TPTP Problem File: LCL968_3.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : LCL968_3 : TPTP v8.2.0. Released v8.2.0.
% Domain : Software Verification
% Problem : Goedel translation of SWV107+1
% Version : [QMLTP] axioms.
% English :
% Refs : [Goe69] Goedel (1969), An Interpretation of the Intuitionistic
% : [RO12] Raths & Otten (2012), The QMLTP Problem Library for Fi
% Source : [QMLTP]
% Names : GSV107+1 [QMLTP]
% : GSV107_1.001_k_constant_rigid.p [QMLTP/4_NX0_SEMs]
% Status : CounterSatisfiable
% Rating : 0.67 v8.2.0
% Syntax : Number of formulae : 194 ( 3 unt; 40 typ; 0 def)
% Number of atoms : 1583 ( 0 equ)
% Maximal formula atoms : 77 ( 10 avg)
% Number of connectives : 1434 ( 5 ~; 23 |; 109 &)
% ( 0 <=>;1297 =>; 0 <=; 0 <~>)
% Maximal formula depth : 54 ( 14 avg)
% Maximal term depth : 9 ( 1 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 58 ( 26 >; 32 *; 0 +; 0 <<)
% Number of predicates : 8 ( 8 usr; 0 prp; 1-3 aty)
% Number of functors : 31 ( 31 usr; 13 con; 0-4 aty)
% Number of variables : 1186 (;1184 !; 2 ?;1186 :)
% SPC : TF0_CSA_NEQ_NAR
% Comments : Generated by embedproblem, version 1.7.14, rigid constant,
% modal_system_D, TFF embedding.
%------------------------------------------------------------------------------
tff('$ki_world_type',type,
'$ki_world': $tType ).
tff('$ki_local_world_decl',type,
'$ki_local_world': '$ki_world' ).
tff('$ki_accessible_decl',type,
'$ki_accessible': ( '$ki_world' * '$ki_world' ) > $o ).
tff(mrel_serial,axiom,
! [W: '$ki_world'] :
? [V: '$ki_world'] : '$ki_accessible'(W,V) ).
tff(typedecl_customqmltpeq,type,
customqmltpeq: ( '$ki_world' * $i * $i ) > $o ).
tff(true_decl,type,
true: '$ki_world' > $o ).
tff(def_decl,type,
def: $i ).
tff(tptp_float_0_0_decl,type,
tptp_float_0_0: $i ).
tff(tptp_minus_1_decl,type,
tptp_minus_1: $i ).
tff(n0_decl,type,
n0: $i ).
tff(n1_decl,type,
n1: $i ).
tff(n2_decl,type,
n2: $i ).
tff(n3_decl,type,
n3: $i ).
tff(n4_decl,type,
n4: $i ).
tff(use_decl,type,
use: $i ).
tff(n5_decl,type,
n5: $i ).
tff(n6_decl,type,
n6: $i ).
tff(n999_decl,type,
n999: $i ).
tff(geq_decl,type,
geq: ( '$ki_world' * $i * $i ) > $o ).
tff(lt_decl,type,
lt: ( '$ki_world' * $i * $i ) > $o ).
tff(leq_decl,type,
leq: ( '$ki_world' * $i * $i ) > $o ).
tff(gt_decl,type,
gt: ( '$ki_world' * $i * $i ) > $o ).
tff(minus_decl,type,
minus: ( $i * $i ) > $i ).
tff(dim_decl,type,
dim: ( $i * $i ) > $i ).
tff(sum_decl,type,
sum: ( $i * $i * $i ) > $i ).
tff(plus_decl,type,
plus: ( $i * $i ) > $i ).
tff(inv_decl,type,
inv: $i > $i ).
tff(uniform_int_rnd_decl,type,
uniform_int_rnd: ( $i * $i ) > $i ).
tff(succ_decl,type,
succ: $i > $i ).
tff(pred_decl,type,
pred: $i > $i ).
tff(tptp_update3_decl,type,
tptp_update3: ( $i * $i * $i * $i ) > $i ).
tff(tptp_update2_decl,type,
tptp_update2: ( $i * $i * $i ) > $i ).
tff(tptp_const_array1_decl,type,
tptp_const_array1: ( $i * $i ) > $i ).
tff(tptp_mmul_decl,type,
tptp_mmul: ( $i * $i ) > $i ).
tff(a_select3_decl,type,
a_select3: ( $i * $i * $i ) > $i ).
tff(tptp_const_array2_decl,type,
tptp_const_array2: ( $i * $i * $i ) > $i ).
tff(a_select2_decl,type,
a_select2: ( $i * $i ) > $i ).
tff(tptp_madd_decl,type,
tptp_madd: ( $i * $i ) > $i ).
tff(tptp_msub_decl,type,
tptp_msub: ( $i * $i ) > $i ).
tff(trans_decl,type,
trans: $i > $i ).
tff('$ki_exists_in_world_$i_decl',type,
'$ki_exists_in_world_$i': ( '$ki_world' * $i ) > $o ).
tff('$ki_exists_in_world_$i_const',axiom,
! [W: '$ki_world',X: $i] : '$ki_exists_in_world_$i'(W,X) ).
tff('$ki_exists_in_world_$i_nonempty',axiom,
! [W: '$ki_world'] :
? [X: $i] : '$ki_exists_in_world_$i'(W,X) ).
tff(reflexivity,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,X,X) ) ) ) ).
tff(symmetry,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> customqmltpeq(W2,X,Y) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> customqmltpeq(W2,Y,X) ) ) ) ) ) ) ) ).
tff(transitivity,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W1,Z)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,X,Y) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,Y,Z) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,X,Z) ) ) ) ) ) ) ) ) ) ).
tff(a_select2_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,a_select2(A,C),a_select2(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(a_select2_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,a_select2(C,A),a_select2(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(a_select3_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,a_select3(A,C,D),a_select3(B,C,D)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(a_select3_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,a_select3(C,A,D),a_select3(C,B,D)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(a_select3_substitution_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,a_select3(C,D,A),a_select3(C,D,B)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(dim_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,dim(A,C),dim(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(dim_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,dim(C,A),dim(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(inv_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> customqmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> customqmltpeq(W2,inv(A),inv(B)) ) ) ) ) ) ) ) ).
tff(minus_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,minus(A,C),minus(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(minus_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,minus(C,A),minus(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(plus_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,plus(A,C),plus(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(plus_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,plus(C,A),plus(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(pred_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> customqmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> customqmltpeq(W2,pred(A),pred(B)) ) ) ) ) ) ) ) ).
tff(succ_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> customqmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> customqmltpeq(W2,succ(A),succ(B)) ) ) ) ) ) ) ) ).
tff(sum_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,sum(A,C,D),sum(B,C,D)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(sum_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,sum(C,A,D),sum(C,B,D)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(sum_substitution_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,sum(C,D,A),sum(C,D,B)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(tptp_const_array1_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,tptp_const_array1(A,C),tptp_const_array1(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(tptp_const_array1_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,tptp_const_array1(C,A),tptp_const_array1(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(tptp_const_array2_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,tptp_const_array2(A,C,D),tptp_const_array2(B,C,D)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(tptp_const_array2_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,tptp_const_array2(C,A,D),tptp_const_array2(C,B,D)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(tptp_const_array2_substitution_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,tptp_const_array2(C,D,A),tptp_const_array2(C,D,B)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(tptp_madd_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,tptp_madd(A,C),tptp_madd(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(tptp_madd_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,tptp_madd(C,A),tptp_madd(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(tptp_mmul_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,tptp_mmul(A,C),tptp_mmul(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(tptp_mmul_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,tptp_mmul(C,A),tptp_mmul(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(tptp_msub_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,tptp_msub(A,C),tptp_msub(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(tptp_msub_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,tptp_msub(C,A),tptp_msub(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(tptp_update2_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,tptp_update2(A,C,D),tptp_update2(B,C,D)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(tptp_update2_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,tptp_update2(C,A,D),tptp_update2(C,B,D)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(tptp_update2_substitution_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,A,B) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,tptp_update2(C,D,A),tptp_update2(C,D,B)) ) ) ) ) ) ) ) ) ) ) ) ).
tff(tptp_update3_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [E: $i] :
( '$ki_exists_in_world_$i'(W3,E)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ( ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,A,B) )
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,tptp_update3(A,C,D,E),tptp_update3(B,C,D,E)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(tptp_update3_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [E: $i] :
( '$ki_exists_in_world_$i'(W3,E)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ( ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,A,B) )
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,tptp_update3(C,A,D,E),tptp_update3(C,B,D,E)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(tptp_update3_substitution_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [E: $i] :
( '$ki_exists_in_world_$i'(W3,E)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ( ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,A,B) )
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,tptp_update3(C,D,A,E),tptp_update3(C,D,B,E)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(tptp_update3_substitution_4,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [E: $i] :
( '$ki_exists_in_world_$i'(W3,E)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ( ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,A,B) )
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,tptp_update3(C,D,E,A),tptp_update3(C,D,E,B)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(trans_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> customqmltpeq(W2,A,B) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> customqmltpeq(W2,trans(A),trans(B)) ) ) ) ) ) ) ) ).
tff(uniform_int_rnd_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,uniform_int_rnd(A,C),uniform_int_rnd(B,C)) ) ) ) ) ) ) ) ) ) ).
tff(uniform_int_rnd_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,uniform_int_rnd(C,A),uniform_int_rnd(C,B)) ) ) ) ) ) ) ) ) ) ).
tff(geq_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> geq(W3,A,C) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> geq(W3,B,C) ) ) ) ) ) ) ) ) ) ).
tff(geq_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> geq(W3,C,A) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> geq(W3,C,B) ) ) ) ) ) ) ) ) ) ).
tff(gt_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> gt(W3,A,C) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> gt(W3,B,C) ) ) ) ) ) ) ) ) ) ).
tff(gt_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> gt(W3,C,A) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> gt(W3,C,B) ) ) ) ) ) ) ) ) ) ).
tff(leq_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> leq(W3,A,C) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> leq(W3,B,C) ) ) ) ) ) ) ) ) ) ).
tff(leq_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> leq(W3,C,A) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> leq(W3,C,B) ) ) ) ) ) ) ) ) ) ).
tff(lt_substitution_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> lt(W3,A,C) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> lt(W3,B,C) ) ) ) ) ) ) ) ) ) ).
tff(lt_substitution_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,A,B) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> lt(W3,C,A) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> lt(W3,C,B) ) ) ) ) ) ) ) ) ) ).
tff(totality,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> gt(W1,X,Y) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> gt(W1,Y,X) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,Y) ) ) ) ) ) ) ).
tff(transitivity_gt,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W1,Z)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> gt(W3,X,Y) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> gt(W3,Y,Z) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> gt(W3,X,Z) ) ) ) ) ) ) ) ) ) ).
tff(irreflexivity_gt,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ~ ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> gt(W1,X,X) ) ) ) ) ).
tff(reflexivity_leq,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> leq(W0,X,X) ) ) ) ).
tff(transitivity_leq,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [Z: $i] :
( '$ki_exists_in_world_$i'(W1,Z)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> leq(W3,X,Y) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> leq(W3,Y,Z) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> leq(W3,X,Z) ) ) ) ) ) ) ) ) ) ).
tff(lt_gt,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> lt(W2,X,Y) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> gt(W2,Y,X) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> gt(W2,Y,X) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> lt(W2,X,Y) ) ) ) ) ) ) ) ) ).
tff(leq_geq,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> geq(W2,X,Y) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,Y,X) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,Y,X) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> geq(W2,X,Y) ) ) ) ) ) ) ) ) ).
tff(leq_gt1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> gt(W2,Y,X) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,X,Y) ) ) ) ) ) ) ) ).
tff(leq_gt2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,X,Y) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ~ ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,X,Y) ) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> gt(W2,Y,X) ) ) ) ) ) ) ) ).
tff(leq_gt_pred,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,X,pred(Y)) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> gt(W2,Y,X) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> gt(W2,Y,X) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,X,pred(Y)) ) ) ) ) ) ) ) ) ).
tff(gt_succ,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> gt(W0,succ(X),X) ) ) ) ).
tff(leq_succ,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,X,Y) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,X,succ(Y)) ) ) ) ) ) ) ) ).
tff(leq_succ_gt_equiv,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,X,Y) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> gt(W2,succ(Y),X) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> gt(W2,succ(Y),X) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,X,Y) ) ) ) ) ) ) ) ) ).
tff(uniform_int_rand_ranges_hi,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W0,C)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,n0,X) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,uniform_int_rnd(C,X),X) ) ) ) ) ) ) ) ).
tff(uniform_int_rand_ranges_lo,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W0,C)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,n0,X) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,n0,uniform_int_rnd(C,X)) ) ) ) ) ) ) ) ).
tff(const_array1_select,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W,I)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [L: $i] :
( '$ki_exists_in_world_$i'(W0,L)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W1,U)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [Val: $i] :
( '$ki_exists_in_world_$i'(W2,Val)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> leq(W4,L,I) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> leq(W4,I,U) ) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,a_select2(tptp_const_array1(dim(L,U),Val),I),Val) ) ) ) ) ) ) ) ) ) ) ) ).
tff(const_array2_select,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W,I)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [L1: $i] :
( '$ki_exists_in_world_$i'(W0,L1)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [U1: $i] :
( '$ki_exists_in_world_$i'(W1,U1)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W2,J)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [L2: $i] :
( '$ki_exists_in_world_$i'(W3,L2)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [U2: $i] :
( '$ki_exists_in_world_$i'(W4,U2)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ! [Val: $i] :
( '$ki_exists_in_world_$i'(W5,Val)
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> ( ( ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,L1,I) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,I,U1) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,L2,J) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,J,U2) ) )
=> ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> customqmltpeq(W7,a_select3(tptp_const_array2(dim(L1,U1),dim(L2,U2),Val),I,J),Val) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(matrix_symm_trans,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [N: $i] :
( '$ki_exists_in_world_$i'(W0,N)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W2,I)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W3,J)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ( ( ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,n0,I) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,I,N) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,n0,J) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,J,N) ) )
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,a_select3(A,I,J),a_select3(A,J,I)) ) ) ) ) ) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W2,I)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W3,J)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ( ( ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,n0,I) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,I,N) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,n0,J) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,J,N) ) )
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,a_select3(trans(A),I,J),a_select3(trans(A),J,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(matrix_symm_inv,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [N: $i] :
( '$ki_exists_in_world_$i'(W0,N)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W2,I)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W3,J)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ( ( ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,n0,I) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,I,N) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,n0,J) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,J,N) ) )
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,a_select3(A,I,J),a_select3(A,J,I)) ) ) ) ) ) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W2,I)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W3,J)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ( ( ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,n0,I) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,I,N) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,n0,J) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,J,N) ) )
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,a_select3(inv(A),I,J),a_select3(inv(A),J,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(matrix_symm_update_diagonal,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [N: $i] :
( '$ki_exists_in_world_$i'(W0,N)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W2,I)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W3,J)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ( ( ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,n0,I) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,I,N) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,n0,J) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> leq(W5,J,N) ) )
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,a_select3(A,I,J),a_select3(A,J,I)) ) ) ) ) ) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W2,I)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W3,J)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [K: $i] :
( '$ki_exists_in_world_$i'(W4,K)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ! [VAL: $i] :
( '$ki_exists_in_world_$i'(W5,VAL)
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> ( ( ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,n0,I) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,I,N) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,n0,J) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,J,N) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,n0,K) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,K,N) ) )
=> ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> customqmltpeq(W7,a_select3(tptp_update3(A,K,K,VAL),I,J),a_select3(tptp_update3(A,K,K,VAL),J,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(matrix_symm_add,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [N: $i] :
( '$ki_exists_in_world_$i'(W1,N)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W3,I)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W4,J)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ( ( ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,I) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,I,N) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,J) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,J,N) ) )
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> customqmltpeq(W6,a_select3(A,I,J),a_select3(A,J,I)) ) ) ) ) ) ) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W3,I)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W4,J)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ( ( ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,I) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,I,N) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,J) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,J,N) ) )
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> customqmltpeq(W6,a_select3(B,I,J),a_select3(B,J,I)) ) ) ) ) ) ) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W3,I)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W4,J)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ( ( ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,I) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,I,N) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,J) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,J,N) ) )
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> customqmltpeq(W6,a_select3(tptp_madd(A,B),I,J),a_select3(tptp_madd(A,B),J,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(matrix_symm_sub,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [N: $i] :
( '$ki_exists_in_world_$i'(W1,N)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W3,I)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W4,J)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ( ( ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,I) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,I,N) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,J) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,J,N) ) )
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> customqmltpeq(W6,a_select3(A,I,J),a_select3(A,J,I)) ) ) ) ) ) ) )
& ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W3,I)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W4,J)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ( ( ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,I) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,I,N) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,J) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,J,N) ) )
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> customqmltpeq(W6,a_select3(B,I,J),a_select3(B,J,I)) ) ) ) ) ) ) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W3,I)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W4,J)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ( ( ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,I) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,I,N) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,J) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,J,N) ) )
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> customqmltpeq(W6,a_select3(tptp_msub(A,B),I,J),a_select3(tptp_msub(A,B),J,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(matrix_symm_aba1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [N: $i] :
( '$ki_exists_in_world_$i'(W1,N)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W3,I)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W4,J)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ( ( ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,I) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,I,N) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,J) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,J,N) ) )
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> customqmltpeq(W6,a_select3(B,I,J),a_select3(B,J,I)) ) ) ) ) ) ) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W3,I)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W4,J)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ( ( ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,I) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,I,N) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,J) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,J,N) ) )
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> customqmltpeq(W6,a_select3(tptp_mmul(A,tptp_mmul(B,trans(A))),I,J),a_select3(tptp_mmul(A,tptp_mmul(B,trans(A))),J,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(matrix_symm_aba2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [N: $i] :
( '$ki_exists_in_world_$i'(W1,N)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [M: $i] :
( '$ki_exists_in_world_$i'(W2,M)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W4,I)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W5,J)
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> ( ( ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,n0,I) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,I,M) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,n0,J) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,J,M) ) )
=> ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> customqmltpeq(W7,a_select3(B,I,J),a_select3(B,J,I)) ) ) ) ) ) ) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W4,I)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W5,J)
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> ( ( ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,n0,I) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,I,N) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,n0,J) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> leq(W7,J,N) ) )
=> ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> customqmltpeq(W7,a_select3(tptp_mmul(A,tptp_mmul(B,trans(A))),I,J),a_select3(tptp_mmul(A,tptp_mmul(B,trans(A))),J,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(matrix_symm_joseph_update,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [A: $i] :
( '$ki_exists_in_world_$i'(W,A)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [B: $i] :
( '$ki_exists_in_world_$i'(W0,B)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [C: $i] :
( '$ki_exists_in_world_$i'(W1,C)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [D: $i] :
( '$ki_exists_in_world_$i'(W2,D)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [E: $i] :
( '$ki_exists_in_world_$i'(W3,E)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [F: $i] :
( '$ki_exists_in_world_$i'(W4,F)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ! [N: $i] :
( '$ki_exists_in_world_$i'(W5,N)
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> ! [M: $i] :
( '$ki_exists_in_world_$i'(W6,M)
=> ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> ( ( ! [W8: '$ki_world'] :
( '$ki_accessible'(W7,W8)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W8,I)
=> ! [W9: '$ki_world'] :
( '$ki_accessible'(W8,W9)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W9,J)
=> ! [W10: '$ki_world'] :
( '$ki_accessible'(W9,W10)
=> ( ( ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,n0,I) )
& ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,I,M) )
& ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,n0,J) )
& ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,J,M) ) )
=> ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> customqmltpeq(W11,a_select3(D,I,J),a_select3(D,J,I)) ) ) ) ) ) ) )
& ! [W8: '$ki_world'] :
( '$ki_accessible'(W7,W8)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W8,I)
=> ! [W9: '$ki_world'] :
( '$ki_accessible'(W8,W9)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W9,J)
=> ! [W10: '$ki_world'] :
( '$ki_accessible'(W9,W10)
=> ( ( ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,n0,I) )
& ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,I,N) )
& ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,n0,J) )
& ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,J,N) ) )
=> ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> customqmltpeq(W11,a_select3(A,I,J),a_select3(A,J,I)) ) ) ) ) ) ) )
& ! [W8: '$ki_world'] :
( '$ki_accessible'(W7,W8)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W8,I)
=> ! [W9: '$ki_world'] :
( '$ki_accessible'(W8,W9)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W9,J)
=> ! [W10: '$ki_world'] :
( '$ki_accessible'(W9,W10)
=> ( ( ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,n0,I) )
& ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,I,N) )
& ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,n0,J) )
& ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,J,N) ) )
=> ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> customqmltpeq(W11,a_select3(F,I,J),a_select3(F,J,I)) ) ) ) ) ) ) ) )
=> ! [W8: '$ki_world'] :
( '$ki_accessible'(W7,W8)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W8,I)
=> ! [W9: '$ki_world'] :
( '$ki_accessible'(W8,W9)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W9,J)
=> ! [W10: '$ki_world'] :
( '$ki_accessible'(W9,W10)
=> ( ( ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,n0,I) )
& ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,I,N) )
& ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,n0,J) )
& ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> leq(W11,J,N) ) )
=> ! [W11: '$ki_world'] :
( '$ki_accessible'(W10,W11)
=> customqmltpeq(W11,a_select3(tptp_madd(A,tptp_mmul(B,tptp_mmul(tptp_madd(tptp_mmul(C,tptp_mmul(D,trans(C))),tptp_mmul(E,tptp_mmul(F,trans(E)))),trans(B)))),I,J),a_select3(tptp_madd(A,tptp_mmul(B,tptp_mmul(tptp_madd(tptp_mmul(C,tptp_mmul(D,trans(C))),tptp_mmul(E,tptp_mmul(F,trans(E)))),trans(B)))),J,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(sum_plus_base,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [Body: $i] :
( '$ki_exists_in_world_$i'(W,Body)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,sum(n0,tptp_minus_1,Body),n0) ) ) ) ).
tff(sum_plus_base_float,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [Body: $i] :
( '$ki_exists_in_world_$i'(W,Body)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,tptp_float_0_0,sum(n0,tptp_minus_1,Body)) ) ) ) ).
tff(succ_tptp_minus_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> customqmltpeq(W,succ(tptp_minus_1),n0) ) ).
tff(succ_plus_1_r,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,plus(X,n1),succ(X)) ) ) ) ).
tff(succ_plus_1_l,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,plus(n1,X),succ(X)) ) ) ) ).
tff(succ_plus_2_r,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,plus(X,n2),succ(succ(X))) ) ) ) ).
tff(succ_plus_2_l,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,plus(n2,X),succ(succ(X))) ) ) ) ).
tff(succ_plus_3_r,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,plus(X,n3),succ(succ(succ(X)))) ) ) ) ).
tff(succ_plus_3_l,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,plus(n3,X),succ(succ(succ(X)))) ) ) ) ).
tff(succ_plus_4_r,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,plus(X,n4),succ(succ(succ(succ(X))))) ) ) ) ).
tff(succ_plus_4_l,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,plus(n4,X),succ(succ(succ(succ(X))))) ) ) ) ).
tff(succ_plus_5_r,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,plus(X,n5),succ(succ(succ(succ(succ(X)))))) ) ) ) ).
tff(succ_plus_5_l,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,plus(n5,X),succ(succ(succ(succ(succ(X)))))) ) ) ) ).
tff(pred_minus_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,minus(X,n1),pred(X)) ) ) ) ).
tff(pred_succ,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,pred(succ(X)),X) ) ) ) ).
tff(succ_pred,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,succ(pred(X)),X) ) ) ) ).
tff(leq_succ_succ,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,succ(X),succ(Y)) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,X,Y) ) ) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,X,Y) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,succ(X),succ(Y)) ) ) ) ) ) ) ) ) ).
tff(leq_succ_gt,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,succ(X),Y) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> gt(W2,Y,X) ) ) ) ) ) ) ) ).
tff(leq_minus,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [Y: $i] :
( '$ki_exists_in_world_$i'(W0,Y)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,minus(X,Y),X) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> leq(W2,n0,Y) ) ) ) ) ) ) ) ).
tff(sel3_update_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W0,U)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [V: $i] :
( '$ki_exists_in_world_$i'(W1,V)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [VAL: $i] :
( '$ki_exists_in_world_$i'(W2,VAL)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> customqmltpeq(W3,a_select3(tptp_update3(X,U,V,VAL),U,V),VAL) ) ) ) ) ) ) ) ) ) ).
tff(sel3_update_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W,I)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W0,J)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W1,U)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [V: $i] :
( '$ki_exists_in_world_$i'(W2,V)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W3,X)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [VAL: $i] :
( '$ki_exists_in_world_$i'(W4,VAL)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ! [VAL2: $i] :
( '$ki_exists_in_world_$i'(W5,VAL2)
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> ( ( ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> ~ ! [W8: '$ki_world'] :
( '$ki_accessible'(W7,W8)
=> customqmltpeq(W8,I,U) ) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> customqmltpeq(W7,J,V) )
& ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> customqmltpeq(W7,a_select3(X,U,V),VAL) ) )
=> ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> customqmltpeq(W7,a_select3(tptp_update3(X,I,J,VAL2),U,V),VAL) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(sel3_update_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W,I)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [J: $i] :
( '$ki_exists_in_world_$i'(W0,J)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W1,U)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [V: $i] :
( '$ki_exists_in_world_$i'(W2,V)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W3,X)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [VAL: $i] :
( '$ki_exists_in_world_$i'(W4,VAL)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ( ( ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> ! [I0: $i] :
( '$ki_exists_in_world_$i'(W6,I0)
=> ! [W7: '$ki_world'] :
( '$ki_accessible'(W6,W7)
=> ! [J0: $i] :
( '$ki_exists_in_world_$i'(W7,J0)
=> ! [W8: '$ki_world'] :
( '$ki_accessible'(W7,W8)
=> ( ( ! [W9: '$ki_world'] :
( '$ki_accessible'(W8,W9)
=> leq(W9,n0,I0) )
& ! [W9: '$ki_world'] :
( '$ki_accessible'(W8,W9)
=> leq(W9,n0,J0) )
& ! [W9: '$ki_world'] :
( '$ki_accessible'(W8,W9)
=> leq(W9,I0,U) )
& ! [W9: '$ki_world'] :
( '$ki_accessible'(W8,W9)
=> leq(W9,J0,V) ) )
=> ! [W9: '$ki_world'] :
( '$ki_accessible'(W8,W9)
=> customqmltpeq(W9,a_select3(X,I0,J0),VAL) ) ) ) ) ) ) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,I) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,I,U) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,J) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,J,V) ) )
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> customqmltpeq(W6,a_select3(tptp_update3(X,U,V,VAL),I,J),VAL) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(sel2_update_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W0,U)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [VAL: $i] :
( '$ki_exists_in_world_$i'(W1,VAL)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> customqmltpeq(W2,a_select2(tptp_update2(X,U,VAL),U),VAL) ) ) ) ) ) ) ) ).
tff(sel2_update_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W,I)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W0,U)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W1,X)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [VAL: $i] :
( '$ki_exists_in_world_$i'(W2,VAL)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ! [VAL2: $i] :
( '$ki_exists_in_world_$i'(W3,VAL2)
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ( ( ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ~ ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> customqmltpeq(W6,I,U) ) )
& ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,a_select2(X,U),VAL) ) )
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> customqmltpeq(W5,a_select2(tptp_update2(X,I,VAL2),U),VAL) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(sel2_update_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [I: $i] :
( '$ki_exists_in_world_$i'(W,I)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ! [U: $i] :
( '$ki_exists_in_world_$i'(W0,U)
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W1,X)
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ! [VAL: $i] :
( '$ki_exists_in_world_$i'(W2,VAL)
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ! [I0: $i] :
( '$ki_exists_in_world_$i'(W4,I0)
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ( ( ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,n0,I0) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> leq(W6,I0,U) ) )
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> customqmltpeq(W6,a_select2(X,I0),VAL) ) ) ) ) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> leq(W4,n0,I) )
& ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> leq(W4,I,U) ) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> customqmltpeq(W4,a_select2(tptp_update2(X,U,VAL),I),VAL) ) ) ) ) ) ) ) ) ) ) ) ).
tff(ttrue,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> true(W) ) ).
tff(defuse,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ~ ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> customqmltpeq(W0,def,use) ) ) ).
tff(gt_5_4,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n5,n4) ) ).
tff(gt_6_4,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n6,n4) ) ).
tff(gt_999_4,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n999,n4) ) ).
tff(gt_6_5,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n6,n5) ) ).
tff(gt_999_5,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n999,n5) ) ).
tff(gt_999_6,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n999,n6) ) ).
tff(gt_4_tptp_minus_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n4,tptp_minus_1) ) ).
tff(gt_5_tptp_minus_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n5,tptp_minus_1) ) ).
tff(gt_6_tptp_minus_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n6,tptp_minus_1) ) ).
tff(gt_999_tptp_minus_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n999,tptp_minus_1) ) ).
tff(gt_0_tptp_minus_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n0,tptp_minus_1) ) ).
tff(gt_1_tptp_minus_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n1,tptp_minus_1) ) ).
tff(gt_2_tptp_minus_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n2,tptp_minus_1) ) ).
tff(gt_3_tptp_minus_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n3,tptp_minus_1) ) ).
tff(gt_4_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n4,n0) ) ).
tff(gt_5_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n5,n0) ) ).
tff(gt_6_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n6,n0) ) ).
tff(gt_999_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n999,n0) ) ).
tff(gt_1_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n1,n0) ) ).
tff(gt_2_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n2,n0) ) ).
tff(gt_3_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n3,n0) ) ).
tff(gt_4_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n4,n1) ) ).
tff(gt_5_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n5,n1) ) ).
tff(gt_6_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n6,n1) ) ).
tff(gt_999_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n999,n1) ) ).
tff(gt_2_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n2,n1) ) ).
tff(gt_3_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n3,n1) ) ).
tff(gt_4_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n4,n2) ) ).
tff(gt_5_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n5,n2) ) ).
tff(gt_6_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n6,n2) ) ).
tff(gt_999_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n999,n2) ) ).
tff(gt_3_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n3,n2) ) ).
tff(gt_4_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n4,n3) ) ).
tff(gt_5_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n5,n3) ) ).
tff(gt_6_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n6,n3) ) ).
tff(gt_999_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> gt(W,n999,n3) ) ).
tff(finite_domain_4,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,n0,X) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,X,n4) ) )
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n0) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n1) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n2) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n3) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n4) ) ) ) ) ) ) ).
tff(finite_domain_5,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,n0,X) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,X,n5) ) )
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n0) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n1) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n2) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n3) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n4) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n5) ) ) ) ) ) ) ).
tff(finite_domain_6,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,n0,X) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,X,n6) ) )
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n0) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n1) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n2) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n3) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n4) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n5) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n6) ) ) ) ) ) ) ).
tff(finite_domain_0,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,n0,X) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,X,n0) ) )
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n0) ) ) ) ) ) ).
tff(finite_domain_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,n0,X) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,X,n1) ) )
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n0) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n1) ) ) ) ) ) ) ).
tff(finite_domain_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,n0,X) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,X,n2) ) )
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n0) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n1) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n2) ) ) ) ) ) ) ).
tff(finite_domain_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ! [X: $i] :
( '$ki_exists_in_world_$i'(W,X)
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,n0,X) )
& ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> leq(W1,X,n3) ) )
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n0) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n1) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n2) )
| ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> customqmltpeq(W1,X,n3) ) ) ) ) ) ) ).
tff(successor_4,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> customqmltpeq(W,succ(succ(succ(succ(n0)))),n4) ) ).
tff(successor_5,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> customqmltpeq(W,succ(succ(succ(succ(succ(n0))))),n5) ) ).
tff(successor_6,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> customqmltpeq(W,succ(succ(succ(succ(succ(succ(n0)))))),n6) ) ).
tff(successor_1,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> customqmltpeq(W,succ(n0),n1) ) ).
tff(successor_2,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> customqmltpeq(W,succ(succ(n0)),n2) ) ).
tff(successor_3,axiom,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> customqmltpeq(W,succ(succ(succ(n0))),n3) ) ).
tff(verify,conjecture,
! [W: '$ki_world'] :
( '$ki_accessible'('$ki_local_world',W)
=> ( ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> geq(W0,minus(n3,n1),n0) )
=> ! [W0: '$ki_world'] :
( '$ki_accessible'(W,W0)
=> ( ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> geq(W1,minus(n6,n1),n0) )
=> ! [W1: '$ki_world'] :
( '$ki_accessible'(W0,W1)
=> ( ( ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> geq(W2,n2,n0) )
& ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> geq(W2,minus(n999,n1),n0) ) )
=> ! [W2: '$ki_world'] :
( '$ki_accessible'(W1,W2)
=> ( ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> geq(W3,minus(n6,n1),n0) )
=> ! [W3: '$ki_world'] :
( '$ki_accessible'(W2,W3)
=> ( ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> geq(W4,minus(n6,n1),n0) )
=> ! [W4: '$ki_world'] :
( '$ki_accessible'(W3,W4)
=> ( ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> geq(W5,minus(n6,n1),n0) )
=> ! [W5: '$ki_world'] :
( '$ki_accessible'(W4,W5)
=> ( ( ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> geq(W6,minus(n3,n1),n0) )
& ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> geq(W6,minus(n999,n1),n0) ) )
=> ! [W6: '$ki_world'] :
( '$ki_accessible'(W5,W6)
=> true(W6) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------