TPTP Problem File: ITP045^2.p
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%------------------------------------------------------------------------------
% File : ITP045^2 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Determinacy problem prob_286__3350038_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Determinacy/prob_286__3350038_1 [Des21]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0
% Syntax : Number of formulae : 333 ( 96 unt; 71 typ; 0 def)
% Number of atoms : 837 ( 375 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 23496 ( 118 ~; 33 |; 56 &;22909 @)
% ( 0 <=>; 380 =>; 0 <=; 0 <~>)
% Maximal formula depth : 44 ( 13 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 2601 (2601 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 67 usr; 11 con; 0-7 aty)
% Number of variables : 1699 ( 57 ^;1480 !; 52 ?;1699 :)
% ( 110 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:16:58.659
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Data_Oconst,type,
const: $tType ).
thf(ty_t_Data_Ocntxt,type,
cntxt: $tType > $tType > $tType > $tType ).
thf(ty_t_Data_Oexpr,type,
expr: $tType > $tType > $tType > $tType ).
thf(ty_t_Data_Oval,type,
val: $tType > $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_v,type,
v: $tType ).
thf(ty_tf_r,type,
r: $tType ).
thf(ty_tf_l,type,
l: $tType ).
% Explicit typings (61)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_c_Data_Ocombine,type,
combine:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Data_Oconst_OF,type,
f: const ).
thf(sy_c_Data_Oconst_OT,type,
t: const ).
thf(sy_c_Data_Oconst_OUnit,type,
unit: const ).
thf(sy_c_Data_Odoms,type,
doms:
!>[L: $tType,R: $tType,V: $tType] : ( ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) > ( set @ L ) ) ).
thf(sy_c_Data_Oexpr_OAssign,type,
assign:
!>[R: $tType,L: $tType,V: $tType] : ( ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) ).
thf(sy_c_Data_Oexpr_OIte,type,
ite:
!>[R: $tType,L: $tType,V: $tType] : ( ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) ).
thf(sy_c_Data_Oexpr_ORead,type,
read:
!>[R: $tType,L: $tType,V: $tType] : ( ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) ).
thf(sy_c_Data_Oexpr_ORef,type,
ref:
!>[R: $tType,L: $tType,V: $tType] : ( ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) ).
thf(sy_c_Data_Oexpr_ORfork,type,
rfork:
!>[R: $tType,L: $tType,V: $tType] : ( ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) ).
thf(sy_c_Data_Oexpr_ORjoin,type,
rjoin:
!>[R: $tType,L: $tType,V: $tType] : ( ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) ).
thf(sy_c_Data_Oexpr_OVE,type,
ve:
!>[R: $tType,L: $tType,V: $tType] : ( ( val @ R @ L @ V ) > ( expr @ R @ L @ V ) ) ).
thf(sy_c_Data_Oplug,type,
plug:
!>[R: $tType,L: $tType,V: $tType] : ( ( cntxt @ R @ L @ V ) > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) ).
thf(sy_c_Data_Oval_OCV,type,
cv:
!>[R: $tType,L: $tType,V: $tType] : ( const > ( val @ R @ L @ V ) ) ).
thf(sy_c_Data_Oval_OLoc,type,
loc:
!>[L: $tType,R: $tType,V: $tType] : ( L > ( val @ R @ L @ V ) ) ).
thf(sy_c_Data_Oval_ORid,type,
rid:
!>[R: $tType,L: $tType,V: $tType] : ( R > ( val @ R @ L @ V ) ) ).
thf(sy_c_Executions_Osubstitution_Oexecution,type,
execution:
!>[R: $tType,L: $tType,V: $tType] : ( ( ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) > ( expr @ R @ L @ V ) > ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > $o ) ).
thf(sy_c_Executions_Osubstitution_Oinductive__invariant,type,
inductive_invariant:
!>[R: $tType,L: $tType,V: $tType] : ( ( ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) > ( ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > $o ) > $o ) ).
thf(sy_c_Executions_Osubstitution_Omaximal__execution,type,
maximal_execution:
!>[R: $tType,L: $tType,V: $tType] : ( ( ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) > ( expr @ R @ L @ V ) > ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > $o ) ).
thf(sy_c_Executions_Osubstitution_Oreachable,type,
reachable:
!>[R: $tType,L: $tType,V: $tType] : ( ( ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) > ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > $o ) ).
thf(sy_c_Executions_Osubstitution_Osteps,type,
steps:
!>[R: $tType,L: $tType,V: $tType] : ( ( ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) > ( set @ ( product_prod @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) ) ) ) ).
thf(sy_c_Executions_Osubstitution_Oterminates__in,type,
terminates_in:
!>[R: $tType,L: $tType,V: $tType] : ( ( ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) > ( expr @ R @ L @ V ) > ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Map_Odom,type,
dom:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Occurrences_OLID_092_060_094sub_062G,type,
lID_G:
!>[R: $tType,L: $tType,V: $tType] : ( ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > ( set @ L ) ) ).
thf(sy_c_Occurrences_OLID_092_060_094sub_062L,type,
lID_L:
!>[L: $tType,R: $tType,V: $tType] : ( ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) > ( set @ L ) ) ).
thf(sy_c_Occurrences_ORID_092_060_094sub_062G,type,
rID_G:
!>[R: $tType,L: $tType,V: $tType] : ( ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > ( set @ R ) ) ).
thf(sy_c_Occurrences_ORID_092_060_094sub_062L,type,
rID_L:
!>[L: $tType,R: $tType,V: $tType] : ( ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) > ( set @ R ) ) ).
thf(sy_c_OperationalSemantics_Osubstitution_Odomains__subsume,type,
domains_subsume:
!>[L: $tType,R: $tType,V: $tType] : ( ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) > $o ) ).
thf(sy_c_OperationalSemantics_Osubstitution_Odomains__subsume__globally,type,
domain398919503obally:
!>[R: $tType,L: $tType,V: $tType] : ( ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > $o ) ).
thf(sy_c_OperationalSemantics_Osubstitution_Orevision__step,type,
revision_step:
!>[R: $tType,L: $tType,V: $tType] : ( ( ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) > R > ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > $o ) ).
thf(sy_c_OperationalSemantics_Osubstitution_Orevision__step__relaxed,type,
revisi1433015572elaxed:
!>[R: $tType,L: $tType,V: $tType] : ( ( ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) > R > ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > $o ) ).
thf(sy_c_OperationalSemantics_Osubstitution_Osubsumes__accessible,type,
subsumes_accessible:
!>[R: $tType,L: $tType,V: $tType] : ( R > R > ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > $o ) ).
thf(sy_c_OperationalSemantics_Osubstitution_Osubsumes__accessible__globally,type,
subsum380726323obally:
!>[R: $tType,L: $tType,V: $tType] : ( ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > $o ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Othe,type,
the:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Renaming_Oeq__states,type,
eq_states:
!>[R: $tType,L: $tType,V: $tType] : ( ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Substitution_Osubstitution,type,
substitution:
!>[R: $tType,L: $tType,V: $tType] : ( ( ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ) ) > $o ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v__092_060E_062,type,
e: cntxt @ r @ l @ v ).
thf(sy_v__092_060sigma_062,type,
sigma: l > ( option @ ( val @ r @ l @ v ) ) ).
thf(sy_v__092_060tau_062,type,
tau: l > ( option @ ( val @ r @ l @ v ) ) ).
thf(sy_v_e,type,
e2: expr @ r @ l @ v ).
thf(sy_v_r,type,
r2: r ).
thf(sy_v_r_H,type,
r3: r ).
thf(sy_v_r_H_H,type,
r4: r ).
thf(sy_v_s_092_060_094sub_0621,type,
s_1: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ).
thf(sy_v_s_092_060_094sub_0622,type,
s_2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ).
thf(sy_v_s_092_060_094sub_0622_H,type,
s_22: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ).
thf(sy_v_subst,type,
subst: ( expr @ r @ l @ v ) > v > ( expr @ r @ l @ v ) > ( expr @ r @ l @ v ) ).
thf(sy_v_x,type,
x: product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ).
% Relevant facts (255)
thf(fact_0_assms_I4_J,axiom,
~ ( member @ r @ r4 @ ( rID_G @ r @ l @ v @ s_2 ) ) ).
% assms(4)
thf(fact_1_assms_I7_J,axiom,
r2 != r4 ).
% assms(7)
thf(fact_2_assms_I6_J,axiom,
r2 != r3 ).
% assms(6)
thf(fact_3_assms_I1_J,axiom,
( s_2
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ s_1 @ r2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ x ) ) ) ).
% assms(1)
thf(fact_4_assms_I3_J,axiom,
( ( s_2 @ r3 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ tau @ ( plug @ r @ l @ v @ e @ ( rfork @ r @ l @ v @ e2 ) ) ) ) ) ) ).
% assms(3)
thf(fact_5_assms_I2_J,axiom,
( s_22
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ s_1 @ r3 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ tau @ ( plug @ r @ l @ v @ e @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ r4 ) ) ) ) ) ) ) @ r4
@ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ ( combine @ l @ ( val @ r @ l @ v ) @ sigma @ tau )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v )
@ ^ [X: l] : ( none @ ( val @ r @ l @ v ) )
@ e2 ) ) ) ) ) ).
% assms(2)
thf(fact_6_revision__step_Ofork,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,E2: expr @ r @ l @ v,R3: r] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rfork @ r @ l @ v @ E2 ) ) ) ) ) )
=> ( ~ ( member @ r @ R3 @ ( rID_G @ r @ l @ v @ S ) )
=> ( revision_step @ r @ l @ v @ subst @ R2 @ S
@ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ R3 ) ) ) ) ) ) ) @ R3
@ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Sigma @ Tau )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v )
@ ^ [X: l] : ( none @ ( val @ r @ l @ v ) )
@ E2 ) ) ) ) ) ) ) ).
% revision_step.fork
thf(fact_7_assms_I5_J,axiom,
( revision_step @ r @ l @ v @ subst @ r2 @ s_22
@ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )
@ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ s_1 @ r3 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ tau @ ( plug @ r @ l @ v @ e @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ r4 ) ) ) ) ) ) ) @ r4
@ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ ( combine @ l @ ( val @ r @ l @ v ) @ sigma @ tau )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v )
@ ^ [X: l] : ( none @ ( val @ r @ l @ v ) )
@ e2 ) ) ) )
@ r2
@ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ x ) ) ) ).
% assms(5)
thf(fact_8_restrictions__inwards,axiom,
! [A: $tType,B: $tType,X2: A,X3: A,F: A > ( option @ B ),Y: B] :
( ( X2 != X3 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ F @ X2 @ ( some @ B @ Y ) ) @ X3 @ ( none @ B ) )
= ( fun_upd @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ F @ X3 @ ( none @ B ) ) @ X2 @ ( some @ B @ Y ) ) ) ) ).
% restrictions_inwards
thf(fact_9_empty__upd__none,axiom,
! [B: $tType,A: $tType,X2: A] :
( ( fun_upd @ A @ ( option @ B )
@ ^ [X: A] : ( none @ B )
@ X2
@ ( none @ B ) )
= ( ^ [X: A] : ( none @ B ) ) ) ).
% empty_upd_none
thf(fact_10_not__None__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( X2
!= ( none @ A ) )
= ( ? [Y2: A] :
( X2
= ( some @ A @ Y2 ) ) ) ) ).
% not_None_eq
thf(fact_11_not__Some__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( ! [Y2: A] :
( X2
!= ( some @ A @ Y2 ) ) )
= ( X2
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_12_map__upd__nonempty,axiom,
! [B: $tType,A: $tType,T: A > ( option @ B ),K: A,X2: B] :
( ( fun_upd @ A @ ( option @ B ) @ T @ K @ ( some @ B @ X2 ) )
!= ( ^ [X: A] : ( none @ B ) ) ) ).
% map_upd_nonempty
thf(fact_13_fork__pseudodeterministic,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,E2: expr @ r @ l @ v,S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rfork @ r @ l @ v @ E2 ) ) ) ) ) )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
= ( ? [R4: r] :
( ~ ( member @ r @ R4 @ ( rID_G @ r @ l @ v @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rfork @ r @ l @ v @ E2 ) ) ) ) ) ) ) )
& ( S2
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ R4 ) ) ) ) ) ) ) @ R4
@ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Sigma @ Tau )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v )
@ ^ [X: l] : ( none @ ( val @ r @ l @ v ) )
@ E2 ) ) ) ) ) ) ) ) ) ).
% fork_pseudodeterministic
thf(fact_14_revision__step__relaxed_Ofork,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,E2: expr @ r @ l @ v,R3: r] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rfork @ r @ l @ v @ E2 ) ) ) ) ) )
=> ( ~ ( member @ r @ R3 @ ( rID_G @ r @ l @ v @ S ) )
=> ( revisi1433015572elaxed @ r @ l @ v @ subst @ R2 @ S
@ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ R3 ) ) ) ) ) ) ) @ R3
@ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Sigma @ Tau )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v )
@ ^ [X: l] : ( none @ ( val @ r @ l @ v ) )
@ E2 ) ) ) ) ) ) ) ).
% revision_step_relaxed.fork
thf(fact_15_val_Oinject_I4_J,axiom,
! [V: $tType,L: $tType,R: $tType,X4: R,Y4: R] :
( ( ( rid @ R @ L @ V @ X4 )
= ( rid @ R @ L @ V @ Y4 ) )
= ( X4 = Y4 ) ) ).
% val.inject(4)
thf(fact_16_injective__cntxt,axiom,
! [C: $tType,B: $tType,A: $tType,E: cntxt @ A @ B @ C,E1: expr @ A @ B @ C,E22: expr @ A @ B @ C] :
( ( ( plug @ A @ B @ C @ E @ E1 )
= ( plug @ A @ B @ C @ E @ E22 ) )
= ( E1 = E22 ) ) ).
% injective_cntxt
thf(fact_17_expr_Oinject_I1_J,axiom,
! [V: $tType,L: $tType,R: $tType,X1: val @ R @ L @ V,Y1: val @ R @ L @ V] :
( ( ( ve @ R @ L @ V @ X1 )
= ( ve @ R @ L @ V @ Y1 ) )
= ( X1 = Y1 ) ) ).
% expr.inject(1)
thf(fact_18_fun__upd__upd,axiom,
! [A: $tType,B: $tType,F: A > B,X2: A,Y: B,Z: B] :
( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X2 @ Y ) @ X2 @ Z )
= ( fun_upd @ A @ B @ F @ X2 @ Z ) ) ).
% fun_upd_upd
thf(fact_19_fun__upd__triv,axiom,
! [B: $tType,A: $tType,F: A > B,X2: A] :
( ( fun_upd @ A @ B @ F @ X2 @ ( F @ X2 ) )
= F ) ).
% fun_upd_triv
thf(fact_20_option_Oinject,axiom,
! [A: $tType,X22: A,Y22: A] :
( ( ( some @ A @ X22 )
= ( some @ A @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_21_fun__upd__apply,axiom,
! [A: $tType,B: $tType] :
( ( fun_upd @ B @ A )
= ( ^ [F2: B > A,X: B,Y2: A,Z2: B] : ( if @ A @ ( Z2 = X ) @ Y2 @ ( F2 @ Z2 ) ) ) ) ).
% fun_upd_apply
thf(fact_22_expr_Oinject_I7_J,axiom,
! [V: $tType,L: $tType,R: $tType,X7: expr @ R @ L @ V,Y7: expr @ R @ L @ V] :
( ( ( rfork @ R @ L @ V @ X7 )
= ( rfork @ R @ L @ V @ Y7 ) )
= ( X7 = Y7 ) ) ).
% expr.inject(7)
thf(fact_23_RID_092_060_094sub_062GI_I1_J,axiom,
! [C: $tType,A: $tType,B: $tType,S: B > ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ),R2: B,V2: product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) )] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) @ V2 ) )
=> ( member @ B @ R2 @ ( rID_G @ B @ A @ C @ S ) ) ) ).
% RID\<^sub>GI(1)
thf(fact_24_in__restricted__global__in__unrestricted__global_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,R3: A,S: A > ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ),R2: A] :
( ( member @ A @ R3 @ ( rID_G @ A @ B @ C @ ( fun_upd @ A @ ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) ) ) )
=> ( member @ A @ R3 @ ( rID_G @ A @ B @ C @ S ) ) ) ).
% in_restricted_global_in_unrestricted_global(1)
thf(fact_25_only__fork__introduces__rids_H,axiom,
! [R2: r,S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R3: r] :
( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
=> ( ! [Sigma2: l > ( option @ ( val @ r @ l @ v ) ),Tau2: l > ( option @ ( val @ r @ l @ v ) ),E3: cntxt @ r @ l @ v,E4: expr @ r @ l @ v] :
( ( S @ R2 )
!= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma2 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau2 @ ( plug @ r @ l @ v @ E3 @ ( rfork @ r @ l @ v @ E4 ) ) ) ) ) )
=> ( ~ ( member @ r @ R3 @ ( rID_G @ r @ l @ v @ S ) )
=> ~ ( member @ r @ R3 @ ( rID_G @ r @ l @ v @ S2 ) ) ) ) ) ).
% only_fork_introduces_rids'
thf(fact_26_in__restricted__global__in__updated__global_I1_J,axiom,
! [C: $tType,A: $tType,B: $tType,R3: A,S: A > ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ),R2: A,Ls: product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) )] :
( ( member @ A @ R3 @ ( rID_G @ A @ B @ C @ ( fun_upd @ A @ ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) ) ) )
=> ( member @ A @ R3 @ ( rID_G @ A @ B @ C @ ( fun_upd @ A @ ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) @ Ls ) ) ) ) ) ).
% in_restricted_global_in_updated_global(1)
thf(fact_27_substitution__axioms,axiom,
substitution @ r @ l @ v @ subst ).
% substitution_axioms
thf(fact_28_expr_Odistinct_I11_J,axiom,
! [V: $tType,L: $tType,R: $tType,X1: val @ R @ L @ V,X7: expr @ R @ L @ V] :
( ( ve @ R @ L @ V @ X1 )
!= ( rfork @ R @ L @ V @ X7 ) ) ).
% expr.distinct(11)
thf(fact_29_fun__upd__idem__iff,axiom,
! [A: $tType,B: $tType,F: A > B,X2: A,Y: B] :
( ( ( fun_upd @ A @ B @ F @ X2 @ Y )
= F )
= ( ( F @ X2 )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_30_fun__upd__twist,axiom,
! [A: $tType,B: $tType,A2: A,C2: A,M: A > B,B2: B,D: B] :
( ( A2 != C2 )
=> ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A2 @ B2 ) @ C2 @ D )
= ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C2 @ D ) @ A2 @ B2 ) ) ) ).
% fun_upd_twist
thf(fact_31_fun__upd__other,axiom,
! [B: $tType,A: $tType,Z: A,X2: A,F: A > B,Y: B] :
( ( Z != X2 )
=> ( ( fun_upd @ A @ B @ F @ X2 @ Y @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_32_fun__upd__same,axiom,
! [B: $tType,A: $tType,F: B > A,X2: B,Y: A] :
( ( fun_upd @ B @ A @ F @ X2 @ Y @ X2 )
= Y ) ).
% fun_upd_same
thf(fact_33_fun__upd__idem,axiom,
! [A: $tType,B: $tType,F: B > A,X2: B,Y: A] :
( ( ( F @ X2 )
= Y )
=> ( ( fun_upd @ B @ A @ F @ X2 @ Y )
= F ) ) ).
% fun_upd_idem
thf(fact_34_fun__upd__eqD,axiom,
! [A: $tType,B: $tType,F: A > B,X2: A,Y: B,G: A > B,Z: B] :
( ( ( fun_upd @ A @ B @ F @ X2 @ Y )
= ( fun_upd @ A @ B @ G @ X2 @ Z ) )
=> ( Y = Z ) ) ).
% fun_upd_eqD
thf(fact_35_fun__upd__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_upd @ A @ B )
= ( ^ [F2: A > B,A3: A,B3: B,X: A] : ( if @ B @ ( X = A3 ) @ B3 @ ( F2 @ X ) ) ) ) ).
% fun_upd_def
thf(fact_36_combine__options__cases,axiom,
! [A: $tType,B: $tType,X2: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
( ( ( X2
= ( none @ A ) )
=> ( P @ X2 @ Y ) )
=> ( ( ( Y
= ( none @ B ) )
=> ( P @ X2 @ Y ) )
=> ( ! [A4: A,B4: B] :
( ( X2
= ( some @ A @ A4 ) )
=> ( ( Y
= ( some @ B @ B4 ) )
=> ( P @ X2 @ Y ) ) )
=> ( P @ X2 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_37_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
! [X5: option @ A] : ( P2 @ X5 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
& ! [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ).
% split_option_all
thf(fact_38_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
? [X5: option @ A] : ( P2 @ X5 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
| ? [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_39_option_Oinducts,axiom,
! [A: $tType,P: ( option @ A ) > $o,Option: option @ A] :
( ( P @ ( none @ A ) )
=> ( ! [X6: A] : ( P @ ( some @ A @ X6 ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_40_option_Oexhaust,axiom,
! [A: $tType,Y: option @ A] :
( ( Y
!= ( none @ A ) )
=> ~ ! [X23: A] :
( Y
!= ( some @ A @ X23 ) ) ) ).
% option.exhaust
thf(fact_41_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X22: A] :
( ( Option
= ( some @ A @ X22 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_42_option_Odistinct_I1_J,axiom,
! [A: $tType,X22: A] :
( ( none @ A )
!= ( some @ A @ X22 ) ) ).
% option.distinct(1)
thf(fact_43_combine_Ocases,axiom,
! [B: $tType,A: $tType,X2: product_prod @ ( A > ( option @ B ) ) @ ( product_prod @ ( A > ( option @ B ) ) @ A )] :
~ ! [F3: A > ( option @ B ),G2: A > ( option @ B ),X6: A] :
( X2
!= ( product_Pair @ ( A > ( option @ B ) ) @ ( product_prod @ ( A > ( option @ B ) ) @ A ) @ F3 @ ( product_Pair @ ( A > ( option @ B ) ) @ A @ G2 @ X6 ) ) ) ).
% combine.cases
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X6: A] :
( ( P @ X6 )
= ( Q @ X6 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X6: A] :
( ( F @ X6 )
= ( G @ X6 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_map__upd__Some__unfold,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),A2: B,B2: A,X2: B,Y: A] :
( ( ( fun_upd @ B @ ( option @ A ) @ M @ A2 @ ( some @ A @ B2 ) @ X2 )
= ( some @ A @ Y ) )
= ( ( ( X2 = A2 )
& ( B2 = Y ) )
| ( ( X2 != A2 )
& ( ( M @ X2 )
= ( some @ A @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_49_map__upd__triv,axiom,
! [A: $tType,B: $tType,T: B > ( option @ A ),K: B,X2: A] :
( ( ( T @ K )
= ( some @ A @ X2 ) )
=> ( ( fun_upd @ B @ ( option @ A ) @ T @ K @ ( some @ A @ X2 ) )
= T ) ) ).
% map_upd_triv
thf(fact_50_map__upd__eqD1,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),A2: A,X2: B,N: A > ( option @ B ),Y: B] :
( ( ( fun_upd @ A @ ( option @ B ) @ M @ A2 @ ( some @ B @ X2 ) )
= ( fun_upd @ A @ ( option @ B ) @ N @ A2 @ ( some @ B @ Y ) ) )
=> ( X2 = Y ) ) ).
% map_upd_eqD1
thf(fact_51_LID__local__store_Oinduct,axiom,
! [V: $tType,R: $tType,L: $tType,P: ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) > $o,A0: product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) )] :
( ! [Sigma2: L > ( option @ ( val @ R @ L @ V ) ),Tau2: L > ( option @ ( val @ R @ L @ V ) ),E4: expr @ R @ L @ V] : ( P @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma2 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau2 @ E4 ) ) )
=> ( P @ A0 ) ) ).
% LID_local_store.induct
thf(fact_52_LID__local__store_Ocases,axiom,
! [V: $tType,L: $tType,R: $tType,X2: product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) )] :
~ ! [Sigma2: L > ( option @ ( val @ R @ L @ V ) ),Tau2: L > ( option @ ( val @ R @ L @ V ) ),E4: expr @ R @ L @ V] :
( X2
!= ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma2 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau2 @ E4 ) ) ) ).
% LID_local_store.cases
thf(fact_53_combine_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( combine @ A @ B )
= ( ^ [F2: A > ( option @ B ),G3: A > ( option @ B ),X: A] :
( if @ ( option @ B )
@ ( ( G3 @ X )
= ( none @ B ) )
@ ( F2 @ X )
@ ( G3 @ X ) ) ) ) ).
% combine.simps
thf(fact_54_combine_Oelims,axiom,
! [B: $tType,A: $tType,X2: A > ( option @ B ),Xa: A > ( option @ B ),Xb: A,Y: option @ B] :
( ( ( combine @ A @ B @ X2 @ Xa @ Xb )
= Y )
=> ( ( ( ( Xa @ Xb )
= ( none @ B ) )
=> ( Y
= ( X2 @ Xb ) ) )
& ( ( ( Xa @ Xb )
!= ( none @ B ) )
=> ( Y
= ( Xa @ Xb ) ) ) ) ) ).
% combine.elims
thf(fact_55_local__steps__commute,axiom,
! [S_2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S_1: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,X2: product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ),S_22: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R3: r,Y: product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) )] :
( ( S_2
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S_1 @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ X2 ) ) )
=> ( ( S_22
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S_1 @ R3 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ Y ) ) )
=> ( ( revision_step @ r @ l @ v @ subst @ R3 @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S_1 @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ X2 ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S_1 @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ X2 ) ) @ R3 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ Y ) ) )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S_1 @ R3 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ Y ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S_1 @ R3 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ Y ) ) @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ X2 ) ) )
=> ? [S_3: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S_32: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( eq_states @ r @ l @ v @ S_3 @ S_32 )
& ( ( revision_step @ r @ l @ v @ subst @ R3 @ S_2 @ S_3 )
| ( S_2 = S_3 ) )
& ( ( revision_step @ r @ l @ v @ subst @ R2 @ S_22 @ S_32 )
| ( S_22 = S_32 ) ) ) ) ) ) ) ).
% local_steps_commute
thf(fact_56_revision__step__relaxed_Ojoin_092_060_094sub_062_092_060epsilon_062,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,R3: r] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rjoin @ r @ l @ v @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ R3 ) ) ) ) ) ) ) )
=> ( ( ( S @ R3 )
= ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) )
=> ( revisi1433015572elaxed @ r @ l @ v @ subst @ R2 @ S
@ ^ [X: r] : ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) ) ) ).
% revision_step_relaxed.join\<^sub>\<epsilon>
thf(fact_57_revision__step_Ojoin_092_060_094sub_062_092_060epsilon_062,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,R3: r] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rjoin @ r @ l @ v @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ R3 ) ) ) ) ) ) ) )
=> ( ( ( S @ R3 )
= ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) )
=> ( revision_step @ r @ l @ v @ subst @ R2 @ S
@ ^ [X: r] : ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) ) ) ).
% revision_step.join\<^sub>\<epsilon>
thf(fact_58_transition__relations__equivalent,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( reachable @ r @ l @ v @ subst @ S )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
= ( revisi1433015572elaxed @ r @ l @ v @ subst @ R2 @ S @ S2 ) ) ) ).
% transition_relations_equivalent
thf(fact_59_SLC__commute,axiom,
! [S_33: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S_34: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R3: r,S_2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,S_22: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( S_33 = S_34 )
=> ( ( revision_step @ r @ l @ v @ subst @ R3 @ S_2 @ S_33 )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ S_22 @ S_34 )
=> ( ( eq_states @ r @ l @ v @ S_33 @ S_34 )
& ( ( revision_step @ r @ l @ v @ subst @ R3 @ S_2 @ S_33 )
| ( S_2 = S_33 ) )
& ( ( revision_step @ r @ l @ v @ subst @ R2 @ S_22 @ S_34 )
| ( S_22 = S_34 ) ) ) ) ) ) ).
% SLC_commute
thf(fact_60_SLC__sym,axiom,
! [R3: r,S_2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,S_22: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ? [S_35: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S_36: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( eq_states @ r @ l @ v @ S_35 @ S_36 )
& ( ( revision_step @ r @ l @ v @ subst @ R3 @ S_2 @ S_35 )
| ( S_2 = S_35 ) )
& ( ( revision_step @ r @ l @ v @ subst @ R2 @ S_22 @ S_36 )
| ( S_22 = S_36 ) ) )
=> ? [S_3: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S_32: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( eq_states @ r @ l @ v @ S_3 @ S_32 )
& ( ( revision_step @ r @ l @ v @ subst @ R2 @ S_22 @ S_3 )
| ( S_22 = S_3 ) )
& ( ( revision_step @ r @ l @ v @ subst @ R3 @ S_2 @ S_32 )
| ( S_2 = S_32 ) ) ) ) ).
% SLC_sym
thf(fact_61_local__determinism,axiom,
! [R2: r,S_1: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S_2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S_22: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( revision_step @ r @ l @ v @ subst @ R2 @ S_1 @ S_2 )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ S_1 @ S_22 )
=> ( eq_states @ r @ l @ v @ S_2 @ S_22 ) ) ) ).
% local_determinism
thf(fact_62_reachability__closed__under__execution__step,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( reachable @ r @ l @ v @ subst @ S )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
=> ( reachable @ r @ l @ v @ subst @ S2 ) ) ) ).
% reachability_closed_under_execution_step
thf(fact_63_only__fork__introduces__rids,axiom,
! [R2: r,S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
=> ( ! [Sigma2: l > ( option @ ( val @ r @ l @ v ) ),Tau2: l > ( option @ ( val @ r @ l @ v ) ),E3: cntxt @ r @ l @ v,E4: expr @ r @ l @ v] :
( ( S @ R2 )
!= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma2 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau2 @ ( plug @ r @ l @ v @ E3 @ ( rfork @ r @ l @ v @ E4 ) ) ) ) ) )
=> ( ord_less_eq @ ( set @ r ) @ ( rID_G @ r @ l @ v @ S2 ) @ ( rID_G @ r @ l @ v @ S ) ) ) ) ).
% only_fork_introduces_rids
thf(fact_64_substitution_Ofork__pseudodeterministic,axiom,
! [L: $tType,V: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,E2: expr @ R @ L @ V,S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rfork @ R @ L @ V @ E2 ) ) ) ) ) )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
= ( ? [R4: R] :
( ~ ( member @ R @ R4 @ ( rID_G @ R @ L @ V @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rfork @ R @ L @ V @ E2 ) ) ) ) ) ) ) )
& ( S2
= ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( rid @ R @ L @ V @ R4 ) ) ) ) ) ) ) @ R4
@ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) )
@ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Sigma @ Tau )
@ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V )
@ ^ [X: L] : ( none @ ( val @ R @ L @ V ) )
@ E2 ) ) ) ) ) ) ) ) ) ) ).
% substitution.fork_pseudodeterministic
thf(fact_65_expr_Oinject_I8_J,axiom,
! [V: $tType,L: $tType,R: $tType,X8: expr @ R @ L @ V,Y8: expr @ R @ L @ V] :
( ( ( rjoin @ R @ L @ V @ X8 )
= ( rjoin @ R @ L @ V @ Y8 ) )
= ( X8 = Y8 ) ) ).
% expr.inject(8)
thf(fact_66_SLC__join_092_060_094sub_062_092_060epsilon_062,axiom,
! [S_1: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,R5: r,S_2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R3: r,S_22: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( ( S_1 @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rjoin @ r @ l @ v @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ R5 ) ) ) ) ) ) ) )
=> ( ( S_2
= ( ^ [X: r] : ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) )
=> ( ( ( S_1 @ R5 )
= ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) )
=> ( ( revision_step @ r @ l @ v @ subst @ R3 @ S_1 @ S_22 )
=> ( ( R2 != R3 )
=> ? [S_3: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S_32: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( eq_states @ r @ l @ v @ S_3 @ S_32 )
& ( ( revision_step @ r @ l @ v @ subst @ R3 @ S_2 @ S_3 )
| ( S_2 = S_3 ) )
& ( ( revision_step @ r @ l @ v @ subst @ R2 @ S_22 @ S_32 )
| ( S_22 = S_32 ) ) ) ) ) ) ) ) ).
% SLC_join\<^sub>\<epsilon>
thf(fact_67_reachable__def,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( reachable @ r @ l @ v @ subst @ S )
= ( ? [E5: expr @ r @ l @ v,S3: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] : ( execution @ r @ l @ v @ subst @ E5 @ S3 @ S ) ) ) ).
% reachable_def
thf(fact_68_rjoin_092_060_094sub_062_092_060epsilon_062__deterministic,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,R3: r,S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rjoin @ r @ l @ v @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ R3 ) ) ) ) ) ) ) )
=> ( ( ( S @ R3 )
= ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
= ( S2
= ( ^ [X: r] : ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) ) ) ) ) ).
% rjoin\<^sub>\<epsilon>_deterministic
thf(fact_69_substitution_Olocal__determinism,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),R2: R,S_1: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S_2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S_22: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S_1 @ S_2 )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S_1 @ S_22 )
=> ( eq_states @ R @ L @ V @ S_2 @ S_22 ) ) ) ) ).
% substitution.local_determinism
thf(fact_70_substitution_OSLC__commute,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S_33: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S_34: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R3: R,S_2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,S_22: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( S_33 = S_34 )
=> ( ( revision_step @ R @ L @ V @ Subst @ R3 @ S_2 @ S_33 )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S_22 @ S_34 )
=> ( ( eq_states @ R @ L @ V @ S_33 @ S_34 )
& ( ( revision_step @ R @ L @ V @ Subst @ R3 @ S_2 @ S_33 )
| ( S_2 = S_33 ) )
& ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S_22 @ S_34 )
| ( S_22 = S_34 ) ) ) ) ) ) ) ).
% substitution.SLC_commute
thf(fact_71_substitution_OSLC__sym,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),R3: R,S_2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,S_22: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ? [S_35: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S_36: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( eq_states @ R @ L @ V @ S_35 @ S_36 )
& ( ( revision_step @ R @ L @ V @ Subst @ R3 @ S_2 @ S_35 )
| ( S_2 = S_35 ) )
& ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S_22 @ S_36 )
| ( S_22 = S_36 ) ) )
=> ? [S_3: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S_32: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( eq_states @ R @ L @ V @ S_3 @ S_32 )
& ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S_22 @ S_3 )
| ( S_22 = S_3 ) )
& ( ( revision_step @ R @ L @ V @ Subst @ R3 @ S_2 @ S_32 )
| ( S_2 = S_32 ) ) ) ) ) ).
% substitution.SLC_sym
thf(fact_72_expr_Odistinct_I13_J,axiom,
! [V: $tType,L: $tType,R: $tType,X1: val @ R @ L @ V,X8: expr @ R @ L @ V] :
( ( ve @ R @ L @ V @ X1 )
!= ( rjoin @ R @ L @ V @ X8 ) ) ).
% expr.distinct(13)
thf(fact_73_expr_Odistinct_I55_J,axiom,
! [V: $tType,L: $tType,R: $tType,X7: expr @ R @ L @ V,X8: expr @ R @ L @ V] :
( ( rfork @ R @ L @ V @ X7 )
!= ( rjoin @ R @ L @ V @ X8 ) ) ).
% expr.distinct(55)
thf(fact_74_substitution_Olocal__steps__commute,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S_2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S_1: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,X2: product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ),S_22: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R3: R,Y: product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( S_2
= ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S_1 @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ X2 ) ) )
=> ( ( S_22
= ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S_1 @ R3 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ Y ) ) )
=> ( ( revision_step @ R @ L @ V @ Subst @ R3 @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S_1 @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ X2 ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S_1 @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ X2 ) ) @ R3 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ Y ) ) )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S_1 @ R3 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ Y ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S_1 @ R3 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ Y ) ) @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ X2 ) ) )
=> ? [S_3: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S_32: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( eq_states @ R @ L @ V @ S_3 @ S_32 )
& ( ( revision_step @ R @ L @ V @ Subst @ R3 @ S_2 @ S_3 )
| ( S_2 = S_3 ) )
& ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S_22 @ S_32 )
| ( S_22 = S_32 ) ) ) ) ) ) ) ) ).
% substitution.local_steps_commute
thf(fact_75_ID__restricted__global__subset__unrestricted_I2_J,axiom,
! [C: $tType,A: $tType,B: $tType,S: B > ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ),R2: B] : ( ord_less_eq @ ( set @ B ) @ ( rID_G @ B @ A @ C @ ( fun_upd @ B @ ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ) ) ) @ ( rID_G @ B @ A @ C @ S ) ) ).
% ID_restricted_global_subset_unrestricted(2)
thf(fact_76_substitution_OSLC__join_092_060_094sub_062_092_060epsilon_062,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S_1: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,R5: R,S_2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R3: R,S_22: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S_1 @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rjoin @ R @ L @ V @ ( ve @ R @ L @ V @ ( rid @ R @ L @ V @ R5 ) ) ) ) ) ) ) )
=> ( ( S_2
= ( ^ [X: R] : ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) )
=> ( ( ( S_1 @ R5 )
= ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) )
=> ( ( revision_step @ R @ L @ V @ Subst @ R3 @ S_1 @ S_22 )
=> ( ( R2 != R3 )
=> ? [S_3: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S_32: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( eq_states @ R @ L @ V @ S_3 @ S_32 )
& ( ( revision_step @ R @ L @ V @ Subst @ R3 @ S_2 @ S_3 )
| ( S_2 = S_3 ) )
& ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S_22 @ S_32 )
| ( S_22 = S_32 ) ) ) ) ) ) ) ) ) ).
% substitution.SLC_join\<^sub>\<epsilon>
thf(fact_77_substitution_Orjoin_092_060_094sub_062_092_060epsilon_062__deterministic,axiom,
! [L: $tType,V: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,R3: R,S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rjoin @ R @ L @ V @ ( ve @ R @ L @ V @ ( rid @ R @ L @ V @ R3 ) ) ) ) ) ) ) )
=> ( ( ( S @ R3 )
= ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
= ( S2
= ( ^ [X: R] : ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) ) ) ) ) ) ).
% substitution.rjoin\<^sub>\<epsilon>_deterministic
thf(fact_78_inductive__invariant__is__execution__invariant,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),P: ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) > $o] :
( ( reachable @ r @ l @ v @ subst @ S )
=> ( ( inductive_invariant @ r @ l @ v @ subst @ P )
=> ( P @ S ) ) ) ).
% inductive_invariant_is_execution_invariant
thf(fact_79_substitution_Orevision__step__relaxed_Ofork,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,E2: expr @ R @ L @ V,R3: R] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rfork @ R @ L @ V @ E2 ) ) ) ) ) )
=> ( ~ ( member @ R @ R3 @ ( rID_G @ R @ L @ V @ S ) )
=> ( revisi1433015572elaxed @ R @ L @ V @ Subst @ R2 @ S
@ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( rid @ R @ L @ V @ R3 ) ) ) ) ) ) ) @ R3
@ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) )
@ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Sigma @ Tau )
@ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V )
@ ^ [X: L] : ( none @ ( val @ R @ L @ V ) )
@ E2 ) ) ) ) ) ) ) ) ).
% substitution.revision_step_relaxed.fork
thf(fact_80_substitution_Orevision__step_Ofork,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,E2: expr @ R @ L @ V,R3: R] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rfork @ R @ L @ V @ E2 ) ) ) ) ) )
=> ( ~ ( member @ R @ R3 @ ( rID_G @ R @ L @ V @ S ) )
=> ( revision_step @ R @ L @ V @ Subst @ R2 @ S
@ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( rid @ R @ L @ V @ R3 ) ) ) ) ) ) ) @ R3
@ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) )
@ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Sigma @ Tau )
@ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V )
@ ^ [X: L] : ( none @ ( val @ R @ L @ V ) )
@ E2 ) ) ) ) ) ) ) ) ).
% substitution.revision_step.fork
thf(fact_81_substitution_Orevision__step__relaxed_Ojoin_092_060_094sub_062_092_060epsilon_062,axiom,
! [L: $tType,V: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,R3: R] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rjoin @ R @ L @ V @ ( ve @ R @ L @ V @ ( rid @ R @ L @ V @ R3 ) ) ) ) ) ) ) )
=> ( ( ( S @ R3 )
= ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) )
=> ( revisi1433015572elaxed @ R @ L @ V @ Subst @ R2 @ S
@ ^ [X: R] : ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) ) ) ) ).
% substitution.revision_step_relaxed.join\<^sub>\<epsilon>
thf(fact_82_substitution_Orevision__step_Ojoin_092_060_094sub_062_092_060epsilon_062,axiom,
! [L: $tType,V: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,R3: R] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rjoin @ R @ L @ V @ ( ve @ R @ L @ V @ ( rid @ R @ L @ V @ R3 ) ) ) ) ) ) ) )
=> ( ( ( S @ R3 )
= ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) )
=> ( revision_step @ R @ L @ V @ Subst @ R2 @ S
@ ^ [X: R] : ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) ) ) ) ).
% substitution.revision_step.join\<^sub>\<epsilon>
thf(fact_83_substitution_Oonly__fork__introduces__rids,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),R2: R,S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
=> ( ! [Sigma2: L > ( option @ ( val @ R @ L @ V ) ),Tau2: L > ( option @ ( val @ R @ L @ V ) ),E3: cntxt @ R @ L @ V,E4: expr @ R @ L @ V] :
( ( S @ R2 )
!= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma2 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau2 @ ( plug @ R @ L @ V @ E3 @ ( rfork @ R @ L @ V @ E4 ) ) ) ) ) )
=> ( ord_less_eq @ ( set @ R ) @ ( rID_G @ R @ L @ V @ S2 ) @ ( rID_G @ R @ L @ V @ S ) ) ) ) ) ).
% substitution.only_fork_introduces_rids
thf(fact_84_reachable__imp___092_060S_062_092_060_094sub_062G,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( reachable @ r @ l @ v @ subst @ S )
=> ( domain398919503obally @ r @ l @ v @ S ) ) ).
% reachable_imp_\<S>\<^sub>G
thf(fact_85_domains__subsume__globally__def,axiom,
( ( domain398919503obally @ r @ l @ v )
= ( ^ [S4: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
! [R6: r,Ls2: product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) )] :
( ( ( S4 @ R6 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ Ls2 ) )
=> ( domains_subsume @ l @ r @ v @ Ls2 ) ) ) ) ).
% domains_subsume_globally_def
thf(fact_86_substitution_Orevision__step_Ocong,axiom,
! [V: $tType,L: $tType,R: $tType] :
( ( revision_step @ R @ L @ V )
= ( revision_step @ R @ L @ V ) ) ).
% substitution.revision_step.cong
thf(fact_87_substitution_Orevision__step__relaxed_Ocong,axiom,
! [V: $tType,L: $tType,R: $tType] :
( ( revisi1433015572elaxed @ R @ L @ V )
= ( revisi1433015572elaxed @ R @ L @ V ) ) ).
% substitution.revision_step_relaxed.cong
thf(fact_88_substitution_Oonly__fork__introduces__rids_H,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),R2: R,S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R3: R] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
=> ( ! [Sigma2: L > ( option @ ( val @ R @ L @ V ) ),Tau2: L > ( option @ ( val @ R @ L @ V ) ),E3: cntxt @ R @ L @ V,E4: expr @ R @ L @ V] :
( ( S @ R2 )
!= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma2 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau2 @ ( plug @ R @ L @ V @ E3 @ ( rfork @ R @ L @ V @ E4 ) ) ) ) ) )
=> ( ~ ( member @ R @ R3 @ ( rID_G @ R @ L @ V @ S ) )
=> ~ ( member @ R @ R3 @ ( rID_G @ R @ L @ V @ S2 ) ) ) ) ) ) ).
% substitution.only_fork_introduces_rids'
thf(fact_89_nice__ind__inv__is__inductive__invariant,axiom,
( inductive_invariant @ r @ l @ v @ subst
@ ^ [S4: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( domain398919503obally @ r @ l @ v @ S4 )
& ( subsum380726323obally @ r @ l @ v @ S4 ) ) ) ).
% nice_ind_inv_is_inductive_invariant
thf(fact_90_step__preserves___092_060S_062_092_060_094sub_062G__and___092_060A_062_092_060_094sub_062G_I2_J,axiom,
! [R2: r,S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
=> ( ( domain398919503obally @ r @ l @ v @ S )
=> ( ( subsum380726323obally @ r @ l @ v @ S )
=> ( subsum380726323obally @ r @ l @ v @ S2 ) ) ) ) ).
% step_preserves_\<S>\<^sub>G_and_\<A>\<^sub>G(2)
thf(fact_91_step__preserves___092_060S_062_092_060_094sub_062G__and___092_060A_062_092_060_094sub_062G_I1_J,axiom,
! [R2: r,S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
=> ( ( domain398919503obally @ r @ l @ v @ S )
=> ( ( subsum380726323obally @ r @ l @ v @ S )
=> ( domain398919503obally @ r @ l @ v @ S2 ) ) ) ) ).
% step_preserves_\<S>\<^sub>G_and_\<A>\<^sub>G(1)
thf(fact_92_valid__stepE,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( member @ ( product_prod @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) ) @ ( product_Pair @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ S @ S2 ) @ ( steps @ r @ l @ v @ subst ) )
=> ? [R7: r] : ( revision_step @ r @ l @ v @ subst @ R7 @ S @ S2 ) ) ).
% valid_stepE
thf(fact_93_substitution_Odomains__subsume__globallyI,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ! [R7: R,Sigma2: L > ( option @ ( val @ R @ L @ V ) ),Tau2: L > ( option @ ( val @ R @ L @ V ) ),E4: expr @ R @ L @ V] :
( ( ( S @ R7 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma2 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau2 @ E4 ) ) ) )
=> ( domains_subsume @ L @ R @ V @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma2 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau2 @ E4 ) ) ) )
=> ( domain398919503obally @ R @ L @ V @ S ) ) ) ).
% substitution.domains_subsume_globallyI
thf(fact_94_substitution_Otransition__relations__equivalent,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( reachable @ R @ L @ V @ Subst @ S )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
= ( revisi1433015572elaxed @ R @ L @ V @ Subst @ R2 @ S @ S2 ) ) ) ) ).
% substitution.transition_relations_equivalent
thf(fact_95_valid__stepI,axiom,
! [R2: r,S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
=> ( member @ ( product_prod @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) ) @ ( product_Pair @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ S @ S2 ) @ ( steps @ r @ l @ v @ subst ) ) ) ).
% valid_stepI
thf(fact_96_domains__subsume__globallyI,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ! [R7: r,Sigma2: l > ( option @ ( val @ r @ l @ v ) ),Tau2: l > ( option @ ( val @ r @ l @ v ) ),E4: expr @ r @ l @ v] :
( ( ( S @ R7 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma2 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau2 @ E4 ) ) ) )
=> ( domains_subsume @ l @ r @ v @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma2 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau2 @ E4 ) ) ) )
=> ( domain398919503obally @ r @ l @ v @ S ) ) ).
% domains_subsume_globallyI
thf(fact_97_maximal__execution__def,axiom,
! [E2: expr @ r @ l @ v,S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( maximal_execution @ r @ l @ v @ subst @ E2 @ S @ S2 )
= ( ( execution @ r @ l @ v @ subst @ E2 @ S @ S2 )
& ~ ? [S5: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] : ( member @ ( product_prod @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) ) @ ( product_Pair @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ S2 @ S5 ) @ ( steps @ r @ l @ v @ subst ) ) ) ) ).
% maximal_execution_def
thf(fact_98_substitution_Osteps_Ocong,axiom,
! [V: $tType,L: $tType,R: $tType] :
( ( steps @ R @ L @ V )
= ( steps @ R @ L @ V ) ) ).
% substitution.steps.cong
thf(fact_99_substitution_Ovalid__stepE,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( member @ ( product_prod @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) ) @ ( product_Pair @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ S @ S2 ) @ ( steps @ R @ L @ V @ Subst ) )
=> ? [R7: R] : ( revision_step @ R @ L @ V @ Subst @ R7 @ S @ S2 ) ) ) ).
% substitution.valid_stepE
thf(fact_100_substitution_Ovalid__stepI,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),R2: R,S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
=> ( member @ ( product_prod @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) ) @ ( product_Pair @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ S @ S2 ) @ ( steps @ R @ L @ V @ Subst ) ) ) ) ).
% substitution.valid_stepI
thf(fact_101_substitution_Oreachable_Ocong,axiom,
! [V: $tType,L: $tType,R: $tType] :
( ( reachable @ R @ L @ V )
= ( reachable @ R @ L @ V ) ) ).
% substitution.reachable.cong
thf(fact_102_substitution_Oinductive__invariant_Ocong,axiom,
! [V: $tType,L: $tType,R: $tType] :
( ( inductive_invariant @ R @ L @ V )
= ( inductive_invariant @ R @ L @ V ) ) ).
% substitution.inductive_invariant.cong
thf(fact_103_substitution_Oexecution_Ocong,axiom,
! [V: $tType,L: $tType,R: $tType] :
( ( execution @ R @ L @ V )
= ( execution @ R @ L @ V ) ) ).
% substitution.execution.cong
thf(fact_104_substitution_Onice__ind__inv__is__inductive__invariant,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( inductive_invariant @ R @ L @ V @ Subst
@ ^ [S4: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( domain398919503obally @ R @ L @ V @ S4 )
& ( subsum380726323obally @ R @ L @ V @ S4 ) ) ) ) ).
% substitution.nice_ind_inv_is_inductive_invariant
thf(fact_105_substitution_Ostep__preserves___092_060S_062_092_060_094sub_062G__and___092_060A_062_092_060_094sub_062G_I2_J,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),R2: R,S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
=> ( ( domain398919503obally @ R @ L @ V @ S )
=> ( ( subsum380726323obally @ R @ L @ V @ S )
=> ( subsum380726323obally @ R @ L @ V @ S2 ) ) ) ) ) ).
% substitution.step_preserves_\<S>\<^sub>G_and_\<A>\<^sub>G(2)
thf(fact_106_substitution_Ostep__preserves___092_060S_062_092_060_094sub_062G__and___092_060A_062_092_060_094sub_062G_I1_J,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),R2: R,S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
=> ( ( domain398919503obally @ R @ L @ V @ S )
=> ( ( subsum380726323obally @ R @ L @ V @ S )
=> ( domain398919503obally @ R @ L @ V @ S2 ) ) ) ) ) ).
% substitution.step_preserves_\<S>\<^sub>G_and_\<A>\<^sub>G(1)
thf(fact_107_substitution_Odomains__subsume__globally__def,axiom,
! [L: $tType,V: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( domain398919503obally @ R @ L @ V @ S )
= ( ! [R6: R,Ls2: product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) )] :
( ( ( S @ R6 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ Ls2 ) )
=> ( domains_subsume @ L @ R @ V @ Ls2 ) ) ) ) ) ).
% substitution.domains_subsume_globally_def
thf(fact_108_substitution_Oreachability__closed__under__execution__step,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( reachable @ R @ L @ V @ Subst @ S )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
=> ( reachable @ R @ L @ V @ Subst @ S2 ) ) ) ) ).
% substitution.reachability_closed_under_execution_step
thf(fact_109_substitution_Oreachable__imp___092_060S_062_092_060_094sub_062G,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( reachable @ R @ L @ V @ Subst @ S )
=> ( domain398919503obally @ R @ L @ V @ S ) ) ) ).
% substitution.reachable_imp_\<S>\<^sub>G
thf(fact_110_substitution_Oinductive__invariant__is__execution__invariant,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),P: ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) > $o] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( reachable @ R @ L @ V @ Subst @ S )
=> ( ( inductive_invariant @ R @ L @ V @ Subst @ P )
=> ( P @ S ) ) ) ) ).
% substitution.inductive_invariant_is_execution_invariant
thf(fact_111_substitution_Oreachable__def,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( reachable @ R @ L @ V @ Subst @ S )
= ( ? [E5: expr @ R @ L @ V,S3: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] : ( execution @ R @ L @ V @ Subst @ E5 @ S3 @ S ) ) ) ) ).
% substitution.reachable_def
thf(fact_112_SLC__join,axiom,
! [S_1: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,R5: r,S_2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),Tau3: l > ( option @ ( val @ r @ l @ v ) ),Sigma3: l > ( option @ ( val @ r @ l @ v ) ),V2: val @ r @ l @ v,R3: r,S_22: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( ( S_1 @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rjoin @ r @ l @ v @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ R5 ) ) ) ) ) ) ) )
=> ( ( S_2
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S_1 @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Tau @ Tau3 ) @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ unit ) ) ) ) ) ) ) @ R5 @ ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) )
=> ( ( ( S_1 @ R5 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma3 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau3 @ ( ve @ r @ l @ v @ V2 ) ) ) ) )
=> ( ( revision_step @ r @ l @ v @ subst @ R3 @ S_1 @ S_22 )
=> ( ( R2 != R3 )
=> ? [S_3: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S_32: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( eq_states @ r @ l @ v @ S_3 @ S_32 )
& ( ( revision_step @ r @ l @ v @ subst @ R3 @ S_2 @ S_3 )
| ( S_2 = S_3 ) )
& ( ( revision_step @ r @ l @ v @ subst @ R2 @ S_22 @ S_32 )
| ( S_22 = S_32 ) ) ) ) ) ) ) ) ).
% SLC_join
thf(fact_113_join__and__local__commute,axiom,
! [S_2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S_1: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),Tau3: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,R5: r,S_22: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R3: r,Ls: product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ),Sigma3: l > ( option @ ( val @ r @ l @ v ) ),V2: val @ r @ l @ v] :
( ( S_2
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S_1 @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Tau @ Tau3 ) @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ unit ) ) ) ) ) ) ) @ R5 @ ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) )
=> ( ( S_22
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S_1 @ R3 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ Ls ) ) )
=> ( ( R2 != R3 )
=> ( ( R3 != R5 )
=> ( ( revision_step @ r @ l @ v @ subst @ R3 @ S_2 @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S_1 @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Tau @ Tau3 ) @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ unit ) ) ) ) ) ) ) @ R5 @ ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ R3 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ Ls ) ) )
=> ( ( ( S_22 @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rjoin @ r @ l @ v @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ R5 ) ) ) ) ) ) ) )
=> ( ( ( S_22 @ R5 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma3 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau3 @ ( ve @ r @ l @ v @ V2 ) ) ) ) )
=> ? [S_3: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S_32: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( eq_states @ r @ l @ v @ S_3 @ S_32 )
& ( ( revision_step @ r @ l @ v @ subst @ R3 @ S_2 @ S_3 )
| ( S_2 = S_3 ) )
& ( ( revision_step @ r @ l @ v @ subst @ R2 @ S_22 @ S_32 )
| ( S_22 = S_32 ) ) ) ) ) ) ) ) ) ) ).
% join_and_local_commute
thf(fact_114_substitution_Omaximal__execution__def,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),E2: expr @ R @ L @ V,S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( maximal_execution @ R @ L @ V @ Subst @ E2 @ S @ S2 )
= ( ( execution @ R @ L @ V @ Subst @ E2 @ S @ S2 )
& ~ ? [S5: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] : ( member @ ( product_prod @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) ) @ ( product_Pair @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ S2 @ S5 ) @ ( steps @ R @ L @ V @ Subst ) ) ) ) ) ).
% substitution.maximal_execution_def
thf(fact_115_substitution_Osubsumes__accessible__globallyI,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ! [R_1: R,Sigma_1: L > ( option @ ( val @ R @ L @ V ) ),Tau_1: L > ( option @ ( val @ R @ L @ V ) ),E_1: expr @ R @ L @ V,R_2: R,Sigma_2: L > ( option @ ( val @ R @ L @ V ) ),Tau_2: L > ( option @ ( val @ R @ L @ V ) ),E_2: expr @ R @ L @ V] :
( ( ( S @ R_1 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma_1 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau_1 @ E_1 ) ) ) )
=> ( ( ( S @ R_2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma_2 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau_2 @ E_2 ) ) ) )
=> ( subsumes_accessible @ R @ L @ V @ R_1 @ R_2 @ S ) ) )
=> ( subsum380726323obally @ R @ L @ V @ S ) ) ) ).
% substitution.subsumes_accessible_globallyI
thf(fact_116_revision__step__relaxed_Ojoin,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,R3: r,Sigma3: l > ( option @ ( val @ r @ l @ v ) ),Tau3: l > ( option @ ( val @ r @ l @ v ) ),V2: val @ r @ l @ v] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rjoin @ r @ l @ v @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ R3 ) ) ) ) ) ) ) )
=> ( ( ( S @ R3 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma3 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau3 @ ( ve @ r @ l @ v @ V2 ) ) ) ) )
=> ( revisi1433015572elaxed @ r @ l @ v @ subst @ R2 @ S @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Tau @ Tau3 ) @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ unit ) ) ) ) ) ) ) @ R3 @ ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) ) ) ) ).
% revision_step_relaxed.join
thf(fact_117_val_Oinject_I1_J,axiom,
! [V: $tType,L: $tType,R: $tType,X1: const,Y1: const] :
( ( ( cv @ R @ L @ V @ X1 )
= ( cv @ R @ L @ V @ Y1 ) )
= ( X1 = Y1 ) ) ).
% val.inject(1)
thf(fact_118_terminates__in__def,axiom,
! [E2: expr @ r @ l @ v,S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( terminates_in @ r @ l @ v @ subst @ E2 @ S2 )
= ( ? [S4: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] : ( maximal_execution @ r @ l @ v @ subst @ E2 @ S4 @ S2 ) ) ) ).
% terminates_in_def
thf(fact_119_revision__step_Ojoin,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,R3: r,Sigma3: l > ( option @ ( val @ r @ l @ v ) ),Tau3: l > ( option @ ( val @ r @ l @ v ) ),V2: val @ r @ l @ v] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rjoin @ r @ l @ v @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ R3 ) ) ) ) ) ) ) )
=> ( ( ( S @ R3 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma3 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau3 @ ( ve @ r @ l @ v @ V2 ) ) ) ) )
=> ( revision_step @ r @ l @ v @ subst @ R2 @ S @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Tau @ Tau3 ) @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ unit ) ) ) ) ) ) ) @ R3 @ ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) ) ) ) ).
% revision_step.join
thf(fact_120_subsumes__accessible__globallyI,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ! [R_1: r,Sigma_1: l > ( option @ ( val @ r @ l @ v ) ),Tau_1: l > ( option @ ( val @ r @ l @ v ) ),E_1: expr @ r @ l @ v,R_2: r,Sigma_2: l > ( option @ ( val @ r @ l @ v ) ),Tau_2: l > ( option @ ( val @ r @ l @ v ) ),E_2: expr @ r @ l @ v] :
( ( ( S @ R_1 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma_1 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau_1 @ E_1 ) ) ) )
=> ( ( ( S @ R_2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma_2 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau_2 @ E_2 ) ) ) )
=> ( subsumes_accessible @ r @ l @ v @ R_1 @ R_2 @ S ) ) )
=> ( subsum380726323obally @ r @ l @ v @ S ) ) ).
% subsumes_accessible_globallyI
thf(fact_121_rjoin__deterministic,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,R3: r,Sigma3: l > ( option @ ( val @ r @ l @ v ) ),Tau3: l > ( option @ ( val @ r @ l @ v ) ),V2: val @ r @ l @ v,S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( rjoin @ r @ l @ v @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ R3 ) ) ) ) ) ) ) )
=> ( ( ( S @ R3 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma3 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau3 @ ( ve @ r @ l @ v @ V2 ) ) ) ) )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
= ( S2
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Tau @ Tau3 ) @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ unit ) ) ) ) ) ) ) @ R3 @ ( none @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) ) ) ) ) ).
% rjoin_deterministic
thf(fact_122_substitution_Omaximal__execution_Ocong,axiom,
! [V: $tType,L: $tType,R: $tType] :
( ( maximal_execution @ R @ L @ V )
= ( maximal_execution @ R @ L @ V ) ) ).
% substitution.maximal_execution.cong
thf(fact_123_val_Odistinct_I5_J,axiom,
! [L: $tType,V: $tType,R: $tType,X1: const,X4: R] :
( ( cv @ R @ L @ V @ X1 )
!= ( rid @ R @ L @ V @ X4 ) ) ).
% val.distinct(5)
thf(fact_124_substitution_Orevision__step_Ojoin,axiom,
! [V: $tType,R: $tType,L: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,R3: R,Sigma3: L > ( option @ ( val @ R @ L @ V ) ),Tau3: L > ( option @ ( val @ R @ L @ V ) ),V2: val @ R @ L @ V] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rjoin @ R @ L @ V @ ( ve @ R @ L @ V @ ( rid @ R @ L @ V @ R3 ) ) ) ) ) ) ) )
=> ( ( ( S @ R3 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma3 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau3 @ ( ve @ R @ L @ V @ V2 ) ) ) ) )
=> ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Tau @ Tau3 ) @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ unit ) ) ) ) ) ) ) @ R3 @ ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) ) ) ) ) ).
% substitution.revision_step.join
thf(fact_125_substitution_Orjoin__deterministic,axiom,
! [V: $tType,R: $tType,L: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,R3: R,Sigma3: L > ( option @ ( val @ R @ L @ V ) ),Tau3: L > ( option @ ( val @ R @ L @ V ) ),V2: val @ R @ L @ V,S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rjoin @ R @ L @ V @ ( ve @ R @ L @ V @ ( rid @ R @ L @ V @ R3 ) ) ) ) ) ) ) )
=> ( ( ( S @ R3 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma3 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau3 @ ( ve @ R @ L @ V @ V2 ) ) ) ) )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
= ( S2
= ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Tau @ Tau3 ) @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ unit ) ) ) ) ) ) ) @ R3 @ ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) ) ) ) ) ) ).
% substitution.rjoin_deterministic
thf(fact_126_substitution_Orevision__step__relaxed_Ojoin,axiom,
! [V: $tType,R: $tType,L: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,R3: R,Sigma3: L > ( option @ ( val @ R @ L @ V ) ),Tau3: L > ( option @ ( val @ R @ L @ V ) ),V2: val @ R @ L @ V] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rjoin @ R @ L @ V @ ( ve @ R @ L @ V @ ( rid @ R @ L @ V @ R3 ) ) ) ) ) ) ) )
=> ( ( ( S @ R3 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma3 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau3 @ ( ve @ R @ L @ V @ V2 ) ) ) ) )
=> ( revisi1433015572elaxed @ R @ L @ V @ Subst @ R2 @ S @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Tau @ Tau3 ) @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ unit ) ) ) ) ) ) ) @ R3 @ ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) ) ) ) ) ).
% substitution.revision_step_relaxed.join
thf(fact_127_substitution_Ojoin__and__local__commute,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S_2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S_1: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),Tau3: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,R5: R,S_22: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R3: R,Ls: product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ),Sigma3: L > ( option @ ( val @ R @ L @ V ) ),V2: val @ R @ L @ V] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( S_2
= ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S_1 @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Tau @ Tau3 ) @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ unit ) ) ) ) ) ) ) @ R5 @ ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) )
=> ( ( S_22
= ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S_1 @ R3 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ Ls ) ) )
=> ( ( R2 != R3 )
=> ( ( R3 != R5 )
=> ( ( revision_step @ R @ L @ V @ Subst @ R3 @ S_2 @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S_1 @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Tau @ Tau3 ) @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ unit ) ) ) ) ) ) ) @ R5 @ ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ R3 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ Ls ) ) )
=> ( ( ( S_22 @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rjoin @ R @ L @ V @ ( ve @ R @ L @ V @ ( rid @ R @ L @ V @ R5 ) ) ) ) ) ) ) )
=> ( ( ( S_22 @ R5 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma3 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau3 @ ( ve @ R @ L @ V @ V2 ) ) ) ) )
=> ? [S_3: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S_32: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( eq_states @ R @ L @ V @ S_3 @ S_32 )
& ( ( revision_step @ R @ L @ V @ Subst @ R3 @ S_2 @ S_3 )
| ( S_2 = S_3 ) )
& ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S_22 @ S_32 )
| ( S_22 = S_32 ) ) ) ) ) ) ) ) ) ) ) ).
% substitution.join_and_local_commute
thf(fact_128_substitution_OSLC__join,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S_1: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,R5: R,S_2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),Tau3: L > ( option @ ( val @ R @ L @ V ) ),Sigma3: L > ( option @ ( val @ R @ L @ V ) ),V2: val @ R @ L @ V,R3: R,S_22: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S_1 @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( rjoin @ R @ L @ V @ ( ve @ R @ L @ V @ ( rid @ R @ L @ V @ R5 ) ) ) ) ) ) ) )
=> ( ( S_2
= ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S_1 @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Tau @ Tau3 ) @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ unit ) ) ) ) ) ) ) @ R5 @ ( none @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) )
=> ( ( ( S_1 @ R5 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma3 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau3 @ ( ve @ R @ L @ V @ V2 ) ) ) ) )
=> ( ( revision_step @ R @ L @ V @ Subst @ R3 @ S_1 @ S_22 )
=> ( ( R2 != R3 )
=> ? [S_3: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S_32: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( eq_states @ R @ L @ V @ S_3 @ S_32 )
& ( ( revision_step @ R @ L @ V @ Subst @ R3 @ S_2 @ S_3 )
| ( S_2 = S_3 ) )
& ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S_22 @ S_32 )
| ( S_22 = S_32 ) ) ) ) ) ) ) ) ) ).
% substitution.SLC_join
thf(fact_129_reachable__imp__identifiers__finite_I1_J,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( reachable @ r @ l @ v @ subst @ S )
=> ( finite_finite @ r @ ( rID_G @ r @ l @ v @ S ) ) ) ).
% reachable_imp_identifiers_finite(1)
thf(fact_130_RID_092_060_094sub_062G__finite__invariant,axiom,
! [R2: r,S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
=> ( ( finite_finite @ r @ ( rID_G @ r @ l @ v @ S ) )
=> ( finite_finite @ r @ ( rID_G @ r @ l @ v @ S2 ) ) ) ) ).
% RID\<^sub>G_finite_invariant
thf(fact_131_reachability__closed__under__execution,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( reachable @ r @ l @ v @ subst @ S )
=> ( ( member @ ( product_prod @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) ) @ ( product_Pair @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ S @ S2 ) @ ( transitive_rtrancl @ ( r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) ) @ ( steps @ r @ l @ v @ subst ) ) )
=> ( reachable @ r @ l @ v @ subst @ S2 ) ) ) ).
% reachability_closed_under_execution
thf(fact_132_revision__step__relaxed_OifTrue,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,E1: expr @ r @ l @ v,E22: expr @ r @ l @ v] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ite @ r @ l @ v @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ t ) ) @ E1 @ E22 ) ) ) ) ) )
=> ( revisi1433015572elaxed @ r @ l @ v @ subst @ R2 @ S @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ E1 ) ) ) ) ) ) ) ).
% revision_step_relaxed.ifTrue
thf(fact_133_empty__map__finite_I3_J,axiom,
! [H: $tType,I: $tType,G4: $tType] :
( finite_finite @ G4
@ ( rID_G @ G4 @ H @ I
@ ^ [X: G4] : ( none @ ( product_prod @ ( H > ( option @ ( val @ G4 @ H @ I ) ) ) @ ( product_prod @ ( H > ( option @ ( val @ G4 @ H @ I ) ) ) @ ( expr @ G4 @ H @ I ) ) ) ) ) ) ).
% empty_map_finite(3)
thf(fact_134_expr_Oinject_I3_J,axiom,
! [V: $tType,L: $tType,R: $tType,X31: expr @ R @ L @ V,X32: expr @ R @ L @ V,X33: expr @ R @ L @ V,Y31: expr @ R @ L @ V,Y32: expr @ R @ L @ V,Y33: expr @ R @ L @ V] :
( ( ( ite @ R @ L @ V @ X31 @ X32 @ X33 )
= ( ite @ R @ L @ V @ Y31 @ Y32 @ Y33 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 )
& ( X33 = Y33 ) ) ) ).
% expr.inject(3)
thf(fact_135_revision__step_OifTrue,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,E1: expr @ r @ l @ v,E22: expr @ r @ l @ v] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ite @ r @ l @ v @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ t ) ) @ E1 @ E22 ) ) ) ) ) )
=> ( revision_step @ r @ l @ v @ subst @ R2 @ S @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ E1 ) ) ) ) ) ) ) ).
% revision_step.ifTrue
thf(fact_136_finite__state__imp__restriction__finite_I1_J,axiom,
! [C: $tType,A: $tType,B: $tType,S: A > ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ),R2: A] :
( ( finite_finite @ A @ ( rID_G @ A @ B @ C @ S ) )
=> ( finite_finite @ A @ ( rID_G @ A @ B @ C @ ( fun_upd @ A @ ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) ) ) ) ) ).
% finite_state_imp_restriction_finite(1)
thf(fact_137_ifTrue__deterministic,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,E1: expr @ r @ l @ v,E22: expr @ r @ l @ v,S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ite @ r @ l @ v @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ t ) ) @ E1 @ E22 ) ) ) ) ) )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
= ( S2
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ E1 ) ) ) ) ) ) ) ) ).
% ifTrue_deterministic
thf(fact_138_expr_Odistinct_I3_J,axiom,
! [V: $tType,L: $tType,R: $tType,X1: val @ R @ L @ V,X31: expr @ R @ L @ V,X32: expr @ R @ L @ V,X33: expr @ R @ L @ V] :
( ( ve @ R @ L @ V @ X1 )
!= ( ite @ R @ L @ V @ X31 @ X32 @ X33 ) ) ).
% expr.distinct(3)
thf(fact_139_expr_Odistinct_I35_J,axiom,
! [V: $tType,L: $tType,R: $tType,X31: expr @ R @ L @ V,X32: expr @ R @ L @ V,X33: expr @ R @ L @ V,X8: expr @ R @ L @ V] :
( ( ite @ R @ L @ V @ X31 @ X32 @ X33 )
!= ( rjoin @ R @ L @ V @ X8 ) ) ).
% expr.distinct(35)
thf(fact_140_expr_Odistinct_I33_J,axiom,
! [V: $tType,L: $tType,R: $tType,X31: expr @ R @ L @ V,X32: expr @ R @ L @ V,X33: expr @ R @ L @ V,X7: expr @ R @ L @ V] :
( ( ite @ R @ L @ V @ X31 @ X32 @ X33 )
!= ( rfork @ R @ L @ V @ X7 ) ) ).
% expr.distinct(33)
thf(fact_141_substitution_Oterminates__in_Ocong,axiom,
! [V: $tType,L: $tType,R: $tType] :
( ( terminates_in @ R @ L @ V )
= ( terminates_in @ R @ L @ V ) ) ).
% substitution.terminates_in.cong
thf(fact_142_const_Odistinct_I3_J,axiom,
unit != t ).
% const.distinct(3)
thf(fact_143_substitution_ORID_092_060_094sub_062G__finite__invariant,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),R2: R,S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
=> ( ( finite_finite @ R @ ( rID_G @ R @ L @ V @ S ) )
=> ( finite_finite @ R @ ( rID_G @ R @ L @ V @ S2 ) ) ) ) ) ).
% substitution.RID\<^sub>G_finite_invariant
thf(fact_144_substitution_Oreachable__imp__identifiers__finite_I1_J,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( reachable @ R @ L @ V @ Subst @ S )
=> ( finite_finite @ R @ ( rID_G @ R @ L @ V @ S ) ) ) ) ).
% substitution.reachable_imp_identifiers_finite(1)
thf(fact_145_substitution_Oempty__map__finite_I3_J,axiom,
! [R: $tType,L: $tType,V: $tType,H: $tType,I: $tType,G4: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( finite_finite @ G4
@ ( rID_G @ G4 @ H @ I
@ ^ [X: G4] : ( none @ ( product_prod @ ( H > ( option @ ( val @ G4 @ H @ I ) ) ) @ ( product_prod @ ( H > ( option @ ( val @ G4 @ H @ I ) ) ) @ ( expr @ G4 @ H @ I ) ) ) ) ) ) ) ).
% substitution.empty_map_finite(3)
thf(fact_146_substitution_Oreachability__closed__under__execution,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( reachable @ R @ L @ V @ Subst @ S )
=> ( ( member @ ( product_prod @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) ) @ ( product_Pair @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ S @ S2 ) @ ( transitive_rtrancl @ ( R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) ) @ ( steps @ R @ L @ V @ Subst ) ) )
=> ( reachable @ R @ L @ V @ Subst @ S2 ) ) ) ) ).
% substitution.reachability_closed_under_execution
thf(fact_147_substitution_Ofinite__state__imp__restriction__finite_I1_J,axiom,
! [V: $tType,L: $tType,R: $tType,C: $tType,A: $tType,B: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: A > ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ),R2: A] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( finite_finite @ A @ ( rID_G @ A @ B @ C @ S ) )
=> ( finite_finite @ A @ ( rID_G @ A @ B @ C @ ( fun_upd @ A @ ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) ) ) ) ) ) ).
% substitution.finite_state_imp_restriction_finite(1)
thf(fact_148_substitution_OifTrue__deterministic,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,E1: expr @ R @ L @ V,E22: expr @ R @ L @ V,S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ite @ R @ L @ V @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ t ) ) @ E1 @ E22 ) ) ) ) ) )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
= ( S2
= ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ E1 ) ) ) ) ) ) ) ) ) ).
% substitution.ifTrue_deterministic
thf(fact_149_substitution_Orevision__step_OifTrue,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,E1: expr @ R @ L @ V,E22: expr @ R @ L @ V] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ite @ R @ L @ V @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ t ) ) @ E1 @ E22 ) ) ) ) ) )
=> ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ E1 ) ) ) ) ) ) ) ) ).
% substitution.revision_step.ifTrue
thf(fact_150_substitution_Orevision__step__relaxed_OifTrue,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,E1: expr @ R @ L @ V,E22: expr @ R @ L @ V] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ite @ R @ L @ V @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ t ) ) @ E1 @ E22 ) ) ) ) ) )
=> ( revisi1433015572elaxed @ R @ L @ V @ Subst @ R2 @ S @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ E1 ) ) ) ) ) ) ) ) ).
% substitution.revision_step_relaxed.ifTrue
thf(fact_151_substitution_Oterminates__in__def,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),E2: expr @ R @ L @ V,S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( terminates_in @ R @ L @ V @ Subst @ E2 @ S2 )
= ( ? [S4: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] : ( maximal_execution @ R @ L @ V @ Subst @ E2 @ S4 @ S2 ) ) ) ) ).
% substitution.terminates_in_def
thf(fact_152_revision__step__relaxed_OifFalse,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,E1: expr @ r @ l @ v,E22: expr @ r @ l @ v] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ite @ r @ l @ v @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ f ) ) @ E1 @ E22 ) ) ) ) ) )
=> ( revisi1433015572elaxed @ r @ l @ v @ subst @ R2 @ S @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ E22 ) ) ) ) ) ) ) ).
% revision_step_relaxed.ifFalse
thf(fact_153_revision__step_OifFalse,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,E1: expr @ r @ l @ v,E22: expr @ r @ l @ v] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ite @ r @ l @ v @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ f ) ) @ E1 @ E22 ) ) ) ) ) )
=> ( revision_step @ r @ l @ v @ subst @ R2 @ S @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ E22 ) ) ) ) ) ) ) ).
% revision_step.ifFalse
thf(fact_154_ifFalse__deterministic,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,E1: expr @ r @ l @ v,E22: expr @ r @ l @ v,S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ite @ r @ l @ v @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ f ) ) @ E1 @ E22 ) ) ) ) ) )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
= ( S2
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ E22 ) ) ) ) ) ) ) ) ).
% ifFalse_deterministic
thf(fact_155_finite__Collect__subsets,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite @ A @ A5 )
=> ( finite_finite @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_156_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite @ A @ ( collect @ A @ P ) )
& ( finite_finite @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_157_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ( finite_finite @ A @ ( collect @ A @ P ) )
| ( finite_finite @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_158_const_Odistinct_I1_J,axiom,
unit != f ).
% const.distinct(1)
thf(fact_159_const_Odistinct_I5_J,axiom,
f != t ).
% const.distinct(5)
thf(fact_160_const_Oinduct,axiom,
! [P: const > $o,Const: const] :
( ( P @ unit )
=> ( ( P @ f )
=> ( ( P @ t )
=> ( P @ Const ) ) ) ) ).
% const.induct
thf(fact_161_const_Oexhaust,axiom,
! [Y: const] :
( ( Y != unit )
=> ( ( Y != f )
=> ( Y = t ) ) ) ).
% const.exhaust
thf(fact_162_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B,R8: A > B > $o] :
( ~ ( finite_finite @ A @ A5 )
=> ( ( finite_finite @ B @ B6 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A5 )
=> ? [Xa2: B] :
( ( member @ B @ Xa2 @ B6 )
& ( R8 @ X6 @ Xa2 ) ) )
=> ? [X6: B] :
( ( member @ B @ X6 @ B6 )
& ~ ( finite_finite @ A
@ ( collect @ A
@ ^ [A3: A] :
( ( member @ A @ A3 @ A5 )
& ( R8 @ A3 @ X6 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_163_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite @ A @ ( collect @ A @ P ) )
=> ? [X_1: A] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_164_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A5 )
& ( ord_less_eq @ A @ A2 @ X6 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A5 )
=> ( ( ord_less_eq @ A @ X6 @ Xa2 )
=> ( X6 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_165_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ? [X6: A] :
( ( member @ A @ X6 @ A5 )
& ( ord_less_eq @ A @ X6 @ A2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A5 )
=> ( ( ord_less_eq @ A @ Xa2 @ X6 )
=> ( X6 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_166_finite__subset,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( finite_finite @ A @ B6 )
=> ( finite_finite @ A @ A5 ) ) ) ).
% finite_subset
thf(fact_167_infinite__super,axiom,
! [A: $tType,S6: set @ A,T2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S6 @ T2 )
=> ( ~ ( finite_finite @ A @ S6 )
=> ~ ( finite_finite @ A @ T2 ) ) ) ).
% infinite_super
thf(fact_168_rev__finite__subset,axiom,
! [A: $tType,B6: set @ A,A5: set @ A] :
( ( finite_finite @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( finite_finite @ A @ A5 ) ) ) ).
% rev_finite_subset
thf(fact_169_finite__update__induct,axiom,
! [B: $tType,A: $tType,F: A > B,C2: B,P: ( A > B ) > $o] :
( ( finite_finite @ A
@ ( collect @ A
@ ^ [A3: A] :
( ( F @ A3 )
!= C2 ) ) )
=> ( ( P
@ ^ [A3: A] : C2 )
=> ( ! [A4: A,B4: B,F3: A > B] :
( ( finite_finite @ A
@ ( collect @ A
@ ^ [C3: A] :
( ( F3 @ C3 )
!= C2 ) ) )
=> ( ( ( F3 @ A4 )
= C2 )
=> ( ( B4 != C2 )
=> ( ( P @ F3 )
=> ( P @ ( fun_upd @ A @ B @ F3 @ A4 @ B4 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_update_induct
thf(fact_170_substitution_Orevision__step_OifFalse,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,E1: expr @ R @ L @ V,E22: expr @ R @ L @ V] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ite @ R @ L @ V @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ f ) ) @ E1 @ E22 ) ) ) ) ) )
=> ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ E22 ) ) ) ) ) ) ) ) ).
% substitution.revision_step.ifFalse
thf(fact_171_substitution_OifFalse__deterministic,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,E1: expr @ R @ L @ V,E22: expr @ R @ L @ V,S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ite @ R @ L @ V @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ f ) ) @ E1 @ E22 ) ) ) ) ) )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
= ( S2
= ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ E22 ) ) ) ) ) ) ) ) ) ).
% substitution.ifFalse_deterministic
thf(fact_172_substitution_Orevision__step__relaxed_OifFalse,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,E1: expr @ R @ L @ V,E22: expr @ R @ L @ V] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ite @ R @ L @ V @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ f ) ) @ E1 @ E22 ) ) ) ) ) )
=> ( revisi1433015572elaxed @ R @ L @ V @ Subst @ R2 @ S @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ E22 ) ) ) ) ) ) ) ) ).
% substitution.revision_step_relaxed.ifFalse
thf(fact_173_reachable__imp__identifiers__available_I1_J,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( reachable @ r @ l @ v @ subst @ S )
=> ( ~ ( finite_finite @ r @ ( top_top @ ( set @ r ) ) )
=> ? [R7: r] :
~ ( member @ r @ R7 @ ( rID_G @ r @ l @ v @ S ) ) ) ) ).
% reachable_imp_identifiers_available(1)
thf(fact_174_local__state__of__finite__restricted__global__state__is__finite_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,S: B > ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ),R3: B,Ls: product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ),R2: B] :
( ( ( S @ R3 )
= ( some @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) @ Ls ) )
=> ( ( finite_finite @ B @ ( rID_G @ B @ A @ C @ ( fun_upd @ B @ ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ) ) ) )
=> ( ( R2 != R3 )
=> ( finite_finite @ B @ ( rID_L @ A @ B @ C @ Ls ) ) ) ) ) ).
% local_state_of_finite_restricted_global_state_is_finite(1)
thf(fact_175_empty__map__finite_I4_J,axiom,
! [J: $tType,M2: $tType,K2: $tType] :
( finite_finite @ J
@ ( lID_G @ K2 @ J @ M2
@ ^ [X: K2] : ( none @ ( product_prod @ ( J > ( option @ ( val @ K2 @ J @ M2 ) ) ) @ ( product_prod @ ( J > ( option @ ( val @ K2 @ J @ M2 ) ) ) @ ( expr @ K2 @ J @ M2 ) ) ) ) ) ) ).
% empty_map_finite(4)
thf(fact_176_RID_092_060_094sub_062L__finite__invariant,axiom,
! [R2: r,S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
=> ( ( finite_finite @ l @ ( lID_G @ r @ l @ v @ S ) )
=> ( finite_finite @ l @ ( lID_G @ r @ l @ v @ S2 ) ) ) ) ).
% RID\<^sub>L_finite_invariant
thf(fact_177_reachable__imp__identifiers__finite_I2_J,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( reachable @ r @ l @ v @ subst @ S )
=> ( finite_finite @ l @ ( lID_G @ r @ l @ v @ S ) ) ) ).
% reachable_imp_identifiers_finite(2)
thf(fact_178_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_179_finite__Collect__not,axiom,
! [A: $tType,P: A > $o] :
( ( finite_finite @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite @ A
@ ( collect @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
= ( finite_finite @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_180_in__restricted__global__in__unrestricted__global_I2_J,axiom,
! [C: $tType,B: $tType,A: $tType,L2: B,S: A > ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ),R2: A] :
( ( member @ B @ L2 @ ( lID_G @ A @ B @ C @ ( fun_upd @ A @ ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) ) ) )
=> ( member @ B @ L2 @ ( lID_G @ A @ B @ C @ S ) ) ) ).
% in_restricted_global_in_unrestricted_global(2)
thf(fact_181_RID_092_060_094sub_062GI_I2_J,axiom,
! [C: $tType,A: $tType,B: $tType,S: B > ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ),R3: B,Ls: product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ),R2: B] :
( ( ( S @ R3 )
= ( some @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) @ Ls ) )
=> ( ( member @ B @ R2 @ ( rID_L @ A @ B @ C @ Ls ) )
=> ( member @ B @ R2 @ ( rID_G @ B @ A @ C @ S ) ) ) ) ).
% RID\<^sub>GI(2)
thf(fact_182_finite__state__imp__restriction__finite_I2_J,axiom,
! [C: $tType,A: $tType,B: $tType,S: A > ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ),R2: A] :
( ( finite_finite @ B @ ( lID_G @ A @ B @ C @ S ) )
=> ( finite_finite @ B @ ( lID_G @ A @ B @ C @ ( fun_upd @ A @ ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) ) ) ) ) ).
% finite_state_imp_restriction_finite(2)
thf(fact_183_in__restricted__global__in__updated__global_I2_J,axiom,
! [C: $tType,A: $tType,B: $tType,L2: B,S: A > ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ),R2: A,Ls: product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) )] :
( ( member @ B @ L2 @ ( lID_G @ A @ B @ C @ ( fun_upd @ A @ ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) ) ) )
=> ( member @ B @ L2 @ ( lID_G @ A @ B @ C @ ( fun_upd @ A @ ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) @ Ls ) ) ) ) ) ).
% in_restricted_global_in_updated_global(2)
thf(fact_184_substitution_Oreachable__imp__identifiers__available_I2_J,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( reachable @ R @ L @ V @ Subst @ S )
=> ( ~ ( finite_finite @ L @ ( top_top @ ( set @ L ) ) )
=> ? [L3: L] :
~ ( member @ L @ L3 @ ( lID_G @ R @ L @ V @ S ) ) ) ) ) ).
% substitution.reachable_imp_identifiers_available(2)
thf(fact_185_substitution_ORID_092_060_094sub_062L__finite__invariant,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),R2: R,S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
=> ( ( finite_finite @ L @ ( lID_G @ R @ L @ V @ S ) )
=> ( finite_finite @ L @ ( lID_G @ R @ L @ V @ S2 ) ) ) ) ) ).
% substitution.RID\<^sub>L_finite_invariant
thf(fact_186_ID__restricted__global__subset__unrestricted_I1_J,axiom,
! [C: $tType,A: $tType,B: $tType,S: B > ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ),R2: B] : ( ord_less_eq @ ( set @ A ) @ ( lID_G @ B @ A @ C @ ( fun_upd @ B @ ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ) ) ) @ ( lID_G @ B @ A @ C @ S ) ) ).
% ID_restricted_global_subset_unrestricted(1)
thf(fact_187_substitution_Oreachable__imp__identifiers__finite_I2_J,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( reachable @ R @ L @ V @ Subst @ S )
=> ( finite_finite @ L @ ( lID_G @ R @ L @ V @ S ) ) ) ) ).
% substitution.reachable_imp_identifiers_finite(2)
thf(fact_188_substitution_Oempty__map__finite_I4_J,axiom,
! [R: $tType,L: $tType,V: $tType,J: $tType,M2: $tType,K2: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( finite_finite @ J
@ ( lID_G @ K2 @ J @ M2
@ ^ [X: K2] : ( none @ ( product_prod @ ( J > ( option @ ( val @ K2 @ J @ M2 ) ) ) @ ( product_prod @ ( J > ( option @ ( val @ K2 @ J @ M2 ) ) ) @ ( expr @ K2 @ J @ M2 ) ) ) ) ) ) ) ).
% substitution.empty_map_finite(4)
thf(fact_189_substitution_Oreachable__imp__identifiers__available_I1_J,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( reachable @ R @ L @ V @ Subst @ S )
=> ( ~ ( finite_finite @ R @ ( top_top @ ( set @ R ) ) )
=> ? [R7: R] :
~ ( member @ R @ R7 @ ( rID_G @ R @ L @ V @ S ) ) ) ) ) ).
% substitution.reachable_imp_identifiers_available(1)
thf(fact_190_substitution_Ofinite__state__imp__restriction__finite_I2_J,axiom,
! [V: $tType,L: $tType,R: $tType,C: $tType,A: $tType,B: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: A > ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ),R2: A] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( finite_finite @ B @ ( lID_G @ A @ B @ C @ S ) )
=> ( finite_finite @ B @ ( lID_G @ A @ B @ C @ ( fun_upd @ A @ ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) ) ) ) ) ) ) ).
% substitution.finite_state_imp_restriction_finite(2)
thf(fact_191_substitution_Olocal__state__of__finite__restricted__global__state__is__finite_I1_J,axiom,
! [V: $tType,L: $tType,R: $tType,C: $tType,B: $tType,A: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: B > ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ),R3: B,Ls: product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ),R2: B] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R3 )
= ( some @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) @ Ls ) )
=> ( ( finite_finite @ B @ ( rID_G @ B @ A @ C @ ( fun_upd @ B @ ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ) ) ) )
=> ( ( R2 != R3 )
=> ( finite_finite @ B @ ( rID_L @ A @ B @ C @ Ls ) ) ) ) ) ) ).
% substitution.local_state_of_finite_restricted_global_state_is_finite(1)
thf(fact_192_only__new__introduces__lids_H,axiom,
! [R2: r,S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),L2: l] :
( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
=> ( ! [Sigma2: l > ( option @ ( val @ r @ l @ v ) ),Tau2: l > ( option @ ( val @ r @ l @ v ) ),E3: cntxt @ r @ l @ v,V3: val @ r @ l @ v] :
( ( S @ R2 )
!= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma2 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau2 @ ( plug @ r @ l @ v @ E3 @ ( ref @ r @ l @ v @ ( ve @ r @ l @ v @ V3 ) ) ) ) ) ) )
=> ( ~ ( member @ l @ L2 @ ( lID_G @ r @ l @ v @ S ) )
=> ~ ( member @ l @ L2 @ ( lID_G @ r @ l @ v @ S2 ) ) ) ) ) ).
% only_new_introduces_lids'
thf(fact_193_local__state__of__finite__restricted__global__state__is__finite_I2_J,axiom,
! [C: $tType,B: $tType,A: $tType,S: B > ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ),R3: B,Ls: product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ),R2: B] :
( ( ( S @ R3 )
= ( some @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) @ Ls ) )
=> ( ( finite_finite @ A @ ( lID_G @ B @ A @ C @ ( fun_upd @ B @ ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ) ) ) )
=> ( ( R2 != R3 )
=> ( finite_finite @ A @ ( lID_L @ A @ B @ C @ Ls ) ) ) ) ) ).
% local_state_of_finite_restricted_global_state_is_finite(2)
thf(fact_194_only__new__introduces__lids,axiom,
! [R2: r,S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
=> ( ! [Sigma2: l > ( option @ ( val @ r @ l @ v ) ),Tau2: l > ( option @ ( val @ r @ l @ v ) ),E3: cntxt @ r @ l @ v,V3: val @ r @ l @ v] :
( ( S @ R2 )
!= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma2 @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau2 @ ( plug @ r @ l @ v @ E3 @ ( ref @ r @ l @ v @ ( ve @ r @ l @ v @ V3 ) ) ) ) ) ) )
=> ( ord_less_eq @ ( set @ l ) @ ( lID_G @ r @ l @ v @ S2 ) @ ( lID_G @ r @ l @ v @ S ) ) ) ) ).
% only_new_introduces_lids
thf(fact_195_expr_Oinject_I4_J,axiom,
! [V: $tType,L: $tType,R: $tType,X4: expr @ R @ L @ V,Y4: expr @ R @ L @ V] :
( ( ( ref @ R @ L @ V @ X4 )
= ( ref @ R @ L @ V @ Y4 ) )
= ( X4 = Y4 ) ) ).
% expr.inject(4)
thf(fact_196_reachable__imp__identifiers__available_I2_J,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( reachable @ r @ l @ v @ subst @ S )
=> ( ~ ( finite_finite @ l @ ( top_top @ ( set @ l ) ) )
=> ? [L3: l] :
~ ( member @ l @ L3 @ ( lID_G @ r @ l @ v @ S ) ) ) ) ).
% reachable_imp_identifiers_available(2)
thf(fact_197_LID_092_060_094sub_062GI,axiom,
! [C: $tType,A: $tType,B: $tType,S: B > ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ),R2: B,Ls: product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ),L2: A] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) @ Ls ) )
=> ( ( member @ A @ L2 @ ( lID_L @ A @ B @ C @ Ls ) )
=> ( member @ A @ L2 @ ( lID_G @ B @ A @ C @ S ) ) ) ) ).
% LID\<^sub>GI
thf(fact_198_expr_Odistinct_I41_J,axiom,
! [V: $tType,L: $tType,R: $tType,X4: expr @ R @ L @ V,X7: expr @ R @ L @ V] :
( ( ref @ R @ L @ V @ X4 )
!= ( rfork @ R @ L @ V @ X7 ) ) ).
% expr.distinct(41)
thf(fact_199_expr_Odistinct_I43_J,axiom,
! [V: $tType,L: $tType,R: $tType,X4: expr @ R @ L @ V,X8: expr @ R @ L @ V] :
( ( ref @ R @ L @ V @ X4 )
!= ( rjoin @ R @ L @ V @ X8 ) ) ).
% expr.distinct(43)
thf(fact_200_expr_Odistinct_I5_J,axiom,
! [V: $tType,L: $tType,R: $tType,X1: val @ R @ L @ V,X4: expr @ R @ L @ V] :
( ( ve @ R @ L @ V @ X1 )
!= ( ref @ R @ L @ V @ X4 ) ) ).
% expr.distinct(5)
thf(fact_201_expr_Odistinct_I27_J,axiom,
! [V: $tType,L: $tType,R: $tType,X31: expr @ R @ L @ V,X32: expr @ R @ L @ V,X33: expr @ R @ L @ V,X4: expr @ R @ L @ V] :
( ( ite @ R @ L @ V @ X31 @ X32 @ X33 )
!= ( ref @ R @ L @ V @ X4 ) ) ).
% expr.distinct(27)
thf(fact_202_LID_092_060_094sub_062GE,axiom,
! [C: $tType,B: $tType,A: $tType,L2: A,S: B > ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) )] :
( ( member @ A @ L2 @ ( lID_G @ B @ A @ C @ S ) )
=> ~ ! [R7: B,Ls3: product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) )] :
( ( ( S @ R7 )
= ( some @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) @ Ls3 ) )
=> ~ ( member @ A @ L2 @ ( lID_L @ A @ B @ C @ Ls3 ) ) ) ) ).
% LID\<^sub>GE
thf(fact_203_substitution_Oonly__new__introduces__lids_H,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),R2: R,S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),L2: L] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
=> ( ! [Sigma2: L > ( option @ ( val @ R @ L @ V ) ),Tau2: L > ( option @ ( val @ R @ L @ V ) ),E3: cntxt @ R @ L @ V,V3: val @ R @ L @ V] :
( ( S @ R2 )
!= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma2 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau2 @ ( plug @ R @ L @ V @ E3 @ ( ref @ R @ L @ V @ ( ve @ R @ L @ V @ V3 ) ) ) ) ) ) )
=> ( ~ ( member @ L @ L2 @ ( lID_G @ R @ L @ V @ S ) )
=> ~ ( member @ L @ L2 @ ( lID_G @ R @ L @ V @ S2 ) ) ) ) ) ) ).
% substitution.only_new_introduces_lids'
thf(fact_204_substitution_Oonly__new__introduces__lids,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),R2: R,S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
=> ( ! [Sigma2: L > ( option @ ( val @ R @ L @ V ) ),Tau2: L > ( option @ ( val @ R @ L @ V ) ),E3: cntxt @ R @ L @ V,V3: val @ R @ L @ V] :
( ( S @ R2 )
!= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma2 @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau2 @ ( plug @ R @ L @ V @ E3 @ ( ref @ R @ L @ V @ ( ve @ R @ L @ V @ V3 ) ) ) ) ) ) )
=> ( ord_less_eq @ ( set @ L ) @ ( lID_G @ R @ L @ V @ S2 ) @ ( lID_G @ R @ L @ V @ S ) ) ) ) ) ).
% substitution.only_new_introduces_lids
thf(fact_205_substitution_Olocal__state__of__finite__restricted__global__state__is__finite_I2_J,axiom,
! [V: $tType,L: $tType,R: $tType,C: $tType,B: $tType,A: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: B > ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ),R3: B,Ls: product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ),R2: B] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R3 )
= ( some @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) @ Ls ) )
=> ( ( finite_finite @ A @ ( lID_G @ B @ A @ C @ ( fun_upd @ B @ ( option @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ) @ S @ R2 @ ( none @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( product_prod @ ( A > ( option @ ( val @ B @ A @ C ) ) ) @ ( expr @ B @ A @ C ) ) ) ) ) ) )
=> ( ( R2 != R3 )
=> ( finite_finite @ A @ ( lID_L @ A @ B @ C @ Ls ) ) ) ) ) ) ).
% substitution.local_state_of_finite_restricted_global_state_is_finite(2)
thf(fact_206_domains__subsume__def,axiom,
( ( domains_subsume @ l @ r @ v )
= ( ^ [Ls2: product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) )] : ( ord_less_eq @ ( set @ l ) @ ( lID_L @ l @ r @ v @ Ls2 ) @ ( doms @ l @ r @ v @ Ls2 ) ) ) ) ).
% domains_subsume_def
thf(fact_207_revision__step_Onew,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,V2: val @ r @ l @ v,L2: l] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ref @ r @ l @ v @ ( ve @ r @ l @ v @ V2 ) ) ) ) ) ) )
=> ( ~ ( member @ l @ L2 @ ( lID_G @ r @ l @ v @ S ) )
=> ( revision_step @ r @ l @ v @ subst @ R2 @ S @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ ( fun_upd @ l @ ( option @ ( val @ r @ l @ v ) ) @ Tau @ L2 @ ( some @ ( val @ r @ l @ v ) @ V2 ) ) @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( loc @ l @ r @ v @ L2 ) ) ) ) ) ) ) ) ) ) ).
% revision_step.new
thf(fact_208_val_Oinject_I3_J,axiom,
! [V: $tType,R: $tType,L: $tType,X34: L,Y3: L] :
( ( ( loc @ L @ R @ V @ X34 )
= ( loc @ L @ R @ V @ Y3 ) )
= ( X34 = Y3 ) ) ).
% val.inject(3)
thf(fact_209_new__pseudodeterministic,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,V2: val @ r @ l @ v,S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ref @ r @ l @ v @ ( ve @ r @ l @ v @ V2 ) ) ) ) ) ) )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
= ( ? [L4: l] :
( ~ ( member @ l @ L4 @ ( lID_G @ r @ l @ v @ S ) )
& ( S2
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ ( fun_upd @ l @ ( option @ ( val @ r @ l @ v ) ) @ Tau @ L4 @ ( some @ ( val @ r @ l @ v ) @ V2 ) ) @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( loc @ l @ r @ v @ L4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% new_pseudodeterministic
thf(fact_210_val_Odistinct_I15_J,axiom,
! [L: $tType,V: $tType,R: $tType,X34: L,X4: R] :
( ( loc @ L @ R @ V @ X34 )
!= ( rid @ R @ L @ V @ X4 ) ) ).
% val.distinct(15)
thf(fact_211_val_Odistinct_I3_J,axiom,
! [R: $tType,V: $tType,L: $tType,X1: const,X34: L] :
( ( cv @ R @ L @ V @ X1 )
!= ( loc @ L @ R @ V @ X34 ) ) ).
% val.distinct(3)
thf(fact_212_substitution_Odomains__subsume__def,axiom,
! [V: $tType,R: $tType,L: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),Ls: product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( domains_subsume @ L @ R @ V @ Ls )
= ( ord_less_eq @ ( set @ L ) @ ( lID_L @ L @ R @ V @ Ls ) @ ( doms @ L @ R @ V @ Ls ) ) ) ) ).
% substitution.domains_subsume_def
thf(fact_213_substitution_Onew__pseudodeterministic,axiom,
! [V: $tType,R: $tType,L: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,V2: val @ R @ L @ V,S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ref @ R @ L @ V @ ( ve @ R @ L @ V @ V2 ) ) ) ) ) ) )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
= ( ? [L4: L] :
( ~ ( member @ L @ L4 @ ( lID_G @ R @ L @ V @ S ) )
& ( S2
= ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ ( fun_upd @ L @ ( option @ ( val @ R @ L @ V ) ) @ Tau @ L4 @ ( some @ ( val @ R @ L @ V ) @ V2 ) ) @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( loc @ L @ R @ V @ L4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% substitution.new_pseudodeterministic
thf(fact_214_substitution_Orevision__step_Onew,axiom,
! [R: $tType,V: $tType,L: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,V2: val @ R @ L @ V,L2: L] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ref @ R @ L @ V @ ( ve @ R @ L @ V @ V2 ) ) ) ) ) ) )
=> ( ~ ( member @ L @ L2 @ ( lID_G @ R @ L @ V @ S ) )
=> ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ ( fun_upd @ L @ ( option @ ( val @ R @ L @ V ) ) @ Tau @ L2 @ ( some @ ( val @ R @ L @ V ) @ V2 ) ) @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( loc @ L @ R @ V @ L2 ) ) ) ) ) ) ) ) ) ) ) ).
% substitution.revision_step.new
thf(fact_215_revision__step__relaxed_Oset,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,L2: l,V2: val @ r @ l @ v] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( assign @ r @ l @ v @ ( ve @ r @ l @ v @ ( loc @ l @ r @ v @ L2 ) ) @ ( ve @ r @ l @ v @ V2 ) ) ) ) ) ) )
=> ( revisi1433015572elaxed @ r @ l @ v @ subst @ R2 @ S @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ ( fun_upd @ l @ ( option @ ( val @ r @ l @ v ) ) @ Tau @ L2 @ ( some @ ( val @ r @ l @ v ) @ V2 ) ) @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ unit ) ) ) ) ) ) ) ) ) ).
% revision_step_relaxed.set
thf(fact_216_substitution_Orevision__step__relaxed_Oset,axiom,
! [R: $tType,L: $tType,V: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,L2: L,V2: val @ R @ L @ V] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( assign @ R @ L @ V @ ( ve @ R @ L @ V @ ( loc @ L @ R @ V @ L2 ) ) @ ( ve @ R @ L @ V @ V2 ) ) ) ) ) ) )
=> ( revisi1433015572elaxed @ R @ L @ V @ Subst @ R2 @ S @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ ( fun_upd @ L @ ( option @ ( val @ R @ L @ V ) ) @ Tau @ L2 @ ( some @ ( val @ R @ L @ V ) @ V2 ) ) @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ unit ) ) ) ) ) ) ) ) ) ) ).
% substitution.revision_step_relaxed.set
thf(fact_217_expr_Oinject_I6_J,axiom,
! [V: $tType,L: $tType,R: $tType,X61: expr @ R @ L @ V,X62: expr @ R @ L @ V,Y61: expr @ R @ L @ V,Y62: expr @ R @ L @ V] :
( ( ( assign @ R @ L @ V @ X61 @ X62 )
= ( assign @ R @ L @ V @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% expr.inject(6)
thf(fact_218_expr_Odistinct_I39_J,axiom,
! [V: $tType,L: $tType,R: $tType,X4: expr @ R @ L @ V,X61: expr @ R @ L @ V,X62: expr @ R @ L @ V] :
( ( ref @ R @ L @ V @ X4 )
!= ( assign @ R @ L @ V @ X61 @ X62 ) ) ).
% expr.distinct(39)
thf(fact_219_expr_Odistinct_I51_J,axiom,
! [V: $tType,L: $tType,R: $tType,X61: expr @ R @ L @ V,X62: expr @ R @ L @ V,X7: expr @ R @ L @ V] :
( ( assign @ R @ L @ V @ X61 @ X62 )
!= ( rfork @ R @ L @ V @ X7 ) ) ).
% expr.distinct(51)
thf(fact_220_expr_Odistinct_I53_J,axiom,
! [V: $tType,L: $tType,R: $tType,X61: expr @ R @ L @ V,X62: expr @ R @ L @ V,X8: expr @ R @ L @ V] :
( ( assign @ R @ L @ V @ X61 @ X62 )
!= ( rjoin @ R @ L @ V @ X8 ) ) ).
% expr.distinct(53)
thf(fact_221_expr_Odistinct_I9_J,axiom,
! [V: $tType,L: $tType,R: $tType,X1: val @ R @ L @ V,X61: expr @ R @ L @ V,X62: expr @ R @ L @ V] :
( ( ve @ R @ L @ V @ X1 )
!= ( assign @ R @ L @ V @ X61 @ X62 ) ) ).
% expr.distinct(9)
thf(fact_222_expr_Odistinct_I31_J,axiom,
! [V: $tType,L: $tType,R: $tType,X31: expr @ R @ L @ V,X32: expr @ R @ L @ V,X33: expr @ R @ L @ V,X61: expr @ R @ L @ V,X62: expr @ R @ L @ V] :
( ( ite @ R @ L @ V @ X31 @ X32 @ X33 )
!= ( assign @ R @ L @ V @ X61 @ X62 ) ) ).
% expr.distinct(31)
thf(fact_223_revision__step_Oset,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,L2: l,V2: val @ r @ l @ v] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( assign @ r @ l @ v @ ( ve @ r @ l @ v @ ( loc @ l @ r @ v @ L2 ) ) @ ( ve @ r @ l @ v @ V2 ) ) ) ) ) ) )
=> ( ( member @ l @ L2 @ ( dom @ l @ ( val @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Sigma @ Tau ) ) )
=> ( revision_step @ r @ l @ v @ subst @ R2 @ S @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ ( fun_upd @ l @ ( option @ ( val @ r @ l @ v ) ) @ Tau @ L2 @ ( some @ ( val @ r @ l @ v ) @ V2 ) ) @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ unit ) ) ) ) ) ) ) ) ) ) ).
% revision_step.set
thf(fact_224_set__deterministic,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,L2: l,V2: val @ r @ l @ v,S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( assign @ r @ l @ v @ ( ve @ r @ l @ v @ ( loc @ l @ r @ v @ L2 ) ) @ ( ve @ r @ l @ v @ V2 ) ) ) ) ) ) )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
= ( ( member @ l @ L2 @ ( dom @ l @ ( val @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Sigma @ Tau ) ) )
& ( S2
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ ( fun_upd @ l @ ( option @ ( val @ r @ l @ v ) ) @ Tau @ L2 @ ( some @ ( val @ r @ l @ v ) @ V2 ) ) @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( cv @ r @ l @ v @ unit ) ) ) ) ) ) ) ) ) ) ) ).
% set_deterministic
thf(fact_225_dom__const,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( dom @ A @ B
@ ^ [X: A] : ( some @ B @ ( F @ X ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% dom_const
thf(fact_226_domI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A2: B,B2: A] :
( ( ( M @ A2 )
= ( some @ A @ B2 ) )
=> ( member @ B @ A2 @ ( dom @ B @ A @ M ) ) ) ).
% domI
thf(fact_227_domD,axiom,
! [A: $tType,B: $tType,A2: A,M: A > ( option @ B )] :
( ( member @ A @ A2 @ ( dom @ A @ B @ M ) )
=> ? [B4: B] :
( ( M @ A2 )
= ( some @ B @ B4 ) ) ) ).
% domD
thf(fact_228_dom__def,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M3: A > ( option @ B )] :
( collect @ A
@ ^ [A3: A] :
( ( M3 @ A3 )
!= ( none @ B ) ) ) ) ) ).
% dom_def
thf(fact_229_domIff,axiom,
! [A: $tType,B: $tType,A2: A,M: A > ( option @ B )] :
( ( member @ A @ A2 @ ( dom @ A @ B @ M ) )
= ( ( M @ A2 )
!= ( none @ B ) ) ) ).
% domIff
thf(fact_230_finite__map__freshness,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B )] :
( ( finite_finite @ A @ ( dom @ A @ B @ F ) )
=> ( ~ ( finite_finite @ A @ ( top_top @ ( set @ A ) ) )
=> ? [X6: A] :
( ( F @ X6 )
= ( none @ B ) ) ) ) ).
% finite_map_freshness
thf(fact_231_substitution_Oset__deterministic,axiom,
! [R: $tType,L: $tType,V: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,L2: L,V2: val @ R @ L @ V,S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( assign @ R @ L @ V @ ( ve @ R @ L @ V @ ( loc @ L @ R @ V @ L2 ) ) @ ( ve @ R @ L @ V @ V2 ) ) ) ) ) ) )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
= ( ( member @ L @ L2 @ ( dom @ L @ ( val @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Sigma @ Tau ) ) )
& ( S2
= ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ ( fun_upd @ L @ ( option @ ( val @ R @ L @ V ) ) @ Tau @ L2 @ ( some @ ( val @ R @ L @ V ) @ V2 ) ) @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ unit ) ) ) ) ) ) ) ) ) ) ) ) ).
% substitution.set_deterministic
thf(fact_232_substitution_Orevision__step_Oset,axiom,
! [R: $tType,L: $tType,V: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,L2: L,V2: val @ R @ L @ V] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( assign @ R @ L @ V @ ( ve @ R @ L @ V @ ( loc @ L @ R @ V @ L2 ) ) @ ( ve @ R @ L @ V @ V2 ) ) ) ) ) ) )
=> ( ( member @ L @ L2 @ ( dom @ L @ ( val @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Sigma @ Tau ) ) )
=> ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ ( fun_upd @ L @ ( option @ ( val @ R @ L @ V ) ) @ Tau @ L2 @ ( some @ ( val @ R @ L @ V ) @ V2 ) ) @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( cv @ R @ L @ V @ unit ) ) ) ) ) ) ) ) ) ) ) ).
% substitution.revision_step.set
thf(fact_233_revision__step_Oget,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,L2: l] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( read @ r @ l @ v @ ( ve @ r @ l @ v @ ( loc @ l @ r @ v @ L2 ) ) ) ) ) ) ) )
=> ( ( member @ l @ L2 @ ( dom @ l @ ( val @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Sigma @ Tau ) ) )
=> ( revision_step @ r @ l @ v @ subst @ R2 @ S @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( the @ ( val @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Sigma @ Tau @ L2 ) ) ) ) ) ) ) ) ) ) ) ).
% revision_step.get
thf(fact_234_revision__step__relaxed_Oget,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,L2: l] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( read @ r @ l @ v @ ( ve @ r @ l @ v @ ( loc @ l @ r @ v @ L2 ) ) ) ) ) ) ) )
=> ( revisi1433015572elaxed @ r @ l @ v @ subst @ R2 @ S @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( the @ ( val @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Sigma @ Tau @ L2 ) ) ) ) ) ) ) ) ) ) ).
% revision_step_relaxed.get
thf(fact_235__092_060S_062_092_060_094sub_062G__imp___092_060A_062__refl,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r] :
( ( domain398919503obally @ r @ l @ v @ S )
=> ( ( member @ r @ R2 @ ( dom @ r @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ S ) )
=> ( subsumes_accessible @ r @ l @ v @ R2 @ R2 @ S ) ) ) ).
% \<S>\<^sub>G_imp_\<A>_refl
thf(fact_236_subsumes__accessible__globally__def,axiom,
( ( subsum380726323obally @ r @ l @ v )
= ( ^ [S4: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
! [R_12: r,R_22: r] :
( ( member @ r @ R_12 @ ( dom @ r @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ S4 ) )
=> ( ( member @ r @ R_22 @ ( dom @ r @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ S4 ) )
=> ( subsumes_accessible @ r @ l @ v @ R_12 @ R_22 @ S4 ) ) ) ) ) ).
% subsumes_accessible_globally_def
thf(fact_237_expr_Oinject_I5_J,axiom,
! [V: $tType,L: $tType,R: $tType,X52: expr @ R @ L @ V,Y5: expr @ R @ L @ V] :
( ( ( read @ R @ L @ V @ X52 )
= ( read @ R @ L @ V @ Y5 ) )
= ( X52 = Y5 ) ) ).
% expr.inject(5)
thf(fact_238_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_239_get__deterministic,axiom,
! [S: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ),R2: r,Sigma: l > ( option @ ( val @ r @ l @ v ) ),Tau: l > ( option @ ( val @ r @ l @ v ) ),E: cntxt @ r @ l @ v,L2: l,S2: r > ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )] :
( ( ( S @ R2 )
= ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( read @ r @ l @ v @ ( ve @ r @ l @ v @ ( loc @ l @ r @ v @ L2 ) ) ) ) ) ) ) )
=> ( ( revision_step @ r @ l @ v @ subst @ R2 @ S @ S2 )
= ( ( member @ l @ L2 @ ( dom @ l @ ( val @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Sigma @ Tau ) ) )
& ( S2
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ Sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ Tau @ ( plug @ r @ l @ v @ E @ ( ve @ r @ l @ v @ ( the @ ( val @ r @ l @ v ) @ ( combine @ l @ ( val @ r @ l @ v ) @ Sigma @ Tau @ L2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% get_deterministic
thf(fact_240_expr_Odistinct_I37_J,axiom,
! [V: $tType,L: $tType,R: $tType,X4: expr @ R @ L @ V,X52: expr @ R @ L @ V] :
( ( ref @ R @ L @ V @ X4 )
!= ( read @ R @ L @ V @ X52 ) ) ).
% expr.distinct(37)
thf(fact_241_option_Oexpand,axiom,
! [A: $tType,Option: option @ A,Option2: option @ A] :
( ( ( Option
= ( none @ A ) )
= ( Option2
= ( none @ A ) ) )
=> ( ( ( Option
!= ( none @ A ) )
=> ( ( Option2
!= ( none @ A ) )
=> ( ( the @ A @ Option )
= ( the @ A @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_242_expr_Odistinct_I47_J,axiom,
! [V: $tType,L: $tType,R: $tType,X52: expr @ R @ L @ V,X7: expr @ R @ L @ V] :
( ( read @ R @ L @ V @ X52 )
!= ( rfork @ R @ L @ V @ X7 ) ) ).
% expr.distinct(47)
thf(fact_243_expr_Odistinct_I49_J,axiom,
! [V: $tType,L: $tType,R: $tType,X52: expr @ R @ L @ V,X8: expr @ R @ L @ V] :
( ( read @ R @ L @ V @ X52 )
!= ( rjoin @ R @ L @ V @ X8 ) ) ).
% expr.distinct(49)
thf(fact_244_expr_Odistinct_I7_J,axiom,
! [V: $tType,L: $tType,R: $tType,X1: val @ R @ L @ V,X52: expr @ R @ L @ V] :
( ( ve @ R @ L @ V @ X1 )
!= ( read @ R @ L @ V @ X52 ) ) ).
% expr.distinct(7)
thf(fact_245_expr_Odistinct_I29_J,axiom,
! [V: $tType,L: $tType,R: $tType,X31: expr @ R @ L @ V,X32: expr @ R @ L @ V,X33: expr @ R @ L @ V,X52: expr @ R @ L @ V] :
( ( ite @ R @ L @ V @ X31 @ X32 @ X33 )
!= ( read @ R @ L @ V @ X52 ) ) ).
% expr.distinct(29)
thf(fact_246_expr_Odistinct_I45_J,axiom,
! [V: $tType,L: $tType,R: $tType,X52: expr @ R @ L @ V,X61: expr @ R @ L @ V,X62: expr @ R @ L @ V] :
( ( read @ R @ L @ V @ X52 )
!= ( assign @ R @ L @ V @ X61 @ X62 ) ) ).
% expr.distinct(45)
thf(fact_247_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_248_option_Osel,axiom,
! [A: $tType,X22: A] :
( ( the @ A @ ( some @ A @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_249_RID_092_060_094sub_062GE,axiom,
! [C: $tType,A: $tType,B: $tType,R2: A,S: A > ( option @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) )] :
( ( member @ A @ R2 @ ( rID_G @ A @ B @ C @ S ) )
=> ( ~ ( member @ A @ R2 @ ( dom @ A @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) @ S ) )
=> ~ ! [R9: A,Ls3: product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) )] :
( ( ( S @ R9 )
= ( some @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( product_prod @ ( B > ( option @ ( val @ A @ B @ C ) ) ) @ ( expr @ A @ B @ C ) ) ) @ Ls3 ) )
=> ~ ( member @ A @ R2 @ ( rID_L @ B @ A @ C @ Ls3 ) ) ) ) ) ).
% RID\<^sub>GE
thf(fact_250_substitution_O_092_060S_062_092_060_094sub_062G__imp___092_060A_062__refl,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( domain398919503obally @ R @ L @ V @ S )
=> ( ( member @ R @ R2 @ ( dom @ R @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ S ) )
=> ( subsumes_accessible @ R @ L @ V @ R2 @ R2 @ S ) ) ) ) ).
% substitution.\<S>\<^sub>G_imp_\<A>_refl
thf(fact_251_substitution_Osubsumes__accessible__globally__def,axiom,
! [V: $tType,L: $tType,R: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( subsum380726323obally @ R @ L @ V @ S )
= ( ! [R_12: R,R_22: R] :
( ( member @ R @ R_12 @ ( dom @ R @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ S ) )
=> ( ( member @ R @ R_22 @ ( dom @ R @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ S ) )
=> ( subsumes_accessible @ R @ L @ V @ R_12 @ R_22 @ S ) ) ) ) ) ) ).
% substitution.subsumes_accessible_globally_def
thf(fact_252_substitution_Orevision__step__relaxed_Oget,axiom,
! [R: $tType,V: $tType,L: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,L2: L] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( read @ R @ L @ V @ ( ve @ R @ L @ V @ ( loc @ L @ R @ V @ L2 ) ) ) ) ) ) ) )
=> ( revisi1433015572elaxed @ R @ L @ V @ Subst @ R2 @ S @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( the @ ( val @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Sigma @ Tau @ L2 ) ) ) ) ) ) ) ) ) ) ) ).
% substitution.revision_step_relaxed.get
thf(fact_253_substitution_Oget__deterministic,axiom,
! [R: $tType,V: $tType,L: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,L2: L,S2: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) )] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( read @ R @ L @ V @ ( ve @ R @ L @ V @ ( loc @ L @ R @ V @ L2 ) ) ) ) ) ) ) )
=> ( ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ S2 )
= ( ( member @ L @ L2 @ ( dom @ L @ ( val @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Sigma @ Tau ) ) )
& ( S2
= ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( the @ ( val @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Sigma @ Tau @ L2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% substitution.get_deterministic
thf(fact_254_substitution_Orevision__step_Oget,axiom,
! [R: $tType,V: $tType,L: $tType,Subst: ( expr @ R @ L @ V ) > V > ( expr @ R @ L @ V ) > ( expr @ R @ L @ V ),S: R > ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ),R2: R,Sigma: L > ( option @ ( val @ R @ L @ V ) ),Tau: L > ( option @ ( val @ R @ L @ V ) ),E: cntxt @ R @ L @ V,L2: L] :
( ( substitution @ R @ L @ V @ Subst )
=> ( ( ( S @ R2 )
= ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( read @ R @ L @ V @ ( ve @ R @ L @ V @ ( loc @ L @ R @ V @ L2 ) ) ) ) ) ) ) )
=> ( ( member @ L @ L2 @ ( dom @ L @ ( val @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Sigma @ Tau ) ) )
=> ( revision_step @ R @ L @ V @ Subst @ R2 @ S @ ( fun_upd @ R @ ( option @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) ) @ S @ R2 @ ( some @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) ) @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( product_prod @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) ) @ Sigma @ ( product_Pair @ ( L > ( option @ ( val @ R @ L @ V ) ) ) @ ( expr @ R @ L @ V ) @ Tau @ ( plug @ R @ L @ V @ E @ ( ve @ R @ L @ V @ ( the @ ( val @ R @ L @ V ) @ ( combine @ L @ ( val @ R @ L @ V ) @ Sigma @ Tau @ L2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% substitution.revision_step.get
% Type constructors (3)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 )
=> ( order @ ( A6 > A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_1,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_2,axiom,
order @ $o ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ s_1 @ r2 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ x ) ) @ r3 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ tau @ ( plug @ r @ l @ v @ e @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ r4 ) ) ) ) ) ) ) @ r4
@ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ ( combine @ l @ ( val @ r @ l @ v ) @ sigma @ tau )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v )
@ ^ [X: l] : ( none @ ( val @ r @ l @ v ) )
@ e2 ) ) ) )
= ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) )
@ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ ( fun_upd @ r @ ( option @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) ) @ s_1 @ r3 @ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ sigma @ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) @ tau @ ( plug @ r @ l @ v @ e @ ( ve @ r @ l @ v @ ( rid @ r @ l @ v @ r4 ) ) ) ) ) ) ) @ r4
@ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) @ ( combine @ l @ ( val @ r @ l @ v ) @ sigma @ tau )
@ ( product_Pair @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v )
@ ^ [X: l] : ( none @ ( val @ r @ l @ v ) )
@ e2 ) ) ) )
@ r2
@ ( some @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( product_prod @ ( l > ( option @ ( val @ r @ l @ v ) ) ) @ ( expr @ r @ l @ v ) ) ) @ x ) ) ) ).
%------------------------------------------------------------------------------