and(t2, and(t3, X))
% 13.61/2.06 = { by axiom 23 (and_associative) R->L }
% 13.61/2.06 and(and(t2, t3), X)
% 13.61/2.06 = { by lemma 45 }
% 13.61/2.06 and(not(t4), X)
% 13.61/2.06 = { by axiom 3 (and_commutative) }
% 13.61/2.06 and(X, not(t4))
% 13.61/2.06
% 13.61/2.06 Lemma 47: and(t2, and(t3, X)) = and(X, not(t4)).
% 13.61/2.06 Proof:
% 13.61/2.06 and(t2, and(t3, X))
% 13.61/2.06 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.06 and(t2, and(X, t3))
% 13.61/2.06 = { by lemma 46 }
% 13.61/2.06 and(X, not(t4))
% 13.61/2.06
% 13.61/2.06 Lemma 48: and(X, or(Y, X)) = X.
% 13.61/2.06 Proof:
% 13.61/2.06 and(X, or(Y, X))
% 13.61/2.06 = { by axiom 7 (or_commutative) R->L }
% 13.61/2.06 and(X, or(X, Y))
% 13.61/2.06 = { by axiom 21 (and_absorption) }
% 13.61/2.06 X
% 13.61/2.06
% 13.61/2.06 Lemma 49: and(t1, t4) = t4.
% 13.61/2.06 Proof:
% 13.61/2.06 and(t1, t4)
% 13.61/2.06 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.06 and(t4, t1)
% 13.61/2.06 = { by axiom 6 (not_involution) R->L }
% 13.61/2.06 and(t4, not(not(t1)))
% 13.61/2.06 = { by lemma 44 R->L }
% 13.61/2.06 and(t4, not(and(t2, not(t1))))
% 13.61/2.06 = { by lemma 43 R->L }
% 13.61/2.06 and(t4, not(and(t2, not(or(t1, and(b, t1))))))
% 13.61/2.06 = { by lemma 34 }
% 13.61/2.06 and(t4, not(and(t2, not(or(t1, not(t3))))))
% 13.61/2.06 = { by lemma 42 }
% 13.61/2.06 and(t4, not(and(t2, and(t3, not(t1)))))
% 13.61/2.06 = { by lemma 47 }
% 13.61/2.06 and(t4, not(and(not(t1), not(t4))))
% 13.61/2.06 = { by axiom 22 (or_demorgan) R->L }
% 13.61/2.06 and(t4, not(not(or(t1, t4))))
% 13.61/2.06 = { by axiom 6 (not_involution) }
% 13.61/2.06 and(t4, or(t1, t4))
% 13.61/2.06 = { by lemma 48 }
% 13.61/2.06 t4
% 13.61/2.06
% 13.61/2.06 Lemma 50: or(and(X, not(Y)), and(not(X), Y)) = xor(X, Y).
% 13.61/2.06 Proof:
% 13.61/2.06 or(and(X, not(Y)), and(not(X), Y))
% 13.61/2.06 = { by axiom 7 (or_commutative) R->L }
% 13.61/2.06 or(and(not(X), Y), and(X, not(Y)))
% 13.61/2.06 = { by axiom 30 (xor_definition) }
% 13.61/2.06 xor(X, Y)
% 13.61/2.06
% 13.61/2.06 Lemma 51: xor(X, ll1) = not(X).
% 13.61/2.06 Proof:
% 13.61/2.06 xor(X, ll1)
% 13.61/2.06 = { by lemma 50 R->L }
% 13.61/2.06 or(and(X, not(ll1)), and(not(X), ll1))
% 13.61/2.06 = { by axiom 2 (ll1_inverse) }
% 13.61/2.06 or(and(X, ll0), and(not(X), ll1))
% 13.61/2.06 = { by axiom 4 (and_boundedness) }
% 13.61/2.06 or(ll0, and(not(X), ll1))
% 13.61/2.06 = { by lemma 32 }
% 13.61/2.06 and(not(X), ll1)
% 13.61/2.06 = { by axiom 5 (and_identity) }
% 13.61/2.06 not(X)
% 13.61/2.06
% 13.61/2.06 Lemma 52: xor(X, not(Y)) = not(xor(X, Y)).
% 13.61/2.06 Proof:
% 13.61/2.06 xor(X, not(Y))
% 13.61/2.06 = { by lemma 51 R->L }
% 13.61/2.06 xor(X, xor(Y, ll1))
% 13.61/2.06 = { by axiom 26 (xor_associative) }
% 13.61/2.06 xor(xor(X, Y), ll1)
% 13.61/2.06 = { by lemma 51 }
% 13.61/2.06 not(xor(X, Y))
% 13.61/2.06
% 13.61/2.06 Lemma 53: xor(not(X), Y) = not(xor(X, Y)).
% 13.61/2.06 Proof:
% 13.61/2.06 xor(not(X), Y)
% 13.61/2.06 = { by axiom 9 (xor_commutative) R->L }
% 13.61/2.06 xor(Y, not(X))
% 13.61/2.06 = { by lemma 52 }
% 13.61/2.06 not(xor(Y, X))
% 13.61/2.06 = { by axiom 9 (xor_commutative) }
% 13.61/2.06 not(xor(X, Y))
% 13.61/2.06
% 13.61/2.06 Lemma 54: or(t2, not(t4)) = t2.
% 13.61/2.06 Proof:
% 13.61/2.06 or(t2, not(t4))
% 13.61/2.06 = { by lemma 45 R->L }
% 13.61/2.06 or(t2, and(t2, t3))
% 13.61/2.06 = { by axiom 24 (or_absorption) }
% 13.61/2.06 t2
% 13.61/2.06
% 13.61/2.06 Lemma 55: and(X, or(Y, not(X))) = and(X, Y).
% 13.61/2.06 Proof:
% 13.61/2.06 and(X, or(Y, not(X)))
% 13.61/2.06 = { by axiom 29 (or_distributive) }
% 13.61/2.06 or(and(X, Y), and(X, not(X)))
% 13.61/2.06 = { by axiom 10 (and_complement) }
% 13.61/2.06 or(and(X, Y), ll0)
% 13.61/2.06 = { by axiom 8 (or_identity) }
% 13.61/2.06 and(X, Y)
% 13.61/2.06
% 13.61/2.06 Lemma 56: and(X, and(Y, not(X))) = ll0.
% 13.61/2.06 Proof:
% 13.61/2.06 and(X, and(Y, not(X)))
% 13.61/2.06 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.06 and(X, and(not(X), Y))
% 13.61/2.06 = { by axiom 23 (and_associative) R->L }
% 13.61/2.06 and(and(X, not(X)), Y)
% 13.61/2.06 = { by axiom 10 (and_complement) }
% 13.61/2.06 and(ll0, Y)
% 13.61/2.06 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.06 and(Y, ll0)
% 13.61/2.06 = { by axiom 4 (and_boundedness) }
% 13.61/2.06 ll0
% 13.61/2.06
% 13.61/2.06 Lemma 57: xor(X, or(X, Y)) = and(Y, not(X)).
% 13.61/2.06 Proof:
% 13.61/2.06 xor(X, or(X, Y))
% 13.61/2.06 = { by axiom 6 (not_involution) R->L }
% 13.61/2.06 xor(X, or(not(not(X)), Y))
% 13.61/2.07 = { by lemma 50 R->L }
% 13.61/2.07 or(and(X, not(or(not(not(X)), Y))), and(not(X), or(not(not(X)), Y)))
% 13.61/2.07 = { by axiom 7 (or_commutative) R->L }
% 13.61/2.07 or(and(X, not(or(not(not(X)), Y))), and(not(X), or(Y, not(not(X)))))
% 13.61/2.07 = { by lemma 55 }
% 13.61/2.07 or(and(X, not(or(not(not(X)), Y))), and(not(X), Y))
% 13.61/2.07 = { by axiom 7 (or_commutative) R->L }
% 13.61/2.07 or(and(X, not(or(Y, not(not(X))))), and(not(X), Y))
% 13.61/2.07 = { by lemma 42 }
% 13.61/2.07 or(and(X, and(not(X), not(Y))), and(not(X), Y))
% 13.61/2.07 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.07 or(and(X, and(not(Y), not(X))), and(not(X), Y))
% 13.61/2.07 = { by lemma 56 }
% 13.61/2.07 or(ll0, and(not(X), Y))
% 13.61/2.07 = { by lemma 32 }
% 13.61/2.07 and(not(X), Y)
% 13.61/2.07 = { by axiom 3 (and_commutative) }
% 13.61/2.07 and(Y, not(X))
% 13.61/2.07
% 13.61/2.07 Lemma 58: xor(X, or(Y, X)) = and(Y, not(X)).
% 13.61/2.07 Proof:
% 13.61/2.07 xor(X, or(Y, X))
% 13.61/2.07 = { by axiom 7 (or_commutative) R->L }
% 13.61/2.07 xor(X, or(X, Y))
% 13.61/2.07 = { by lemma 57 }
% 13.61/2.07 and(Y, not(X))
% 13.61/2.07
% 13.61/2.07 Lemma 59: not(xor(t4, t2)) = and(t4, t2).
% 13.61/2.07 Proof:
% 13.61/2.07 not(xor(t4, t2))
% 13.61/2.07 = { by lemma 53 R->L }
% 13.61/2.07 xor(not(t4), t2)
% 13.61/2.07 = { by lemma 54 R->L }
% 13.61/2.07 xor(not(t4), or(t2, not(t4)))
% 13.61/2.07 = { by lemma 58 }
% 13.61/2.07 and(t2, not(not(t4)))
% 13.61/2.07 = { by axiom 6 (not_involution) }
% 13.61/2.07 and(t2, t4)
% 13.61/2.07 = { by axiom 3 (and_commutative) }
% 13.61/2.07 and(t4, t2)
% 13.61/2.07
% 13.61/2.07 Lemma 60: xor(X, and(X, Y)) = and(X, not(Y)).
% 13.61/2.07 Proof:
% 13.61/2.07 xor(X, and(X, Y))
% 13.61/2.07 = { by axiom 6 (not_involution) R->L }
% 13.61/2.07 not(not(xor(X, and(X, Y))))
% 13.61/2.07 = { by lemma 52 R->L }
% 13.61/2.07 not(xor(X, not(and(X, Y))))
% 13.61/2.07 = { by lemma 53 R->L }
% 13.61/2.07 xor(not(X), not(and(X, Y)))
% 13.61/2.07 = { by axiom 25 (and_demorgan) }
% 13.61/2.07 xor(not(X), or(not(X), not(Y)))
% 13.61/2.07 = { by lemma 57 }
% 13.61/2.07 and(not(Y), not(not(X)))
% 13.61/2.07 = { by axiom 22 (or_demorgan) R->L }
% 13.61/2.07 not(or(Y, not(X)))
% 13.61/2.07 = { by lemma 42 }
% 13.61/2.07 and(X, not(Y))
% 13.61/2.07
% 13.61/2.07 Lemma 61: and(a, and(X, t1)) = and(X, not(t2)).
% 13.61/2.07 Proof:
% 13.61/2.07 and(a, and(X, t1))
% 13.61/2.07 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.07 and(a, and(t1, X))
% 13.61/2.07 = { by axiom 23 (and_associative) R->L }
% 13.61/2.07 and(and(a, t1), X)
% 13.61/2.07 = { by lemma 38 }
% 13.61/2.07 and(not(t2), X)
% 13.61/2.07 = { by axiom 3 (and_commutative) }
% 13.61/2.07 and(X, not(t2))
% 13.61/2.07
% 13.61/2.07 Lemma 62: and(a, and(t1, X)) = and(X, not(t2)).
% 13.61/2.07 Proof:
% 13.61/2.07 and(a, and(t1, X))
% 13.61/2.07 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.07 and(a, and(X, t1))
% 13.61/2.07 = { by lemma 61 }
% 13.61/2.07 and(X, not(t2))
% 13.61/2.07
% 13.61/2.07 Lemma 63: and(t4, t2) = not(t3).
% 13.61/2.07 Proof:
% 13.61/2.07 and(t4, t2)
% 13.61/2.07 = { by lemma 59 R->L }
% 13.61/2.07 not(xor(t4, t2))
% 13.61/2.07 = { by axiom 9 (xor_commutative) R->L }
% 13.61/2.07 not(xor(t2, t4))
% 13.61/2.07 = { by lemma 52 R->L }
% 13.61/2.07 xor(t2, not(t4))
% 13.61/2.07 = { by lemma 45 R->L }
% 13.61/2.07 xor(t2, and(t2, t3))
% 13.61/2.07 = { by lemma 60 }
% 13.61/2.07 and(t2, not(t3))
% 13.61/2.07 = { by lemma 42 R->L }
% 13.61/2.07 not(or(t3, not(t2)))
% 13.61/2.07 = { by lemma 38 R->L }
% 13.61/2.07 not(or(t3, and(a, t1)))
% 13.61/2.07 = { by axiom 21 (and_absorption) R->L }
% 13.61/2.07 not(or(t3, and(and(a, or(a, t3)), t1)))
% 13.61/2.07 = { by lemma 36 }
% 13.61/2.07 not(or(t3, and(and(a, t3), t1)))
% 13.61/2.07 = { by axiom 23 (and_associative) }
% 13.61/2.07 not(or(t3, and(a, and(t3, t1))))
% 13.61/2.07 = { by axiom 3 (and_commutative) }
% 13.61/2.07 not(or(t3, and(a, and(t1, t3))))
% 13.61/2.07 = { by lemma 62 }
% 13.61/2.07 not(or(t3, and(t3, not(t2))))
% 13.61/2.07 = { by axiom 24 (or_absorption) }
% 13.61/2.07 not(t3)
% 13.61/2.07
% 13.61/2.07 Lemma 64: and(t5, t4) = not(t6).
% 13.61/2.07 Proof:
% 13.61/2.07 and(t5, t4)
% 13.61/2.07 = { by axiom 6 (not_involution) R->L }
% 13.61/2.07 not(not(and(t5, t4)))
% 13.61/2.07 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.07 not(not(and(t4, t5)))
% 13.61/2.07 = { by axiom 17 (t6) }
% 13.61/2.07 not(t6)
% 13.61/2.07
% 13.61/2.07 Lemma 65: xor(X, ll0) = X.
% 13.61/2.07 Proof:
% 13.61/2.07 xor(X, ll0)
% 13.61/2.07 = { by lemma 50 R->L }
% 13.61/2.07 or(and(X, not(ll0)), and(not(X), ll0))
% 13.61/2.07 = { by axiom 1 (ll0_inverse) }
% 13.61/2.07 or(and(X, ll1), and(not(X), ll0))
% 13.61/2.07 = { by axiom 5 (and_identity) }
% 13.61/2.07 or(X, and(not(X), ll0))
% 13.61/2.07 = { by axiom 4 (and_boundedness) }
% 13.61/2.07 or(X, ll0)
% 13.61/2.07 = { by axiom 8 (or_identity) }
% 13.61/2.07 X
% 13.61/2.07
% 13.61/2.07 Lemma 66: and(b, and(t1, X)) = and(X, not(t3)).
% 13.61/2.07 Proof:
% 13.61/2.07 and(b, and(t1, X))
% 13.61/2.07 = { by axiom 23 (and_associative) R->L }
% 13.61/2.07 and(and(b, t1), X)
% 13.61/2.07 = { by lemma 34 }
% 13.61/2.07 and(not(t3), X)
% 13.61/2.07 = { by axiom 3 (and_commutative) }
% 13.61/2.07 and(X, not(t3))
% 13.61/2.07
% 13.61/2.07 Lemma 67: not(or(a, t3)) = and(b, not(a)).
% 13.61/2.07 Proof:
% 13.61/2.07 not(or(a, t3))
% 13.61/2.07 = { by axiom 22 (or_demorgan) }
% 13.61/2.07 and(not(a), not(t3))
% 13.61/2.07 = { by lemma 66 R->L }
% 13.61/2.07 and(b, and(t1, not(a)))
% 13.61/2.07 = { by lemma 42 R->L }
% 13.61/2.07 and(b, not(or(a, not(t1))))
% 13.61/2.07 = { by lemma 37 R->L }
% 13.61/2.07 and(b, not(or(a, and(a, b))))
% 13.61/2.07 = { by axiom 24 (or_absorption) }
% 13.61/2.07 and(b, not(a))
% 13.61/2.07
% 13.61/2.07 Lemma 68: or(a, not(t3)) = or(a, b).
% 13.61/2.07 Proof:
% 13.61/2.07 or(a, not(t3))
% 13.61/2.07 = { by lemma 36 R->L }
% 13.61/2.07 or(a, not(or(a, t3)))
% 13.61/2.07 = { by lemma 67 }
% 13.61/2.07 or(a, and(b, not(a)))
% 13.61/2.07 = { by lemma 33 }
% 13.61/2.07 or(a, b)
% 13.61/2.07
% 13.61/2.07 Lemma 69: and(b, not(t2)) = ll0.
% 13.61/2.07 Proof:
% 13.61/2.07 and(b, not(t2))
% 13.61/2.07 = { by lemma 61 R->L }
% 13.61/2.07 and(a, and(b, t1))
% 13.61/2.07 = { by lemma 34 }
% 13.61/2.07 and(a, not(t3))
% 13.61/2.07 = { by lemma 35 }
% 13.61/2.07 ll0
% 13.61/2.07
% 13.61/2.07 Lemma 70: or(b, not(t2)) = or(a, b).
% 13.61/2.07 Proof:
% 13.61/2.07 or(b, not(t2))
% 13.61/2.07 = { by axiom 8 (or_identity) R->L }
% 13.61/2.07 or(b, not(or(t2, ll0)))
% 13.61/2.07 = { by lemma 69 R->L }
% 13.61/2.07 or(b, not(or(t2, and(b, not(t2)))))
% 13.61/2.07 = { by lemma 33 }
% 13.61/2.07 or(b, not(or(t2, b)))
% 13.61/2.07 = { by axiom 7 (or_commutative) }
% 13.61/2.07 or(b, not(or(b, t2)))
% 13.61/2.07 = { by axiom 22 (or_demorgan) }
% 13.61/2.07 or(b, and(not(b), not(t2)))
% 13.61/2.07 = { by lemma 62 R->L }
% 13.61/2.07 or(b, and(a, and(t1, not(b))))
% 13.61/2.07 = { by lemma 42 R->L }
% 13.61/2.07 or(b, and(a, not(or(b, not(t1)))))
% 13.61/2.07 = { by lemma 37 R->L }
% 13.61/2.07 or(b, and(a, not(or(b, and(a, b)))))
% 13.61/2.07 = { by lemma 43 }
% 13.61/2.07 or(b, and(a, not(b)))
% 13.61/2.07 = { by lemma 33 }
% 13.61/2.07 or(b, a)
% 13.61/2.07 = { by axiom 7 (or_commutative) }
% 13.61/2.07 or(a, b)
% 13.61/2.07
% 13.61/2.07 Lemma 71: and(cin, not(t6)) = ll0.
% 13.61/2.07 Proof:
% 13.61/2.07 and(cin, not(t6))
% 13.61/2.07 = { by lemma 64 R->L }
% 13.61/2.07 and(cin, and(t5, t4))
% 13.61/2.07 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.07 and(cin, and(t4, t5))
% 13.61/2.07 = { by axiom 23 (and_associative) R->L }
% 13.61/2.07 and(and(cin, t4), t5)
% 13.61/2.07 = { by lemma 39 R->L }
% 13.61/2.07 and(and(cin, t4), not(and(cin, t4)))
% 13.61/2.07 = { by axiom 10 (and_complement) }
% 13.61/2.07 ll0
% 13.61/2.07
% 13.61/2.07 Lemma 72: and(t2, not(b)) = xor(b, t2).
% 13.61/2.07 Proof:
% 13.61/2.07 and(t2, not(b))
% 13.61/2.07 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.07 and(not(b), t2)
% 13.61/2.07 = { by lemma 32 R->L }
% 13.61/2.07 or(ll0, and(not(b), t2))
% 13.61/2.07 = { by lemma 69 R->L }
% 13.61/2.07 or(and(b, not(t2)), and(not(b), t2))
% 13.61/2.07 = { by lemma 50 }
% 13.61/2.07 xor(b, t2)
% 13.61/2.07
% 13.61/2.07 Lemma 73: and(t3, not(a)) = xor(a, t3).
% 13.61/2.07 Proof:
% 13.61/2.07 and(t3, not(a))
% 13.61/2.07 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.07 and(not(a), t3)
% 13.61/2.07 = { by lemma 32 R->L }
% 13.61/2.07 or(ll0, and(not(a), t3))
% 13.61/2.07 = { by lemma 35 R->L }
% 13.61/2.07 or(and(a, not(t3)), and(not(a), t3))
% 13.61/2.07 = { by lemma 50 }
% 13.61/2.07 xor(a, t3)
% 13.61/2.07
% 13.61/2.07 Lemma 74: and(t5, not(t4)) = not(t4).
% 13.61/2.07 Proof:
% 13.61/2.07 and(t5, not(t4))
% 13.61/2.07 = { by lemma 42 R->L }
% 13.61/2.07 not(or(t4, not(t5)))
% 13.61/2.07 = { by lemma 40 R->L }
% 13.61/2.07 not(or(t4, and(cin, t4)))
% 13.61/2.07 = { by lemma 43 }
% 13.61/2.07 not(t4)
% 13.61/2.07
% 13.61/2.07 Lemma 75: not(or(b, t4)) = xor(b, t2).
% 13.61/2.07 Proof:
% 13.61/2.07 not(or(b, t4))
% 13.61/2.07 = { by axiom 22 (or_demorgan) }
% 13.61/2.07 and(not(b), not(t4))
% 13.61/2.08 = { by lemma 47 R->L }
% 13.61/2.08 and(t2, and(t3, not(b)))
% 13.61/2.08 = { by lemma 42 R->L }
% 13.61/2.08 and(t2, not(or(b, not(t3))))
% 13.61/2.08 = { by lemma 34 R->L }
% 13.61/2.08 and(t2, not(or(b, and(b, t1))))
% 13.61/2.08 = { by axiom 24 (or_absorption) }
% 13.61/2.08 and(t2, not(b))
% 13.61/2.08 = { by lemma 72 }
% 13.61/2.08 xor(b, t2)
% 13.61/2.08
% 13.61/2.08 Lemma 76: and(t6, and(X, t7)) = and(X, not(sum)).
% 13.61/2.08 Proof:
% 13.61/2.08 and(t6, and(X, t7))
% 13.61/2.08 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.08 and(t6, and(t7, X))
% 13.61/2.08 = { by axiom 23 (and_associative) R->L }
% 13.61/2.08 and(and(t6, t7), X)
% 13.61/2.08 = { by axiom 6 (not_involution) R->L }
% 13.61/2.08 and(not(not(and(t6, t7))), X)
% 13.61/2.08 = { by axiom 19 (sum) }
% 13.61/2.08 and(not(sum), X)
% 13.61/2.08 = { by axiom 3 (and_commutative) }
% 13.61/2.08 and(X, not(sum))
% 13.61/2.08
% 13.61/2.08 Lemma 77: not(or(cin, sum)) = xor(cin, t6).
% 13.61/2.08 Proof:
% 13.61/2.08 not(or(cin, sum))
% 13.61/2.08 = { by axiom 22 (or_demorgan) }
% 13.61/2.08 and(not(cin), not(sum))
% 13.61/2.08 = { by lemma 76 R->L }
% 13.61/2.08 and(t6, and(not(cin), t7))
% 13.61/2.08 = { by axiom 3 (and_commutative) }
% 13.61/2.08 and(t6, and(t7, not(cin)))
% 13.61/2.08 = { by lemma 42 R->L }
% 13.61/2.08 and(t6, not(or(cin, not(t7))))
% 13.61/2.08 = { by lemma 41 R->L }
% 13.61/2.08 and(t6, not(or(cin, and(cin, t5))))
% 13.61/2.08 = { by axiom 24 (or_absorption) }
% 13.61/2.08 and(t6, not(cin))
% 13.61/2.08 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.08 and(not(cin), t6)
% 13.61/2.08 = { by lemma 32 R->L }
% 13.61/2.08 or(ll0, and(not(cin), t6))
% 13.61/2.08 = { by lemma 71 R->L }
% 13.61/2.08 or(and(cin, not(t6)), and(not(cin), t6))
% 13.61/2.08 = { by lemma 50 }
% 13.61/2.08 xor(cin, t6)
% 13.61/2.08
% 13.61/2.08 Lemma 78: and(cin, and(t5, X)) = and(X, not(t7)).
% 13.61/2.08 Proof:
% 13.61/2.08 and(cin, and(t5, X))
% 13.61/2.08 = { by axiom 23 (and_associative) R->L }
% 13.61/2.08 and(and(cin, t5), X)
% 13.61/2.08 = { by lemma 41 }
% 13.61/2.08 and(not(t7), X)
% 13.61/2.08 = { by axiom 3 (and_commutative) }
% 13.61/2.08 and(X, not(t7))
% 13.61/2.08
% 13.61/2.08 Lemma 79: and(t1, and(t5, X)) = and(X, not(carry)).
% 13.61/2.08 Proof:
% 13.61/2.08 and(t1, and(t5, X))
% 13.61/2.08 = { by axiom 23 (and_associative) R->L }
% 13.61/2.08 and(and(t1, t5), X)
% 13.61/2.08 = { by axiom 6 (not_involution) R->L }
% 13.61/2.08 and(not(not(and(t1, t5))), X)
% 13.61/2.08 = { by axiom 14 (carry) }
% 13.61/2.08 and(not(carry), X)
% 13.61/2.08 = { by axiom 3 (and_commutative) }
% 13.61/2.08 and(X, not(carry))
% 13.61/2.08
% 13.61/2.08 Lemma 80: not(and(X, not(Y))) = or(Y, not(X)).
% 13.61/2.08 Proof:
% 13.61/2.08 not(and(X, not(Y)))
% 13.61/2.08 = { by axiom 3 (and_commutative) R->L }
% 13.61/2.08 not(and(not(Y), X))
% 13.61/2.08 = { by axiom 25 (and_demorgan) }
% 13.61/2.08 or(not(not(Y)), not(X))
% 13.61/2.08 = { by axiom 6 (not_involution) }
% 13.61/2.08 or(Y, not(X))
% 13.61/2.08
% 13.61/2.08 Lemma 81: xor(X, xor(Y, X)) = Y.
% 13.61/2.08 Proof:
% 13.61/2.08 xor(X, xor(Y, X))
% 13.61/2.08 = { by axiom 9 (xor_commutative) R->L }
% 13.61/2.08 xor(xor(Y, X), X)
% 13.61/2.08 = { by axiom 26 (xor_associative) R->L }
% 13.61/2.08 xor(Y, xor(X, X))
% 13.61/2.08 = { by axiom 6 (not_involution) R->L }
% 13.61/2.08 xor(Y, xor(X, not(not(X))))
% 13.61/2.08 = { by lemma 50 R->L }
% 13.61/2.08 xor(Y, or(and(X, not(not(not(X)))), and(not(X), not(not(X)))))
% 13.61/2.08 = { by axiom 10 (and_complement) }
% 13.61/2.08 xor(Y, or(and(X, not(not(not(X)))), ll0))
% 13.61/2.08 = { by axiom 8 (or_identity) }
% 13.61/2.08 xor(Y, and(X, not(not(not(X)))))
% 13.61/2.08 = { by axiom 6 (not_involution) }
% 13.61/2.08 xor(Y, and(X, not(X)))
% 13.61/2.08 = { by axiom 10 (and_complement) }
% 13.61/2.08 xor(Y, ll0)
% 13.61/2.08 = { by lemma 65 }
% 13.61/2.08 Y
% 13.61/2.08
% 13.61/2.08 Lemma 82: xor(X, xor(X, Y)) = Y.
% 13.61/2.08 Proof:
% 13.61/2.08 xor(X, xor(X, Y))
% 13.61/2.08 = { by axiom 9 (xor_commutative) R->L }
% 13.61/2.08 xor(X, xor(Y, X))
% 13.61/2.08 = { by lemma 81 }
% 13.61/2.08 Y
% 13.61/2.08
% 13.61/2.08 Lemma 83: or(X, not(or(X, Y))) = or(X, not(Y)).
% 13.61/2.08 Proof:
% 13.61/2.08 or(X, not(or(X, Y)))
% 13.61/2.08 = { by axiom 7 (or_commutative) R->L }
% 13.61/2.08 or(X, not(or(Y, X)))
% 13.61/2.08 = { by axiom 22 (or_demorgan) }
% 13.80/2.08 or(X, and(not(Y), not(X)))
% 13.80/2.08 = { by lemma 33 }
% 13.80/2.08 or(X, not(Y))
% 13.80/2.08
% 13.80/2.08 Goal 1 (prove_circuit): tuple(sum, carry) = tuple(sum_def, carry_def).
% 13.80/2.08 Proof:
% 13.80/2.08 tuple(sum, carry)
% 13.80/2.08 = { by lemma 65 R->L }
% 13.80/2.08 tuple(xor(sum, ll0), carry)
% 13.80/2.08 = { by lemma 56 R->L }
% 13.80/2.08 tuple(xor(sum, and(sum, and(not(cin), not(sum)))), carry)
% 13.80/2.08 = { by axiom 22 (or_demorgan) R->L }
% 13.80/2.08 tuple(xor(sum, and(sum, not(or(cin, sum)))), carry)
% 13.80/2.08 = { by lemma 77 }
% 13.80/2.08 tuple(xor(sum, and(sum, xor(cin, t6))), carry)
% 13.80/2.08 = { by lemma 60 }
% 13.80/2.08 tuple(and(sum, not(xor(cin, t6))), carry)
% 13.80/2.08 = { by lemma 58 R->L }
% 13.80/2.08 tuple(xor(xor(cin, t6), or(sum, xor(cin, t6))), carry)
% 13.80/2.08 = { by lemma 77 R->L }
% 13.80/2.08 tuple(xor(xor(cin, t6), or(sum, not(or(cin, sum)))), carry)
% 13.80/2.08 = { by axiom 7 (or_commutative) R->L }
% 13.80/2.08 tuple(xor(xor(cin, t6), or(sum, not(or(sum, cin)))), carry)
% 13.80/2.08 = { by lemma 83 }
% 13.80/2.08 tuple(xor(xor(cin, t6), or(sum, not(cin))), carry)
% 13.80/2.08 = { by lemma 80 R->L }
% 13.80/2.08 tuple(xor(xor(cin, t6), not(and(cin, not(sum)))), carry)
% 13.80/2.08 = { by lemma 76 R->L }
% 13.80/2.08 tuple(xor(xor(cin, t6), not(and(t6, and(cin, t7)))), carry)
% 13.80/2.08 = { by axiom 8 (or_identity) R->L }
% 13.80/2.08 tuple(xor(xor(cin, t6), not(and(t6, and(cin, or(t7, ll0))))), carry)
% 13.80/2.08 = { by lemma 71 R->L }
% 13.80/2.08 tuple(xor(xor(cin, t6), not(and(t6, and(cin, or(t7, and(cin, not(t6))))))), carry)
% 13.80/2.08 = { by lemma 64 R->L }
% 13.80/2.08 tuple(xor(xor(cin, t6), not(and(t6, and(cin, or(t7, and(cin, and(t5, t4))))))), carry)
% 13.80/2.08 = { by lemma 78 }
% 13.80/2.08 tuple(xor(xor(cin, t6), not(and(t6, and(cin, or(t7, and(t4, not(t7))))))), carry)
% 13.80/2.08 = { by lemma 33 }
% 13.80/2.08 tuple(xor(xor(cin, t6), not(and(t6, and(cin, or(t7, t4))))), carry)
% 13.80/2.08 = { by axiom 7 (or_commutative) }
% 13.80/2.08 tuple(xor(xor(cin, t6), not(and(t6, and(cin, or(t4, t7))))), carry)
% 13.80/2.08 = { by axiom 6 (not_involution) R->L }
% 13.80/2.08 tuple(xor(xor(cin, t6), not(and(t6, and(cin, not(not(or(t4, t7))))))), carry)
% 13.80/2.08 = { by axiom 22 (or_demorgan) }
% 13.80/2.08 tuple(xor(xor(cin, t6), not(and(t6, and(cin, not(and(not(t4), not(t7))))))), carry)
% 13.80/2.08 = { by lemma 78 R->L }
% 13.80/2.08 tuple(xor(xor(cin, t6), not(and(t6, and(cin, not(and(cin, and(t5, not(t4)))))))), carry)
% 13.80/2.09 = { by lemma 74 }
% 13.80/2.09 tuple(xor(xor(cin, t6), not(and(t6, and(cin, not(and(cin, not(t4))))))), carry)
% 13.80/2.09 = { by lemma 80 }
% 13.80/2.09 tuple(xor(xor(cin, t6), not(and(t6, and(cin, or(t4, not(cin)))))), carry)
% 13.80/2.09 = { by lemma 55 }
% 13.80/2.09 tuple(xor(xor(cin, t6), not(and(t6, and(cin, t4)))), carry)
% 13.80/2.09 = { by lemma 40 }
% 13.80/2.09 tuple(xor(xor(cin, t6), not(and(t6, not(t5)))), carry)
% 13.80/2.09 = { by lemma 42 R->L }
% 13.80/2.09 tuple(xor(xor(cin, t6), not(not(or(t5, not(t6))))), carry)
% 13.80/2.09 = { by lemma 64 R->L }
% 13.80/2.09 tuple(xor(xor(cin, t6), not(not(or(t5, and(t5, t4))))), carry)
% 13.80/2.09 = { by axiom 24 (or_absorption) }
% 13.80/2.09 tuple(xor(xor(cin, t6), not(not(t5))), carry)
% 13.80/2.09 = { by axiom 6 (not_involution) }
% 13.80/2.09 tuple(xor(xor(cin, t6), t5), carry)
% 13.80/2.09 = { by axiom 26 (xor_associative) R->L }
% 13.80/2.09 tuple(xor(cin, xor(t6, t5)), carry)
% 13.80/2.09 = { by axiom 9 (xor_commutative) }
% 13.80/2.09 tuple(xor(cin, xor(t5, t6)), carry)
% 13.80/2.09 = { by axiom 6 (not_involution) R->L }
% 13.80/2.09 tuple(xor(cin, xor(t5, not(not(t6)))), carry)
% 13.80/2.09 = { by lemma 64 R->L }
% 13.80/2.09 tuple(xor(cin, xor(t5, not(and(t5, t4)))), carry)
% 13.80/2.09 = { by axiom 6 (not_involution) R->L }
% 13.80/2.09 tuple(xor(cin, xor(t5, not(and(t5, not(not(t4)))))), carry)
% 13.80/2.09 = { by lemma 58 R->L }
% 13.80/2.09 tuple(xor(cin, xor(t5, not(xor(not(t4), or(t5, not(t4)))))), carry)
% 13.80/2.09 = { by lemma 74 R->L }
% 13.80/2.09 tuple(xor(cin, xor(t5, not(xor(not(t4), or(t5, and(t5, not(t4))))))), carry)
% 13.80/2.09 = { by axiom 24 (or_absorption) }
% 13.80/2.09 tuple(xor(cin, xor(t5, not(xor(not(t4), t5)))), carry)
% 13.80/2.09 = { by lemma 53 }
% 13.80/2.09 tuple(xor(cin, xor(t5, not(not(xor(t4, t5))))), carry)
% 13.80/2.09 = { by axiom 9 (xor_commutative) }
% 13.80/2.09 tuple(xor(cin, xor(t5, not(not(xor(t5, t4))))), carry)
% 13.80/2.09 = { by axiom 6 (not_involution) }
% 13.80/2.09 tuple(xor(cin, xor(t5, xor(t5, t4))), carry)
% 13.80/2.09 = { by lemma 82 }
% 13.80/2.09 tuple(xor(cin, t4), carry)
% 13.80/2.09 = { by lemma 82 R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(a, t4))), carry)
% 13.80/2.09 = { by lemma 81 R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(a, xor(t2, xor(t4, t2))))), carry)
% 13.80/2.09 = { by axiom 6 (not_involution) R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(a, xor(t2, not(not(xor(t4, t2))))))), carry)
% 13.80/2.09 = { by lemma 59 }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(a, xor(t2, not(and(t4, t2)))))), carry)
% 13.80/2.09 = { by lemma 63 }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(a, xor(t2, not(not(t3)))))), carry)
% 13.80/2.09 = { by axiom 6 (not_involution) }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(a, xor(t2, t3)))), carry)
% 13.80/2.09 = { by axiom 9 (xor_commutative) R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(a, xor(t3, t2)))), carry)
% 13.80/2.09 = { by axiom 26 (xor_associative) }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), t2))), carry)
% 13.80/2.09 = { by axiom 6 (not_involution) R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), not(not(t2))))), carry)
% 13.80/2.09 = { by lemma 54 R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), not(not(or(t2, not(t4))))))), carry)
% 13.80/2.09 = { by lemma 42 }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), not(and(t4, not(t2)))))), carry)
% 13.80/2.09 = { by lemma 60 R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), not(xor(t4, and(t4, t2)))))), carry)
% 13.80/2.09 = { by lemma 63 }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), not(xor(t4, not(t3)))))), carry)
% 13.80/2.09 = { by lemma 43 R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), not(xor(t4, not(or(t3, and(t2, t3)))))))), carry)
% 13.80/2.09 = { by lemma 45 }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), not(xor(t4, not(or(t3, not(t4)))))))), carry)
% 13.80/2.09 = { by lemma 42 }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), not(xor(t4, and(t4, not(t3))))))), carry)
% 13.80/2.09 = { by lemma 66 R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), not(xor(t4, and(b, and(t1, t4))))))), carry)
% 13.80/2.09 = { by lemma 49 }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), not(xor(t4, and(b, t4)))))), carry)
% 13.80/2.09 = { by axiom 3 (and_commutative) R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), not(xor(t4, and(t4, b)))))), carry)
% 13.80/2.09 = { by lemma 60 }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), not(and(t4, not(b)))))), carry)
% 13.80/2.09 = { by lemma 80 }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), or(b, not(t4))))), carry)
% 13.80/2.09 = { by lemma 83 R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), or(b, not(or(b, t4)))))), carry)
% 13.80/2.09 = { by lemma 75 }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), or(b, xor(b, t2))))), carry)
% 13.80/2.09 = { by lemma 72 R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), or(b, and(t2, not(b)))))), carry)
% 13.80/2.09 = { by lemma 42 R->L }
% 13.80/2.09 tuple(xor(cin, xor(a, xor(xor(a, t3), or(b, not(or(b, not(t2))))))), carry)
% 13.80/2.10 = { by lemma 70 }
% 13.80/2.10 tuple(xor(cin, xor(a, xor(xor(a, t3), or(b, not(or(a, b)))))), carry)
% 13.80/2.10 = { by lemma 68 R->L }
% 13.80/2.10 tuple(xor(cin, xor(a, xor(xor(a, t3), or(b, not(or(a, not(t3))))))), carry)
% 13.80/2.10 = { by lemma 42 }
% 13.80/2.10 tuple(xor(cin, xor(a, xor(xor(a, t3), or(b, and(t3, not(a)))))), carry)
% 13.80/2.10 = { by lemma 73 }
% 13.80/2.10 tuple(xor(cin, xor(a, xor(xor(a, t3), or(b, xor(a, t3))))), carry)
% 13.80/2.10 = { by lemma 58 }
% 13.80/2.10 tuple(xor(cin, xor(a, and(b, not(xor(a, t3))))), carry)
% 13.80/2.10 = { by lemma 73 R->L }
% 13.80/2.10 tuple(xor(cin, xor(a, and(b, not(and(t3, not(a)))))), carry)
% 13.80/2.10 = { by lemma 80 }
% 13.80/2.10 tuple(xor(cin, xor(a, and(b, or(a, not(t3))))), carry)
% 13.80/2.10 = { by lemma 68 }
% 13.80/2.10 tuple(xor(cin, xor(a, and(b, or(a, b)))), carry)
% 13.80/2.10 = { by lemma 48 }
% 13.80/2.10 tuple(xor(cin, xor(a, b)), carry)
% 13.80/2.10 = { by axiom 9 (xor_commutative) R->L }
% 13.80/2.10 tuple(xor(cin, xor(b, a)), carry)
% 13.80/2.10 = { by lemma 81 R->L }
% 13.80/2.10 tuple(xor(cin, xor(b, xor(xor(b, cin), xor(a, xor(b, cin))))), carry)
% 13.80/2.10 = { by axiom 26 (xor_associative) }
% 13.80/2.10 tuple(xor(cin, xor(b, xor(xor(b, cin), xor(xor(a, b), cin)))), carry)
% 13.80/2.10 = { by axiom 27 (sum_def) }
% 13.80/2.10 tuple(xor(cin, xor(b, xor(xor(b, cin), sum_def))), carry)
% 13.80/2.10 = { by axiom 26 (xor_associative) R->L }
% 13.80/2.10 tuple(xor(cin, xor(b, xor(b, xor(cin, sum_def)))), carry)
% 13.80/2.10 = { by lemma 82 }
% 13.80/2.10 tuple(xor(cin, xor(cin, sum_def)), carry)
% 13.80/2.10 = { by lemma 82 }
% 13.80/2.10 tuple(sum_def, carry)
% 13.80/2.10 = { by axiom 5 (and_identity) R->L }
% 13.80/2.10 tuple(sum_def, and(carry, ll1))
% 13.80/2.10 = { by axiom 20 (or_complement) R->L }
% 13.80/2.10 tuple(sum_def, and(carry, or(cin, not(cin))))
% 13.80/2.10 = { by axiom 29 (or_distributive) }
% 13.80/2.10 tuple(sum_def, or(and(carry, cin), and(carry, not(cin))))
% 13.80/2.10 = { by axiom 3 (and_commutative) R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), and(carry, not(cin))))
% 13.80/2.10 = { by lemma 57 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), xor(cin, or(cin, carry))))
% 13.80/2.10 = { by axiom 6 (not_involution) R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), xor(cin, not(not(or(cin, carry))))))
% 13.80/2.10 = { by axiom 22 (or_demorgan) }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), xor(cin, not(and(not(cin), not(carry))))))
% 13.80/2.10 = { by lemma 79 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), xor(cin, not(and(t1, and(t5, not(cin)))))))
% 13.80/2.10 = { by lemma 42 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), xor(cin, not(and(t1, not(or(cin, not(t5))))))))
% 13.80/2.10 = { by lemma 40 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), xor(cin, not(and(t1, not(or(cin, and(cin, t4))))))))
% 13.80/2.10 = { by axiom 24 (or_absorption) }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), xor(cin, not(and(t1, not(cin))))))
% 13.80/2.10 = { by lemma 80 }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), xor(cin, or(cin, not(t1)))))
% 13.80/2.10 = { by lemma 57 }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), and(not(t1), not(cin))))
% 13.80/2.10 = { by axiom 22 (or_demorgan) R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), not(or(t1, cin))))
% 13.80/2.10 = { by axiom 7 (or_commutative) }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), not(or(cin, t1))))
% 13.80/2.10 = { by axiom 22 (or_demorgan) }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), and(not(cin), not(t1))))
% 13.80/2.10 = { by lemma 37 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, carry), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 32 R->L }
% 13.80/2.10 tuple(sum_def, or(or(ll0, and(cin, carry)), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 71 R->L }
% 13.80/2.10 tuple(sum_def, or(or(and(cin, not(t6)), and(cin, carry)), and(not(cin), and(a, b))))
% 13.80/2.10 = { by axiom 29 (or_distributive) R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(not(t6), carry)), and(not(cin), and(a, b))))
% 13.80/2.10 = { by axiom 7 (or_commutative) }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(carry, not(t6))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 64 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(carry, and(t5, t4))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 49 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(carry, and(t5, and(t1, t4)))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by axiom 3 (and_commutative) R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(carry, and(t5, and(t4, t1)))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by axiom 23 (and_associative) R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(carry, and(and(t5, t4), t1))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 64 }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(carry, and(not(t6), t1))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by axiom 3 (and_commutative) }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(carry, and(t1, not(t6)))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 64 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(carry, and(t1, and(t5, t4)))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 79 }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(carry, and(t4, not(carry)))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 33 }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(carry, t4)), and(not(cin), and(a, b))))
% 13.80/2.10 = { by axiom 7 (or_commutative) }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(t4, carry)), and(not(cin), and(a, b))))
% 13.80/2.10 = { by axiom 6 (not_involution) R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, not(not(or(t4, carry)))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by axiom 22 (or_demorgan) }
% 13.80/2.10 tuple(sum_def, or(and(cin, not(and(not(t4), not(carry)))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 79 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, not(and(t1, and(t5, not(t4))))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 74 }
% 13.80/2.10 tuple(sum_def, or(and(cin, not(and(t1, not(t4)))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 80 }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(t4, not(t1))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 44 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(t4, and(t2, not(t1)))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 37 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(t4, and(t2, and(a, b)))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by axiom 3 (and_commutative) R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(t4, and(t2, and(b, a)))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 55 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(t4, and(t2, and(b, or(a, not(b)))))), and(not(cin), and(a, b))))
% 13.80/2.10 = { by lemma 80 R->L }
% 13.80/2.10 tuple(sum_def, or(and(cin, or(t4, and(t2, and(b, not(and(b, not(a))))))), and(not(cin), and(a, b))))
% 13.80/2.11 = { by lemma 67 R->L }
% 13.80/2.11 tuple(sum_def, or(and(cin, or(t4, and(t2, and(b, not(not(or(a, t3))))))), and(not(cin), and(a, b))))
% 13.80/2.11 = { by axiom 6 (not_involution) }
% 13.80/2.11 tuple(sum_def, or(and(cin, or(t4, and(t2, and(b, or(a, t3))))), and(not(cin), and(a, b))))
% 13.80/2.11 = { by lemma 36 }
% 13.80/2.11 tuple(sum_def, or(and(cin, or(t4, and(t2, and(b, t3)))), and(not(cin), and(a, b))))
% 13.80/2.11 = { by lemma 46 }
% 13.80/2.11 tuple(sum_def, or(and(cin, or(t4, and(b, not(t4)))), and(not(cin), and(a, b))))
% 13.80/2.11 = { by lemma 33 }
% 13.80/2.11 tuple(sum_def, or(and(cin, or(t4, b)), and(not(cin), and(a, b))))
% 13.80/2.11 = { by axiom 7 (or_commutative) }
% 13.80/2.11 tuple(sum_def, or(and(cin, or(b, t4)), and(not(cin), and(a, b))))
% 13.80/2.11 = { by axiom 6 (not_involution) R->L }
% 13.80/2.11 tuple(sum_def, or(and(cin, not(not(or(b, t4)))), and(not(cin), and(a, b))))
% 13.80/2.11 = { by lemma 75 }
% 13.80/2.11 tuple(sum_def, or(and(cin, not(xor(b, t2))), and(not(cin), and(a, b))))
% 13.80/2.11 = { by lemma 72 R->L }
% 13.80/2.11 tuple(sum_def, or(and(cin, not(and(t2, not(b)))), and(not(cin), and(a, b))))
% 13.80/2.11 = { by lemma 80 }
% 13.80/2.11 tuple(sum_def, or(and(cin, or(b, not(t2))), and(not(cin), and(a, b))))
% 13.80/2.11 = { by lemma 70 }
% 13.80/2.11 tuple(sum_def, or(and(cin, or(a, b)), and(not(cin), and(a, b))))
% 13.80/2.11 = { by axiom 31 (carry_def) }
% 13.80/2.11 tuple(sum_def, carry_def)
% 13.80/2.11 % SZS output end Proof
% 13.80/2.11
% 13.80/2.11 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------