%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : LCL452^7 : TPTP v9.3.1. Released v5.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Mon Sep 7 11:34:13 AM UTC 2026
% Result : CounterSatisfiable 0.12s 0.44s
% Output : Saturation 0.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : LCL452^7 : TPTP v9.3.1. Released v5.5.0.
% 0.00/0.04 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.08/0.35 % Computer : n016.cluster.edu
% 0.08/0.35 % Model : x86_64 x86_64
% 0.08/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.35 % Memory : 8046.5625MB
% 0.08/0.35 % OS : Linux 6.8.0-71-generic
% 0.12/0.35 % CPULimit : 300
% 0.12/0.35 % WCLimit : 300
% 0.12/0.35 % DateTime : Fri Sep 4 18:40:28 UTC 2026
% 0.12/0.36 % CPUTime :
% 0.12/0.36 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.12/0.38 ---- Original DTF file ---
% 0.12/0.38 thf(spec,logic,$$dhol).
% 0.12/0.38 %------------------------------------------------------------------------------
% 0.12/0.38 % File : LCL452^7 : TPTP v9.3.1. Released v5.5.0.
% 0.12/0.38 % Domain : Logic Calculi
% 0.12/0.38 % Problem : Prove Lukasiewicz's cn2 axiom from Hilbert's axiomatization
% 0.12/0.38 % Version : [Ben12] axioms.
% 0.12/0.38 % English :
% 0.12/0.38
% 0.12/0.38 % Refs : [HB34] Hilbert & Bernays (1934), Grundlagen der Mathematick
% 0.12/0.38 % : [Goe69] Goedel (1969), An Interpretation of the Intuitionistic
% 0.12/0.38 % : [Hal] Halleck (URL), John Halleck's Logic Systems
% 0.12/0.38 % : [Ben12] Benzmueller (2012), Email to Geoff Sutcliffe
% 0.12/0.38 % Source : [Ben12]
% 0.12/0.38 % Names : s4-cumul-GLC452+1 [Ben12]
% 0.12/0.38
% 0.12/0.38 % Status : Theorem
% 0.12/0.38 % Rating : 1.00 v5.5.0
% 0.12/0.38 % Syntax : Number of formulae : 181 ( 38 unt; 74 typ; 32 def)
% 0.12/0.38 % Number of atoms : 999 ( 36 equ; 0 cnn)
% 0.12/0.38 % Maximal formula atoms : 36 ( 9 avg)
% 0.12/0.38 % Number of connectives : 1504 ( 5 ~; 5 |; 9 &;1475 @)
% 0.12/0.38 % ( 0 <=>; 10 =>; 0 <=; 0 <~>)
% 0.12/0.38 % Maximal formula depth : 23 ( 9 avg)
% 0.12/0.38 % Number of types : 3 ( 1 usr)
% 0.12/0.38 % Number of type conns : 223 ( 223 >; 0 *; 0 +; 0 <<)
% 0.12/0.38 % Number of symbols : 75 ( 73 usr; 1 con; 0-3 aty)
% 0.12/0.38 % Number of variables : 258 ( 203 ^; 48 !; 7 ?; 258 :)
% 0.12/0.38 % SPC : TH0_THM_EQU_NAR_NDT
% 0.12/0.38
% 0.12/0.38 % Comments : Goedel translation of LCL452+1
% 0.12/0.38 %------------------------------------------------------------------------------
% 0.12/0.38 %----Include axioms for Modal logic S4 under cumulative domains
% 0.12/0.38 include('Axioms/LCL015^0.ax').
% 0.12/0.38 include('Axioms/LCL013^5.ax').
% 0.12/0.38 include('Axioms/LCL015^1.ax').
% 0.12/0.38 %------------------------------------------------------------------------------
% 0.12/0.38 thf(kn1_type,type,
% 0.12/0.38 kn1: $i > $o ).
% 0.12/0.38
% 0.12/0.38 thf(kn2_type,type,
% 0.12/0.38 kn2: $i > $o ).
% 0.12/0.38
% 0.12/0.38 thf(kn3_type,type,
% 0.12/0.38 kn3: $i > $o ).
% 0.12/0.38
% 0.12/0.38 thf(cn1_type,type,
% 0.12/0.38 cn1: $i > $o ).
% 0.12/0.38
% 0.12/0.38 thf(cn3_type,type,
% 0.12/0.38 cn3: $i > $o ).
% 0.12/0.38
% 0.12/0.38 thf(r1_type,type,
% 0.12/0.38 r1: $i > $o ).
% 0.12/0.38
% 0.12/0.38 thf(r2_type,type,
% 0.12/0.38 r2: $i > $o ).
% 0.12/0.38
% 0.12/0.38 thf(r3_type,type,
% 0.12/0.39 r3: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(r4_type,type,
% 0.12/0.39 r4: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(r5_type,type,
% 0.12/0.39 r5: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(op_and_type,type,
% 0.12/0.39 op_and: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(op_implies_or_type,type,
% 0.12/0.39 op_implies_or: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(op_implies_and_type,type,
% 0.12/0.39 op_implies_and: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(modus_ponens_type,type,
% 0.12/0.39 modus_ponens: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(modus_tollens_type,type,
% 0.12/0.39 modus_tollens: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(implies_1_type,type,
% 0.12/0.39 implies_1: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(implies_2_type,type,
% 0.12/0.39 implies_2: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(implies_3_type,type,
% 0.12/0.39 implies_3: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(and_1_type,type,
% 0.12/0.39 and_1: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(and_2_type,type,
% 0.12/0.39 and_2: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(and_3_type,type,
% 0.12/0.39 and_3: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(or_1_type,type,
% 0.12/0.39 or_1: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(or_2_type,type,
% 0.12/0.39 or_2: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(or_3_type,type,
% 0.12/0.39 or_3: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(equivalence_1_type,type,
% 0.12/0.39 equivalence_1: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(equivalence_2_type,type,
% 0.12/0.39 equivalence_2: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(equivalence_3_type,type,
% 0.12/0.39 equivalence_3: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(substitution_of_equivalents_type,type,
% 0.12/0.39 substitution_of_equivalents: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(op_or_type,type,
% 0.12/0.39 op_or: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(op_implies_type,type,
% 0.12/0.39 op_implies: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(op_equiv_type,type,
% 0.12/0.39 op_equiv: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(cn2_type,type,
% 0.12/0.39 cn2: $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(is_a_theorem_type,type,
% 0.12/0.39 is_a_theorem: mu > $i > $o ).
% 0.12/0.39
% 0.12/0.39 thf(not_type,type,
% 0.12/0.39 not: mu > mu ).
% 0.12/0.39
% 0.12/0.39 thf(existence_of_not_ax,axiom,
% 0.12/0.39 ! [V: $i,V1: mu] : ( exists_in_world @ ( not @ V1 ) @ V ) ).
% 0.12/0.39
% 0.12/0.39 thf(or_type,type,
% 0.12/0.39 or: mu > mu > mu ).
% 0.12/0.39
% 0.12/0.39 thf(existence_of_or_ax,axiom,
% 0.12/0.39 ! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( or @ V2 @ V1 ) @ V ) ).
% 0.12/0.39
% 0.12/0.39 thf(implies_type,type,
% 0.12/0.39 implies: mu > mu > mu ).
% 0.12/0.39
% 0.12/0.39 thf(existence_of_implies_ax,axiom,
% 0.12/0.39 ! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( implies @ V2 @ V1 ) @ V ) ).
% 0.12/0.39
% 0.12/0.39 thf(and_type,type,
% 0.12/0.39 and: mu > mu > mu ).
% 0.12/0.39
% 0.12/0.39 thf(existence_of_and_ax,axiom,
% 0.12/0.39 ! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( and @ V2 @ V1 ) @ V ) ).
% 0.12/0.39
% 0.12/0.39 thf(equiv_type,type,
% 0.12/0.39 equiv: mu > mu > mu ).
% 0.12/0.39
% 0.12/0.39 thf(existence_of_equiv_ax,axiom,
% 0.12/0.39 ! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( equiv @ V2 @ V1 ) @ V ) ).
% 0.12/0.39
% 0.12/0.39 thf(reflexivity,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] : ( mbox_s4 @ ( qmltpeq @ X @ X ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(symmetry,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ X @ Y ) ) @ ( mbox_s4 @ ( qmltpeq @ Y @ X ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(transitivity,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Z: mu] : ( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( qmltpeq @ X @ Y ) ) @ ( mbox_s4 @ ( qmltpeq @ Y @ Z ) ) ) @ ( mbox_s4 @ ( qmltpeq @ X @ Z ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(and_substitution_1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [A: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [B: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( and @ A @ C ) @ ( and @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(and_substitution_2,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [A: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [B: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( and @ C @ A ) @ ( and @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(equiv_substitution_1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [A: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [B: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( equiv @ A @ C ) @ ( equiv @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(equiv_substitution_2,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [A: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [B: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( equiv @ C @ A ) @ ( equiv @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(implies_substitution_1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [A: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [B: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( implies @ A @ C ) @ ( implies @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(implies_substitution_2,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [A: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [B: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( implies @ C @ A ) @ ( implies @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(not_substitution_1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [A: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [B: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( not @ A ) @ ( not @ B ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(or_substitution_1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [A: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [B: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( or @ A @ C ) @ ( or @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(or_substitution_2,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [A: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [B: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( or @ C @ A ) @ ( or @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(is_a_theorem_substitution_1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [A: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [B: mu] : ( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( is_a_theorem @ A ) ) ) @ ( mbox_s4 @ ( is_a_theorem @ B ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(modus_ponens,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ modus_ponens )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( is_a_theorem @ X ) ) @ ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ Y ) ) ) ) @ ( mbox_s4 @ ( is_a_theorem @ Y ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( is_a_theorem @ X ) ) @ ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ Y ) ) ) ) @ ( mbox_s4 @ ( is_a_theorem @ Y ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ modus_ponens ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(substitution_of_equivalents,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ substitution_of_equivalents )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( is_a_theorem @ ( equiv @ X @ Y ) ) ) @ ( mbox_s4 @ ( qmltpeq @ X @ Y ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( is_a_theorem @ ( equiv @ X @ Y ) ) ) @ ( mbox_s4 @ ( qmltpeq @ X @ Y ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ substitution_of_equivalents ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(modus_tollens,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ modus_tollens )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ ( not @ Y ) @ ( not @ X ) ) @ ( implies @ X @ Y ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ ( not @ Y ) @ ( not @ X ) ) @ ( implies @ X @ Y ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ modus_tollens ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(implies_1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ implies_1 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ ( implies @ Y @ X ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ ( implies @ Y @ X ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ implies_1 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(implies_2,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ implies_2 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ ( implies @ X @ Y ) ) @ ( implies @ X @ Y ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ ( implies @ X @ Y ) ) @ ( implies @ X @ Y ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ implies_2 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(implies_3,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ implies_3 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Z: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ Y ) @ ( implies @ ( implies @ Y @ Z ) @ ( implies @ X @ Z ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Z: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ Y ) @ ( implies @ ( implies @ Y @ Z ) @ ( implies @ X @ Z ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ implies_3 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(and_1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ and_1 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( and @ X @ Y ) @ X ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( and @ X @ Y ) @ X ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ and_1 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(and_2,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ and_2 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( and @ X @ Y ) @ Y ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( and @ X @ Y ) @ Y ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ and_2 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(and_3,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ and_3 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ ( implies @ Y @ ( and @ X @ Y ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ ( implies @ Y @ ( and @ X @ Y ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ and_3 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(or_1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ or_1 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ ( or @ X @ Y ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ ( or @ X @ Y ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ or_1 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(or_2,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ or_2 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ Y @ ( or @ X @ Y ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ Y @ ( or @ X @ Y ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ or_2 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(or_3,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ or_3 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Z: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ Z ) @ ( implies @ ( implies @ Y @ Z ) @ ( implies @ ( or @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Z: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ Z ) @ ( implies @ ( implies @ Y @ Z ) @ ( implies @ ( or @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ or_3 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(equivalence_1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ equivalence_1 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( equiv @ X @ Y ) @ ( implies @ X @ Y ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( equiv @ X @ Y ) @ ( implies @ X @ Y ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ equivalence_1 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(equivalence_2,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ equivalence_2 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( equiv @ X @ Y ) @ ( implies @ Y @ X ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( equiv @ X @ Y ) @ ( implies @ Y @ X ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ equivalence_2 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(equivalence_3,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ equivalence_3 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ Y ) @ ( implies @ ( implies @ Y @ X ) @ ( equiv @ X @ Y ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ Y ) @ ( implies @ ( implies @ Y @ X ) @ ( equiv @ X @ Y ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ equivalence_3 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(kn1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ kn1 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ P @ ( and @ P @ P ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ P @ ( and @ P @ P ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ kn1 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(kn2,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ kn2 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( and @ P @ Q ) @ P ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( and @ P @ Q ) @ P ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ kn2 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(kn3,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ kn3 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ P @ Q ) @ ( implies @ ( not @ ( and @ Q @ R ) ) @ ( not @ ( and @ R @ P ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ P @ Q ) @ ( implies @ ( not @ ( and @ Q @ R ) ) @ ( not @ ( and @ R @ P ) ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ kn3 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(cn1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ cn1 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ P @ Q ) @ ( implies @ ( implies @ Q @ R ) @ ( implies @ P @ R ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ P @ Q ) @ ( implies @ ( implies @ Q @ R ) @ ( implies @ P @ R ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ cn1 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(cn2,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ cn2 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ P @ ( implies @ ( not @ P ) @ Q ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ P @ ( implies @ ( not @ P ) @ Q ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ cn2 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(cn3,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ cn3 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ ( not @ P ) @ P ) @ P ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ ( not @ P ) @ P ) @ P ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ cn3 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(r1,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ r1 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( or @ P @ P ) @ P ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( or @ P @ P ) @ P ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ r1 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(r2,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ r2 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ Q @ ( or @ P @ Q ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ Q @ ( or @ P @ Q ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ r2 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(r3,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ r3 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( or @ P @ Q ) @ ( or @ Q @ P ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( or @ P @ Q ) @ ( or @ Q @ P ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ r3 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(r4,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ r4 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( or @ P @ ( or @ Q @ R ) ) @ ( or @ Q @ ( or @ P @ R ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( or @ P @ ( or @ Q @ R ) ) @ ( or @ Q @ ( or @ P @ R ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ r4 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(r5,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mand
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ r5 )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ Q @ R ) @ ( implies @ ( or @ P @ Q ) @ ( or @ P @ R ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [P: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Q: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ Q @ R ) @ ( implies @ ( or @ P @ Q ) @ ( or @ P @ R ) ) ) ) ) ) ) ) ) ) )
% 0.12/0.39 @ ( mbox_s4 @ r5 ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(op_or,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ op_or )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( qmltpeq @ ( or @ X @ Y ) @ ( not @ ( and @ ( not @ X ) @ ( not @ Y ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(op_and,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ op_and )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( qmltpeq @ ( and @ X @ Y ) @ ( not @ ( or @ ( not @ X ) @ ( not @ Y ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(op_implies_and,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ op_implies_and )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( qmltpeq @ ( implies @ X @ Y ) @ ( not @ ( and @ X @ ( not @ Y ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(op_implies_or,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ op_implies_or )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( qmltpeq @ ( implies @ X @ Y ) @ ( or @ ( not @ X ) @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(op_equiv,axiom,
% 0.12/0.39 ( mvalid
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mimplies @ ( mbox_s4 @ op_equiv )
% 0.12/0.39 @ ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [X: mu] :
% 0.12/0.39 ( mbox_s4
% 0.12/0.39 @ ( mforall_ind
% 0.12/0.39 @ ^ [Y: mu] : ( mbox_s4 @ ( qmltpeq @ ( equiv @ X @ Y ) @ ( and @ ( implies @ X @ Y ) @ ( implies @ Y @ X ) ) ) ) ) ) ) ) ) ) ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_op_or,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ op_or ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_op_implies_and,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ op_implies_and ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_op_equiv,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ op_equiv ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_modus_ponens,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ modus_ponens ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_modus_tollens,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ modus_tollens ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_implies_1,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ implies_1 ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_implies_2,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ implies_2 ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_implies_3,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ implies_3 ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_and_1,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ and_1 ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_and_2,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ and_2 ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_and_3,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ and_3 ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_or_1,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ or_1 ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_or_2,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ or_2 ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_or_3,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ or_3 ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_equivalence_1,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ equivalence_1 ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_equivalence_2,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ equivalence_2 ) ).
% 0.12/0.39
% 0.12/0.39 thf(hilbert_equivalence_3,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ equivalence_3 ) ).
% 0.12/0.39
% 0.12/0.39 thf(substitution_of_equivalents_001,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ substitution_of_equivalents ) ).
% 0.12/0.39
% 0.12/0.39 thf(luka_op_or,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ op_or ) ).
% 0.12/0.39
% 0.12/0.39 thf(luka_op_implies,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ op_implies ) ).
% 0.12/0.39
% 0.12/0.39 thf(luka_op_equiv,axiom,
% 0.12/0.39 mvalid @ ( mbox_s4 @ op_equiv ) ).
% 0.12/0.39
% 0.12/0.39 thf(luka_cn2,conjecture,
% 0.12/0.39 mvalid @ ( mbox_s4 @ cn2 ) ).
% 0.12/0.39
% 0.12/0.39 %------------------------------------------------------------------------------
% 0.12/0.39 ------------------------
% 0.12/0.39 /starexec/sw/sutcliffe/jdk-11.0.2/bin/java: Command not found.
% 0.12/0.39 ---- Embedded in THF ---
% 0.12/0.39 ------------------------
% 0.12/0.41 ---- Cleaned THF ---
% 0.12/0.41 ------------------------
% 0.12/0.44 % (1027186)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_1027167 for (2999ds/2Mi)
% 0.12/0.44 % (1027186)First to succeed.
% 0.12/0.44 % SZS status Satisfiable for DTF2THF_1027167
% 0.12/0.44 % (1027186)# SZS output start Saturation.
% 0.12/0.44 % (1027186)# SZS output end Saturation.
% 0.12/0.44 % (1027186)------------------------------
% 0.12/0.44 % (1027186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.12/0.44 % (1027186)Termination reason: Satisfiable
% 0.12/0.44
% 0.12/0.44 % (1027186)Memory used [KB]: 5373
% 0.12/0.44 % (1027186)Time elapsed: 0.002 s
% 0.12/0.44 % (1027186)------------------------------
% 0.12/0.44 % (1027186)------------------------------
% 0.12/0.44 % (1027182)Success in time 0.009 s
%------------------------------------------------------------------------------