TPTP Problem File: ALG018^7.p
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%------------------------------------------------------------------------------
% File : ALG018^7 : TPTP v8.2.0. Released v5.5.0.
% Domain : General Algebra
% Problem : Groups 4: CPROPS-SORTED-DISCRIMINANT-PROBLEM-1
% Version : [Ben12] axioms.
% English :
% Refs : [Goe69] Goedel (1969), An Interpretation of the Intuitionistic
% : [CM+04] Colton et al. (2004), Automatic Generation of Classifi
% : [Ben12] Benzmueller (2012), Email to Geoff Sutcliffe
% Source : [Ben12]
% Names : s4-cumul-GAL018+1 [Ben12]
% Status : Theorem
% Rating : 1.00 v5.5.0
% Syntax : Number of formulae : 98 ( 37 unt; 42 typ; 32 def)
% Number of atoms : 371 ( 36 equ; 0 cnn)
% Maximal formula atoms : 69 ( 6 avg)
% Number of connectives : 529 ( 5 ~; 5 |; 9 &; 500 @)
% ( 0 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 190 ( 190 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 50 usr; 10 con; 0-3 aty)
% Number of variables : 142 ( 91 ^; 44 !; 7 ?; 142 :)
% SPC : TH0_THM_EQU_NAR
% Comments : Goedel translation of ALG018+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(sorti2_type,type,
sorti2: mu > $i > $o ).
thf(sorti1_type,type,
sorti1: mu > $i > $o ).
thf(op2_type,type,
op2: mu > mu > mu ).
thf(existence_of_op2_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( op2 @ V2 @ V1 ) @ V ) ).
thf(op1_type,type,
op1: mu > mu > mu ).
thf(existence_of_op1_ax,axiom,
! [V: $i,V2: mu,V1: mu] : ( exists_in_world @ ( op1 @ V2 @ V1 ) @ V ) ).
thf(j_type,type,
j: mu > mu ).
thf(existence_of_j_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( j @ V1 ) @ V ) ).
thf(h_type,type,
h: mu > mu ).
thf(existence_of_h_ax,axiom,
! [V: $i,V1: mu] : ( exists_in_world @ ( h @ 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(h_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 @ ( h @ A ) @ ( h @ B ) ) ) ) ) ) ) ) ) ) ).
thf(j_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 @ ( j @ A ) @ ( j @ B ) ) ) ) ) ) ) ) ) ) ).
thf(op1_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 @ ( op1 @ A @ C ) @ ( op1 @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(op1_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 @ ( op1 @ C @ A ) @ ( op1 @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(op2_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 @ ( op2 @ A @ C ) @ ( op2 @ B @ C ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(op2_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 @ ( op2 @ C @ A ) @ ( op2 @ C @ B ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(sorti1_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 @ ( sorti1 @ A ) ) ) @ ( mbox_s4 @ ( sorti1 @ B ) ) ) ) ) ) ) ) ) ).
thf(sorti2_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 @ ( sorti2 @ A ) ) ) @ ( mbox_s4 @ ( sorti2 @ B ) ) ) ) ) ) ) ) ) ).
thf(ax1,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [U: mu] :
( mbox_s4
@ ( mimplies @ ( mbox_s4 @ ( sorti1 @ U ) )
@ ( mbox_s4
@ ( mforall_ind
@ ^ [V: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( sorti1 @ V ) ) @ ( mbox_s4 @ ( sorti1 @ ( op1 @ U @ V ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax2,axiom,
( mvalid
@ ( mbox_s4
@ ( mforall_ind
@ ^ [U: mu] :
( mbox_s4
@ ( mimplies @ ( mbox_s4 @ ( sorti2 @ U ) )
@ ( mbox_s4
@ ( mforall_ind
@ ^ [V: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( sorti2 @ V ) ) @ ( mbox_s4 @ ( sorti2 @ ( op2 @ U @ V ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax3,axiom,
( mvalid
@ ( mexists_ind
@ ^ [U: mu] :
( mand @ ( mbox_s4 @ ( sorti1 @ U ) )
@ ( mbox_s4
@ ( mforall_ind
@ ^ [V: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( sorti1 @ V ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op1 @ V @ V ) @ U ) ) ) ) ) ) ) ) ) ).
thf(ax4,axiom,
( mvalid
@ ( mbox_s4
@ ( mnot
@ ( mexists_ind
@ ^ [U: mu] :
( mand @ ( mbox_s4 @ ( sorti2 @ U ) )
@ ( mbox_s4
@ ( mforall_ind
@ ^ [V: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( sorti2 @ V ) ) @ ( mbox_s4 @ ( qmltpeq @ ( op2 @ V @ V ) @ U ) ) ) ) ) ) ) ) ) ) ) ).
thf(co1,conjecture,
( mvalid
@ ( mbox_s4
@ ( mimplies
@ ( mand
@ ( mbox_s4
@ ( mforall_ind
@ ^ [U: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( sorti1 @ U ) ) @ ( mbox_s4 @ ( sorti2 @ ( h @ U ) ) ) ) ) ) )
@ ( mbox_s4
@ ( mforall_ind
@ ^ [V: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( sorti2 @ V ) ) @ ( mbox_s4 @ ( sorti1 @ ( j @ V ) ) ) ) ) ) ) )
@ ( mbox_s4
@ ( mnot
@ ( mand
@ ( mbox_s4
@ ( mforall_ind
@ ^ [W: mu] :
( mbox_s4
@ ( mimplies @ ( mbox_s4 @ ( sorti1 @ W ) )
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( sorti1 @ X ) ) @ ( mbox_s4 @ ( qmltpeq @ ( h @ ( op1 @ W @ X ) ) @ ( op2 @ ( h @ W ) @ ( h @ X ) ) ) ) ) ) ) ) ) ) ) )
@ ( mand
@ ( mbox_s4
@ ( mforall_ind
@ ^ [Y: mu] :
( mbox_s4
@ ( mimplies @ ( mbox_s4 @ ( sorti2 @ Y ) )
@ ( mbox_s4
@ ( mforall_ind
@ ^ [Z: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( sorti2 @ Z ) ) @ ( mbox_s4 @ ( qmltpeq @ ( j @ ( op2 @ Y @ Z ) ) @ ( op1 @ ( j @ Y ) @ ( j @ Z ) ) ) ) ) ) ) ) ) ) ) )
@ ( mand
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X1: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( sorti2 @ X1 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( h @ ( j @ X1 ) ) @ X1 ) ) ) ) ) )
@ ( mbox_s4
@ ( mforall_ind
@ ^ [X2: mu] : ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( sorti1 @ X2 ) ) @ ( mbox_s4 @ ( qmltpeq @ ( j @ ( h @ X2 ) ) @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
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