TSTP Solution File: KLE140+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE140+1 : TPTP v8.1.2. Released v4.0.0.
% 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 : n007.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 05:36:00 EDT 2023
% Result : Theorem 10.49s 1.83s
% Output : Proof 10.49s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : KLE140+1 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n007.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 : Tue Aug 29 11:51:28 EDT 2023
% 0.13/0.34 % CPUTime :
% 10.49/1.83 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 10.49/1.83
% 10.49/1.83 % SZS status Theorem
% 10.49/1.83
% 10.49/1.83 % SZS output start Proof
% 10.49/1.84 Take the following subset of the input axioms:
% 10.49/1.84 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 10.49/1.84 fof(additive_commutativity, axiom, ![B2, A3]: addition(A3, B2)=addition(B2, A3)).
% 10.49/1.84 fof(distributivity2, axiom, ![B2, A3, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 10.49/1.84 fof(goals, conjecture, ![X0, X1]: (leq(X0, X1) => leq(strong_iteration(X0), strong_iteration(X1)))).
% 10.49/1.85 fof(idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 10.49/1.85 fof(infty_coinduction, axiom, ![A2, B2, C2]: (leq(C2, addition(multiplication(A2, C2), B2)) => leq(C2, multiplication(strong_iteration(A2), B2)))).
% 10.49/1.85 fof(infty_unfold1, axiom, ![A3]: strong_iteration(A3)=addition(multiplication(A3, strong_iteration(A3)), one)).
% 10.49/1.85 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 10.49/1.85 fof(order, axiom, ![B2, A2_2]: (leq(A2_2, B2) <=> addition(A2_2, B2)=B2)).
% 10.49/1.85
% 10.49/1.85 Now clausify the problem and encode Horn clauses using encoding 3 of
% 10.49/1.85 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 10.49/1.85 We repeatedly replace C & s=t => u=v by the two clauses:
% 10.49/1.85 fresh(y, y, x1...xn) = u
% 10.49/1.85 C => fresh(s, t, x1...xn) = v
% 10.49/1.85 where fresh is a fresh function symbol and x1..xn are the free
% 10.49/1.85 variables of u and v.
% 10.49/1.85 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 10.49/1.85 input problem has no model of domain size 1).
% 10.49/1.85
% 10.49/1.85 The encoding turns the above axioms into the following unit equations and goals:
% 10.49/1.85
% 10.49/1.85 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 10.49/1.85 Axiom 2 (idempotence): addition(X, X) = X.
% 10.49/1.85 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 10.49/1.85 Axiom 4 (goals): leq(x0, x1) = true.
% 10.49/1.85 Axiom 5 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 10.49/1.85 Axiom 6 (order_1): fresh(X, X, Y, Z) = Z.
% 10.49/1.85 Axiom 7 (order): fresh5(X, X, Y, Z) = true.
% 10.49/1.85 Axiom 8 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 10.49/1.85 Axiom 9 (infty_coinduction): fresh4(X, X, Y, Z, W) = true.
% 10.49/1.85 Axiom 10 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 10.49/1.85 Axiom 11 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 10.49/1.85 Axiom 12 (order): fresh5(addition(X, Y), Y, X, Y) = leq(X, Y).
% 10.49/1.85 Axiom 13 (infty_coinduction): fresh4(leq(X, addition(multiplication(Y, X), Z)), true, Y, Z, X) = leq(X, multiplication(strong_iteration(Y), Z)).
% 10.49/1.85
% 10.49/1.85 Goal 1 (goals_1): leq(strong_iteration(x0), strong_iteration(x1)) = true.
% 10.49/1.85 Proof:
% 10.49/1.85 leq(strong_iteration(x0), strong_iteration(x1))
% 10.49/1.85 = { by axiom 1 (multiplicative_right_identity) R->L }
% 10.49/1.85 leq(strong_iteration(x0), multiplication(strong_iteration(x1), one))
% 10.49/1.85 = { by axiom 13 (infty_coinduction) R->L }
% 10.49/1.85 fresh4(leq(strong_iteration(x0), addition(multiplication(x1, strong_iteration(x0)), one)), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 3 (additive_commutativity) }
% 10.49/1.85 fresh4(leq(strong_iteration(x0), addition(one, multiplication(x1, strong_iteration(x0)))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 6 (order_1) R->L }
% 10.49/1.85 fresh4(leq(strong_iteration(x0), addition(one, multiplication(fresh(true, true, x0, x1), strong_iteration(x0)))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 4 (goals) R->L }
% 10.49/1.85 fresh4(leq(strong_iteration(x0), addition(one, multiplication(fresh(leq(x0, x1), true, x0, x1), strong_iteration(x0)))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 11 (order_1) }
% 10.49/1.85 fresh4(leq(strong_iteration(x0), addition(one, multiplication(addition(x0, x1), strong_iteration(x0)))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 3 (additive_commutativity) R->L }
% 10.49/1.85 fresh4(leq(strong_iteration(x0), addition(one, multiplication(addition(x1, x0), strong_iteration(x0)))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 10 (distributivity2) }
% 10.49/1.85 fresh4(leq(strong_iteration(x0), addition(one, addition(multiplication(x1, strong_iteration(x0)), multiplication(x0, strong_iteration(x0))))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 3 (additive_commutativity) R->L }
% 10.49/1.85 fresh4(leq(strong_iteration(x0), addition(one, addition(multiplication(x0, strong_iteration(x0)), multiplication(x1, strong_iteration(x0))))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 5 (additive_associativity) }
% 10.49/1.85 fresh4(leq(strong_iteration(x0), addition(addition(one, multiplication(x0, strong_iteration(x0))), multiplication(x1, strong_iteration(x0)))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 3 (additive_commutativity) R->L }
% 10.49/1.85 fresh4(leq(strong_iteration(x0), addition(addition(multiplication(x0, strong_iteration(x0)), one), multiplication(x1, strong_iteration(x0)))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 8 (infty_unfold1) R->L }
% 10.49/1.85 fresh4(leq(strong_iteration(x0), addition(strong_iteration(x0), multiplication(x1, strong_iteration(x0)))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 12 (order) R->L }
% 10.49/1.85 fresh4(fresh5(addition(strong_iteration(x0), addition(strong_iteration(x0), multiplication(x1, strong_iteration(x0)))), addition(strong_iteration(x0), multiplication(x1, strong_iteration(x0))), strong_iteration(x0), addition(strong_iteration(x0), multiplication(x1, strong_iteration(x0)))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 5 (additive_associativity) }
% 10.49/1.85 fresh4(fresh5(addition(addition(strong_iteration(x0), strong_iteration(x0)), multiplication(x1, strong_iteration(x0))), addition(strong_iteration(x0), multiplication(x1, strong_iteration(x0))), strong_iteration(x0), addition(strong_iteration(x0), multiplication(x1, strong_iteration(x0)))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 2 (idempotence) }
% 10.49/1.85 fresh4(fresh5(addition(strong_iteration(x0), multiplication(x1, strong_iteration(x0))), addition(strong_iteration(x0), multiplication(x1, strong_iteration(x0))), strong_iteration(x0), addition(strong_iteration(x0), multiplication(x1, strong_iteration(x0)))), true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 7 (order) }
% 10.49/1.85 fresh4(true, true, x1, one, strong_iteration(x0))
% 10.49/1.85 = { by axiom 9 (infty_coinduction) }
% 10.49/1.85 true
% 10.49/1.85 % SZS output end Proof
% 10.49/1.85
% 10.49/1.85 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------