%------------------------------------------------------------------------------ % File : LCL460^7 : TPTP v8.1.2. Released v5.5.0. % Domain : Logic Calculi % Problem : Prove Rosser's kn2 axiom from Hilbert's axiomatization % Version : [Ben12] axioms. % English : % Refs : [HB34] Hilbert & Bernays (1934), Grundlagen der Mathematick % : [Goe69] Goedel (1969), An Interpretation of the Intuitionistic % : [Hal] Halleck (URL), John Halleck's Logic Systems % : [Ben12] Benzmueller (2012), Email to Geoff Sutcliffe % Source : [Ben12] % Names : s4-cumul-GLC460+1 [Ben12] % Status : Theorem % Rating : 0.46 v8.1.0, 0.45 v7.5.0, 0.57 v7.4.0, 0.56 v7.3.0, 0.67 v7.2.0, 0.62 v7.0.0, 0.57 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.71 v5.5.0 % Syntax : Number of formulae : 180 ( 38 unt; 73 typ; 32 def) % Number of atoms : 999 ( 36 equ; 0 cnn) % Maximal formula atoms : 36 ( 9 avg) % Number of connectives : 1504 ( 5 ~; 5 |; 9 &;1475 @) % ( 0 <=>; 10 =>; 0 <=; 0 <~>) % Maximal formula depth : 23 ( 9 avg) % Number of types : 3 ( 1 usr) % Number of type conns : 222 ( 222 >; 0 *; 0 +; 0 <<) % Number of symbols : 81 ( 79 usr; 8 con; 0-3 aty) % Number of variables : 258 ( 203 ^ 48 !; 7 ?; 258 :) % SPC : TH0_THM_EQU_NAR % Comments : Goedel translation of LCL460+1 %------------------------------------------------------------------------------ %----Include axioms for Modal logic S4 under cumulative domains include('Axioms/LCL015^0.ax'). include('Axioms/LCL013^5.ax'). include('Axioms/LCL015^1.ax'). %------------------------------------------------------------------------------ thf(kn1_type,type, kn1: $i > $o ). thf(kn3_type,type, kn3: $i > $o ). thf(cn1_type,type, cn1: $i > $o ). thf(cn2_type,type, cn2: $i > $o ). thf(cn3_type,type, cn3: $i > $o ). thf(r1_type,type, r1: $i > $o ). thf(r2_type,type, r2: $i > $o ). thf(r3_type,type, r3: $i > $o ). thf(r4_type,type, r4: $i > $o ). thf(r5_type,type, r5: $i > $o ). thf(op_and_type,type, op_and: $i > $o ). thf(op_implies_or_type,type, op_implies_or: $i > $o ). thf(modus_ponens_type,type, modus_ponens: $i > $o ). thf(modus_tollens_type,type, modus_tollens: $i > $o ). thf(implies_1_type,type, implies_1: $i > $o ). thf(implies_2_type,type, implies_2: $i > $o ). thf(implies_3_type,type, implies_3: $i > $o ). thf(and_1_type,type, and_1: $i > $o ). thf(and_2_type,type, and_2: $i > $o ). thf(and_3_type,type, and_3: $i > $o ). thf(or_1_type,type, or_1: $i > $o ). thf(or_2_type,type, or_2: $i > $o ). thf(or_3_type,type, or_3: $i > $o ). thf(equivalence_1_type,type, equivalence_1: $i > $o ). thf(equivalence_2_type,type, equivalence_2: $i > $o ). thf(equivalence_3_type,type, equivalence_3: $i > $o ). thf(substitution_of_equivalents_type,type, substitution_of_equivalents: $i > $o ). thf(op_or_type,type, op_or: $i > $o ). thf(op_implies_and_type,type, op_implies_and: $i > $o ). thf(op_equiv_type,type, op_equiv: $i > $o ). thf(kn2_type,type, kn2: $i > $o ). thf(is_a_theorem_type,type, is_a_theorem: mu > $i > $o ). thf(not_type,type, not: mu > mu ). thf(existence_of_not_ax,axiom, ! [V: $i,V1: mu] : ( exists_in_world @ ( not @ V1 ) @ V ) ). thf(or_type,type, or: mu > mu > mu ). thf(existence_of_or_ax,axiom, ! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( or @ V2 @ V1 ) @ V ) ). thf(implies_type,type, implies: mu > mu > mu ). thf(existence_of_implies_ax,axiom, ! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( implies @ V2 @ V1 ) @ V ) ). thf(and_type,type, and: mu > mu > mu ). thf(existence_of_and_ax,axiom, ! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( and @ V2 @ V1 ) @ V ) ). thf(equiv_type,type, equiv: mu > mu > mu ). thf(existence_of_equiv_ax,axiom, ! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( equiv @ V2 @ V1 ) @ V ) ). thf(reflexivity,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( qmltpeq @ X @ X ) ) ) ) ) ). thf(symmetry,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ X @ Y ) ) @ ( mbox_s4 @ ( qmltpeq @ Y @ X ) ) ) ) ) ) ) ) ) ). thf(transitivity,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Z: mu] : ( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( qmltpeq @ X @ Y ) ) @ ( mbox_s4 @ ( qmltpeq @ Y @ Z ) ) ) @ ( mbox_s4 @ ( qmltpeq @ X @ Z ) ) ) ) ) ) ) ) ) ) ) ). thf(and_substitution_1,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [A: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [B: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( and @ A @ C ) @ ( and @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ). thf(and_substitution_2,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [A: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [B: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( and @ C @ A ) @ ( and @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ). thf(equiv_substitution_1,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [A: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [B: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( equiv @ A @ C ) @ ( equiv @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ). thf(equiv_substitution_2,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [A: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [B: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( equiv @ C @ A ) @ ( equiv @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ). thf(implies_substitution_1,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [A: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [B: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( implies @ A @ C ) @ ( implies @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ). thf(implies_substitution_2,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [A: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [B: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( implies @ C @ A ) @ ( implies @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ). thf(not_substitution_1,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [A: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [B: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( not @ A ) @ ( not @ B ) ) ) ) ) ) ) ) ) ) ). thf(or_substitution_1,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [A: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [B: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( or @ A @ C ) @ ( or @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ). thf(or_substitution_2,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [A: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [B: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [C: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( qmltpeq @ ( or @ C @ A ) @ ( or @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ). thf(is_a_theorem_substitution_1,axiom, ( mvalid @ ( mbox_s4 @ ( mforall_ind @ ^ [A: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [B: mu] : ( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( qmltpeq @ A @ B ) ) @ ( mbox_s4 @ ( is_a_theorem @ A ) ) ) @ ( mbox_s4 @ ( is_a_theorem @ B ) ) ) ) ) ) ) ) ) ). thf(modus_ponens,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ modus_ponens ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [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 ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [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 ) ) ) ) ) ) ) ) @ ( mbox_s4 @ modus_ponens ) ) ) ) ) ). thf(substitution_of_equivalents,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ substitution_of_equivalents ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( is_a_theorem @ ( equiv @ X @ Y ) ) ) @ ( mbox_s4 @ ( qmltpeq @ X @ Y ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( is_a_theorem @ ( equiv @ X @ Y ) ) ) @ ( mbox_s4 @ ( qmltpeq @ X @ Y ) ) ) ) ) ) ) ) @ ( mbox_s4 @ substitution_of_equivalents ) ) ) ) ) ). thf(modus_tollens,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ modus_tollens ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ ( not @ Y ) @ ( not @ X ) ) @ ( implies @ X @ Y ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ ( not @ Y ) @ ( not @ X ) ) @ ( implies @ X @ Y ) ) ) ) ) ) ) ) @ ( mbox_s4 @ modus_tollens ) ) ) ) ) ). thf(implies_1,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ implies_1 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ ( implies @ Y @ X ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ ( implies @ Y @ X ) ) ) ) ) ) ) ) @ ( mbox_s4 @ implies_1 ) ) ) ) ) ). thf(implies_2,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ implies_2 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ ( implies @ X @ Y ) ) @ ( implies @ X @ Y ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ ( implies @ X @ Y ) ) @ ( implies @ X @ Y ) ) ) ) ) ) ) ) @ ( mbox_s4 @ implies_2 ) ) ) ) ) ). thf(implies_3,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ implies_3 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Z: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ Y ) @ ( implies @ ( implies @ Y @ Z ) @ ( implies @ X @ Z ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Z: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ Y ) @ ( implies @ ( implies @ Y @ Z ) @ ( implies @ X @ Z ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ implies_3 ) ) ) ) ) ). thf(and_1,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ and_1 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( and @ X @ Y ) @ X ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( and @ X @ Y ) @ X ) ) ) ) ) ) ) @ ( mbox_s4 @ and_1 ) ) ) ) ) ). thf(and_2,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ and_2 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( and @ X @ Y ) @ Y ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( and @ X @ Y ) @ Y ) ) ) ) ) ) ) @ ( mbox_s4 @ and_2 ) ) ) ) ) ). thf(and_3,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ and_3 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ ( implies @ Y @ ( and @ X @ Y ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ ( implies @ Y @ ( and @ X @ Y ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ and_3 ) ) ) ) ) ). thf(or_1,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ or_1 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ ( or @ X @ Y ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ X @ ( or @ X @ Y ) ) ) ) ) ) ) ) @ ( mbox_s4 @ or_1 ) ) ) ) ) ). thf(or_2,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ or_2 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ Y @ ( or @ X @ Y ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ Y @ ( or @ X @ Y ) ) ) ) ) ) ) ) @ ( mbox_s4 @ or_2 ) ) ) ) ) ). thf(or_3,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ or_3 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Z: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ Z ) @ ( implies @ ( implies @ Y @ Z ) @ ( implies @ ( or @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Z: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ Z ) @ ( implies @ ( implies @ Y @ Z ) @ ( implies @ ( or @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ or_3 ) ) ) ) ) ). thf(equivalence_1,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ equivalence_1 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( equiv @ X @ Y ) @ ( implies @ X @ Y ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( equiv @ X @ Y ) @ ( implies @ X @ Y ) ) ) ) ) ) ) ) @ ( mbox_s4 @ equivalence_1 ) ) ) ) ) ). thf(equivalence_2,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ equivalence_2 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( equiv @ X @ Y ) @ ( implies @ Y @ X ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( equiv @ X @ Y ) @ ( implies @ Y @ X ) ) ) ) ) ) ) ) @ ( mbox_s4 @ equivalence_2 ) ) ) ) ) ). thf(equivalence_3,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ equivalence_3 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ Y ) @ ( implies @ ( implies @ Y @ X ) @ ( equiv @ X @ Y ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ X @ Y ) @ ( implies @ ( implies @ Y @ X ) @ ( equiv @ X @ Y ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ equivalence_3 ) ) ) ) ) ). thf(kn1,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ kn1 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ P @ ( and @ P @ P ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ P @ ( and @ P @ P ) ) ) ) ) ) @ ( mbox_s4 @ kn1 ) ) ) ) ) ). thf(kn2,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ kn2 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( and @ P @ Q ) @ P ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( and @ P @ Q ) @ P ) ) ) ) ) ) ) @ ( mbox_s4 @ kn2 ) ) ) ) ) ). thf(kn3,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ kn3 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ P @ Q ) @ ( implies @ ( not @ ( and @ Q @ R ) ) @ ( not @ ( and @ R @ P ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ P @ Q ) @ ( implies @ ( not @ ( and @ Q @ R ) ) @ ( not @ ( and @ R @ P ) ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ kn3 ) ) ) ) ) ). thf(cn1,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ cn1 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ P @ Q ) @ ( implies @ ( implies @ Q @ R ) @ ( implies @ P @ R ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ P @ Q ) @ ( implies @ ( implies @ Q @ R ) @ ( implies @ P @ R ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ cn1 ) ) ) ) ) ). thf(cn2,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ cn2 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ P @ ( implies @ ( not @ P ) @ Q ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ P @ ( implies @ ( not @ P ) @ Q ) ) ) ) ) ) ) ) @ ( mbox_s4 @ cn2 ) ) ) ) ) ). thf(cn3,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ cn3 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ ( not @ P ) @ P ) @ P ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ ( not @ P ) @ P ) @ P ) ) ) ) ) @ ( mbox_s4 @ cn3 ) ) ) ) ) ). thf(r1,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ r1 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( or @ P @ P ) @ P ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( or @ P @ P ) @ P ) ) ) ) ) @ ( mbox_s4 @ r1 ) ) ) ) ) ). thf(r2,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ r2 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ Q @ ( or @ P @ Q ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ Q @ ( or @ P @ Q ) ) ) ) ) ) ) ) @ ( mbox_s4 @ r2 ) ) ) ) ) ). thf(r3,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ r3 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( or @ P @ Q ) @ ( or @ Q @ P ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( or @ P @ Q ) @ ( or @ Q @ P ) ) ) ) ) ) ) ) @ ( mbox_s4 @ r3 ) ) ) ) ) ). thf(r4,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ r4 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( or @ P @ ( or @ Q @ R ) ) @ ( or @ Q @ ( or @ P @ R ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( or @ P @ ( or @ Q @ R ) ) @ ( or @ Q @ ( or @ P @ R ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ r4 ) ) ) ) ) ). thf(r5,axiom, ( mvalid @ ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ r5 ) @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ Q @ R ) @ ( implies @ ( or @ P @ Q ) @ ( or @ P @ R ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( mforall_ind @ ^ [P: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Q: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [R: mu] : ( mbox_s4 @ ( is_a_theorem @ ( implies @ ( implies @ Q @ R ) @ ( implies @ ( or @ P @ Q ) @ ( or @ P @ R ) ) ) ) ) ) ) ) ) ) ) @ ( mbox_s4 @ r5 ) ) ) ) ) ). thf(op_or,axiom, ( mvalid @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ op_or ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( qmltpeq @ ( or @ X @ Y ) @ ( not @ ( and @ ( not @ X ) @ ( not @ Y ) ) ) ) ) ) ) ) ) ) ) ) ). thf(op_and,axiom, ( mvalid @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ op_and ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( qmltpeq @ ( and @ X @ Y ) @ ( not @ ( or @ ( not @ X ) @ ( not @ Y ) ) ) ) ) ) ) ) ) ) ) ) ). thf(op_implies_and,axiom, ( mvalid @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ op_implies_and ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( qmltpeq @ ( implies @ X @ Y ) @ ( not @ ( and @ X @ ( not @ Y ) ) ) ) ) ) ) ) ) ) ) ) ). thf(op_implies_or,axiom, ( mvalid @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ op_implies_or ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( qmltpeq @ ( implies @ X @ Y ) @ ( or @ ( not @ X ) @ Y ) ) ) ) ) ) ) ) ) ) ). thf(op_equiv,axiom, ( mvalid @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ op_equiv ) @ ( mbox_s4 @ ( mforall_ind @ ^ [X: mu] : ( mbox_s4 @ ( mforall_ind @ ^ [Y: mu] : ( mbox_s4 @ ( qmltpeq @ ( equiv @ X @ Y ) @ ( and @ ( implies @ X @ Y ) @ ( implies @ Y @ X ) ) ) ) ) ) ) ) ) ) ) ). thf(hilbert_op_or,axiom, mvalid @ ( mbox_s4 @ op_or ) ). thf(hilbert_op_implies_and,axiom, mvalid @ ( mbox_s4 @ op_implies_and ) ). thf(hilbert_op_equiv,axiom, mvalid @ ( mbox_s4 @ op_equiv ) ). thf(hilbert_modus_ponens,axiom, mvalid @ ( mbox_s4 @ modus_ponens ) ). thf(hilbert_modus_tollens,axiom, mvalid @ ( mbox_s4 @ modus_tollens ) ). thf(hilbert_implies_1,axiom, mvalid @ ( mbox_s4 @ implies_1 ) ). thf(hilbert_implies_2,axiom, mvalid @ ( mbox_s4 @ implies_2 ) ). thf(hilbert_implies_3,axiom, mvalid @ ( mbox_s4 @ implies_3 ) ). thf(hilbert_and_1,axiom, mvalid @ ( mbox_s4 @ and_1 ) ). thf(hilbert_and_2,axiom, mvalid @ ( mbox_s4 @ and_2 ) ). thf(hilbert_and_3,axiom, mvalid @ ( mbox_s4 @ and_3 ) ). thf(hilbert_or_1,axiom, mvalid @ ( mbox_s4 @ or_1 ) ). thf(hilbert_or_2,axiom, mvalid @ ( mbox_s4 @ or_2 ) ). thf(hilbert_or_3,axiom, mvalid @ ( mbox_s4 @ or_3 ) ). thf(hilbert_equivalence_1,axiom, mvalid @ ( mbox_s4 @ equivalence_1 ) ). thf(hilbert_equivalence_2,axiom, mvalid @ ( mbox_s4 @ equivalence_2 ) ). thf(hilbert_equivalence_3,axiom, mvalid @ ( mbox_s4 @ equivalence_3 ) ). thf(substitution_of_equivalents_001,axiom, mvalid @ ( mbox_s4 @ substitution_of_equivalents ) ). thf(rosser_op_or,axiom, mvalid @ ( mbox_s4 @ op_or ) ). thf(rosser_op_implies_and,axiom, mvalid @ ( mbox_s4 @ op_implies_and ) ). thf(rosser_op_equiv,axiom, mvalid @ ( mbox_s4 @ op_equiv ) ). thf(rosser_kn2,conjecture, mvalid @ ( mbox_s4 @ kn2 ) ). %------------------------------------------------------------------------------