TSTP Solution File: RNG008-2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : RNG008-2 : TPTP v8.1.2. Released v1.0.0.
% 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 : n015.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 13:58:46 EDT 2023
% Result : Unsatisfiable 14.71s 2.34s
% Output : Proof 16.64s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : RNG008-2 : TPTP v8.1.2. Released v1.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n015.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Sun Aug 27 02:18:38 EDT 2023
% 0.12/0.34 % CPUTime :
% 14.71/2.34 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 14.71/2.34
% 14.71/2.34 % SZS status Unsatisfiable
% 14.71/2.34
% 16.04/2.49 % SZS output start Proof
% 16.04/2.49 Take the following subset of the input axioms:
% 16.04/2.49 fof(a_times_b_is_c, hypothesis, product(a, b, c)).
% 16.04/2.49 fof(addition_is_well_defined, axiom, ![X, Y, U, V]: (~sum(X, Y, U) | (~sum(X, Y, V) | U=V))).
% 16.04/2.49 fof(additive_identity1, axiom, ![X2]: sum(additive_identity, X2, X2)).
% 16.04/2.49 fof(additive_identity2, axiom, ![X2]: sum(X2, additive_identity, X2)).
% 16.04/2.49 fof(associativity_of_addition1, axiom, ![Z, W, X2, Y2, U2, V5]: (~sum(X2, Y2, U2) | (~sum(Y2, Z, V5) | (~sum(U2, Z, W) | sum(X2, V5, W))))).
% 16.04/2.49 fof(associativity_of_addition2, axiom, ![X2, Y2, Z2, W2, U2, V5]: (~sum(X2, Y2, U2) | (~sum(Y2, Z2, V5) | (~sum(X2, V5, W2) | sum(U2, Z2, W2))))).
% 16.04/2.49 fof(cancellation1, axiom, ![X2, Y2, Z2, W2]: (~sum(X2, Y2, Z2) | (~sum(X2, W2, Z2) | Y2=W2))).
% 16.04/2.49 fof(cancellation2, axiom, ![X2, Y2, Z2, W2]: (~sum(X2, Y2, Z2) | (~sum(W2, Y2, Z2) | X2=W2))).
% 16.04/2.49 fof(closure_of_addition, axiom, ![X2, Y2]: sum(X2, Y2, add(X2, Y2))).
% 16.04/2.49 fof(closure_of_multiplication, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 16.04/2.50 fof(commutativity_of_addition, axiom, ![X2, Y2, Z2]: (~sum(X2, Y2, Z2) | sum(Y2, X2, Z2))).
% 16.04/2.50 fof(distributivity1, axiom, ![V1, V2, V3, V4, X2, Y2, Z2]: (~product(X2, Y2, V1) | (~product(X2, Z2, V2) | (~sum(Y2, Z2, V3) | (~product(X2, V3, V4) | sum(V1, V2, V4)))))).
% 16.04/2.50 fof(distributivity3, axiom, ![X2, Y2, Z2, V1_2, V2_2, V3_2, V4_2]: (~product(Y2, X2, V1_2) | (~product(Z2, X2, V2_2) | (~sum(Y2, Z2, V3_2) | (~product(V3_2, X2, V4_2) | sum(V1_2, V2_2, V4_2)))))).
% 16.04/2.50 fof(distributivity4, axiom, ![X2, Y2, Z2, V1_2, V2_2, V3_2, V4_2]: (~product(Y2, X2, V1_2) | (~product(Z2, X2, V2_2) | (~sum(Y2, Z2, V3_2) | (~sum(V1_2, V2_2, V4_2) | product(V3_2, X2, V4_2)))))).
% 16.04/2.50 fof(left_inverse, axiom, ![X2]: sum(additive_inverse(X2), X2, additive_identity)).
% 16.04/2.50 fof(prove_b_times_a_is_c, negated_conjecture, ~product(b, a, c)).
% 16.04/2.50 fof(right_inverse, axiom, ![X2]: sum(X2, additive_inverse(X2), additive_identity)).
% 16.04/2.50 fof(x_squared_is_x, hypothesis, ![X2]: product(X2, X2, X2)).
% 16.04/2.50
% 16.04/2.50 Now clausify the problem and encode Horn clauses using encoding 3 of
% 16.04/2.50 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 16.04/2.50 We repeatedly replace C & s=t => u=v by the two clauses:
% 16.04/2.50 fresh(y, y, x1...xn) = u
% 16.04/2.50 C => fresh(s, t, x1...xn) = v
% 16.04/2.50 where fresh is a fresh function symbol and x1..xn are the free
% 16.04/2.50 variables of u and v.
% 16.04/2.50 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 16.04/2.50 input problem has no model of domain size 1).
% 16.04/2.50
% 16.04/2.50 The encoding turns the above axioms into the following unit equations and goals:
% 16.04/2.50
% 16.04/2.50 Axiom 1 (additive_identity2): sum(X, additive_identity, X) = true.
% 16.04/2.50 Axiom 2 (additive_identity1): sum(additive_identity, X, X) = true.
% 16.04/2.50 Axiom 3 (x_squared_is_x): product(X, X, X) = true.
% 16.04/2.50 Axiom 4 (a_times_b_is_c): product(a, b, c) = true.
% 16.04/2.50 Axiom 5 (right_inverse): sum(X, additive_inverse(X), additive_identity) = true.
% 16.04/2.50 Axiom 6 (left_inverse): sum(additive_inverse(X), X, additive_identity) = true.
% 16.04/2.50 Axiom 7 (cancellation2): fresh(X, X, Y, Z) = Z.
% 16.04/2.50 Axiom 8 (addition_is_well_defined): fresh7(X, X, Y, Z) = Z.
% 16.04/2.50 Axiom 9 (cancellation1): fresh3(X, X, Y, Z) = Z.
% 16.04/2.50 Axiom 10 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 16.04/2.50 Axiom 11 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 16.04/2.50 Axiom 12 (associativity_of_addition1): fresh37(X, X, Y, Z, W) = true.
% 16.04/2.50 Axiom 13 (associativity_of_addition2): fresh35(X, X, Y, Z, W) = true.
% 16.04/2.50 Axiom 14 (distributivity1): fresh29(X, X, Y, Z, W) = true.
% 16.04/2.50 Axiom 15 (distributivity3): fresh21(X, X, Y, Z, W) = true.
% 16.04/2.50 Axiom 16 (distributivity4): fresh17(X, X, Y, Z, W) = true.
% 16.04/2.50 Axiom 17 (commutativity_of_addition): fresh9(X, X, Y, Z, W) = true.
% 16.04/2.50 Axiom 18 (addition_is_well_defined): fresh8(X, X, Y, Z, W, V) = W.
% 16.04/2.50 Axiom 19 (cancellation1): fresh4(X, X, Y, Z, W, V) = Z.
% 16.04/2.50 Axiom 20 (cancellation2): fresh2(X, X, Y, Z, W, V) = Y.
% 16.04/2.50 Axiom 21 (distributivity4): fresh15(X, X, Y, Z, W, V, U) = product(V, Z, U).
% 16.04/2.50 Axiom 22 (associativity_of_addition1): fresh13(X, X, Y, Z, W, V, U) = sum(Y, V, U).
% 16.04/2.50 Axiom 23 (associativity_of_addition2): fresh12(X, X, Y, Z, W, V, U) = sum(W, V, U).
% 16.04/2.50 Axiom 24 (associativity_of_addition1): fresh36(X, X, Y, Z, W, V, U, T) = fresh37(sum(Y, Z, W), true, Y, U, T).
% 16.04/2.50 Axiom 25 (associativity_of_addition2): fresh34(X, X, Y, Z, W, V, U, T) = fresh35(sum(Y, Z, W), true, W, V, T).
% 16.04/2.50 Axiom 26 (distributivity1): fresh27(X, X, Y, Z, W, V, U, T) = sum(Z, V, T).
% 16.04/2.50 Axiom 27 (distributivity3): fresh19(X, X, Y, Z, W, V, U, T) = sum(Z, V, T).
% 16.04/2.50 Axiom 28 (commutativity_of_addition): fresh9(sum(X, Y, Z), true, X, Y, Z) = sum(Y, X, Z).
% 16.04/2.50 Axiom 29 (distributivity1): fresh28(X, X, Y, Z, W, V, U, T, S) = fresh29(sum(Z, V, T), true, W, U, S).
% 16.04/2.50 Axiom 30 (distributivity3): fresh20(X, X, Y, Z, W, V, U, T, S) = fresh21(sum(Y, V, T), true, W, U, S).
% 16.04/2.50 Axiom 31 (distributivity4): fresh16(X, X, Y, Z, W, V, U, T, S) = fresh17(sum(Y, V, T), true, Z, T, S).
% 16.04/2.50 Axiom 32 (addition_is_well_defined): fresh8(sum(X, Y, Z), true, X, Y, W, Z) = fresh7(sum(X, Y, W), true, W, Z).
% 16.04/2.50 Axiom 33 (cancellation1): fresh4(sum(X, Y, Z), true, X, W, Z, Y) = fresh3(sum(X, W, Z), true, W, Y).
% 16.04/2.50 Axiom 34 (cancellation2): fresh2(sum(X, Y, Z), true, W, Y, Z, X) = fresh(sum(W, Y, Z), true, W, X).
% 16.04/2.50 Axiom 35 (distributivity4): fresh14(X, X, Y, Z, W, V, U, T, S) = fresh15(sum(W, U, S), true, Y, Z, V, T, S).
% 16.04/2.50 Axiom 36 (associativity_of_addition1): fresh36(sum(X, Y, Z), true, W, V, X, Y, U, Z) = fresh13(sum(V, Y, U), true, W, V, X, U, Z).
% 16.04/2.50 Axiom 37 (associativity_of_addition2): fresh34(sum(X, Y, Z), true, W, X, V, Y, Z, U) = fresh12(sum(W, Z, U), true, W, X, V, Y, U).
% 16.04/2.50 Axiom 38 (distributivity1): fresh26(X, X, Y, Z, W, V, U, T, S) = fresh27(product(Y, Z, W), true, Z, W, V, U, T, S).
% 16.04/2.50 Axiom 39 (distributivity3): fresh18(X, X, Y, Z, W, V, U, T, S) = fresh19(product(Y, Z, W), true, Y, W, V, U, T, S).
% 16.04/2.50 Axiom 40 (distributivity1): fresh26(product(X, Y, Z), true, X, W, V, U, T, Y, Z) = fresh28(product(X, U, T), true, X, W, V, U, T, Y, Z).
% 16.04/2.50 Axiom 41 (distributivity3): fresh18(product(X, Y, Z), true, W, Y, V, U, T, X, Z) = fresh20(product(U, Y, T), true, W, Y, V, U, T, X, Z).
% 16.04/2.50 Axiom 42 (distributivity4): fresh14(product(X, Y, Z), true, W, Y, V, X, Z, U, T) = fresh16(product(W, Y, V), true, W, Y, V, X, Z, U, T).
% 16.04/2.50
% 16.04/2.50 Lemma 43: fresh3(sum(X, Y, additive_identity), true, Y, additive_inverse(X)) = Y.
% 16.04/2.50 Proof:
% 16.04/2.50 fresh3(sum(X, Y, additive_identity), true, Y, additive_inverse(X))
% 16.04/2.50 = { by axiom 33 (cancellation1) R->L }
% 16.04/2.50 fresh4(sum(X, additive_inverse(X), additive_identity), true, X, Y, additive_identity, additive_inverse(X))
% 16.04/2.50 = { by axiom 5 (right_inverse) }
% 16.04/2.50 fresh4(true, true, X, Y, additive_identity, additive_inverse(X))
% 16.04/2.50 = { by axiom 19 (cancellation1) }
% 16.04/2.50 Y
% 16.04/2.50
% 16.04/2.50 Lemma 44: sum(X, Y, add(Y, X)) = true.
% 16.04/2.50 Proof:
% 16.04/2.50 sum(X, Y, add(Y, X))
% 16.04/2.50 = { by axiom 28 (commutativity_of_addition) R->L }
% 16.04/2.50 fresh9(sum(Y, X, add(Y, X)), true, Y, X, add(Y, X))
% 16.04/2.50 = { by axiom 10 (closure_of_addition) }
% 16.04/2.50 fresh9(true, true, Y, X, add(Y, X))
% 16.04/2.50 = { by axiom 17 (commutativity_of_addition) }
% 16.04/2.50 true
% 16.04/2.50
% 16.04/2.50 Lemma 45: fresh7(sum(X, additive_identity, Y), true, Y, X) = Y.
% 16.04/2.50 Proof:
% 16.04/2.50 fresh7(sum(X, additive_identity, Y), true, Y, X)
% 16.04/2.50 = { by axiom 32 (addition_is_well_defined) R->L }
% 16.04/2.50 fresh8(sum(X, additive_identity, X), true, X, additive_identity, Y, X)
% 16.04/2.50 = { by axiom 1 (additive_identity2) }
% 16.04/2.50 fresh8(true, true, X, additive_identity, Y, X)
% 16.04/2.50 = { by axiom 18 (addition_is_well_defined) }
% 16.04/2.50 Y
% 16.04/2.50
% 16.04/2.50 Lemma 46: fresh(sum(X, Y, Y), true, X, additive_identity) = X.
% 16.04/2.50 Proof:
% 16.04/2.50 fresh(sum(X, Y, Y), true, X, additive_identity)
% 16.04/2.50 = { by axiom 34 (cancellation2) R->L }
% 16.04/2.50 fresh2(sum(additive_identity, Y, Y), true, X, Y, Y, additive_identity)
% 16.04/2.50 = { by axiom 2 (additive_identity1) }
% 16.04/2.50 fresh2(true, true, X, Y, Y, additive_identity)
% 16.04/2.50 = { by axiom 20 (cancellation2) }
% 16.04/2.50 X
% 16.04/2.50
% 16.04/2.50 Lemma 47: fresh18(X, X, Y, Z, multiply(Y, Z), W, V, U, T) = sum(multiply(Y, Z), V, T).
% 16.04/2.50 Proof:
% 16.04/2.50 fresh18(X, X, Y, Z, multiply(Y, Z), W, V, U, T)
% 16.04/2.50 = { by axiom 39 (distributivity3) }
% 16.04/2.50 fresh19(product(Y, Z, multiply(Y, Z)), true, Y, multiply(Y, Z), W, V, U, T)
% 16.04/2.50 = { by axiom 11 (closure_of_multiplication) }
% 16.04/2.50 fresh19(true, true, Y, multiply(Y, Z), W, V, U, T)
% 16.04/2.50 = { by axiom 27 (distributivity3) }
% 16.04/2.50 sum(multiply(Y, Z), V, T)
% 16.04/2.50
% 16.04/2.50 Lemma 48: fresh18(product(X, Y, Z), true, W, Y, V, Y, Y, X, Z) = fresh21(sum(W, Y, X), true, V, Y, Z).
% 16.04/2.50 Proof:
% 16.04/2.50 fresh18(product(X, Y, Z), true, W, Y, V, Y, Y, X, Z)
% 16.04/2.50 = { by axiom 41 (distributivity3) }
% 16.04/2.50 fresh20(product(Y, Y, Y), true, W, Y, V, Y, Y, X, Z)
% 16.04/2.50 = { by axiom 3 (x_squared_is_x) }
% 16.04/2.50 fresh20(true, true, W, Y, V, Y, Y, X, Z)
% 16.04/2.50 = { by axiom 30 (distributivity3) }
% 16.04/2.50 fresh21(sum(W, Y, X), true, V, Y, Z)
% 16.04/2.50
% 16.04/2.50 Lemma 49: multiply(additive_identity, X) = additive_identity.
% 16.04/2.50 Proof:
% 16.04/2.50 multiply(additive_identity, X)
% 16.04/2.50 = { by lemma 46 R->L }
% 16.04/2.50 fresh(sum(multiply(additive_identity, X), X, X), true, multiply(additive_identity, X), additive_identity)
% 16.04/2.51 = { by lemma 47 R->L }
% 16.04/2.51 fresh(fresh18(true, true, additive_identity, X, multiply(additive_identity, X), X, X, X, X), true, multiply(additive_identity, X), additive_identity)
% 16.04/2.51 = { by axiom 3 (x_squared_is_x) R->L }
% 16.04/2.51 fresh(fresh18(product(X, X, X), true, additive_identity, X, multiply(additive_identity, X), X, X, X, X), true, multiply(additive_identity, X), additive_identity)
% 16.04/2.51 = { by lemma 48 }
% 16.04/2.51 fresh(fresh21(sum(additive_identity, X, X), true, multiply(additive_identity, X), X, X), true, multiply(additive_identity, X), additive_identity)
% 16.04/2.51 = { by axiom 2 (additive_identity1) }
% 16.04/2.51 fresh(fresh21(true, true, multiply(additive_identity, X), X, X), true, multiply(additive_identity, X), additive_identity)
% 16.04/2.51 = { by axiom 15 (distributivity3) }
% 16.04/2.51 fresh(true, true, multiply(additive_identity, X), additive_identity)
% 16.04/2.51 = { by axiom 7 (cancellation2) }
% 16.04/2.51 additive_identity
% 16.04/2.51
% 16.04/2.51 Lemma 50: fresh21(sum(X, Y, Z), true, multiply(X, Y), Y, multiply(Z, Y)) = sum(multiply(X, Y), Y, multiply(Z, Y)).
% 16.04/2.51 Proof:
% 16.04/2.51 fresh21(sum(X, Y, Z), true, multiply(X, Y), Y, multiply(Z, Y))
% 16.04/2.51 = { by lemma 48 R->L }
% 16.04/2.51 fresh18(product(Z, Y, multiply(Z, Y)), true, X, Y, multiply(X, Y), Y, Y, Z, multiply(Z, Y))
% 16.04/2.51 = { by axiom 11 (closure_of_multiplication) }
% 16.04/2.51 fresh18(true, true, X, Y, multiply(X, Y), Y, Y, Z, multiply(Z, Y))
% 16.04/2.51 = { by lemma 47 }
% 16.04/2.51 sum(multiply(X, Y), Y, multiply(Z, Y))
% 16.04/2.51
% 16.04/2.51 Lemma 51: fresh34(X, X, Y, Z, add(Y, Z), W, V, U) = true.
% 16.04/2.51 Proof:
% 16.04/2.51 fresh34(X, X, Y, Z, add(Y, Z), W, V, U)
% 16.04/2.51 = { by axiom 25 (associativity_of_addition2) }
% 16.04/2.51 fresh35(sum(Y, Z, add(Y, Z)), true, add(Y, Z), W, U)
% 16.04/2.51 = { by axiom 10 (closure_of_addition) }
% 16.04/2.51 fresh35(true, true, add(Y, Z), W, U)
% 16.04/2.51 = { by axiom 13 (associativity_of_addition2) }
% 16.04/2.51 true
% 16.04/2.51
% 16.04/2.51 Lemma 52: sum(add(X, additive_inverse(Y)), Y, X) = true.
% 16.04/2.51 Proof:
% 16.04/2.51 sum(add(X, additive_inverse(Y)), Y, X)
% 16.04/2.51 = { by axiom 23 (associativity_of_addition2) R->L }
% 16.04/2.51 fresh12(true, true, X, additive_inverse(Y), add(X, additive_inverse(Y)), Y, X)
% 16.04/2.51 = { by axiom 1 (additive_identity2) R->L }
% 16.04/2.51 fresh12(sum(X, additive_identity, X), true, X, additive_inverse(Y), add(X, additive_inverse(Y)), Y, X)
% 16.04/2.51 = { by axiom 37 (associativity_of_addition2) R->L }
% 16.04/2.51 fresh34(sum(additive_inverse(Y), Y, additive_identity), true, X, additive_inverse(Y), add(X, additive_inverse(Y)), Y, additive_identity, X)
% 16.04/2.51 = { by axiom 6 (left_inverse) }
% 16.04/2.51 fresh34(true, true, X, additive_inverse(Y), add(X, additive_inverse(Y)), Y, additive_identity, X)
% 16.04/2.51 = { by lemma 51 }
% 16.04/2.51 true
% 16.04/2.51
% 16.04/2.51 Lemma 53: fresh7(sum(X, Y, Z), true, Z, add(X, Y)) = Z.
% 16.04/2.51 Proof:
% 16.04/2.51 fresh7(sum(X, Y, Z), true, Z, add(X, Y))
% 16.04/2.51 = { by axiom 32 (addition_is_well_defined) R->L }
% 16.04/2.51 fresh8(sum(X, Y, add(X, Y)), true, X, Y, Z, add(X, Y))
% 16.04/2.51 = { by axiom 10 (closure_of_addition) }
% 16.04/2.51 fresh8(true, true, X, Y, Z, add(X, Y))
% 16.04/2.51 = { by axiom 18 (addition_is_well_defined) }
% 16.04/2.51 Z
% 16.04/2.51
% 16.04/2.51 Lemma 54: add(X, Y) = add(Y, X).
% 16.04/2.51 Proof:
% 16.04/2.51 add(X, Y)
% 16.04/2.51 = { by lemma 53 R->L }
% 16.04/2.51 fresh7(sum(Y, X, add(X, Y)), true, add(X, Y), add(Y, X))
% 16.04/2.51 = { by lemma 44 }
% 16.04/2.51 fresh7(true, true, add(X, Y), add(Y, X))
% 16.04/2.51 = { by axiom 8 (addition_is_well_defined) }
% 16.04/2.51 add(Y, X)
% 16.04/2.51
% 16.04/2.51 Lemma 55: fresh26(X, X, Y, Z, multiply(Y, Z), W, V, U, T) = sum(multiply(Y, Z), V, T).
% 16.04/2.51 Proof:
% 16.04/2.51 fresh26(X, X, Y, Z, multiply(Y, Z), W, V, U, T)
% 16.04/2.51 = { by axiom 38 (distributivity1) }
% 16.04/2.51 fresh27(product(Y, Z, multiply(Y, Z)), true, Z, multiply(Y, Z), W, V, U, T)
% 16.04/2.51 = { by axiom 11 (closure_of_multiplication) }
% 16.04/2.51 fresh27(true, true, Z, multiply(Y, Z), W, V, U, T)
% 16.04/2.51 = { by axiom 26 (distributivity1) }
% 16.04/2.51 sum(multiply(Y, Z), V, T)
% 16.04/2.51
% 16.04/2.51 Lemma 56: fresh26(product(X, Y, Z), true, X, W, V, X, X, Y, Z) = fresh29(sum(W, X, Y), true, V, X, Z).
% 16.04/2.51 Proof:
% 16.04/2.51 fresh26(product(X, Y, Z), true, X, W, V, X, X, Y, Z)
% 16.04/2.51 = { by axiom 40 (distributivity1) }
% 16.04/2.51 fresh28(product(X, X, X), true, X, W, V, X, X, Y, Z)
% 16.04/2.51 = { by axiom 3 (x_squared_is_x) }
% 16.04/2.51 fresh28(true, true, X, W, V, X, X, Y, Z)
% 16.04/2.51 = { by axiom 29 (distributivity1) }
% 16.04/2.51 fresh29(sum(W, X, Y), true, V, X, Z)
% 16.04/2.51
% 16.04/2.51 Lemma 57: multiply(X, additive_identity) = additive_identity.
% 16.04/2.51 Proof:
% 16.04/2.51 multiply(X, additive_identity)
% 16.04/2.51 = { by lemma 46 R->L }
% 16.04/2.51 fresh(sum(multiply(X, additive_identity), X, X), true, multiply(X, additive_identity), additive_identity)
% 16.04/2.51 = { by lemma 55 R->L }
% 16.04/2.51 fresh(fresh26(true, true, X, additive_identity, multiply(X, additive_identity), X, X, X, X), true, multiply(X, additive_identity), additive_identity)
% 16.04/2.51 = { by axiom 3 (x_squared_is_x) R->L }
% 16.04/2.51 fresh(fresh26(product(X, X, X), true, X, additive_identity, multiply(X, additive_identity), X, X, X, X), true, multiply(X, additive_identity), additive_identity)
% 16.04/2.51 = { by lemma 56 }
% 16.04/2.51 fresh(fresh29(sum(additive_identity, X, X), true, multiply(X, additive_identity), X, X), true, multiply(X, additive_identity), additive_identity)
% 16.04/2.51 = { by axiom 2 (additive_identity1) }
% 16.04/2.51 fresh(fresh29(true, true, multiply(X, additive_identity), X, X), true, multiply(X, additive_identity), additive_identity)
% 16.04/2.51 = { by axiom 14 (distributivity1) }
% 16.04/2.51 fresh(true, true, multiply(X, additive_identity), additive_identity)
% 16.04/2.51 = { by axiom 7 (cancellation2) }
% 16.04/2.51 additive_identity
% 16.04/2.51
% 16.04/2.51 Lemma 58: fresh26(X, X, Y, Z, W, V, multiply(Y, V), U, multiply(Y, U)) = fresh29(sum(Z, V, U), true, W, multiply(Y, V), multiply(Y, U)).
% 16.04/2.51 Proof:
% 16.04/2.51 fresh26(X, X, Y, Z, W, V, multiply(Y, V), U, multiply(Y, U))
% 16.04/2.51 = { by axiom 38 (distributivity1) }
% 16.04/2.51 fresh27(product(Y, Z, W), true, Z, W, V, multiply(Y, V), U, multiply(Y, U))
% 16.04/2.51 = { by axiom 38 (distributivity1) R->L }
% 16.04/2.51 fresh26(true, true, Y, Z, W, V, multiply(Y, V), U, multiply(Y, U))
% 16.04/2.51 = { by axiom 11 (closure_of_multiplication) R->L }
% 16.04/2.51 fresh26(product(Y, U, multiply(Y, U)), true, Y, Z, W, V, multiply(Y, V), U, multiply(Y, U))
% 16.04/2.51 = { by axiom 40 (distributivity1) }
% 16.04/2.51 fresh28(product(Y, V, multiply(Y, V)), true, Y, Z, W, V, multiply(Y, V), U, multiply(Y, U))
% 16.04/2.51 = { by axiom 11 (closure_of_multiplication) }
% 16.04/2.51 fresh28(true, true, Y, Z, W, V, multiply(Y, V), U, multiply(Y, U))
% 16.04/2.51 = { by axiom 29 (distributivity1) }
% 16.04/2.51 fresh29(sum(Z, V, U), true, W, multiply(Y, V), multiply(Y, U))
% 16.04/2.51
% 16.04/2.51 Lemma 59: fresh36(sum(X, Y, Z), true, W, V, X, Y, add(V, Y), Z) = sum(W, add(Y, V), Z).
% 16.04/2.51 Proof:
% 16.04/2.51 fresh36(sum(X, Y, Z), true, W, V, X, Y, add(V, Y), Z)
% 16.04/2.51 = { by axiom 36 (associativity_of_addition1) }
% 16.04/2.51 fresh13(sum(V, Y, add(V, Y)), true, W, V, X, add(V, Y), Z)
% 16.04/2.51 = { by axiom 10 (closure_of_addition) }
% 16.04/2.51 fresh13(true, true, W, V, X, add(V, Y), Z)
% 16.04/2.51 = { by axiom 22 (associativity_of_addition1) }
% 16.04/2.51 sum(W, add(V, Y), Z)
% 16.04/2.51 = { by lemma 54 }
% 16.04/2.51 sum(W, add(Y, V), Z)
% 16.04/2.51
% 16.04/2.51 Lemma 60: fresh36(X, X, Y, Z, add(Y, Z), W, V, U) = true.
% 16.04/2.51 Proof:
% 16.04/2.51 fresh36(X, X, Y, Z, add(Y, Z), W, V, U)
% 16.04/2.51 = { by axiom 24 (associativity_of_addition1) }
% 16.04/2.51 fresh37(sum(Y, Z, add(Y, Z)), true, Y, V, U)
% 16.04/2.51 = { by axiom 10 (closure_of_addition) }
% 16.04/2.51 fresh37(true, true, Y, V, U)
% 16.04/2.51 = { by axiom 12 (associativity_of_addition1) }
% 16.04/2.51 true
% 16.04/2.51
% 16.04/2.51 Lemma 61: sum(X, add(Y, additive_inverse(add(X, Y))), additive_identity) = true.
% 16.04/2.51 Proof:
% 16.04/2.51 sum(X, add(Y, additive_inverse(add(X, Y))), additive_identity)
% 16.04/2.51 = { by lemma 54 R->L }
% 16.04/2.51 sum(X, add(additive_inverse(add(X, Y)), Y), additive_identity)
% 16.04/2.51 = { by lemma 59 R->L }
% 16.04/2.51 fresh36(sum(add(X, Y), additive_inverse(add(X, Y)), additive_identity), true, X, Y, add(X, Y), additive_inverse(add(X, Y)), add(Y, additive_inverse(add(X, Y))), additive_identity)
% 16.04/2.51 = { by axiom 5 (right_inverse) }
% 16.04/2.51 fresh36(true, true, X, Y, add(X, Y), additive_inverse(add(X, Y)), add(Y, additive_inverse(add(X, Y))), additive_identity)
% 16.04/2.51 = { by lemma 60 }
% 16.04/2.51 true
% 16.04/2.51
% 16.04/2.51 Lemma 62: additive_inverse(X) = X.
% 16.04/2.51 Proof:
% 16.04/2.51 additive_inverse(X)
% 16.04/2.51 = { by lemma 43 R->L }
% 16.04/2.51 fresh3(sum(additive_inverse(additive_inverse(X)), additive_inverse(X), additive_identity), true, additive_inverse(X), additive_inverse(additive_inverse(additive_inverse(X))))
% 16.04/2.51 = { by axiom 6 (left_inverse) }
% 16.04/2.51 fresh3(true, true, additive_inverse(X), additive_inverse(additive_inverse(additive_inverse(X))))
% 16.04/2.51 = { by axiom 9 (cancellation1) }
% 16.04/2.51 additive_inverse(additive_inverse(additive_inverse(X)))
% 16.04/2.51 = { by axiom 8 (addition_is_well_defined) R->L }
% 16.04/2.51 fresh7(true, true, add(additive_identity, additive_inverse(additive_inverse(additive_inverse(X)))), additive_inverse(additive_inverse(additive_inverse(X))))
% 16.04/2.51 = { by lemma 44 R->L }
% 16.04/2.51 fresh7(sum(additive_inverse(additive_inverse(additive_inverse(X))), additive_identity, add(additive_identity, additive_inverse(additive_inverse(additive_inverse(X))))), true, add(additive_identity, additive_inverse(additive_inverse(additive_inverse(X)))), additive_inverse(additive_inverse(additive_inverse(X))))
% 16.04/2.51 = { by lemma 45 }
% 16.64/2.51 add(additive_identity, additive_inverse(additive_inverse(additive_inverse(X))))
% 16.64/2.51 = { by lemma 49 R->L }
% 16.64/2.51 add(multiply(additive_identity, additive_inverse(additive_inverse(X))), additive_inverse(additive_inverse(additive_inverse(X))))
% 16.64/2.51 = { by axiom 20 (cancellation2) R->L }
% 16.64/2.51 fresh2(true, true, add(multiply(additive_identity, additive_inverse(additive_inverse(X))), additive_inverse(additive_inverse(additive_inverse(X)))), additive_inverse(additive_inverse(X)), multiply(additive_identity, additive_inverse(additive_inverse(X))), multiply(additive_inverse(X), additive_inverse(additive_inverse(X))))
% 16.64/2.52 = { by axiom 15 (distributivity3) R->L }
% 16.64/2.52 fresh2(fresh21(true, true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), additive_inverse(additive_inverse(X)), multiply(additive_identity, additive_inverse(additive_inverse(X)))), true, add(multiply(additive_identity, additive_inverse(additive_inverse(X))), additive_inverse(additive_inverse(additive_inverse(X)))), additive_inverse(additive_inverse(X)), multiply(additive_identity, additive_inverse(additive_inverse(X))), multiply(additive_inverse(X), additive_inverse(additive_inverse(X))))
% 16.64/2.52 = { by axiom 5 (right_inverse) R->L }
% 16.64/2.52 fresh2(fresh21(sum(additive_inverse(X), additive_inverse(additive_inverse(X)), additive_identity), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), additive_inverse(additive_inverse(X)), multiply(additive_identity, additive_inverse(additive_inverse(X)))), true, add(multiply(additive_identity, additive_inverse(additive_inverse(X))), additive_inverse(additive_inverse(additive_inverse(X)))), additive_inverse(additive_inverse(X)), multiply(additive_identity, additive_inverse(additive_inverse(X))), multiply(additive_inverse(X), additive_inverse(additive_inverse(X))))
% 16.64/2.52 = { by lemma 50 }
% 16.64/2.52 fresh2(sum(multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), additive_inverse(additive_inverse(X)), multiply(additive_identity, additive_inverse(additive_inverse(X)))), true, add(multiply(additive_identity, additive_inverse(additive_inverse(X))), additive_inverse(additive_inverse(additive_inverse(X)))), additive_inverse(additive_inverse(X)), multiply(additive_identity, additive_inverse(additive_inverse(X))), multiply(additive_inverse(X), additive_inverse(additive_inverse(X))))
% 16.64/2.52 = { by axiom 34 (cancellation2) }
% 16.64/2.52 fresh(sum(add(multiply(additive_identity, additive_inverse(additive_inverse(X))), additive_inverse(additive_inverse(additive_inverse(X)))), additive_inverse(additive_inverse(X)), multiply(additive_identity, additive_inverse(additive_inverse(X)))), true, add(multiply(additive_identity, additive_inverse(additive_inverse(X))), additive_inverse(additive_inverse(additive_inverse(X)))), multiply(additive_inverse(X), additive_inverse(additive_inverse(X))))
% 16.64/2.52 = { by lemma 52 }
% 16.64/2.52 fresh(true, true, add(multiply(additive_identity, additive_inverse(additive_inverse(X))), additive_inverse(additive_inverse(additive_inverse(X)))), multiply(additive_inverse(X), additive_inverse(additive_inverse(X))))
% 16.64/2.52 = { by axiom 7 (cancellation2) }
% 16.64/2.52 multiply(additive_inverse(X), additive_inverse(additive_inverse(X)))
% 16.64/2.52 = { by axiom 19 (cancellation1) R->L }
% 16.64/2.52 fresh4(true, true, additive_inverse(X), multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), additive_identity, X)
% 16.64/2.52 = { by axiom 6 (left_inverse) R->L }
% 16.64/2.52 fresh4(sum(additive_inverse(X), X, additive_identity), true, additive_inverse(X), multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), additive_identity, X)
% 16.64/2.52 = { by axiom 33 (cancellation1) }
% 16.64/2.52 fresh3(sum(additive_inverse(X), multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), additive_identity), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 7 (cancellation2) R->L }
% 16.64/2.52 fresh3(sum(additive_inverse(X), multiply(additive_inverse(X), fresh(true, true, add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), additive_inverse(additive_inverse(X)))), additive_identity), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by lemma 51 R->L }
% 16.64/2.52 fresh3(sum(additive_inverse(X), multiply(additive_inverse(X), fresh(fresh34(true, true, additive_inverse(add(additive_inverse(X), additive_inverse(X))), additive_inverse(X), add(additive_inverse(add(additive_inverse(X), additive_inverse(X))), additive_inverse(X)), additive_inverse(X), add(additive_inverse(X), additive_inverse(X)), additive_identity), true, add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), additive_inverse(additive_inverse(X)))), additive_identity), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 10 (closure_of_addition) R->L }
% 16.64/2.52 fresh3(sum(additive_inverse(X), multiply(additive_inverse(X), fresh(fresh34(sum(additive_inverse(X), additive_inverse(X), add(additive_inverse(X), additive_inverse(X))), true, additive_inverse(add(additive_inverse(X), additive_inverse(X))), additive_inverse(X), add(additive_inverse(add(additive_inverse(X), additive_inverse(X))), additive_inverse(X)), additive_inverse(X), add(additive_inverse(X), additive_inverse(X)), additive_identity), true, add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), additive_inverse(additive_inverse(X)))), additive_identity), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 37 (associativity_of_addition2) }
% 16.64/2.52 fresh3(sum(additive_inverse(X), multiply(additive_inverse(X), fresh(fresh12(sum(additive_inverse(add(additive_inverse(X), additive_inverse(X))), add(additive_inverse(X), additive_inverse(X)), additive_identity), true, additive_inverse(add(additive_inverse(X), additive_inverse(X))), additive_inverse(X), add(additive_inverse(add(additive_inverse(X), additive_inverse(X))), additive_inverse(X)), additive_inverse(X), additive_identity), true, add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), additive_inverse(additive_inverse(X)))), additive_identity), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 6 (left_inverse) }
% 16.64/2.52 fresh3(sum(additive_inverse(X), multiply(additive_inverse(X), fresh(fresh12(true, true, additive_inverse(add(additive_inverse(X), additive_inverse(X))), additive_inverse(X), add(additive_inverse(add(additive_inverse(X), additive_inverse(X))), additive_inverse(X)), additive_inverse(X), additive_identity), true, add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), additive_inverse(additive_inverse(X)))), additive_identity), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 23 (associativity_of_addition2) }
% 16.64/2.52 fresh3(sum(additive_inverse(X), multiply(additive_inverse(X), fresh(sum(add(additive_inverse(add(additive_inverse(X), additive_inverse(X))), additive_inverse(X)), additive_inverse(X), additive_identity), true, add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), additive_inverse(additive_inverse(X)))), additive_identity), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by lemma 54 }
% 16.64/2.52 fresh3(sum(additive_inverse(X), multiply(additive_inverse(X), fresh(sum(add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), additive_inverse(X), additive_identity), true, add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), additive_inverse(additive_inverse(X)))), additive_identity), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 34 (cancellation2) R->L }
% 16.64/2.52 fresh3(sum(additive_inverse(X), multiply(additive_inverse(X), fresh2(sum(additive_inverse(additive_inverse(X)), additive_inverse(X), additive_identity), true, add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), additive_inverse(X), additive_identity, additive_inverse(additive_inverse(X)))), additive_identity), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 6 (left_inverse) }
% 16.64/2.52 fresh3(sum(additive_inverse(X), multiply(additive_inverse(X), fresh2(true, true, add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), additive_inverse(X), additive_identity, additive_inverse(additive_inverse(X)))), additive_identity), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 20 (cancellation2) }
% 16.64/2.52 fresh3(sum(additive_inverse(X), multiply(additive_inverse(X), add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X))))), additive_identity), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by lemma 57 R->L }
% 16.64/2.52 fresh3(sum(additive_inverse(X), multiply(additive_inverse(X), add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X))))), multiply(additive_inverse(X), additive_identity)), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 26 (distributivity1) R->L }
% 16.64/2.52 fresh3(fresh27(true, true, additive_inverse(X), additive_inverse(X), add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), multiply(additive_inverse(X), add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X))))), additive_identity, multiply(additive_inverse(X), additive_identity)), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 3 (x_squared_is_x) R->L }
% 16.64/2.52 fresh3(fresh27(product(additive_inverse(X), additive_inverse(X), additive_inverse(X)), true, additive_inverse(X), additive_inverse(X), add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), multiply(additive_inverse(X), add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X))))), additive_identity, multiply(additive_inverse(X), additive_identity)), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 38 (distributivity1) R->L }
% 16.64/2.52 fresh3(fresh26(Y, Y, additive_inverse(X), additive_inverse(X), additive_inverse(X), add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), multiply(additive_inverse(X), add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X))))), additive_identity, multiply(additive_inverse(X), additive_identity)), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by lemma 58 }
% 16.64/2.52 fresh3(fresh29(sum(additive_inverse(X), add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X)))), additive_identity), true, additive_inverse(X), multiply(additive_inverse(X), add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X))))), multiply(additive_inverse(X), additive_identity)), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by lemma 61 }
% 16.64/2.52 fresh3(fresh29(true, true, additive_inverse(X), multiply(additive_inverse(X), add(additive_inverse(X), additive_inverse(add(additive_inverse(X), additive_inverse(X))))), multiply(additive_inverse(X), additive_identity)), true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 14 (distributivity1) }
% 16.64/2.52 fresh3(true, true, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))), X)
% 16.64/2.52 = { by axiom 9 (cancellation1) }
% 16.64/2.52 X
% 16.64/2.52
% 16.64/2.52 Lemma 63: add(X, add(Y, Z)) = add(Y, add(X, Z)).
% 16.64/2.52 Proof:
% 16.64/2.52 add(X, add(Y, Z))
% 16.64/2.52 = { by lemma 53 R->L }
% 16.64/2.52 fresh7(sum(Y, add(X, Z), add(X, add(Y, Z))), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 16.64/2.52 = { by lemma 54 R->L }
% 16.64/2.52 fresh7(sum(Y, add(X, Z), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 16.64/2.52 = { by lemma 59 R->L }
% 16.64/2.52 fresh7(fresh36(sum(add(Y, Z), X, add(add(Y, Z), X)), true, Y, Z, add(Y, Z), X, add(Z, X), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 16.64/2.52 = { by axiom 10 (closure_of_addition) }
% 16.64/2.52 fresh7(fresh36(true, true, Y, Z, add(Y, Z), X, add(Z, X), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 16.64/2.52 = { by axiom 24 (associativity_of_addition1) }
% 16.64/2.52 fresh7(fresh37(sum(Y, Z, add(Y, Z)), true, Y, add(Z, X), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 16.64/2.52 = { by axiom 24 (associativity_of_addition1) R->L }
% 16.64/2.52 fresh7(fresh36(W, W, Y, Z, add(Y, Z), V, add(Z, X), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 16.64/2.52 = { by lemma 60 }
% 16.64/2.52 fresh7(true, true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 16.64/2.52 = { by axiom 8 (addition_is_well_defined) }
% 16.64/2.52 add(Y, add(X, Z))
% 16.64/2.52
% 16.64/2.52 Lemma 64: add(X, X) = additive_identity.
% 16.64/2.52 Proof:
% 16.64/2.52 add(X, X)
% 16.64/2.52 = { by axiom 7 (cancellation2) R->L }
% 16.64/2.52 fresh(true, true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.52 = { by axiom 15 (distributivity3) R->L }
% 16.64/2.52 fresh(fresh21(true, true, multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.52 = { by lemma 60 R->L }
% 16.64/2.52 fresh(fresh21(fresh36(true, true, additive_identity, X, add(additive_identity, X), additive_inverse(X), additive_identity, add(add(additive_identity, X), additive_inverse(X))), true, multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.52 = { by axiom 10 (closure_of_addition) R->L }
% 16.64/2.52 fresh(fresh21(fresh36(sum(add(additive_identity, X), additive_inverse(X), add(add(additive_identity, X), additive_inverse(X))), true, additive_identity, X, add(additive_identity, X), additive_inverse(X), additive_identity, add(add(additive_identity, X), additive_inverse(X))), true, multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.52 = { by axiom 36 (associativity_of_addition1) }
% 16.64/2.52 fresh(fresh21(fresh13(sum(X, additive_inverse(X), additive_identity), true, additive_identity, X, add(additive_identity, X), additive_identity, add(add(additive_identity, X), additive_inverse(X))), true, multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.52 = { by axiom 5 (right_inverse) }
% 16.64/2.52 fresh(fresh21(fresh13(true, true, additive_identity, X, add(additive_identity, X), additive_identity, add(add(additive_identity, X), additive_inverse(X))), true, multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.52 = { by axiom 22 (associativity_of_addition1) }
% 16.64/2.52 fresh(fresh21(sum(additive_identity, additive_identity, add(add(additive_identity, X), additive_inverse(X))), true, multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.52 = { by lemma 54 }
% 16.64/2.53 fresh(fresh21(sum(additive_identity, additive_identity, add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.53 = { by axiom 30 (distributivity3) R->L }
% 16.64/2.53 fresh(fresh20(true, true, additive_identity, add(additive_inverse(X), add(additive_identity, X)), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), additive_identity, multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(additive_inverse(X), add(additive_identity, X)), add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.53 = { by axiom 11 (closure_of_multiplication) R->L }
% 16.64/2.53 fresh(fresh20(product(additive_identity, add(additive_inverse(X), add(additive_identity, X)), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X)))), true, additive_identity, add(additive_inverse(X), add(additive_identity, X)), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), additive_identity, multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(additive_inverse(X), add(additive_identity, X)), add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.53 = { by axiom 41 (distributivity3) R->L }
% 16.64/2.53 fresh(fresh18(product(add(additive_inverse(X), add(additive_identity, X)), add(additive_inverse(X), add(additive_identity, X)), add(additive_inverse(X), add(additive_identity, X))), true, additive_identity, add(additive_inverse(X), add(additive_identity, X)), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), additive_identity, multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(additive_inverse(X), add(additive_identity, X)), add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.53 = { by axiom 3 (x_squared_is_x) }
% 16.64/2.53 fresh(fresh18(true, true, additive_identity, add(additive_inverse(X), add(additive_identity, X)), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), additive_identity, multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(additive_inverse(X), add(additive_identity, X)), add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.53 = { by lemma 47 }
% 16.64/2.53 fresh(sum(multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(additive_inverse(X), add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.53 = { by lemma 62 }
% 16.64/2.53 fresh(sum(multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), add(X, add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.53 = { by lemma 62 }
% 16.64/2.53 fresh(sum(multiply(additive_identity, add(additive_inverse(X), add(additive_identity, X))), multiply(additive_identity, add(X, add(additive_identity, X))), add(X, add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.53 = { by lemma 62 }
% 16.64/2.53 fresh(sum(multiply(additive_identity, add(X, add(additive_identity, X))), multiply(additive_identity, add(X, add(additive_identity, X))), add(X, add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.53 = { by lemma 49 }
% 16.64/2.53 fresh(sum(multiply(additive_identity, add(X, add(additive_identity, X))), additive_identity, add(X, add(additive_identity, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.53 = { by lemma 63 }
% 16.64/2.53 fresh(sum(multiply(additive_identity, add(X, add(additive_identity, X))), additive_identity, add(additive_identity, add(X, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.53 = { by lemma 63 }
% 16.64/2.53 fresh(sum(multiply(additive_identity, add(additive_identity, add(X, X))), additive_identity, add(additive_identity, add(X, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), add(X, X))
% 16.64/2.53 = { by axiom 34 (cancellation2) R->L }
% 16.64/2.53 fresh2(sum(add(X, X), additive_identity, add(additive_identity, add(X, X))), true, multiply(additive_identity, add(additive_identity, add(X, X))), additive_identity, add(additive_identity, add(X, X)), add(X, X))
% 16.64/2.53 = { by lemma 44 }
% 16.64/2.53 fresh2(true, true, multiply(additive_identity, add(additive_identity, add(X, X))), additive_identity, add(additive_identity, add(X, X)), add(X, X))
% 16.64/2.53 = { by axiom 20 (cancellation2) }
% 16.64/2.53 multiply(additive_identity, add(additive_identity, add(X, X)))
% 16.64/2.53 = { by lemma 49 }
% 16.64/2.53 additive_identity
% 16.64/2.53
% 16.64/2.53 Lemma 65: add(X, additive_identity) = X.
% 16.64/2.53 Proof:
% 16.64/2.53 add(X, additive_identity)
% 16.64/2.53 = { by lemma 45 R->L }
% 16.64/2.53 fresh7(sum(X, additive_identity, add(X, additive_identity)), true, add(X, additive_identity), X)
% 16.64/2.53 = { by axiom 10 (closure_of_addition) }
% 16.64/2.53 fresh7(true, true, add(X, additive_identity), X)
% 16.64/2.53 = { by axiom 8 (addition_is_well_defined) }
% 16.64/2.53 X
% 16.64/2.53
% 16.64/2.53 Lemma 66: add(add(X, Y), Z) = add(X, add(Z, Y)).
% 16.64/2.53 Proof:
% 16.64/2.53 add(add(X, Y), Z)
% 16.64/2.53 = { by lemma 54 R->L }
% 16.64/2.53 add(Z, add(X, Y))
% 16.64/2.53 = { by lemma 63 }
% 16.64/2.53 add(X, add(Z, Y))
% 16.64/2.53
% 16.64/2.53 Lemma 67: add(X, multiply(Y, X)) = multiply(add(X, Y), X).
% 16.64/2.53 Proof:
% 16.64/2.53 add(X, multiply(Y, X))
% 16.64/2.53 = { by lemma 54 R->L }
% 16.64/2.53 add(multiply(Y, X), X)
% 16.64/2.53 = { by axiom 8 (addition_is_well_defined) R->L }
% 16.64/2.53 fresh7(true, true, multiply(add(Y, X), X), add(multiply(Y, X), X))
% 16.64/2.53 = { by axiom 15 (distributivity3) R->L }
% 16.64/2.53 fresh7(fresh21(true, true, multiply(Y, X), X, multiply(add(Y, X), X)), true, multiply(add(Y, X), X), add(multiply(Y, X), X))
% 16.64/2.53 = { by axiom 10 (closure_of_addition) R->L }
% 16.64/2.53 fresh7(fresh21(sum(Y, X, add(Y, X)), true, multiply(Y, X), X, multiply(add(Y, X), X)), true, multiply(add(Y, X), X), add(multiply(Y, X), X))
% 16.64/2.53 = { by lemma 50 }
% 16.64/2.53 fresh7(sum(multiply(Y, X), X, multiply(add(Y, X), X)), true, multiply(add(Y, X), X), add(multiply(Y, X), X))
% 16.64/2.53 = { by lemma 53 }
% 16.64/2.53 multiply(add(Y, X), X)
% 16.64/2.53 = { by lemma 54 }
% 16.64/2.53 multiply(add(X, Y), X)
% 16.64/2.53
% 16.64/2.53 Lemma 68: multiply(add(X, Y), X) = multiply(X, add(X, Y)).
% 16.64/2.53 Proof:
% 16.64/2.53 multiply(add(X, Y), X)
% 16.64/2.53 = { by axiom 20 (cancellation2) R->L }
% 16.64/2.53 fresh2(true, true, multiply(add(X, Y), X), multiply(add(X, Y), Y), add(X, Y), add(add(X, Y), additive_inverse(multiply(add(X, Y), Y))))
% 16.64/2.53 = { by lemma 52 R->L }
% 16.64/2.53 fresh2(sum(add(add(X, Y), additive_inverse(multiply(add(X, Y), Y))), multiply(add(X, Y), Y), add(X, Y)), true, multiply(add(X, Y), X), multiply(add(X, Y), Y), add(X, Y), add(add(X, Y), additive_inverse(multiply(add(X, Y), Y))))
% 16.64/2.53 = { by axiom 34 (cancellation2) }
% 16.64/2.53 fresh(sum(multiply(add(X, Y), X), multiply(add(X, Y), Y), add(X, Y)), true, multiply(add(X, Y), X), add(add(X, Y), additive_inverse(multiply(add(X, Y), Y))))
% 16.64/2.53 = { by lemma 62 }
% 16.64/2.53 fresh(sum(multiply(add(X, Y), X), multiply(add(X, Y), Y), add(X, Y)), true, multiply(add(X, Y), X), add(add(X, Y), multiply(add(X, Y), Y)))
% 16.64/2.53 = { by lemma 55 R->L }
% 16.64/2.53 fresh(fresh26(true, true, add(X, Y), X, multiply(add(X, Y), X), Y, multiply(add(X, Y), Y), add(X, Y), add(X, Y)), true, multiply(add(X, Y), X), add(add(X, Y), multiply(add(X, Y), Y)))
% 16.64/2.53 = { by axiom 3 (x_squared_is_x) R->L }
% 16.64/2.53 fresh(fresh26(product(add(X, Y), add(X, Y), add(X, Y)), true, add(X, Y), X, multiply(add(X, Y), X), Y, multiply(add(X, Y), Y), add(X, Y), add(X, Y)), true, multiply(add(X, Y), X), add(add(X, Y), multiply(add(X, Y), Y)))
% 16.64/2.53 = { by axiom 40 (distributivity1) }
% 16.64/2.53 fresh(fresh28(product(add(X, Y), Y, multiply(add(X, Y), Y)), true, add(X, Y), X, multiply(add(X, Y), X), Y, multiply(add(X, Y), Y), add(X, Y), add(X, Y)), true, multiply(add(X, Y), X), add(add(X, Y), multiply(add(X, Y), Y)))
% 16.64/2.53 = { by axiom 11 (closure_of_multiplication) }
% 16.64/2.53 fresh(fresh28(true, true, add(X, Y), X, multiply(add(X, Y), X), Y, multiply(add(X, Y), Y), add(X, Y), add(X, Y)), true, multiply(add(X, Y), X), add(add(X, Y), multiply(add(X, Y), Y)))
% 16.64/2.53 = { by axiom 29 (distributivity1) }
% 16.64/2.53 fresh(fresh29(sum(X, Y, add(X, Y)), true, multiply(add(X, Y), X), multiply(add(X, Y), Y), add(X, Y)), true, multiply(add(X, Y), X), add(add(X, Y), multiply(add(X, Y), Y)))
% 16.64/2.53 = { by axiom 10 (closure_of_addition) }
% 16.64/2.53 fresh(fresh29(true, true, multiply(add(X, Y), X), multiply(add(X, Y), Y), add(X, Y)), true, multiply(add(X, Y), X), add(add(X, Y), multiply(add(X, Y), Y)))
% 16.64/2.53 = { by axiom 14 (distributivity1) }
% 16.64/2.53 fresh(true, true, multiply(add(X, Y), X), add(add(X, Y), multiply(add(X, Y), Y)))
% 16.64/2.53 = { by axiom 7 (cancellation2) }
% 16.64/2.53 add(add(X, Y), multiply(add(X, Y), Y))
% 16.64/2.53 = { by lemma 66 }
% 16.64/2.53 add(X, add(multiply(add(X, Y), Y), Y))
% 16.64/2.53 = { by lemma 54 }
% 16.64/2.53 add(X, add(Y, multiply(add(X, Y), Y)))
% 16.64/2.53 = { by lemma 67 }
% 16.64/2.53 add(X, multiply(add(Y, add(X, Y)), Y))
% 16.64/2.53 = { by lemma 63 }
% 16.64/2.53 add(X, multiply(add(X, add(Y, Y)), Y))
% 16.64/2.53 = { by lemma 64 }
% 16.64/2.53 add(X, multiply(add(X, additive_identity), Y))
% 16.64/2.53 = { by lemma 65 }
% 16.64/2.53 add(X, multiply(X, Y))
% 16.64/2.53 = { by lemma 54 R->L }
% 16.64/2.53 add(multiply(X, Y), X)
% 16.64/2.53 = { by axiom 8 (addition_is_well_defined) R->L }
% 16.64/2.53 fresh7(true, true, multiply(X, add(X, Y)), add(multiply(X, Y), X))
% 16.64/2.53 = { by axiom 14 (distributivity1) R->L }
% 16.64/2.53 fresh7(fresh29(true, true, multiply(X, Y), X, multiply(X, add(Y, X))), true, multiply(X, add(X, Y)), add(multiply(X, Y), X))
% 16.64/2.53 = { by axiom 10 (closure_of_addition) R->L }
% 16.64/2.53 fresh7(fresh29(sum(Y, X, add(Y, X)), true, multiply(X, Y), X, multiply(X, add(Y, X))), true, multiply(X, add(X, Y)), add(multiply(X, Y), X))
% 16.64/2.53 = { by lemma 56 R->L }
% 16.64/2.53 fresh7(fresh26(product(X, add(Y, X), multiply(X, add(Y, X))), true, X, Y, multiply(X, Y), X, X, add(Y, X), multiply(X, add(Y, X))), true, multiply(X, add(X, Y)), add(multiply(X, Y), X))
% 16.64/2.53 = { by axiom 11 (closure_of_multiplication) }
% 16.64/2.53 fresh7(fresh26(true, true, X, Y, multiply(X, Y), X, X, add(Y, X), multiply(X, add(Y, X))), true, multiply(X, add(X, Y)), add(multiply(X, Y), X))
% 16.64/2.53 = { by lemma 55 }
% 16.64/2.53 fresh7(sum(multiply(X, Y), X, multiply(X, add(Y, X))), true, multiply(X, add(X, Y)), add(multiply(X, Y), X))
% 16.64/2.53 = { by lemma 54 }
% 16.64/2.53 fresh7(sum(multiply(X, Y), X, multiply(X, add(X, Y))), true, multiply(X, add(X, Y)), add(multiply(X, Y), X))
% 16.64/2.53 = { by lemma 53 }
% 16.64/2.53 multiply(X, add(X, Y))
% 16.64/2.53
% 16.64/2.53 Goal 1 (prove_b_times_a_is_c): product(b, a, c) = true.
% 16.64/2.53 Proof:
% 16.64/2.53 product(b, a, c)
% 16.64/2.53 = { by axiom 8 (addition_is_well_defined) R->L }
% 16.64/2.53 product(fresh7(true, true, add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.53 = { by lemma 51 R->L }
% 16.64/2.53 product(fresh7(fresh34(true, true, a, a, add(a, a), add(b, add(a, a)), add(a, add(b, add(a, a))), add(a, add(a, add(b, add(a, a))))), true, add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.53 = { by axiom 10 (closure_of_addition) R->L }
% 16.64/2.53 product(fresh7(fresh34(sum(a, add(b, add(a, a)), add(a, add(b, add(a, a)))), true, a, a, add(a, a), add(b, add(a, a)), add(a, add(b, add(a, a))), add(a, add(a, add(b, add(a, a))))), true, add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.53 = { by axiom 37 (associativity_of_addition2) }
% 16.64/2.53 product(fresh7(fresh12(sum(a, add(a, add(b, add(a, a))), add(a, add(a, add(b, add(a, a))))), true, a, a, add(a, a), add(b, add(a, a)), add(a, add(a, add(b, add(a, a))))), true, add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.53 = { by axiom 10 (closure_of_addition) }
% 16.64/2.53 product(fresh7(fresh12(true, true, a, a, add(a, a), add(b, add(a, a)), add(a, add(a, add(b, add(a, a))))), true, add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.53 = { by axiom 23 (associativity_of_addition2) }
% 16.64/2.53 product(fresh7(sum(add(a, a), add(b, add(a, a)), add(a, add(a, add(b, add(a, a))))), true, add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.54 = { by lemma 54 }
% 16.64/2.54 product(fresh7(sum(add(a, a), add(b, add(a, a)), add(add(a, add(b, add(a, a))), a)), true, add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.54 = { by lemma 62 R->L }
% 16.64/2.54 product(fresh7(sum(add(a, a), add(b, additive_inverse(add(a, a))), add(add(a, add(b, add(a, a))), a)), true, add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.54 = { by axiom 32 (addition_is_well_defined) R->L }
% 16.64/2.54 product(fresh8(sum(add(a, a), add(b, additive_inverse(add(a, a))), b), true, add(a, a), add(b, additive_inverse(add(a, a))), add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.54 = { by lemma 59 R->L }
% 16.64/2.54 product(fresh8(fresh36(sum(additive_identity, b, b), true, add(a, a), additive_inverse(add(a, a)), additive_identity, b, add(additive_inverse(add(a, a)), b), b), true, add(a, a), add(b, additive_inverse(add(a, a))), add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.54 = { by axiom 2 (additive_identity1) }
% 16.64/2.54 product(fresh8(fresh36(true, true, add(a, a), additive_inverse(add(a, a)), additive_identity, b, add(additive_inverse(add(a, a)), b), b), true, add(a, a), add(b, additive_inverse(add(a, a))), add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.54 = { by axiom 24 (associativity_of_addition1) }
% 16.64/2.54 product(fresh8(fresh37(sum(add(a, a), additive_inverse(add(a, a)), additive_identity), true, add(a, a), add(additive_inverse(add(a, a)), b), b), true, add(a, a), add(b, additive_inverse(add(a, a))), add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.54 = { by axiom 5 (right_inverse) }
% 16.64/2.54 product(fresh8(fresh37(true, true, add(a, a), add(additive_inverse(add(a, a)), b), b), true, add(a, a), add(b, additive_inverse(add(a, a))), add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.54 = { by axiom 12 (associativity_of_addition1) }
% 16.64/2.54 product(fresh8(true, true, add(a, a), add(b, additive_inverse(add(a, a))), add(add(a, add(b, add(a, a))), a), b), a, c)
% 16.64/2.54 = { by axiom 18 (addition_is_well_defined) }
% 16.64/2.54 product(add(add(a, add(b, add(a, a))), a), a, c)
% 16.64/2.54 = { by lemma 66 }
% 16.64/2.54 product(add(a, add(a, add(b, add(a, a)))), a, c)
% 16.64/2.54 = { by lemma 64 }
% 16.64/2.54 product(add(a, add(a, add(b, additive_identity))), a, c)
% 16.64/2.54 = { by lemma 65 }
% 16.64/2.54 product(add(a, add(a, b)), a, c)
% 16.64/2.54 = { by lemma 62 R->L }
% 16.64/2.54 product(add(a, add(a, b)), a, additive_inverse(c))
% 16.64/2.54 = { by axiom 9 (cancellation1) R->L }
% 16.64/2.54 product(add(a, add(a, b)), a, fresh3(true, true, multiply(a, add(a, add(a, b))), additive_inverse(c)))
% 16.64/2.54 = { by axiom 14 (distributivity1) R->L }
% 16.64/2.54 product(add(a, add(a, b)), a, fresh3(fresh29(true, true, c, multiply(a, add(a, additive_inverse(add(b, a)))), multiply(a, additive_identity)), true, multiply(a, add(a, add(a, b))), additive_inverse(c)))
% 16.64/2.54 = { by lemma 61 R->L }
% 16.64/2.54 product(add(a, add(a, b)), a, fresh3(fresh29(sum(b, add(a, additive_inverse(add(b, a))), additive_identity), true, c, multiply(a, add(a, additive_inverse(add(b, a)))), multiply(a, additive_identity)), true, multiply(a, add(a, add(a, b))), additive_inverse(c)))
% 16.64/2.54 = { by lemma 58 R->L }
% 16.64/2.54 product(add(a, add(a, b)), a, fresh3(fresh26(X, X, a, b, c, add(a, additive_inverse(add(b, a))), multiply(a, add(a, additive_inverse(add(b, a)))), additive_identity, multiply(a, additive_identity)), true, multiply(a, add(a, add(a, b))), additive_inverse(c)))
% 16.64/2.54 = { by axiom 38 (distributivity1) }
% 16.64/2.54 product(add(a, add(a, b)), a, fresh3(fresh27(product(a, b, c), true, b, c, add(a, additive_inverse(add(b, a))), multiply(a, add(a, additive_inverse(add(b, a)))), additive_identity, multiply(a, additive_identity)), true, multiply(a, add(a, add(a, b))), additive_inverse(c)))
% 16.64/2.54 = { by axiom 4 (a_times_b_is_c) }
% 16.64/2.54 product(add(a, add(a, b)), a, fresh3(fresh27(true, true, b, c, add(a, additive_inverse(add(b, a))), multiply(a, add(a, additive_inverse(add(b, a)))), additive_identity, multiply(a, additive_identity)), true, multiply(a, add(a, add(a, b))), additive_inverse(c)))
% 16.64/2.54 = { by axiom 26 (distributivity1) }
% 16.64/2.54 product(add(a, add(a, b)), a, fresh3(sum(c, multiply(a, add(a, additive_inverse(add(b, a)))), multiply(a, additive_identity)), true, multiply(a, add(a, add(a, b))), additive_inverse(c)))
% 16.64/2.54 = { by lemma 62 }
% 16.64/2.54 product(add(a, add(a, b)), a, fresh3(sum(c, multiply(a, add(a, add(b, a))), multiply(a, additive_identity)), true, multiply(a, add(a, add(a, b))), additive_inverse(c)))
% 16.64/2.54 = { by lemma 57 }
% 16.64/2.54 product(add(a, add(a, b)), a, fresh3(sum(c, multiply(a, add(a, add(b, a))), additive_identity), true, multiply(a, add(a, add(a, b))), additive_inverse(c)))
% 16.64/2.54 = { by lemma 54 }
% 16.64/2.54 product(add(a, add(a, b)), a, fresh3(sum(c, multiply(a, add(a, add(a, b))), additive_identity), true, multiply(a, add(a, add(a, b))), additive_inverse(c)))
% 16.64/2.54 = { by lemma 43 }
% 16.64/2.54 product(add(a, add(a, b)), a, multiply(a, add(a, add(a, b))))
% 16.64/2.54 = { by lemma 68 R->L }
% 16.64/2.54 product(add(a, add(a, b)), a, multiply(add(a, add(a, b)), a))
% 16.64/2.54 = { by lemma 67 R->L }
% 16.64/2.54 product(add(a, add(a, b)), a, add(a, multiply(add(a, b), a)))
% 16.64/2.54 = { by lemma 54 R->L }
% 16.64/2.54 product(add(a, add(a, b)), a, add(multiply(add(a, b), a), a))
% 16.64/2.54 = { by axiom 21 (distributivity4) R->L }
% 16.64/2.54 fresh15(true, true, add(a, b), a, a, add(a, add(a, b)), add(multiply(add(a, b), a), a))
% 16.64/2.54 = { by axiom 10 (closure_of_addition) R->L }
% 16.64/2.54 fresh15(sum(multiply(add(a, b), a), a, add(multiply(add(a, b), a), a)), true, add(a, b), a, a, add(a, add(a, b)), add(multiply(add(a, b), a), a))
% 16.64/2.54 = { by axiom 35 (distributivity4) R->L }
% 16.64/2.54 fresh14(true, true, add(a, b), a, multiply(add(a, b), a), a, a, add(a, add(a, b)), add(multiply(add(a, b), a), a))
% 16.64/2.54 = { by lemma 54 }
% 16.64/2.54 fresh14(true, true, add(a, b), a, multiply(add(a, b), a), a, a, add(a, add(a, b)), add(a, multiply(add(a, b), a)))
% 16.64/2.54 = { by axiom 3 (x_squared_is_x) R->L }
% 16.64/2.54 fresh14(product(a, a, a), true, add(a, b), a, multiply(add(a, b), a), a, a, add(a, add(a, b)), add(a, multiply(add(a, b), a)))
% 16.64/2.54 = { by axiom 42 (distributivity4) }
% 16.64/2.54 fresh16(product(add(a, b), a, multiply(add(a, b), a)), true, add(a, b), a, multiply(add(a, b), a), a, a, add(a, add(a, b)), add(a, multiply(add(a, b), a)))
% 16.64/2.54 = { by axiom 11 (closure_of_multiplication) }
% 16.64/2.54 fresh16(true, true, add(a, b), a, multiply(add(a, b), a), a, a, add(a, add(a, b)), add(a, multiply(add(a, b), a)))
% 16.64/2.54 = { by lemma 67 }
% 16.64/2.54 fresh16(true, true, add(a, b), a, multiply(add(a, b), a), a, a, add(a, add(a, b)), multiply(add(a, add(a, b)), a))
% 16.64/2.54 = { by lemma 68 }
% 16.64/2.54 fresh16(true, true, add(a, b), a, multiply(add(a, b), a), a, a, add(a, add(a, b)), multiply(a, add(a, add(a, b))))
% 16.64/2.54 = { by lemma 54 R->L }
% 16.64/2.54 fresh16(true, true, add(a, b), a, multiply(add(a, b), a), a, a, add(add(a, b), a), multiply(a, add(a, add(a, b))))
% 16.64/2.54 = { by axiom 31 (distributivity4) }
% 16.64/2.54 fresh17(sum(add(a, b), a, add(add(a, b), a)), true, a, add(add(a, b), a), multiply(a, add(a, add(a, b))))
% 16.64/2.54 = { by axiom 10 (closure_of_addition) }
% 16.64/2.54 fresh17(true, true, a, add(add(a, b), a), multiply(a, add(a, add(a, b))))
% 16.64/2.54 = { by axiom 16 (distributivity4) }
% 16.64/2.54 true
% 16.64/2.54 % SZS output end Proof
% 16.64/2.54
% 16.64/2.54 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------