TSTP Solution File: SEU220+3 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SEU220+3 : 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 : n002.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 17:51:38 EDT 2023
% Result : Theorem 25.94s 3.74s
% Output : Proof 25.94s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU220+3 : 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
% 0.14/0.34 % Computer : n002.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Wed Aug 23 14:24:16 EDT 2023
% 0.14/0.34 % CPUTime :
% 25.94/3.74 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 25.94/3.74
% 25.94/3.74 % SZS status Theorem
% 25.94/3.74
% 25.94/3.75 % SZS output start Proof
% 25.94/3.75 Take the following subset of the input axioms:
% 25.94/3.76 fof(dt_k2_funct_1, axiom, ![A2]: ((relation(A2) & function(A2)) => (relation(function_inverse(A2)) & function(function_inverse(A2))))).
% 25.94/3.76 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)))))).
% 25.94/3.76 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)))))))))).
% 25.94/3.76 fof(t55_funct_1, axiom, ![A2_2]: ((relation(A2_2) & function(A2_2)) => (one_to_one(A2_2) => (relation_rng(A2_2)=relation_dom(function_inverse(A2_2)) & relation_dom(A2_2)=relation_rng(function_inverse(A2_2)))))).
% 25.94/3.76 fof(t57_funct_1, conjecture, ![A, B2]: ((relation(B2) & function(B2)) => ((one_to_one(B2) & in(A, relation_rng(B2))) => (A=apply(B2, apply(function_inverse(B2), A)) & A=apply(relation_composition(function_inverse(B2), B2), A))))).
% 25.94/3.76
% 25.94/3.76 Now clausify the problem and encode Horn clauses using encoding 3 of
% 25.94/3.76 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 25.94/3.76 We repeatedly replace C & s=t => u=v by the two clauses:
% 25.94/3.76 fresh(y, y, x1...xn) = u
% 25.94/3.76 C => fresh(s, t, x1...xn) = v
% 25.94/3.76 where fresh is a fresh function symbol and x1..xn are the free
% 25.94/3.76 variables of u and v.
% 25.94/3.76 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 25.94/3.76 input problem has no model of domain size 1).
% 25.94/3.76
% 25.94/3.76 The encoding turns the above axioms into the following unit equations and goals:
% 25.94/3.76
% 25.94/3.76 Axiom 1 (t57_funct_1_2): relation(b) = true2.
% 25.94/3.76 Axiom 2 (t57_funct_1_1): function(b) = true2.
% 25.94/3.76 Axiom 3 (t57_funct_1_3): one_to_one(b) = true2.
% 25.94/3.76 Axiom 4 (t57_funct_1): in(a, relation_rng(b)) = true2.
% 25.94/3.76 Axiom 5 (t55_funct_1_1): fresh43(X, X, Y) = relation_rng(Y).
% 25.94/3.76 Axiom 6 (dt_k2_funct_1): fresh36(X, X, Y) = function(function_inverse(Y)).
% 25.94/3.76 Axiom 7 (dt_k2_funct_1): fresh35(X, X, Y) = true2.
% 25.94/3.76 Axiom 8 (dt_k2_funct_1_1): fresh34(X, X, Y) = relation(function_inverse(Y)).
% 25.94/3.76 Axiom 9 (dt_k2_funct_1_1): fresh33(X, X, Y) = true2.
% 25.94/3.76 Axiom 10 (t55_funct_1_1): fresh4(X, X, Y) = relation_dom(function_inverse(Y)).
% 25.94/3.76 Axiom 11 (t55_funct_1_1): fresh42(X, X, Y) = fresh43(function(Y), true2, Y).
% 25.94/3.76 Axiom 12 (dt_k2_funct_1): fresh36(relation(X), true2, X) = fresh35(function(X), true2, X).
% 25.94/3.76 Axiom 13 (dt_k2_funct_1_1): fresh34(relation(X), true2, X) = fresh33(function(X), true2, X).
% 25.94/3.76 Axiom 14 (t55_funct_1_1): fresh42(one_to_one(X), true2, X) = fresh4(relation(X), true2, X).
% 25.94/3.76 Axiom 15 (t23_funct_1): fresh85(X, X, Y, Z, W) = apply(relation_composition(Z, W), Y).
% 25.94/3.76 Axiom 16 (t54_funct_1): fresh51(X, X, Y, Z, W) = Z.
% 25.94/3.76 Axiom 17 (t23_funct_1): fresh12(X, X, Y, Z, W) = apply(W, apply(Z, Y)).
% 25.94/3.76 Axiom 18 (t23_funct_1): fresh84(X, X, Y, Z, W) = fresh85(function(Z), true2, Y, Z, W).
% 25.94/3.76 Axiom 19 (t23_funct_1): fresh83(X, X, Y, Z, W) = fresh84(function(W), true2, Y, Z, W).
% 25.94/3.76 Axiom 20 (t23_funct_1): fresh82(X, X, Y, Z, W) = fresh83(relation(Z), true2, Y, Z, W).
% 25.94/3.76 Axiom 21 (t54_funct_1): fresh50(X, X, Y, Z, W, V) = fresh51(Z, function_inverse(Y), Y, W, V).
% 25.94/3.76 Axiom 22 (t54_funct_1): fresh49(X, X, Y, Z, W, V) = apply(Y, V).
% 25.94/3.76 Axiom 23 (t54_funct_1): fresh47(X, X, Y, Z, W, V) = fresh48(function(Y), true2, Y, Z, W, V).
% 25.94/3.76 Axiom 24 (t54_funct_1): fresh46(X, X, Y, Z, W, V) = fresh47(function(Z), true2, Y, Z, W, V).
% 25.94/3.76 Axiom 25 (t54_funct_1): fresh45(X, X, Y, Z, W, V) = fresh46(relation(Y), true2, Y, Z, W, V).
% 25.94/3.76 Axiom 26 (t54_funct_1): fresh44(X, X, Y, Z, W, V) = fresh45(relation(Z), true2, Y, Z, W, V).
% 25.94/3.76 Axiom 27 (t54_funct_1): fresh48(X, X, Y, Z, W, V) = fresh49(V, apply(Z, W), Y, Z, W, V).
% 25.94/3.76 Axiom 28 (t23_funct_1): fresh82(relation(X), true2, Y, Z, X) = fresh12(in(Y, relation_dom(Z)), true2, Y, Z, X).
% 25.94/3.76 Axiom 29 (t54_funct_1): fresh44(one_to_one(X), true2, X, Y, Z, W) = fresh50(in(Z, relation_rng(X)), true2, X, Y, Z, W).
% 25.94/3.76
% 25.94/3.76 Lemma 30: function(function_inverse(b)) = true2.
% 25.94/3.76 Proof:
% 25.94/3.76 function(function_inverse(b))
% 25.94/3.76 = { by axiom 6 (dt_k2_funct_1) R->L }
% 25.94/3.76 fresh36(true2, true2, b)
% 25.94/3.76 = { by axiom 1 (t57_funct_1_2) R->L }
% 25.94/3.76 fresh36(relation(b), true2, b)
% 25.94/3.76 = { by axiom 12 (dt_k2_funct_1) }
% 25.94/3.76 fresh35(function(b), true2, b)
% 25.94/3.76 = { by axiom 2 (t57_funct_1_1) }
% 25.94/3.76 fresh35(true2, true2, b)
% 25.94/3.76 = { by axiom 7 (dt_k2_funct_1) }
% 25.94/3.76 true2
% 25.94/3.76
% 25.94/3.76 Lemma 31: relation(function_inverse(b)) = true2.
% 25.94/3.76 Proof:
% 25.94/3.76 relation(function_inverse(b))
% 25.94/3.76 = { by axiom 8 (dt_k2_funct_1_1) R->L }
% 25.94/3.76 fresh34(true2, true2, b)
% 25.94/3.76 = { by axiom 1 (t57_funct_1_2) R->L }
% 25.94/3.76 fresh34(relation(b), true2, b)
% 25.94/3.76 = { by axiom 13 (dt_k2_funct_1_1) }
% 25.94/3.76 fresh33(function(b), true2, b)
% 25.94/3.76 = { by axiom 2 (t57_funct_1_1) }
% 25.94/3.76 fresh33(true2, true2, b)
% 25.94/3.76 = { by axiom 9 (dt_k2_funct_1_1) }
% 25.94/3.76 true2
% 25.94/3.76
% 25.94/3.76 Lemma 32: apply(b, apply(function_inverse(b), a)) = a.
% 25.94/3.76 Proof:
% 25.94/3.76 apply(b, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 22 (t54_funct_1) R->L }
% 25.94/3.76 fresh49(apply(function_inverse(b), a), apply(function_inverse(b), a), b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 27 (t54_funct_1) R->L }
% 25.94/3.76 fresh48(true2, true2, b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 2 (t57_funct_1_1) R->L }
% 25.94/3.76 fresh48(function(b), true2, b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 23 (t54_funct_1) R->L }
% 25.94/3.76 fresh47(true2, true2, b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by lemma 30 R->L }
% 25.94/3.76 fresh47(function(function_inverse(b)), true2, b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 24 (t54_funct_1) R->L }
% 25.94/3.76 fresh46(true2, true2, b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 1 (t57_funct_1_2) R->L }
% 25.94/3.76 fresh46(relation(b), true2, b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 25 (t54_funct_1) R->L }
% 25.94/3.76 fresh45(true2, true2, b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by lemma 31 R->L }
% 25.94/3.76 fresh45(relation(function_inverse(b)), true2, b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 26 (t54_funct_1) R->L }
% 25.94/3.76 fresh44(true2, true2, b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 3 (t57_funct_1_3) R->L }
% 25.94/3.76 fresh44(one_to_one(b), true2, b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 29 (t54_funct_1) }
% 25.94/3.76 fresh50(in(a, relation_rng(b)), true2, b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 4 (t57_funct_1) }
% 25.94/3.76 fresh50(true2, true2, b, function_inverse(b), a, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 21 (t54_funct_1) }
% 25.94/3.76 fresh51(function_inverse(b), function_inverse(b), b, a, apply(function_inverse(b), a))
% 25.94/3.76 = { by axiom 16 (t54_funct_1) }
% 25.94/3.76 a
% 25.94/3.76
% 25.94/3.76 Goal 1 (t57_funct_1_4): tuple4(a, a) = tuple4(apply(relation_composition(function_inverse(b), b), a), apply(b, apply(function_inverse(b), a))).
% 25.94/3.76 Proof:
% 25.94/3.76 tuple4(a, a)
% 25.94/3.76 = { by lemma 32 R->L }
% 25.94/3.76 tuple4(a, apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by lemma 32 R->L }
% 25.94/3.76 tuple4(apply(b, apply(function_inverse(b), a)), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 17 (t23_funct_1) R->L }
% 25.94/3.76 tuple4(fresh12(true2, true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 4 (t57_funct_1) R->L }
% 25.94/3.76 tuple4(fresh12(in(a, relation_rng(b)), true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 5 (t55_funct_1_1) R->L }
% 25.94/3.76 tuple4(fresh12(in(a, fresh43(true2, true2, b)), true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 2 (t57_funct_1_1) R->L }
% 25.94/3.76 tuple4(fresh12(in(a, fresh43(function(b), true2, b)), true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 11 (t55_funct_1_1) R->L }
% 25.94/3.76 tuple4(fresh12(in(a, fresh42(true2, true2, b)), true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 3 (t57_funct_1_3) R->L }
% 25.94/3.76 tuple4(fresh12(in(a, fresh42(one_to_one(b), true2, b)), true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 14 (t55_funct_1_1) }
% 25.94/3.76 tuple4(fresh12(in(a, fresh4(relation(b), true2, b)), true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 1 (t57_funct_1_2) }
% 25.94/3.76 tuple4(fresh12(in(a, fresh4(true2, true2, b)), true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 10 (t55_funct_1_1) }
% 25.94/3.76 tuple4(fresh12(in(a, relation_dom(function_inverse(b))), true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 28 (t23_funct_1) R->L }
% 25.94/3.76 tuple4(fresh82(relation(b), true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 1 (t57_funct_1_2) }
% 25.94/3.76 tuple4(fresh82(true2, true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 20 (t23_funct_1) }
% 25.94/3.76 tuple4(fresh83(relation(function_inverse(b)), true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by lemma 31 }
% 25.94/3.76 tuple4(fresh83(true2, true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 19 (t23_funct_1) }
% 25.94/3.76 tuple4(fresh84(function(b), true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 2 (t57_funct_1_1) }
% 25.94/3.76 tuple4(fresh84(true2, true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 18 (t23_funct_1) }
% 25.94/3.76 tuple4(fresh85(function(function_inverse(b)), true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by lemma 30 }
% 25.94/3.76 tuple4(fresh85(true2, true2, a, function_inverse(b), b), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 = { by axiom 15 (t23_funct_1) }
% 25.94/3.76 tuple4(apply(relation_composition(function_inverse(b), b), a), apply(b, apply(function_inverse(b), a)))
% 25.94/3.76 % SZS output end Proof
% 25.94/3.76
% 25.94/3.76 RESULT: Theorem (the conjecture is true).
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