TSTP Solution File: SEU023+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SEU023+1 : TPTP v8.1.2. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 17:50:49 EDT 2023
% Result : Theorem 71.07s 9.55s
% Output : Proof 71.07s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU023+1 : TPTP v8.1.2. Released v3.2.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n016.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Wed Aug 23 13:38:07 EDT 2023
% 0.13/0.35 % CPUTime :
% 71.07/9.55 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 71.07/9.55
% 71.07/9.55 % SZS status Theorem
% 71.07/9.55
% 71.07/9.55 % SZS output start Proof
% 71.07/9.55 Take the following subset of the input axioms:
% 71.07/9.56 fof(dt_k2_funct_1, axiom, ![A2]: ((relation(A2) & function(A2)) => (relation(function_inverse(A2)) & function(function_inverse(A2))))).
% 71.07/9.56 fof(t23_funct_1, axiom, ![B, A2_2]: ((relation(B) & function(B)) => ![C]: ((relation(C) & function(C)) => (in(A2_2, relation_dom(B)) => apply(relation_composition(B, C), A2_2)=apply(C, apply(B, A2_2)))))).
% 71.07/9.56 fof(t54_funct_1, axiom, ![A2_2]: ((relation(A2_2) & function(A2_2)) => (one_to_one(A2_2) => ![B2]: ((relation(B2) & function(B2)) => (B2=function_inverse(A2_2) <=> (relation_dom(B2)=relation_rng(A2_2) & ![D, C2]: (((in(C2, relation_rng(A2_2)) & D=apply(B2, C2)) => (in(D, relation_dom(A2_2)) & C2=apply(A2_2, D))) & ((in(D, relation_dom(A2_2)) & C2=apply(A2_2, D)) => (in(C2, relation_rng(A2_2)) & D=apply(B2, C2)))))))))).
% 71.07/9.56 fof(t56_funct_1, conjecture, ![A, B2]: ((relation(B2) & function(B2)) => ((one_to_one(B2) & in(A, relation_dom(B2))) => (A=apply(function_inverse(B2), apply(B2, A)) & A=apply(relation_composition(B2, function_inverse(B2)), A))))).
% 71.07/9.56
% 71.07/9.56 Now clausify the problem and encode Horn clauses using encoding 3 of
% 71.07/9.56 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 71.07/9.56 We repeatedly replace C & s=t => u=v by the two clauses:
% 71.07/9.56 fresh(y, y, x1...xn) = u
% 71.07/9.56 C => fresh(s, t, x1...xn) = v
% 71.07/9.56 where fresh is a fresh function symbol and x1..xn are the free
% 71.07/9.56 variables of u and v.
% 71.07/9.56 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 71.07/9.56 input problem has no model of domain size 1).
% 71.07/9.56
% 71.07/9.56 The encoding turns the above axioms into the following unit equations and goals:
% 71.07/9.56
% 71.07/9.56 Axiom 1 (t56_funct_1_2): relation(b) = true2.
% 71.07/9.56 Axiom 2 (t56_funct_1_1): function(b) = true2.
% 71.07/9.56 Axiom 3 (t56_funct_1_3): one_to_one(b) = true2.
% 71.07/9.56 Axiom 4 (t56_funct_1): in(a, relation_dom(b)) = true2.
% 71.07/9.56 Axiom 5 (dt_k2_funct_1): fresh34(X, X, Y) = function(function_inverse(Y)).
% 71.07/9.56 Axiom 6 (dt_k2_funct_1): fresh33(X, X, Y) = true2.
% 71.07/9.56 Axiom 7 (dt_k2_funct_1_1): fresh32(X, X, Y) = relation(function_inverse(Y)).
% 71.07/9.56 Axiom 8 (dt_k2_funct_1_1): fresh31(X, X, Y) = true2.
% 71.07/9.56 Axiom 9 (dt_k2_funct_1): fresh34(relation(X), true2, X) = fresh33(function(X), true2, X).
% 71.07/9.56 Axiom 10 (dt_k2_funct_1_1): fresh32(relation(X), true2, X) = fresh31(function(X), true2, X).
% 71.07/9.56 Axiom 11 (t23_funct_1): fresh79(X, X, Y, Z, W) = apply(relation_composition(Z, W), Y).
% 71.07/9.56 Axiom 12 (t54_funct_1_2): fresh61(X, X, Y, Z, W) = Z.
% 71.07/9.56 Axiom 13 (t23_funct_1): fresh10(X, X, Y, Z, W) = apply(W, apply(Z, Y)).
% 71.07/9.56 Axiom 14 (t23_funct_1): fresh78(X, X, Y, Z, W) = fresh79(function(Z), true2, Y, Z, W).
% 71.07/9.56 Axiom 15 (t23_funct_1): fresh77(X, X, Y, Z, W) = fresh78(function(W), true2, Y, Z, W).
% 71.07/9.56 Axiom 16 (t23_funct_1): fresh76(X, X, Y, Z, W) = fresh77(relation(Z), true2, Y, Z, W).
% 71.07/9.56 Axiom 17 (t54_funct_1_2): fresh60(X, X, Y, Z, W, V) = fresh61(Z, function_inverse(Y), Z, W, V).
% 71.07/9.56 Axiom 18 (t54_funct_1_2): fresh59(X, X, Y, Z, W, V) = apply(Z, V).
% 71.07/9.56 Axiom 19 (t54_funct_1_2): fresh57(X, X, Y, Z, W, V) = fresh58(function(Y), true2, Y, Z, W, V).
% 71.07/9.56 Axiom 20 (t54_funct_1_2): fresh56(X, X, Y, Z, W, V) = fresh57(function(Z), true2, Y, Z, W, V).
% 71.07/9.56 Axiom 21 (t54_funct_1_2): fresh55(X, X, Y, Z, W, V) = fresh56(relation(Y), true2, Y, Z, W, V).
% 71.07/9.56 Axiom 22 (t54_funct_1_2): fresh54(X, X, Y, Z, W, V) = fresh55(relation(Z), true2, Y, Z, W, V).
% 71.07/9.56 Axiom 23 (t54_funct_1_2): fresh58(X, X, Y, Z, W, V) = fresh59(V, apply(Y, W), Y, Z, W, V).
% 71.07/9.56 Axiom 24 (t23_funct_1): fresh76(relation(X), true2, Y, Z, X) = fresh10(in(Y, relation_dom(Z)), true2, Y, Z, X).
% 71.07/9.56 Axiom 25 (t54_funct_1_2): fresh54(one_to_one(X), true2, X, Y, Z, W) = fresh60(in(Z, relation_dom(X)), true2, X, Y, Z, W).
% 71.07/9.56
% 71.07/9.56 Lemma 26: function(function_inverse(b)) = true2.
% 71.07/9.56 Proof:
% 71.07/9.56 function(function_inverse(b))
% 71.07/9.56 = { by axiom 5 (dt_k2_funct_1) R->L }
% 71.07/9.56 fresh34(true2, true2, b)
% 71.07/9.56 = { by axiom 1 (t56_funct_1_2) R->L }
% 71.07/9.56 fresh34(relation(b), true2, b)
% 71.07/9.56 = { by axiom 9 (dt_k2_funct_1) }
% 71.07/9.56 fresh33(function(b), true2, b)
% 71.07/9.56 = { by axiom 2 (t56_funct_1_1) }
% 71.07/9.56 fresh33(true2, true2, b)
% 71.07/9.56 = { by axiom 6 (dt_k2_funct_1) }
% 71.07/9.56 true2
% 71.07/9.56
% 71.07/9.56 Lemma 27: relation(function_inverse(b)) = true2.
% 71.07/9.56 Proof:
% 71.07/9.56 relation(function_inverse(b))
% 71.07/9.56 = { by axiom 7 (dt_k2_funct_1_1) R->L }
% 71.07/9.56 fresh32(true2, true2, b)
% 71.07/9.56 = { by axiom 1 (t56_funct_1_2) R->L }
% 71.07/9.56 fresh32(relation(b), true2, b)
% 71.07/9.56 = { by axiom 10 (dt_k2_funct_1_1) }
% 71.07/9.56 fresh31(function(b), true2, b)
% 71.07/9.56 = { by axiom 2 (t56_funct_1_1) }
% 71.07/9.56 fresh31(true2, true2, b)
% 71.07/9.56 = { by axiom 8 (dt_k2_funct_1_1) }
% 71.07/9.56 true2
% 71.07/9.56
% 71.07/9.56 Lemma 28: apply(function_inverse(b), apply(b, a)) = a.
% 71.07/9.56 Proof:
% 71.07/9.56 apply(function_inverse(b), apply(b, a))
% 71.07/9.56 = { by axiom 18 (t54_funct_1_2) R->L }
% 71.07/9.56 fresh59(apply(b, a), apply(b, a), b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by axiom 23 (t54_funct_1_2) R->L }
% 71.07/9.56 fresh58(true2, true2, b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by axiom 2 (t56_funct_1_1) R->L }
% 71.07/9.56 fresh58(function(b), true2, b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by axiom 19 (t54_funct_1_2) R->L }
% 71.07/9.56 fresh57(true2, true2, b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by lemma 26 R->L }
% 71.07/9.56 fresh57(function(function_inverse(b)), true2, b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by axiom 20 (t54_funct_1_2) R->L }
% 71.07/9.56 fresh56(true2, true2, b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by axiom 1 (t56_funct_1_2) R->L }
% 71.07/9.56 fresh56(relation(b), true2, b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by axiom 21 (t54_funct_1_2) R->L }
% 71.07/9.56 fresh55(true2, true2, b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by lemma 27 R->L }
% 71.07/9.56 fresh55(relation(function_inverse(b)), true2, b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by axiom 22 (t54_funct_1_2) R->L }
% 71.07/9.56 fresh54(true2, true2, b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by axiom 3 (t56_funct_1_3) R->L }
% 71.07/9.56 fresh54(one_to_one(b), true2, b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by axiom 25 (t54_funct_1_2) }
% 71.07/9.56 fresh60(in(a, relation_dom(b)), true2, b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by axiom 4 (t56_funct_1) }
% 71.07/9.56 fresh60(true2, true2, b, function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by axiom 17 (t54_funct_1_2) }
% 71.07/9.56 fresh61(function_inverse(b), function_inverse(b), function_inverse(b), a, apply(b, a))
% 71.07/9.56 = { by axiom 12 (t54_funct_1_2) }
% 71.07/9.56 a
% 71.07/9.56
% 71.07/9.56 Goal 1 (t56_funct_1_4): tuple4(a, a) = tuple4(apply(function_inverse(b), apply(b, a)), apply(relation_composition(b, function_inverse(b)), a)).
% 71.07/9.56 Proof:
% 71.07/9.56 tuple4(a, a)
% 71.07/9.56 = { by lemma 28 R->L }
% 71.07/9.56 tuple4(a, apply(function_inverse(b), apply(b, a)))
% 71.07/9.56 = { by lemma 28 R->L }
% 71.07/9.56 tuple4(apply(function_inverse(b), apply(b, a)), apply(function_inverse(b), apply(b, a)))
% 71.07/9.56 = { by axiom 13 (t23_funct_1) R->L }
% 71.07/9.56 tuple4(apply(function_inverse(b), apply(b, a)), fresh10(true2, true2, a, b, function_inverse(b)))
% 71.07/9.56 = { by axiom 4 (t56_funct_1) R->L }
% 71.07/9.56 tuple4(apply(function_inverse(b), apply(b, a)), fresh10(in(a, relation_dom(b)), true2, a, b, function_inverse(b)))
% 71.07/9.56 = { by axiom 24 (t23_funct_1) R->L }
% 71.07/9.56 tuple4(apply(function_inverse(b), apply(b, a)), fresh76(relation(function_inverse(b)), true2, a, b, function_inverse(b)))
% 71.07/9.56 = { by lemma 27 }
% 71.07/9.56 tuple4(apply(function_inverse(b), apply(b, a)), fresh76(true2, true2, a, b, function_inverse(b)))
% 71.07/9.56 = { by axiom 16 (t23_funct_1) }
% 71.07/9.56 tuple4(apply(function_inverse(b), apply(b, a)), fresh77(relation(b), true2, a, b, function_inverse(b)))
% 71.07/9.56 = { by axiom 1 (t56_funct_1_2) }
% 71.07/9.56 tuple4(apply(function_inverse(b), apply(b, a)), fresh77(true2, true2, a, b, function_inverse(b)))
% 71.07/9.56 = { by axiom 15 (t23_funct_1) }
% 71.07/9.56 tuple4(apply(function_inverse(b), apply(b, a)), fresh78(function(function_inverse(b)), true2, a, b, function_inverse(b)))
% 71.07/9.56 = { by lemma 26 }
% 71.07/9.56 tuple4(apply(function_inverse(b), apply(b, a)), fresh78(true2, true2, a, b, function_inverse(b)))
% 71.07/9.56 = { by axiom 14 (t23_funct_1) }
% 71.07/9.56 tuple4(apply(function_inverse(b), apply(b, a)), fresh79(function(b), true2, a, b, function_inverse(b)))
% 71.07/9.56 = { by axiom 2 (t56_funct_1_1) }
% 71.07/9.56 tuple4(apply(function_inverse(b), apply(b, a)), fresh79(true2, true2, a, b, function_inverse(b)))
% 71.07/9.56 = { by axiom 11 (t23_funct_1) }
% 71.07/9.56 tuple4(apply(function_inverse(b), apply(b, a)), apply(relation_composition(b, function_inverse(b)), a))
% 71.07/9.56 % SZS output end Proof
% 71.07/9.56
% 71.07/9.56 RESULT: Theorem (the conjecture is true).
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