TPTP Problem File: LCL449^7.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL449^7 : TPTP v8.2.0. Released v5.5.0.
% Domain : Logic Calculi
% Problem : Congruence of equiv, to admit substitution of equivalents
% 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
% : [Rab06] Rabe (2006), Towards Determining the Subset Relation b
% : [Ben12] Benzmueller (2012), Email to Geoff Sutcliffe
% Source : [Ben12]
% Names : s4-cumul-GLC449+1 [Ben12]
% Status : Theorem
% Rating : 1.00 v5.5.0
% Syntax : Number of formulae : 177 ( 38 unt; 73 typ; 32 def)
% Number of atoms : 1055 ( 36 equ; 0 cnn)
% Maximal formula atoms : 60 ( 10 avg)
% Number of connectives : 1602 ( 5 ~; 5 |; 9 &;1573 @)
% ( 0 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 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 : 271 ( 216 ^; 48 !; 7 ?; 271 :)
% SPC : TH0_THM_EQU_NAR
% Comments : Goedel translation of LCL449+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(substitution_of_equivalents_type,type,
substitution_of_equivalents: $i > $o ).
thf(kn1_type,type,
kn1: $i > $o ).
thf(kn2_type,type,
kn2: $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(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(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(is_a_theorem_type,type,
is_a_theorem: mu > $i > $o ).
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(not_type,type,
not: mu > mu ).
thf(existence_of_not_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( not @ 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(make_subs_of_equiv,axiom,
( mvalid
@ ( mbox_s4
@ ( mimplies
@ ( mand
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X: mu] : ( mbox_s4 @ ( is_a_theorem @ ( equiv @ X @ X ) ) ) ) )
@ ( mand
@ ( 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 @ ( is_a_theorem @ ( equiv @ ( not @ X ) @ ( not @ Y ) ) ) ) ) ) ) ) ) )
@ ( mand
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X1: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [X2: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [Y1: mu] :
( mbox_s4
@ ( mforall_ind
@ ^ [Y2: mu] : ( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( is_a_theorem @ ( equiv @ X1 @ X2 ) ) ) @ ( mbox_s4 @ ( is_a_theorem @ ( equiv @ Y1 @ Y2 ) ) ) ) @ ( mbox_s4 @ ( is_a_theorem @ ( equiv @ ( and @ X1 @ Y1 ) @ ( and @ X2 @ Y2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( 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 @ ( equiv @ X @ Y ) ) ) ) @ ( mbox_s4 @ ( is_a_theorem @ Y ) ) ) ) ) ) ) ) ) ) )
@ ( 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 ) ) ) ) ) ) ) ) ) ) ) ).
thf(subs_of_equiv,conjecture,
( mvalid
@ ( 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 ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------