0.09/0.09 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.09/0.10 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof 0.09/0.30 % Computer : n007.cluster.edu 0.09/0.30 % Model : x86_64 x86_64 0.09/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.09/0.30 % Memory : 8042.1875MB 0.09/0.30 % OS : Linux 3.10.0-693.el7.x86_64 0.09/0.30 % CPULimit : 1200 0.09/0.30 % WCLimit : 120 0.09/0.30 % DateTime : Tue Jul 13 14:06:19 EDT 2021 0.09/0.30 % CPUTime : 726.51/92.70 % SZS status Theorem 726.51/92.70 726.82/92.73 % SZS output start Proof 726.82/92.73 Take the following subset of the input axioms: 726.82/92.73 fof(aA1, axiom, ![X, Y]: s_e(X, Y, Y, X)). 726.82/92.73 fof(aA4, axiom, ![X, Y, V, W]: (s_e(Y, ext(X, Y, W, V), W, V) & s_t(X, Y, ext(X, Y, W, V)))). 726.82/92.73 fof(aSatz2_14, axiom, ![Xa, Xc, Xb, Xd]: (~s_e(Xa, Xb, Xc, Xd) | s_e(Xb, Xa, Xd, Xc))). 726.82/92.73 fof(aSatz2_2, axiom, ![Xa, Xc, Xb, Xd]: (~s_e(Xa, Xb, Xc, Xd) | s_e(Xc, Xd, Xa, Xb))). 726.82/92.74 fof(aSatz3_1, axiom, ![Xa, Xb]: s_t(Xa, Xb, Xb)). 726.82/92.74 fof(aSatz3_2, axiom, ![Xa, Xc, Xb]: (~s_t(Xa, Xb, Xc) | s_t(Xc, Xb, Xa))). 726.82/92.74 fof(aSatz3_4, axiom, ![Xa, Xc, Xb]: (~s_t(Xb, Xa, Xc) | (Xa=Xb | ~s_t(Xa, Xb, Xc)))). 726.82/92.74 fof(aSatz3_6b, axiom, ![Xa, Xc, Xb, Xd]: (~s_t(Xa, Xb, Xc) | (s_t(Xa, Xb, Xd) | ~s_t(Xa, Xc, Xd)))). 726.82/92.74 fof(aSatz4_6, axiom, ![Xa, Xc, Xb, Xa1, Xb1, Xc1]: (~s_e3(Xa, Xb, Xc, Xa1, Xb1, Xc1) | (s_t(Xa1, Xb1, Xc1) | ~s_t(Xa, Xb, Xc)))). 726.82/92.74 fof(aSatz5_5a, axiom, ![Xa, Xc, Xb, Xd]: ((ins(Xc, Xd, Xa, Xb)=ext(Xa, Xb, insert(Xa, Xb, Xc, Xd), Xd) | ~le(Xa, Xb, Xc, Xd)) & ((~le(Xa, Xb, Xc, Xd) | s_e(Xa, ins(Xc, Xd, Xa, Xb), Xc, Xd)) & (~le(Xa, Xb, Xc, Xd) | s_t(Xa, Xb, ins(Xc, Xd, Xa, Xb)))))). 726.82/92.74 fof(aSatz5_5b, axiom, ![Xa, Xc, Xb, Xd, Xe]: (le(Xa, Xb, Xc, Xd) | (~s_t(Xa, Xb, Xe) | ~s_e(Xa, Xe, Xc, Xd)))). 726.82/92.74 fof(aSatz5_8, axiom, ![Xa, Xc, Xb, Xd, Xe, Xf]: (~le(Xa, Xb, Xc, Xd) | (~le(Xc, Xd, Xe, Xf) | le(Xa, Xb, Xe, Xf)))). 726.82/92.74 fof(aSatz5_9, conjecture, ![Xa, Xc, Xb, Xd]: (s_e(Xa, Xb, Xc, Xd) | (~le(Xc, Xd, Xa, Xb) | ~le(Xa, Xb, Xc, Xd)))). 726.82/92.74 fof(d_Defn4_4, axiom, ![Xa1, Xb1, Xa2, Xa3, Xb2, Xb3]: ((~s_e3(Xa1, Xa2, Xa3, Xb1, Xb2, Xb3) | s_e(Xa2, Xa3, Xb2, Xb3)) & ((s_e3(Xa1, Xa2, Xa3, Xb1, Xb2, Xb3) | (~s_e(Xa2, Xa3, Xb2, Xb3) | (~s_e(Xa1, Xa3, Xb1, Xb3) | ~s_e(Xa1, Xa2, Xb1, Xb2)))) & ((s_e(Xa1, Xa3, Xb1, Xb3) | ~s_e3(Xa1, Xa2, Xa3, Xb1, Xb2, Xb3)) & (~s_e3(Xa1, Xa2, Xa3, Xb1, Xb2, Xb3) | s_e(Xa1, Xa2, Xb1, Xb2)))))). 726.82/92.74 fof(d_Defn5_4, axiom, ![Y, Xa, Xc, Xb, Xd]: ((~s_e(Xa, Xb, Xc, Y) | (le(Xa, Xb, Xc, Xd) | ~s_t(Xc, Y, Xd))) & ((s_e(Xa, Xb, Xc, insert(Xa, Xb, Xc, Xd)) | ~le(Xa, Xb, Xc, Xd)) & (~le(Xa, Xb, Xc, Xd) | s_t(Xc, insert(Xa, Xb, Xc, Xd), Xd))))). 726.82/92.74 fof(d_insert, axiom, ![Xa, Xb, Xa1, Xc1]: insert(Xa, Xb, Xa1, Xc1)=ext(ext(Xc1, Xa1, alpha, gamma), Xa1, Xa, Xb)). 726.82/92.74 726.82/92.74 Now clausify the problem and encode Horn clauses using encoding 3 of 726.82/92.74 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 726.82/92.74 We repeatedly replace C & s=t => u=v by the two clauses: 726.82/92.74 fresh(y, y, x1...xn) = u 726.82/92.74 C => fresh(s, t, x1...xn) = v 726.82/92.74 where fresh is a fresh function symbol and x1..xn are the free 726.82/92.74 variables of u and v. 726.82/92.74 A predicate p(X) is encoded as p(X)=true (this is sound, because the 726.82/92.74 input problem has no model of domain size 1). 726.82/92.74 726.82/92.74 The encoding turns the above axioms into the following unit equations and goals: 726.82/92.74 726.82/92.74 Axiom 1 (aSatz3_1): s_t(X, Y, Y) = true2. 726.82/92.74 Axiom 2 (aA1): s_e(X, Y, Y, X) = true2. 726.82/92.74 Axiom 3 (aSatz5_9): le(xa, xb, xc, xd) = true2. 726.82/92.74 Axiom 4 (aSatz5_9_1): le(xc, xd, xa, xb) = true2. 726.82/92.74 Axiom 5 (aSatz3_4): fresh5(X, X, Y, Z) = Z. 726.82/92.74 Axiom 6 (aSatz3_2): fresh67(X, X, Y, Z, W) = true2. 726.82/92.74 Axiom 7 (aSatz3_6b): fresh59(X, X, Y, Z, W) = true2. 726.82/92.74 Axiom 8 (aSatz4_6): fresh42(X, X, Y, Z, W) = true2. 726.82/92.74 Axiom 9 (aSatz3_4): fresh6(X, X, Y, Z, W) = Y. 726.82/92.74 Axiom 10 (aSatz2_14): fresh76(X, X, Y, Z, W, V) = true2. 726.82/92.74 Axiom 11 (aSatz2_2): fresh74(X, X, Y, Z, W, V) = true2. 726.82/92.74 Axiom 12 (aSatz3_6b): fresh60(X, X, Y, Z, W, V) = s_t(Y, Z, V). 726.82/92.74 Axiom 13 (aSatz5_5a_1): fresh40(X, X, Y, Z, W, V) = true2. 726.82/92.74 Axiom 14 (aSatz5_5a_2): fresh39(X, X, Y, Z, W, V) = true2. 726.82/92.74 Axiom 15 (aSatz5_5b): fresh37(X, X, Y, Z, W, V) = true2. 726.82/92.74 Axiom 16 (aSatz5_8): fresh34(X, X, Y, Z, W, V) = true2. 726.82/92.74 Axiom 17 (d_Defn5_4_2): fresh7(X, X, Y, Z, W, V) = true2. 726.82/92.74 Axiom 18 (aSatz5_5b): fresh38(X, X, Y, Z, W, V, U) = le(Y, Z, W, V). 726.82/92.74 Axiom 19 (aA4): s_e(X, ext(Y, X, Z, W), Z, W) = true2. 726.82/92.74 Axiom 20 (d_insert): insert(X, Y, Z, W) = ext(ext(W, Z, alpha, gamma), Z, X, Y). 726.82/92.74 Axiom 21 (d_Defn4_4): fresh118(X, X, Y, Z, W, V, U, T) = true2. 726.82/92.74 Axiom 22 (aSatz3_2): fresh67(s_t(X, Y, Z), true2, X, Y, Z) = s_t(Z, Y, X). 726.82/92.74 Axiom 23 (aSatz4_6): fresh43(X, X, Y, Z, W, V, U, T) = s_t(V, U, T). 726.82/92.74 Axiom 24 (aSatz5_8): fresh35(X, X, Y, Z, W, V, U, T) = le(Y, Z, U, T). 726.82/92.74 Axiom 25 (d_Defn4_4): fresh14(X, X, Y, Z, W, V, U, T) = s_e3(U, Y, Z, T, W, V). 726.82/92.74 Axiom 26 (aSatz3_4): fresh6(s_t(X, Y, Z), true2, Y, X, Z) = fresh5(s_t(Y, X, Z), true2, Y, X). 726.82/92.74 Axiom 27 (aSatz3_6b): fresh60(s_t(X, Y, Z), true2, X, W, Y, Z) = fresh59(s_t(X, W, Y), true2, X, W, Z). 726.82/92.74 Axiom 28 (aSatz2_14): fresh76(s_e(X, Y, Z, W), true2, X, Y, Z, W) = s_e(Y, X, W, Z). 726.82/92.74 Axiom 29 (aSatz2_2): fresh74(s_e(X, Y, Z, W), true2, X, Y, Z, W) = s_e(Z, W, X, Y). 726.82/92.74 Axiom 30 (aSatz5_5a_1): fresh40(le(X, Y, Z, W), true2, X, Y, Z, W) = s_e(X, ins(Z, W, X, Y), Z, W). 726.82/92.74 Axiom 31 (aSatz5_5a_2): fresh39(le(X, Y, Z, W), true2, X, Y, Z, W) = s_t(X, Y, ins(Z, W, X, Y)). 726.82/92.74 Axiom 32 (aSatz5_5b): fresh38(s_t(X, Y, Z), true2, X, Y, W, V, Z) = fresh37(s_e(X, Z, W, V), true2, X, Y, W, V). 726.82/92.74 Axiom 33 (d_Defn5_4_2): fresh7(le(X, Y, Z, W), true2, X, Y, Z, W) = s_t(Z, insert(X, Y, Z, W), W). 726.82/92.74 Axiom 34 (d_Defn4_4): fresh117(X, X, Y, Z, W, V, U, T) = fresh118(s_e(Y, Z, W, V), true2, Y, Z, W, V, U, T). 726.82/92.74 Axiom 35 (aSatz5_8): fresh35(le(X, Y, Z, W), true2, V, U, X, Y, Z, W) = fresh34(le(V, U, X, Y), true2, V, U, Z, W). 726.82/92.74 Axiom 36 (d_Defn4_4): fresh117(s_e(X, Y, Z, W), true2, V, Y, U, W, X, Z) = fresh14(s_e(X, V, Z, U), true2, V, Y, U, W, X, Z). 726.82/92.74 Axiom 37 (aSatz4_6): fresh43(s_e3(X, Y, Z, W, V, U), true2, X, Y, Z, W, V, U) = fresh42(s_t(X, Y, Z), true2, W, V, U). 726.82/92.74 726.82/92.74 Lemma 38: s_t(xa, xb, ins(xc, xd, xa, xb)) = true2. 726.82/92.74 Proof: 726.82/92.74 s_t(xa, xb, ins(xc, xd, xa, xb)) 726.82/92.74 = { by axiom 31 (aSatz5_5a_2) R->L } 726.82/92.74 fresh39(le(xa, xb, xc, xd), true2, xa, xb, xc, xd) 726.82/92.74 = { by axiom 3 (aSatz5_9) } 726.82/92.74 fresh39(true2, true2, xa, xb, xc, xd) 726.82/92.74 = { by axiom 14 (aSatz5_5a_2) } 726.82/92.74 true2 726.82/92.74 726.82/92.74 Lemma 39: s_e(xa, ins(xc, xd, xa, xb), xc, xd) = true2. 726.82/92.74 Proof: 726.82/92.74 s_e(xa, ins(xc, xd, xa, xb), xc, xd) 726.82/92.74 = { by axiom 30 (aSatz5_5a_1) R->L } 726.82/92.74 fresh40(le(xa, xb, xc, xd), true2, xa, xb, xc, xd) 726.82/92.74 = { by axiom 3 (aSatz5_9) } 726.82/92.74 fresh40(true2, true2, xa, xb, xc, xd) 726.82/92.74 = { by axiom 13 (aSatz5_5a_1) } 726.82/92.74 true2 726.82/92.74 726.82/92.74 Lemma 40: s_e(ext(X, Y, Z, W), Y, W, Z) = true2. 726.82/92.74 Proof: 726.82/92.74 s_e(ext(X, Y, Z, W), Y, W, Z) 726.82/92.74 = { by axiom 28 (aSatz2_14) R->L } 726.82/92.74 fresh76(s_e(Y, ext(X, Y, Z, W), Z, W), true2, Y, ext(X, Y, Z, W), Z, W) 726.82/92.74 = { by axiom 19 (aA4) } 726.82/92.74 fresh76(true2, true2, Y, ext(X, Y, Z, W), Z, W) 726.82/92.74 = { by axiom 10 (aSatz2_14) } 726.82/92.74 true2 726.82/92.74 726.82/92.74 Lemma 41: s_t(xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), xb) = true2. 726.82/92.74 Proof: 726.82/92.74 s_t(xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), xb) 726.82/92.74 = { by axiom 33 (d_Defn5_4_2) R->L } 726.82/92.74 fresh7(le(xa, ins(xc, xd, xa, xb), xa, xb), true2, xa, ins(xc, xd, xa, xb), xa, xb) 726.82/92.74 = { by axiom 24 (aSatz5_8) R->L } 726.82/92.74 fresh7(fresh35(true2, true2, xa, ins(xc, xd, xa, xb), xc, xd, xa, xb), true2, xa, ins(xc, xd, xa, xb), xa, xb) 726.82/92.74 = { by axiom 4 (aSatz5_9_1) R->L } 726.82/92.74 fresh7(fresh35(le(xc, xd, xa, xb), true2, xa, ins(xc, xd, xa, xb), xc, xd, xa, xb), true2, xa, ins(xc, xd, xa, xb), xa, xb) 726.82/92.74 = { by axiom 35 (aSatz5_8) } 726.82/92.74 fresh7(fresh34(le(xa, ins(xc, xd, xa, xb), xc, xd), true2, xa, ins(xc, xd, xa, xb), xa, xb), true2, xa, ins(xc, xd, xa, xb), xa, xb) 726.82/92.74 = { by axiom 18 (aSatz5_5b) R->L } 726.82/92.74 fresh7(fresh34(fresh38(true2, true2, xa, ins(xc, xd, xa, xb), xc, xd, ins(xc, xd, xa, xb)), true2, xa, ins(xc, xd, xa, xb), xa, xb), true2, xa, ins(xc, xd, xa, xb), xa, xb) 726.82/92.74 = { by axiom 1 (aSatz3_1) R->L } 726.82/92.74 fresh7(fresh34(fresh38(s_t(xa, ins(xc, xd, xa, xb), ins(xc, xd, xa, xb)), true2, xa, ins(xc, xd, xa, xb), xc, xd, ins(xc, xd, xa, xb)), true2, xa, ins(xc, xd, xa, xb), xa, xb), true2, xa, ins(xc, xd, xa, xb), xa, xb) 726.82/92.74 = { by axiom 32 (aSatz5_5b) } 726.82/92.74 fresh7(fresh34(fresh37(s_e(xa, ins(xc, xd, xa, xb), xc, xd), true2, xa, ins(xc, xd, xa, xb), xc, xd), true2, xa, ins(xc, xd, xa, xb), xa, xb), true2, xa, ins(xc, xd, xa, xb), xa, xb) 726.82/92.74 = { by lemma 39 } 726.82/92.74 fresh7(fresh34(fresh37(true2, true2, xa, ins(xc, xd, xa, xb), xc, xd), true2, xa, ins(xc, xd, xa, xb), xa, xb), true2, xa, ins(xc, xd, xa, xb), xa, xb) 726.82/92.74 = { by axiom 15 (aSatz5_5b) } 726.82/92.74 fresh7(fresh34(true2, true2, xa, ins(xc, xd, xa, xb), xa, xb), true2, xa, ins(xc, xd, xa, xb), xa, xb) 726.82/92.74 = { by axiom 16 (aSatz5_8) } 726.82/92.74 fresh7(true2, true2, xa, ins(xc, xd, xa, xb), xa, xb) 726.82/92.74 = { by axiom 17 (d_Defn5_4_2) } 726.82/92.74 true2 726.82/92.74 726.82/92.74 Lemma 42: s_t(ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa) = true2. 726.82/92.74 Proof: 726.82/92.74 s_t(ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa) 726.82/92.74 = { by axiom 22 (aSatz3_2) R->L } 726.82/92.74 fresh67(s_t(xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb)), true2, xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb)) 726.82/92.74 = { by axiom 12 (aSatz3_6b) R->L } 726.82/92.74 fresh67(fresh60(true2, true2, xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), xb, ins(xc, xd, xa, xb)), true2, xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb)) 726.82/92.74 = { by lemma 38 R->L } 726.82/92.74 fresh67(fresh60(s_t(xa, xb, ins(xc, xd, xa, xb)), true2, xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), xb, ins(xc, xd, xa, xb)), true2, xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb)) 726.82/92.74 = { by axiom 27 (aSatz3_6b) } 726.82/92.74 fresh67(fresh59(s_t(xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), xb), true2, xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb)), true2, xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb)) 726.82/92.74 = { by lemma 41 } 726.82/92.74 fresh67(fresh59(true2, true2, xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb)), true2, xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb)) 726.82/92.74 = { by axiom 7 (aSatz3_6b) } 726.82/92.74 fresh67(true2, true2, xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb)) 726.82/92.74 = { by axiom 6 (aSatz3_2) } 726.82/92.75 true2 726.82/92.75 726.82/92.75 Goal 1 (aSatz5_9_2): s_e(xa, xb, xc, xd) = true2. 726.82/92.75 Proof: 726.82/92.75 s_e(xa, xb, xc, xd) 726.82/92.75 = { by axiom 5 (aSatz3_4) R->L } 726.82/92.75 s_e(xa, fresh5(true2, true2, ins(xc, xd, xa, xb), xb), xc, xd) 726.82/92.75 = { by axiom 6 (aSatz3_2) R->L } 726.82/92.75 s_e(xa, fresh5(fresh67(true2, true2, xa, xb, ins(xc, xd, xa, xb)), true2, ins(xc, xd, xa, xb), xb), xc, xd) 726.82/92.75 = { by lemma 38 R->L } 726.82/92.75 s_e(xa, fresh5(fresh67(s_t(xa, xb, ins(xc, xd, xa, xb)), true2, xa, xb, ins(xc, xd, xa, xb)), true2, ins(xc, xd, xa, xb), xb), xc, xd) 726.82/92.75 = { by axiom 22 (aSatz3_2) } 726.82/92.75 s_e(xa, fresh5(s_t(ins(xc, xd, xa, xb), xb, xa), true2, ins(xc, xd, xa, xb), xb), xc, xd) 726.82/92.75 = { by axiom 26 (aSatz3_4) R->L } 726.82/92.75 s_e(xa, fresh6(s_t(xb, ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.75 = { by axiom 9 (aSatz3_4) R->L } 726.82/92.75 s_e(xa, fresh6(s_t(xb, fresh6(true2, true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.75 = { by axiom 8 (aSatz4_6) R->L } 726.82/92.75 s_e(xa, fresh6(s_t(xb, fresh6(fresh42(true2, true2, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.75 = { by lemma 42 R->L } 726.82/92.75 s_e(xa, fresh6(s_t(xb, fresh6(fresh42(s_t(ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), true2, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.75 = { by axiom 37 (aSatz4_6) R->L } 726.82/92.75 s_e(xa, fresh6(s_t(xb, fresh6(fresh43(s_e3(ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.75 = { by axiom 20 (d_insert) } 726.82/92.75 s_e(xa, fresh6(s_t(xb, fresh6(fresh43(s_e3(ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.75 = { by axiom 20 (d_insert) } 726.82/92.75 s_e(xa, fresh6(s_t(xb, fresh6(fresh43(s_e3(ins(xc, xd, xa, xb), ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.75 = { by axiom 25 (d_Defn4_4) R->L } 726.82/92.75 s_e(xa, fresh6(s_t(xb, fresh6(fresh43(fresh14(true2, true2, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ins(xc, xd, xa, xb), xa, ins(xc, xd, xa, xb), ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb))), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.75 = { by axiom 2 (aA1) R->L } 726.82/92.75 s_e(xa, fresh6(s_t(xb, fresh6(fresh43(fresh14(s_e(ins(xc, xd, xa, xb), ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), ins(xc, xd, xa, xb)), true2, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ins(xc, xd, xa, xb), xa, ins(xc, xd, xa, xb), ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb))), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.75 = { by axiom 36 (d_Defn4_4) R->L } 726.82/92.75 s_e(xa, fresh6(s_t(xb, fresh6(fresh43(fresh117(s_e(ins(xc, xd, xa, xb), xa, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa), true2, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ins(xc, xd, xa, xb), xa, ins(xc, xd, xa, xb), ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb))), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by axiom 29 (aSatz2_2) R->L } 726.82/92.76 s_e(xa, fresh6(s_t(xb, fresh6(fresh43(fresh117(fresh74(s_e(ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ins(xc, xd, xa, xb), xa), true2, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ins(xc, xd, xa, xb), xa), true2, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ins(xc, xd, xa, xb), xa, ins(xc, xd, xa, xb), ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb))), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by lemma 40 } 726.82/92.76 s_e(xa, fresh6(s_t(xb, fresh6(fresh43(fresh117(fresh74(true2, true2, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ins(xc, xd, xa, xb), xa), true2, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ins(xc, xd, xa, xb), xa, ins(xc, xd, xa, xb), ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb))), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by axiom 11 (aSatz2_2) } 726.82/92.76 s_e(xa, fresh6(s_t(xb, fresh6(fresh43(fresh117(true2, true2, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ins(xc, xd, xa, xb), xa, ins(xc, xd, xa, xb), ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb))), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by axiom 34 (d_Defn4_4) } 726.82/92.76 s_e(xa, fresh6(s_t(xb, fresh6(fresh43(fresh118(s_e(ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ins(xc, xd, xa, xb), xa), true2, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ins(xc, xd, xa, xb), xa, ins(xc, xd, xa, xb), ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb))), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by lemma 40 } 726.82/92.76 s_e(xa, fresh6(s_t(xb, fresh6(fresh43(fresh118(true2, true2, ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb)), xa, ins(xc, xd, xa, xb), xa, ins(xc, xd, xa, xb), ext(ext(xb, xa, alpha, gamma), xa, xa, ins(xc, xd, xa, xb))), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by axiom 21 (d_Defn4_4) } 726.82/92.76 s_e(xa, fresh6(s_t(xb, fresh6(fresh43(true2, true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by axiom 23 (aSatz4_6) } 726.82/92.76 s_e(xa, fresh6(s_t(xb, fresh6(s_t(insert(xa, ins(xc, xd, xa, xb), xa, xb), ins(xc, xd, xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by axiom 26 (aSatz3_4) } 726.82/92.76 s_e(xa, fresh6(s_t(xb, fresh5(s_t(ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb)), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by lemma 42 } 726.82/92.76 s_e(xa, fresh6(s_t(xb, fresh5(true2, true2, ins(xc, xd, xa, xb), insert(xa, ins(xc, xd, xa, xb), xa, xb)), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by axiom 5 (aSatz3_4) } 726.82/92.76 s_e(xa, fresh6(s_t(xb, insert(xa, ins(xc, xd, xa, xb), xa, xb), xa), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by axiom 22 (aSatz3_2) R->L } 726.82/92.76 s_e(xa, fresh6(fresh67(s_t(xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), xb), true2, xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), xb), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by lemma 41 } 726.82/92.76 s_e(xa, fresh6(fresh67(true2, true2, xa, insert(xa, ins(xc, xd, xa, xb), xa, xb), xb), true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by axiom 6 (aSatz3_2) } 726.82/92.76 s_e(xa, fresh6(true2, true2, ins(xc, xd, xa, xb), xb, xa), xc, xd) 726.82/92.76 = { by axiom 9 (aSatz3_4) } 726.82/92.76 s_e(xa, ins(xc, xd, xa, xb), xc, xd) 726.82/92.76 = { by lemma 39 } 726.82/92.76 true2 726.82/92.76 % SZS output end Proof 726.82/92.76 726.82/92.76 RESULT: Theorem (the conjecture is true). 727.16/92.84 EOF