TSTP Solution File: GRA002+4 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : GRA002+4 : TPTP v8.1.2. Bugfixed 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 : n021.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 00:05:03 EDT 2023
% Result : Theorem 0.19s 0.49s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRA002+4 : TPTP v8.1.2. Bugfixed v3.2.0.
% 0.00/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n021.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Sun Aug 27 03:44:42 EDT 2023
% 0.19/0.34 % CPUTime :
% 0.19/0.49 Command-line arguments: --no-flatten-goal
% 0.19/0.49
% 0.19/0.49 % SZS status Theorem
% 0.19/0.49
% 0.19/0.50 % SZS output start Proof
% 0.19/0.50 Take the following subset of the input axioms:
% 0.19/0.50 fof(graph_has_them_all, axiom, ![Things, InThese]: less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))).
% 0.19/0.50 fof(maximal_path_length, conjecture, complete => ![V1, V2, P]: (shortest_path(V1, V2, P) => less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))).
% 0.19/0.50 fof(path_length_sequential_pairs, axiom, ![V1_2, V2_2, P2]: (path(V1_2, V2_2, P2) => number_of_in(sequential_pairs, P2)=minus(length_of(P2), n1))).
% 0.19/0.50 fof(shortest_path_defn, axiom, ![SP, V1_2, V2_2]: (shortest_path(V1_2, V2_2, SP) <=> (path(V1_2, V2_2, SP) & (V1_2!=V2_2 & ![P2]: (path(V1_2, V2_2, P2) => less_or_equal(length_of(SP), length_of(P2))))))).
% 0.19/0.50 fof(triangles_and_sequential_pairs, lemma, complete => ![V1_2, V2_2, P2]: (shortest_path(V1_2, V2_2, P2) => number_of_in(sequential_pairs, P2)=number_of_in(triangles, P2))).
% 0.19/0.50
% 0.19/0.50 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.50 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.50 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.50 fresh(y, y, x1...xn) = u
% 0.19/0.50 C => fresh(s, t, x1...xn) = v
% 0.19/0.50 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.50 variables of u and v.
% 0.19/0.50 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.50 input problem has no model of domain size 1).
% 0.19/0.50
% 0.19/0.50 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.50
% 0.19/0.50 Axiom 1 (maximal_path_length): complete = true2.
% 0.19/0.50 Axiom 2 (path_length_sequential_pairs): fresh25(X, X, Y) = number_of_in(sequential_pairs, Y).
% 0.19/0.50 Axiom 3 (triangles_and_sequential_pairs): fresh3(X, X, Y) = number_of_in(sequential_pairs, Y).
% 0.19/0.50 Axiom 4 (triangles_and_sequential_pairs): fresh2(X, X, Y) = number_of_in(triangles, Y).
% 0.19/0.50 Axiom 5 (maximal_path_length_1): shortest_path(v1, v2, p) = true2.
% 0.19/0.50 Axiom 6 (shortest_path_defn_3): fresh10(X, X, Y, Z, W) = true2.
% 0.19/0.50 Axiom 7 (path_length_sequential_pairs): fresh25(path(X, Y, Z), true2, Z) = minus(length_of(Z), n1).
% 0.19/0.50 Axiom 8 (triangles_and_sequential_pairs): fresh3(shortest_path(X, Y, Z), true2, Z) = fresh2(complete, true2, Z).
% 0.19/0.50 Axiom 9 (graph_has_them_all): less_or_equal(number_of_in(X, Y), number_of_in(X, graph)) = true2.
% 0.19/0.50 Axiom 10 (shortest_path_defn_3): fresh10(shortest_path(X, Y, Z), true2, X, Y, Z) = path(X, Y, Z).
% 0.19/0.50
% 0.19/0.50 Lemma 11: shortest_path(v1, v2, p) = complete.
% 0.19/0.50 Proof:
% 0.19/0.50 shortest_path(v1, v2, p)
% 0.19/0.50 = { by axiom 5 (maximal_path_length_1) }
% 0.19/0.50 true2
% 0.19/0.50 = { by axiom 1 (maximal_path_length) R->L }
% 0.19/0.50 complete
% 0.19/0.50
% 0.19/0.50 Goal 1 (maximal_path_length_2): less_or_equal(minus(length_of(p), n1), number_of_in(triangles, graph)) = true2.
% 0.19/0.50 Proof:
% 0.19/0.50 less_or_equal(minus(length_of(p), n1), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 7 (path_length_sequential_pairs) R->L }
% 0.19/0.50 less_or_equal(fresh25(path(v1, v2, p), true2, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 1 (maximal_path_length) R->L }
% 0.19/0.50 less_or_equal(fresh25(path(v1, v2, p), complete, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 10 (shortest_path_defn_3) R->L }
% 0.19/0.50 less_or_equal(fresh25(fresh10(shortest_path(v1, v2, p), true2, v1, v2, p), complete, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 1 (maximal_path_length) R->L }
% 0.19/0.50 less_or_equal(fresh25(fresh10(shortest_path(v1, v2, p), complete, v1, v2, p), complete, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by lemma 11 }
% 0.19/0.50 less_or_equal(fresh25(fresh10(complete, complete, v1, v2, p), complete, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 6 (shortest_path_defn_3) }
% 0.19/0.50 less_or_equal(fresh25(true2, complete, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 1 (maximal_path_length) R->L }
% 0.19/0.50 less_or_equal(fresh25(complete, complete, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 2 (path_length_sequential_pairs) }
% 0.19/0.50 less_or_equal(number_of_in(sequential_pairs, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 3 (triangles_and_sequential_pairs) R->L }
% 0.19/0.50 less_or_equal(fresh3(complete, complete, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by lemma 11 R->L }
% 0.19/0.50 less_or_equal(fresh3(shortest_path(v1, v2, p), complete, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 1 (maximal_path_length) }
% 0.19/0.50 less_or_equal(fresh3(shortest_path(v1, v2, p), true2, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 8 (triangles_and_sequential_pairs) }
% 0.19/0.50 less_or_equal(fresh2(complete, true2, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 1 (maximal_path_length) R->L }
% 0.19/0.50 less_or_equal(fresh2(complete, complete, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 4 (triangles_and_sequential_pairs) }
% 0.19/0.50 less_or_equal(number_of_in(triangles, p), number_of_in(triangles, graph))
% 0.19/0.50 = { by axiom 9 (graph_has_them_all) }
% 0.19/0.50 true2
% 0.19/0.50 % SZS output end Proof
% 0.19/0.50
% 0.19/0.50 RESULT: Theorem (the conjecture is true).
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