TPTP Problem File: ITP080^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP080^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Irreducible problem prob_414__6626236_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 : Irreducible/prob_414__6626236_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 432 ( 123 unt; 80 typ; 0 def)
% Number of atoms : 1081 ( 546 equ; 0 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 4835 ( 250 ~; 18 |; 109 &;3799 @)
% ( 0 <=>; 659 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 441 ( 441 >; 0 *; 0 +; 0 <<)
% Number of symbols : 66 ( 65 usr; 16 con; 0-9 aty)
% Number of variables : 1493 ( 16 ^;1379 !; 98 ?;1493 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:39:25.169
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J_J_J,type,
list_l1129649930D_node: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J_J,type,
list_P738500740D_node: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J_J,type,
set_Pr1040144478D_node: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J,type,
produc1453890942D_node: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__node_Mtf__edgeD_J_J,type,
list_P561207620_edgeD: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_Itf__val_J_J,type,
option_list_val: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__node_Mtf__val_J,type,
produc1432036078de_val: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__node_J_J,type,
list_list_node: $tType ).
thf(ty_n_t__List__Olist_Itf__node_J,type,
list_node: $tType ).
thf(ty_n_t__List__Olist_Itf__val_J,type,
list_val: $tType ).
thf(ty_n_t__Set__Oset_Itf__node_J,type,
set_node: $tType ).
thf(ty_n_t__Set__Oset_Itf__val_J,type,
set_val: $tType ).
thf(ty_n_tf__node,type,
node: $tType ).
thf(ty_n_tf__val,type,
val: $tType ).
thf(ty_n_tf__g,type,
g: $tType ).
% Explicit typings (65)
thf(sy_c_Finite__Set_Ofinite_001tf__node,type,
finite_finite_node: set_node > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__val,type,
finite_finite_val: set_val > $o ).
thf(sy_c_Graph__path_Ograph__Entry_OEntryPath_001tf__g_001tf__node_001tf__edgeD,type,
graph_1994935542_edgeD: ( g > list_node ) > ( g > $o ) > ( g > node > list_P561207620_edgeD ) > ( g > node ) > g > list_node > $o ).
thf(sy_c_Graph__path_Ograph__Entry_OisIdom_001tf__g_001tf__node_001tf__edgeD,type,
graph_1670286392_edgeD: ( g > list_node ) > ( g > $o ) > ( g > node > list_P561207620_edgeD ) > ( g > node ) > g > node > node > $o ).
thf(sy_c_Graph__path_Ograph__Entry__base_Odominates_001tf__g_001tf__node_001tf__edgeD,type,
graph_436675702_edgeD: ( g > list_node ) > ( g > $o ) > ( g > node > list_P561207620_edgeD ) > ( g > node ) > g > node > node > $o ).
thf(sy_c_Graph__path_Ograph__path__base_OinEdges_001tf__g_001tf__node_001tf__edgeD,type,
graph_1947481694_edgeD: ( g > node > list_P561207620_edgeD ) > g > node > list_P738500740D_node ).
thf(sy_c_Graph__path_Ograph__path__base_Opath2_001tf__g_001tf__node_001tf__edgeD,type,
graph_1012773594_edgeD: ( g > list_node ) > ( g > $o ) > ( g > node > list_P561207620_edgeD ) > g > node > list_node > node > $o ).
thf(sy_c_Graph__path_Ograph__path__base_Opredecessors_001tf__g_001tf__node_001tf__edgeD,type,
graph_272749361_edgeD: ( g > node > list_P561207620_edgeD ) > g > node > list_node ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J,type,
append2096883353D_node: list_P738500740D_node > list_P738500740D_node > list_P738500740D_node ).
thf(sy_c_List_Oappend_001tf__node,type,
append_node: list_node > list_node > list_node ).
thf(sy_c_List_Oappend_001tf__val,type,
append_val: list_val > list_val > list_val ).
thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J,type,
distin1098566007D_node: list_P738500740D_node > $o ).
thf(sy_c_List_Odistinct_001tf__node,type,
distinct_node: list_node > $o ).
thf(sy_c_List_Odistinct_001tf__val,type,
distinct_val: list_val > $o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J_J,type,
cons_l1288865338D_node: list_P738500740D_node > list_l1129649930D_node > list_l1129649930D_node ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__node_J,type,
cons_list_node: list_node > list_list_node > list_list_node ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J,type,
cons_P1018517044D_node: produc1453890942D_node > list_P738500740D_node > list_P738500740D_node ).
thf(sy_c_List_Olist_OCons_001tf__node,type,
cons_node: node > list_node > list_node ).
thf(sy_c_List_Olist_OCons_001tf__val,type,
cons_val: val > list_val > list_val ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J_J,type,
nil_li1626782346D_node: list_l1129649930D_node ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__node_J,type,
nil_list_node: list_list_node ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J,type,
nil_Pr1769730692D_node: list_P738500740D_node ).
thf(sy_c_List_Olist_ONil_001tf__node,type,
nil_node: list_node ).
thf(sy_c_List_Olist_ONil_001tf__val,type,
nil_val: list_val ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J,type,
hd_Pro1395892457D_node: list_P738500740D_node > produc1453890942D_node ).
thf(sy_c_List_Olist_Ohd_001tf__node,type,
hd_node: list_node > node ).
thf(sy_c_List_Olist_Ohd_001tf__val,type,
hd_val: list_val > val ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J,type,
set_Pr1238794387D_node: list_P738500740D_node > set_Pr1040144478D_node ).
thf(sy_c_List_Olist_Oset_001tf__node,type,
set_node2: list_node > set_node ).
thf(sy_c_List_Olist_Oset_001tf__val,type,
set_val2: list_val > set_val ).
thf(sy_c_List_Olist_Otl_001t__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J,type,
tl_Pro1633633005D_node: list_P738500740D_node > list_P738500740D_node ).
thf(sy_c_List_Olist_Otl_001tf__node,type,
tl_node: list_node > list_node ).
thf(sy_c_List_Olist_Otl_001tf__val,type,
tl_val: list_val > list_val ).
thf(sy_c_Minimality_Ograph__Entry_Oreducible_001tf__g_001tf__node_001tf__edgeD,type,
graph_589078910_edgeD: ( g > list_node ) > ( g > $o ) > ( g > node > list_P561207620_edgeD ) > ( g > node ) > g > $o ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OallDefs_001tf__g_001tf__node_001tf__val,type,
sSA_CF139593942de_val: ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > node > set_val ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OdefAss_001tf__g_001tf__node_001tf__edgeD_001tf__val,type,
sSA_CF1156973626eD_val: ( g > list_node ) > ( g > $o ) > ( g > node > list_P561207620_edgeD ) > ( g > node ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > node > val > $o ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OphiDefs_001tf__g_001tf__node_001tf__val,type,
sSA_CF370335846de_val: ( g > produc1432036078de_val > option_list_val ) > g > node > set_val ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OphiUses_001tf__g_001tf__node_001tf__edgeD_001tf__val,type,
sSA_CF848637139eD_val: ( g > list_node ) > ( g > node > list_P561207620_edgeD ) > ( g > produc1432036078de_val > option_list_val ) > g > node > set_val ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_OdefNode_001tf__g_001tf__node_001tf__val,type,
sSA_CF551432799de_val: ( g > list_node ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > val > node ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_OphiArg_001tf__g_001tf__node_001tf__val,type,
sSA_CF1252180629de_val: ( g > list_node ) > ( g > node > set_val ) > ( g > produc1432036078de_val > option_list_val ) > g > val > val > $o ).
thf(sy_c_SSA__CFG_OCFG__base_OdefAss_H_001tf__g_001tf__node_001tf__edgeD_001tf__val,type,
sSA_CF1558836456eD_val: ( g > list_node ) > ( g > $o ) > ( g > node > list_P561207620_edgeD ) > ( g > node ) > ( g > node > set_val ) > g > node > val > $o ).
thf(sy_c_Set_OCollect_001tf__node,type,
collect_node: ( node > $o ) > set_node ).
thf(sy_c_Set_OCollect_001tf__val,type,
collect_val: ( val > $o ) > set_val ).
thf(sy_c_Sublist_Osuffix_001t__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J,type,
suffix1143830554D_node: list_P738500740D_node > list_P738500740D_node > $o ).
thf(sy_c_Sublist_Osuffix_001tf__node,type,
suffix_node: list_node > list_node > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__node_Mt__Product____Type__Oprod_Itf__edgeD_Mtf__node_J_J,type,
member1797643303D_node: produc1453890942D_node > set_Pr1040144478D_node > $o ).
thf(sy_c_member_001tf__node,type,
member_node: node > set_node > $o ).
thf(sy_c_member_001tf__val,type,
member_val: val > set_val > $o ).
thf(sy_v_Entry,type,
entry: g > node ).
thf(sy_v__092_060alpha_062n,type,
alpha_n: g > list_node ).
thf(sy_v__092_060phi_062_092_060_094sub_062r,type,
phi_r: val ).
thf(sy_v_defs,type,
defs: g > node > set_val ).
thf(sy_v_g,type,
g2: g ).
thf(sy_v_inEdges_H,type,
inEdges: g > node > list_P561207620_edgeD ).
thf(sy_v_invar,type,
invar: g > $o ).
thf(sy_v_m,type,
m: node ).
thf(sy_v_ms,type,
ms: list_node ).
thf(sy_v_n,type,
n: node ).
thf(sy_v_ns,type,
ns: list_node ).
thf(sy_v_phis,type,
phis: g > produc1432036078de_val > option_list_val ).
thf(sy_v_pred_092_060_094sub_062_092_060phi_062_092_060_094sub_062r____,type,
pred_phi_r: node ).
thf(sy_v_r,type,
r: val ).
thf(sy_v_rs_H____,type,
rs: list_node ).
thf(sy_v_rs____,type,
rs2: list_node ).
thf(sy_v_s,type,
s: val ).
% Relevant facts (351)
thf(fact_0_old_Opath2__hd,axiom,
! [G: g,N: node,Ns: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( N
= ( hd_node @ Ns ) ) ) ).
% old.path2_hd
thf(fact_1_False,axiom,
r != phi_r ).
% False
thf(fact_2_old_OEntry__in__graph,axiom,
! [G: g] : ( member_node @ ( entry @ G ) @ ( set_node2 @ ( alpha_n @ G ) ) ) ).
% old.Entry_in_graph
thf(fact_3_rs_H__props_I2_J,axiom,
graph_1994935542_edgeD @ alpha_n @ invar @ inEdges @ entry @ g2 @ rs ).
% rs'_props(2)
thf(fact_4_rs_H__props_I1_J,axiom,
graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ g2 @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ r ) @ rs @ pred_phi_r ).
% rs'_props(1)
thf(fact_5_old_OEntryPath__distinct,axiom,
! [G: g,Ns: list_node] :
( ( graph_1994935542_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ Ns )
=> ( distinct_node @ Ns ) ) ).
% old.EntryPath_distinct
thf(fact_6_old_O_092_060alpha_062n__distinct,axiom,
! [G: g] :
( ( invar @ G )
=> ( distinct_node @ ( alpha_n @ G ) ) ) ).
% old.\<alpha>n_distinct
thf(fact_7_rs_H__props_I3_J,axiom,
member_val @ r @ ( sSA_CF848637139eD_val @ alpha_n @ inEdges @ phis @ g2 @ pred_phi_r ) ).
% rs'_props(3)
thf(fact_8_assms_I10_J,axiom,
sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ g2 @ phi_r @ r ).
% assms(10)
thf(fact_9_allDefs__disjoint_H,axiom,
! [N: node,G: g,M: node,V: val] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( member_node @ M @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) )
=> ( ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ M ) )
=> ( N = M ) ) ) ) ) ).
% allDefs_disjoint'
thf(fact_10_assms_I7_J,axiom,
graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ g2 @ n @ ns @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ r ) ).
% assms(7)
thf(fact_11_defNode__eq,axiom,
! [N: node,G: g,V: val] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) )
=> ( ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ G @ V )
= N ) ) ) ).
% defNode_eq
thf(fact_12_rs_H__props_I4_J,axiom,
member_node @ pred_phi_r @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ g2 @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_r ) ) ) ).
% rs'_props(4)
thf(fact_13_defs__in__allDefs,axiom,
! [V: val,G: g,N: node] :
( ( member_val @ V @ ( defs @ G @ N ) )
=> ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) ) ) ).
% defs_in_allDefs
thf(fact_14_FormalSSA__Misc_Oin__set__tlD,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ ( tl_val @ Xs ) ) )
=> ( member_val @ X @ ( set_val2 @ Xs ) ) ) ).
% FormalSSA_Misc.in_set_tlD
thf(fact_15_FormalSSA__Misc_Oin__set__tlD,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ ( tl_node @ Xs ) ) )
=> ( member_node @ X @ ( set_node2 @ Xs ) ) ) ).
% FormalSSA_Misc.in_set_tlD
thf(fact_16_old_Opath2__hd__in___092_060alpha_062n,axiom,
! [G: g,N: node,Ns: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) ) ) ).
% old.path2_hd_in_\<alpha>n
thf(fact_17_old_Opath2__hd__in__ns,axiom,
! [G: g,N: node,Ns: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( member_node @ N @ ( set_node2 @ Ns ) ) ) ).
% old.path2_hd_in_ns
thf(fact_18_old_Opath2__in___092_060alpha_062n,axiom,
! [G: g,N: node,Ns: list_node,M: node,L: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( member_node @ L @ ( set_node2 @ Ns ) )
=> ( member_node @ L @ ( set_node2 @ ( alpha_n @ G ) ) ) ) ) ).
% old.path2_in_\<alpha>n
thf(fact_19_old_Oinvar,axiom,
! [G: g] : ( invar @ G ) ).
% old.invar
thf(fact_20_old_Opath2__tl__in___092_060alpha_062n,axiom,
! [G: g,N: node,Ns: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( member_node @ M @ ( set_node2 @ ( alpha_n @ G ) ) ) ) ).
% old.path2_tl_in_\<alpha>n
thf(fact_21_old_Opath2__last__in__ns,axiom,
! [G: g,N: node,Ns: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( member_node @ M @ ( set_node2 @ Ns ) ) ) ).
% old.path2_last_in_ns
thf(fact_22_old_Osuccessor__is__node,axiom,
! [N: node,G: g,N2: node] :
( ( member_node @ N @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ N2 ) ) )
=> ( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( invar @ G )
=> ( member_node @ N2 @ ( set_node2 @ ( alpha_n @ G ) ) ) ) ) ) ).
% old.successor_is_node
thf(fact_23_old_Opredecessor__is__node,axiom,
! [N: node,G: g,N2: node] :
( ( member_node @ N @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ N2 ) ) )
=> ( ( invar @ G )
=> ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) ) ) ) ).
% old.predecessor_is_node
thf(fact_24_old_Opath2__forget__hd,axiom,
! [G: g,N: node,Ns: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( hd_node @ Ns ) @ Ns @ M ) ) ).
% old.path2_forget_hd
thf(fact_25_old_OEntry__reachesE,axiom,
! [N: node,G: g] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( invar @ G )
=> ~ ! [Ns2: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( entry @ G ) @ Ns2 @ N )
=> ~ ( graph_1994935542_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ Ns2 ) ) ) ) ).
% old.Entry_reachesE
thf(fact_26__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062rs_H_Apred_092_060_094sub_062_092_060phi_062_092_060_094sub_062r_O_A_092_060lbrakk_062g_A_092_060turnstile_062_AdefNode_Ag_Ar_Nrs_H_092_060rightarrow_062pred_092_060_094sub_062_092_060phi_062_092_060_094sub_062r_059_Aold_OEntryPath_Ag_Ars_H_059_Ar_A_092_060in_062_AphiUses_Ag_Apred_092_060_094sub_062_092_060phi_062_092_060_094sub_062r_059_Apred_092_060_094sub_062_092_060phi_062_092_060_094sub_062r_A_092_060in_062_Aset_A_Iold_Opredecessors_Ag_A_IdefNode_Ag_A_092_060phi_062_092_060_094sub_062r_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Rs: list_node,Pred_phi_r: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ g2 @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ r ) @ Rs @ Pred_phi_r )
=> ( ( graph_1994935542_edgeD @ alpha_n @ invar @ inEdges @ entry @ g2 @ Rs )
=> ( ( member_val @ r @ ( sSA_CF848637139eD_val @ alpha_n @ inEdges @ phis @ g2 @ Pred_phi_r ) )
=> ~ ( member_node @ Pred_phi_r @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ g2 @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_r ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>rs' pred\<^sub>\<phi>\<^sub>r. \<lbrakk>g \<turnstile> defNode g r-rs'\<rightarrow>pred\<^sub>\<phi>\<^sub>r; old.EntryPath g rs'; r \<in> phiUses g pred\<^sub>\<phi>\<^sub>r; pred\<^sub>\<phi>\<^sub>r \<in> set (old.predecessors g (defNode g \<phi>\<^sub>r))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_27_old_OEntry__reaches,axiom,
! [N: node,G: g] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( invar @ G )
=> ? [Ns2: list_node] : ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( entry @ G ) @ Ns2 @ N ) ) ) ).
% old.Entry_reaches
thf(fact_28_old_Oidom__ex,axiom,
! [G: g,N: node] :
( ( invar @ G )
=> ( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( N
!= ( entry @ G ) )
=> ? [X2: node] :
( ( graph_1670286392_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ X2 )
& ! [Y: node] :
( ( graph_1670286392_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ Y )
=> ( Y = X2 ) ) ) ) ) ) ).
% old.idom_ex
thf(fact_29_FormalSSA__Misc_Odistinct__hd__tl,axiom,
! [Xs: list_val] :
( ( distinct_val @ Xs )
=> ~ ( member_val @ ( hd_val @ Xs ) @ ( set_val2 @ ( tl_val @ Xs ) ) ) ) ).
% FormalSSA_Misc.distinct_hd_tl
thf(fact_30_FormalSSA__Misc_Odistinct__hd__tl,axiom,
! [Xs: list_node] :
( ( distinct_node @ Xs )
=> ~ ( member_node @ ( hd_node @ Xs ) @ ( set_node2 @ ( tl_node @ Xs ) ) ) ) ).
% FormalSSA_Misc.distinct_hd_tl
thf(fact_31_defAss_H__extend,axiom,
! [G: g,M: node,V: val,N: node,Ns: list_node] :
( ( sSA_CF1558836456eD_val @ alpha_n @ invar @ inEdges @ entry @ defs @ G @ M @ V )
=> ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ! [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ ( tl_node @ Ns ) ) )
=> ~ ( member_val @ V @ ( defs @ G @ X2 ) ) )
=> ( sSA_CF1558836456eD_val @ alpha_n @ invar @ inEdges @ entry @ defs @ G @ N @ V ) ) ) ) ).
% defAss'_extend
thf(fact_32_defAss_H__def,axiom,
! [G: g,M: node,V: val] :
( ( sSA_CF1558836456eD_val @ alpha_n @ invar @ inEdges @ entry @ defs @ G @ M @ V )
= ( ! [Ns3: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( entry @ G ) @ Ns3 @ M )
=> ? [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ Ns3 ) )
& ( member_val @ V @ ( defs @ G @ X3 ) ) ) ) ) ) ).
% defAss'_def
thf(fact_33_defAss_HI,axiom,
! [G: g,M: node,V: val] :
( ! [Ns2: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( entry @ G ) @ Ns2 @ M )
=> ? [X4: node] :
( ( member_node @ X4 @ ( set_node2 @ Ns2 ) )
& ( member_val @ V @ ( defs @ G @ X4 ) ) ) )
=> ( sSA_CF1558836456eD_val @ alpha_n @ invar @ inEdges @ entry @ defs @ G @ M @ V ) ) ).
% defAss'I
thf(fact_34_defAss_HE,axiom,
! [G: g,M: node,V: val,Ns: list_node] :
( ( sSA_CF1558836456eD_val @ alpha_n @ invar @ inEdges @ entry @ defs @ G @ M @ V )
=> ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( entry @ G ) @ Ns @ M )
=> ~ ! [N3: node] :
( ( member_node @ N3 @ ( set_node2 @ Ns ) )
=> ~ ( member_val @ V @ ( defs @ G @ N3 ) ) ) ) ) ).
% defAss'E
thf(fact_35_defAss__extend,axiom,
! [G: g,M: node,V: val,N: node,Ns: list_node] :
( ( sSA_CF1156973626eD_val @ alpha_n @ invar @ inEdges @ entry @ defs @ phis @ G @ M @ V )
=> ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ! [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ ( tl_node @ Ns ) ) )
=> ~ ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ X2 ) ) )
=> ( sSA_CF1156973626eD_val @ alpha_n @ invar @ inEdges @ entry @ defs @ phis @ G @ N @ V ) ) ) ) ).
% defAss_extend
thf(fact_36_assms_I8_J,axiom,
graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ g2 @ m @ ms @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ s ) ).
% assms(8)
thf(fact_37_defAss__def,axiom,
! [G: g,M: node,V: val] :
( ( sSA_CF1156973626eD_val @ alpha_n @ invar @ inEdges @ entry @ defs @ phis @ G @ M @ V )
= ( ! [Ns3: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( entry @ G ) @ Ns3 @ M )
=> ? [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ Ns3 ) )
& ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ X3 ) ) ) ) ) ) ).
% defAss_def
thf(fact_38_defAssI,axiom,
! [G: g,M: node,V: val] :
( ! [Ns2: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( entry @ G ) @ Ns2 @ M )
=> ? [X4: node] :
( ( member_node @ X4 @ ( set_node2 @ Ns2 ) )
& ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ X4 ) ) ) )
=> ( sSA_CF1156973626eD_val @ alpha_n @ invar @ inEdges @ entry @ defs @ phis @ G @ M @ V ) ) ).
% defAssI
thf(fact_39_defAssD,axiom,
! [G: g,M: node,V: val,Ns: list_node] :
( ( sSA_CF1156973626eD_val @ alpha_n @ invar @ inEdges @ entry @ defs @ phis @ G @ M @ V )
=> ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( entry @ G ) @ Ns @ M )
=> ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Ns ) )
& ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ X2 ) ) ) ) ) ).
% defAssD
thf(fact_40_phiDefs__in__allDefs,axiom,
! [V: val,G: g,N: node] :
( ( member_val @ V @ ( sSA_CF370335846de_val @ phis @ G @ N ) )
=> ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) ) ) ).
% phiDefs_in_allDefs
thf(fact_41_defAss__dominating,axiom,
! [N: node,G: g,V: val] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( sSA_CF1156973626eD_val @ alpha_n @ invar @ inEdges @ entry @ defs @ phis @ G @ N @ V )
= ( ? [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ ( alpha_n @ G ) ) )
& ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ X3 @ N )
& ( member_val @ V @ ( sSA_CF139593942de_val @ defs @ phis @ G @ X3 ) ) ) ) ) ) ).
% defAss_dominating
thf(fact_42_old_Osuccessor__in___092_060alpha_062n,axiom,
! [G: g,N: node] :
( ( ( graph_272749361_edgeD @ inEdges @ G @ N )
!= nil_node )
=> ( ( invar @ G )
=> ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) ) ) ) ).
% old.successor_in_\<alpha>n
thf(fact_43_phiUses__finite,axiom,
! [N: node,G: g] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( finite_finite_val @ ( sSA_CF848637139eD_val @ alpha_n @ inEdges @ phis @ G @ N ) ) ) ).
% phiUses_finite
thf(fact_44_old_Opath2__split__last__prop,axiom,
! [G: g,N: node,Ns: list_node,M: node,P: node > $o] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ? [X4: node] :
( ( member_node @ X4 @ ( set_node2 @ Ns ) )
& ( P @ X4 ) )
=> ~ ! [N4: node,Ns4: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N4 @ Ns4 @ M )
=> ( ( P @ N4 )
=> ( ! [X4: node] :
( ( member_node @ X4 @ ( set_node2 @ ( tl_node @ Ns4 ) ) )
=> ~ ( P @ X4 ) )
=> ~ ( suffix_node @ Ns4 @ Ns ) ) ) ) ) ) ).
% old.path2_split_last_prop
thf(fact_45_allDefs__finite,axiom,
! [N: node,G: g] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( finite_finite_val @ ( sSA_CF139593942de_val @ defs @ phis @ G @ N ) ) ) ).
% allDefs_finite
thf(fact_46_old_Opath2__not__Nil,axiom,
! [G: g,N: node,Ns: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( Ns != nil_node ) ) ).
% old.path2_not_Nil
thf(fact_47_mem__Collect__eq,axiom,
! [A: node,P: node > $o] :
( ( member_node @ A @ ( collect_node @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A: val,P: val > $o] :
( ( member_val @ A @ ( collect_val @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_node] :
( ( collect_node
@ ^ [X3: node] : ( member_node @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_val] :
( ( collect_val
@ ^ [X3: val] : ( member_val @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_old_Opath2__not__Nil2,axiom,
! [G: g,N: node,M: node] :
~ ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ nil_node @ M ) ).
% old.path2_not_Nil2
thf(fact_52_old_Opath2__app,axiom,
! [G: g,N: node,Ns: list_node,M: node,Ms: list_node,L: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ M @ Ms @ L )
=> ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( append_node @ Ns @ ( tl_node @ Ms ) ) @ L ) ) ) ).
% old.path2_app
thf(fact_53_defs__finite,axiom,
! [G: g,N: node] : ( finite_finite_val @ ( defs @ G @ N ) ) ).
% defs_finite
thf(fact_54_phiDefs__finite,axiom,
! [G: g,N: node] : ( finite_finite_val @ ( sSA_CF370335846de_val @ phis @ G @ N ) ) ).
% phiDefs_finite
thf(fact_55_strict__dom__trans_H,axiom,
! [N: node,M: node,G: g,M2: node] :
( ( ( N != M )
& ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M ) )
=> ( ( ( M != M2 )
& ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ M @ M2 ) )
=> ( ( N != M2 )
& ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M2 ) ) ) ) ).
% strict_dom_trans'
thf(fact_56_old_Ostrict__dom__trans,axiom,
! [G: g,N: node,M: node,M2: node] :
( ( invar @ G )
=> ( ( ( N != M )
& ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M ) )
=> ( ( ( M != M2 )
& ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ M @ M2 ) )
=> ( ( N != M2 )
& ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M2 ) ) ) ) ) ).
% old.strict_dom_trans
thf(fact_57_old_Odominates__trans,axiom,
! [G: g,N: node,N2: node,N5: node] :
( ( invar @ G )
=> ( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ N2 )
=> ( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N2 @ N5 )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ N5 ) ) ) ) ).
% old.dominates_trans
thf(fact_58_old_Odominates__antitrans,axiom,
! [G: g,N_1: node,M: node,N_2: node] :
( ( invar @ G )
=> ( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N_1 @ M )
=> ( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N_2 @ M )
=> ( ~ ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N_1 @ N_2 )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N_2 @ N_1 ) ) ) ) ) ).
% old.dominates_antitrans
thf(fact_59_old_Odominates__antisymm,axiom,
! [G: g,N: node,N2: node] :
( ( invar @ G )
=> ( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ N2 )
=> ( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N2 @ N )
=> ( N = N2 ) ) ) ) ).
% old.dominates_antisymm
thf(fact_60_dominates__trans_H,axiom,
! [G: g,N: node,N2: node,N5: node] :
( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ N2 )
=> ( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N2 @ N5 )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ N5 ) ) ) ).
% dominates_trans'
thf(fact_61_dominates__antisymm_H,axiom,
! [G: g,N: node,N2: node] :
( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ N2 )
=> ( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N2 @ N )
=> ( N = N2 ) ) ) ).
% dominates_antisymm'
thf(fact_62_dominates__refl_H,axiom,
! [N: node,G: g] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ N ) ) ).
% dominates_refl'
thf(fact_63_old_Odominates__path,axiom,
! [G: g,N: node,M: node] :
( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M )
=> ( ( invar @ G )
=> ~ ! [Ns2: list_node] :
~ ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns2 @ M ) ) ) ).
% old.dominates_path
thf(fact_64_old_Odominates__mid,axiom,
! [G: g,N: node,X: node,M: node,Ns: list_node] :
( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ X )
=> ( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ X @ M )
=> ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( invar @ G )
=> ( member_node @ X @ ( set_node2 @ Ns ) ) ) ) ) ) ).
% old.dominates_mid
thf(fact_65_old_Odominates__def,axiom,
! [G: g,N: node,M: node] :
( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M )
= ( ( member_node @ M @ ( set_node2 @ ( alpha_n @ G ) ) )
& ! [Ns3: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( entry @ G ) @ Ns3 @ M )
=> ( member_node @ N @ ( set_node2 @ Ns3 ) ) ) ) ) ).
% old.dominates_def
thf(fact_66_old_OdominatesE,axiom,
! [G: g,N: node,M: node] :
( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M )
=> ~ ( ( member_node @ M @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ~ ! [Ns5: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( entry @ G ) @ Ns5 @ M )
=> ( member_node @ N @ ( set_node2 @ Ns5 ) ) ) ) ) ).
% old.dominatesE
thf(fact_67_old_Odominates__unsnoc,axiom,
! [G: g,N: node,M: node,M2: node] :
( ( invar @ G )
=> ( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M )
=> ( ( member_node @ M2 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ M ) ) )
=> ( ( N != M )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M2 ) ) ) ) ) ).
% old.dominates_unsnoc
thf(fact_68_non__dominated__predecessor,axiom,
! [N: node,G: g] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( N
!= ( entry @ G ) )
=> ~ ! [M3: node] :
( ( member_node @ M3 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ N ) ) )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M3 ) ) ) ) ).
% non_dominated_predecessor
thf(fact_69_old_OisIdom__def,axiom,
! [G: g,N: node,M: node] :
( ( graph_1670286392_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M )
= ( ( M != N )
& ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ M @ N )
& ! [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( ( X3 != N )
& ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ X3 @ N ) )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ X3 @ M ) ) ) ) ) ).
% old.isIdom_def
thf(fact_70_old_OEntryPath__suffix,axiom,
! [G: g,Ns: list_node,Ns6: list_node] :
( ( graph_1994935542_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ Ns )
=> ( ( suffix_node @ Ns6 @ Ns )
=> ( ( Ns6 != nil_node )
=> ( graph_1994935542_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ Ns6 ) ) ) ) ).
% old.EntryPath_suffix
thf(fact_71_old_Odominates__unsnoc_H,axiom,
! [G: g,N: node,M: node,M2: node,Ms: list_node] :
( ( invar @ G )
=> ( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M )
=> ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ M2 @ Ms @ M )
=> ( ! [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ ( tl_node @ Ms ) ) )
=> ( X2 != N ) )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M2 ) ) ) ) ) ).
% old.dominates_unsnoc'
thf(fact_72_old_Odominates__extend,axiom,
! [G: g,N: node,M: node,M2: node,Ms: list_node] :
( ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M )
=> ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ M2 @ Ms @ M )
=> ( ~ ( member_node @ N @ ( set_node2 @ ( tl_node @ Ms ) ) )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M2 ) ) ) ) ).
% old.dominates_extend
thf(fact_73_old_OEntry__no__predecessor,axiom,
! [G: g] :
( ( graph_272749361_edgeD @ inEdges @ G @ ( entry @ G ) )
= nil_node ) ).
% old.Entry_no_predecessor
thf(fact_74_old_Odominates__refl,axiom,
! [G: g,N: node] :
( ( invar @ G )
=> ( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ N ) ) ) ).
% old.dominates_refl
thf(fact_75_old_OEntry__dominates,axiom,
! [G: g,N: node] :
( ( invar @ G )
=> ( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ ( entry @ G ) @ N ) ) ) ).
% old.Entry_dominates
thf(fact_76_old_OdominatesI,axiom,
! [M: node,G: g,N: node] :
( ( member_node @ M @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ! [Ns2: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( entry @ G ) @ Ns2 @ M )
=> ( member_node @ N @ ( set_node2 @ Ns2 ) ) )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ N @ M ) ) ) ).
% old.dominatesI
thf(fact_77_old_OEntry__iff__unreachable,axiom,
! [G: g,N: node] :
( ( invar @ G )
=> ( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( ( graph_272749361_edgeD @ inEdges @ G @ N )
= nil_node )
= ( N
= ( entry @ G ) ) ) ) ) ).
% old.Entry_iff_unreachable
thf(fact_78_old_Oreducible__def,axiom,
! [G: g] :
( ( graph_589078910_edgeD @ alpha_n @ invar @ inEdges @ entry @ G )
= ( ! [N6: node,Ns3: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N6 @ Ns3 @ N6 )
=> ? [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ Ns3 ) )
& ! [Y2: node] :
( ( member_node @ Y2 @ ( set_node2 @ Ns3 ) )
=> ( graph_436675702_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ X3 @ Y2 ) ) ) ) ) ) ).
% old.reducible_def
thf(fact_79_in__hd__or__tl__conv,axiom,
! [L: list_val,X: val] :
( ( L != nil_val )
=> ( ( ( X
= ( hd_val @ L ) )
| ( member_val @ X @ ( set_val2 @ ( tl_val @ L ) ) ) )
= ( member_val @ X @ ( set_val2 @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_80_in__hd__or__tl__conv,axiom,
! [L: list_P738500740D_node,X: produc1453890942D_node] :
( ( L != nil_Pr1769730692D_node )
=> ( ( ( X
= ( hd_Pro1395892457D_node @ L ) )
| ( member1797643303D_node @ X @ ( set_Pr1238794387D_node @ ( tl_Pro1633633005D_node @ L ) ) ) )
= ( member1797643303D_node @ X @ ( set_Pr1238794387D_node @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_81_in__hd__or__tl__conv,axiom,
! [L: list_node,X: node] :
( ( L != nil_node )
=> ( ( ( X
= ( hd_node @ L ) )
| ( member_node @ X @ ( set_node2 @ ( tl_node @ L ) ) ) )
= ( member_node @ X @ ( set_node2 @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_82_old_Opath2__cases,axiom,
! [G: g,N: node,Ns: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( ( Ns
= ( cons_node @ N @ nil_node ) )
=> ( M != N ) )
=> ~ ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( hd_node @ ( tl_node @ Ns ) ) @ ( tl_node @ Ns ) @ M )
=> ~ ( member_node @ N @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ ( hd_node @ ( tl_node @ Ns ) ) ) ) ) ) ) ) ).
% old.path2_cases
thf(fact_83_old_Opath2__snoc,axiom,
! [G: g,N: node,Ns: list_node,M: node,M2: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( member_node @ M @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ M2 ) ) )
=> ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( append_node @ Ns @ ( cons_node @ M2 @ nil_node ) ) @ M2 ) ) ) ).
% old.path2_snoc
thf(fact_84_old_Opath2__rev__induct,axiom,
! [G: g,N: node,Ns: list_node,M: node,P: node > list_node > node > $o] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( P @ N @ ( cons_node @ N @ nil_node ) @ N ) )
=> ( ! [Ns2: list_node,M4: node,M3: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns2 @ M4 )
=> ( ( P @ N @ Ns2 @ M4 )
=> ( ( member_node @ M4 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ M3 ) ) )
=> ( P @ N @ ( append_node @ Ns2 @ ( cons_node @ M3 @ nil_node ) ) @ M3 ) ) ) )
=> ( P @ N @ Ns @ M ) ) ) ) ).
% old.path2_rev_induct
thf(fact_85_tl__append2,axiom,
! [Xs: list_node,Ys: list_node] :
( ( Xs != nil_node )
=> ( ( tl_node @ ( append_node @ Xs @ Ys ) )
= ( append_node @ ( tl_node @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_86_tl__append2,axiom,
! [Xs: list_P738500740D_node,Ys: list_P738500740D_node] :
( ( Xs != nil_Pr1769730692D_node )
=> ( ( tl_Pro1633633005D_node @ ( append2096883353D_node @ Xs @ Ys ) )
= ( append2096883353D_node @ ( tl_Pro1633633005D_node @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_87_hd__append2,axiom,
! [Xs: list_node,Ys: list_node] :
( ( Xs != nil_node )
=> ( ( hd_node @ ( append_node @ Xs @ Ys ) )
= ( hd_node @ Xs ) ) ) ).
% hd_append2
thf(fact_88_hd__append2,axiom,
! [Xs: list_P738500740D_node,Ys: list_P738500740D_node] :
( ( Xs != nil_Pr1769730692D_node )
=> ( ( hd_Pro1395892457D_node @ ( append2096883353D_node @ Xs @ Ys ) )
= ( hd_Pro1395892457D_node @ Xs ) ) ) ).
% hd_append2
thf(fact_89_same__suffix__nil,axiom,
! [Ys: list_P738500740D_node,Xs: list_P738500740D_node] :
( ( suffix1143830554D_node @ ( append2096883353D_node @ Ys @ Xs ) @ Xs )
= ( Ys = nil_Pr1769730692D_node ) ) ).
% same_suffix_nil
thf(fact_90_same__suffix__nil,axiom,
! [Ys: list_node,Xs: list_node] :
( ( suffix_node @ ( append_node @ Ys @ Xs ) @ Xs )
= ( Ys = nil_node ) ) ).
% same_suffix_nil
thf(fact_91_old_Opath2__induct,axiom,
! [G: g,N: node,Ns: list_node,M: node,P: node > list_node > node > $o] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( ( invar @ G )
=> ( P @ M @ ( cons_node @ M @ nil_node ) @ M ) )
=> ( ! [Ns2: list_node,N4: node,N3: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N3 @ Ns2 @ M )
=> ( ( P @ N3 @ Ns2 @ M )
=> ( ( member_node @ N4 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ N3 ) ) )
=> ( P @ N4 @ ( cons_node @ N4 @ Ns2 ) @ M ) ) ) )
=> ( P @ N @ Ns @ M ) ) ) ) ).
% old.path2_induct
thf(fact_92_old_Oelem__set__implies__elem__tl__app__cons,axiom,
! [X: val,Xs: list_val,Ys: list_val,Y3: val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ( member_val @ X @ ( set_val2 @ ( tl_val @ ( append_val @ Ys @ ( cons_val @ Y3 @ Xs ) ) ) ) ) ) ).
% old.elem_set_implies_elem_tl_app_cons
thf(fact_93_old_Oelem__set__implies__elem__tl__app__cons,axiom,
! [X: node,Xs: list_node,Ys: list_node,Y3: node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ( member_node @ X @ ( set_node2 @ ( tl_node @ ( append_node @ Ys @ ( cons_node @ Y3 @ Xs ) ) ) ) ) ) ).
% old.elem_set_implies_elem_tl_app_cons
thf(fact_94_list_Oinject,axiom,
! [X21: node,X22: list_node,Y21: node,Y22: list_node] :
( ( ( cons_node @ X21 @ X22 )
= ( cons_node @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_95_same__append__eq,axiom,
! [Xs: list_node,Ys: list_node,Zs: list_node] :
( ( ( append_node @ Xs @ Ys )
= ( append_node @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_96_append__same__eq,axiom,
! [Ys: list_node,Xs: list_node,Zs: list_node] :
( ( ( append_node @ Ys @ Xs )
= ( append_node @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_97_append__assoc,axiom,
! [Xs: list_node,Ys: list_node,Zs: list_node] :
( ( append_node @ ( append_node @ Xs @ Ys ) @ Zs )
= ( append_node @ Xs @ ( append_node @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_98_append_Oassoc,axiom,
! [A: list_node,B: list_node,C: list_node] :
( ( append_node @ ( append_node @ A @ B ) @ C )
= ( append_node @ A @ ( append_node @ B @ C ) ) ) ).
% append.assoc
thf(fact_99_suffix__order_Odual__order_Orefl,axiom,
! [A: list_node] : ( suffix_node @ A @ A ) ).
% suffix_order.dual_order.refl
thf(fact_100_suffix__order_Oorder__refl,axiom,
! [X: list_node] : ( suffix_node @ X @ X ) ).
% suffix_order.order_refl
thf(fact_101_old_Opath2__split_I2_J,axiom,
! [G: g,N: node,Ns: list_node,N2: node,Ns6: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( append_node @ Ns @ ( cons_node @ N2 @ Ns6 ) ) @ M )
=> ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N2 @ ( cons_node @ N2 @ Ns6 ) @ M ) ) ).
% old.path2_split(2)
thf(fact_102_List_Ofinite__set,axiom,
! [Xs: list_node] : ( finite_finite_node @ ( set_node2 @ Xs ) ) ).
% List.finite_set
thf(fact_103_List_Ofinite__set,axiom,
! [Xs: list_val] : ( finite_finite_val @ ( set_val2 @ Xs ) ) ).
% List.finite_set
thf(fact_104_append_Oright__neutral,axiom,
! [A: list_node] :
( ( append_node @ A @ nil_node )
= A ) ).
% append.right_neutral
thf(fact_105_append_Oright__neutral,axiom,
! [A: list_P738500740D_node] :
( ( append2096883353D_node @ A @ nil_Pr1769730692D_node )
= A ) ).
% append.right_neutral
thf(fact_106_empty__append__eq__id,axiom,
( ( append_node @ nil_node )
= ( ^ [X3: list_node] : X3 ) ) ).
% empty_append_eq_id
thf(fact_107_empty__append__eq__id,axiom,
( ( append2096883353D_node @ nil_Pr1769730692D_node )
= ( ^ [X3: list_P738500740D_node] : X3 ) ) ).
% empty_append_eq_id
thf(fact_108_append__is__Nil__conv,axiom,
! [Xs: list_node,Ys: list_node] :
( ( ( append_node @ Xs @ Ys )
= nil_node )
= ( ( Xs = nil_node )
& ( Ys = nil_node ) ) ) ).
% append_is_Nil_conv
thf(fact_109_append__is__Nil__conv,axiom,
! [Xs: list_P738500740D_node,Ys: list_P738500740D_node] :
( ( ( append2096883353D_node @ Xs @ Ys )
= nil_Pr1769730692D_node )
= ( ( Xs = nil_Pr1769730692D_node )
& ( Ys = nil_Pr1769730692D_node ) ) ) ).
% append_is_Nil_conv
thf(fact_110_Nil__is__append__conv,axiom,
! [Xs: list_node,Ys: list_node] :
( ( nil_node
= ( append_node @ Xs @ Ys ) )
= ( ( Xs = nil_node )
& ( Ys = nil_node ) ) ) ).
% Nil_is_append_conv
thf(fact_111_Nil__is__append__conv,axiom,
! [Xs: list_P738500740D_node,Ys: list_P738500740D_node] :
( ( nil_Pr1769730692D_node
= ( append2096883353D_node @ Xs @ Ys ) )
= ( ( Xs = nil_Pr1769730692D_node )
& ( Ys = nil_Pr1769730692D_node ) ) ) ).
% Nil_is_append_conv
thf(fact_112_self__append__conv2,axiom,
! [Ys: list_node,Xs: list_node] :
( ( Ys
= ( append_node @ Xs @ Ys ) )
= ( Xs = nil_node ) ) ).
% self_append_conv2
thf(fact_113_self__append__conv2,axiom,
! [Ys: list_P738500740D_node,Xs: list_P738500740D_node] :
( ( Ys
= ( append2096883353D_node @ Xs @ Ys ) )
= ( Xs = nil_Pr1769730692D_node ) ) ).
% self_append_conv2
thf(fact_114_append__self__conv2,axiom,
! [Xs: list_node,Ys: list_node] :
( ( ( append_node @ Xs @ Ys )
= Ys )
= ( Xs = nil_node ) ) ).
% append_self_conv2
thf(fact_115_append__self__conv2,axiom,
! [Xs: list_P738500740D_node,Ys: list_P738500740D_node] :
( ( ( append2096883353D_node @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr1769730692D_node ) ) ).
% append_self_conv2
thf(fact_116_self__append__conv,axiom,
! [Xs: list_node,Ys: list_node] :
( ( Xs
= ( append_node @ Xs @ Ys ) )
= ( Ys = nil_node ) ) ).
% self_append_conv
thf(fact_117_self__append__conv,axiom,
! [Xs: list_P738500740D_node,Ys: list_P738500740D_node] :
( ( Xs
= ( append2096883353D_node @ Xs @ Ys ) )
= ( Ys = nil_Pr1769730692D_node ) ) ).
% self_append_conv
thf(fact_118_append__self__conv,axiom,
! [Xs: list_node,Ys: list_node] :
( ( ( append_node @ Xs @ Ys )
= Xs )
= ( Ys = nil_node ) ) ).
% append_self_conv
thf(fact_119_append__self__conv,axiom,
! [Xs: list_P738500740D_node,Ys: list_P738500740D_node] :
( ( ( append2096883353D_node @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr1769730692D_node ) ) ).
% append_self_conv
thf(fact_120_append__Nil2,axiom,
! [Xs: list_node] :
( ( append_node @ Xs @ nil_node )
= Xs ) ).
% append_Nil2
thf(fact_121_append__Nil2,axiom,
! [Xs: list_P738500740D_node] :
( ( append2096883353D_node @ Xs @ nil_Pr1769730692D_node )
= Xs ) ).
% append_Nil2
thf(fact_122_suffix__bot_Obot_Oextremum__unique,axiom,
! [A: list_P738500740D_node] :
( ( suffix1143830554D_node @ A @ nil_Pr1769730692D_node )
= ( A = nil_Pr1769730692D_node ) ) ).
% suffix_bot.bot.extremum_unique
thf(fact_123_suffix__bot_Obot_Oextremum__unique,axiom,
! [A: list_node] :
( ( suffix_node @ A @ nil_node )
= ( A = nil_node ) ) ).
% suffix_bot.bot.extremum_unique
thf(fact_124_suffix__Nil,axiom,
! [Xs: list_P738500740D_node] :
( ( suffix1143830554D_node @ Xs @ nil_Pr1769730692D_node )
= ( Xs = nil_Pr1769730692D_node ) ) ).
% suffix_Nil
thf(fact_125_suffix__Nil,axiom,
! [Xs: list_node] :
( ( suffix_node @ Xs @ nil_node )
= ( Xs = nil_node ) ) ).
% suffix_Nil
thf(fact_126_same__suffix__suffix,axiom,
! [Ys: list_node,Xs: list_node,Zs: list_node] :
( ( suffix_node @ ( append_node @ Ys @ Xs ) @ ( append_node @ Zs @ Xs ) )
= ( suffix_node @ Ys @ Zs ) ) ).
% same_suffix_suffix
thf(fact_127_old_Opath2__split_I1_J,axiom,
! [G: g,N: node,Ns: list_node,N2: node,Ns6: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( append_node @ Ns @ ( cons_node @ N2 @ Ns6 ) ) @ M )
=> ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( append_node @ Ns @ ( cons_node @ N2 @ nil_node ) ) @ N2 ) ) ).
% old.path2_split(1)
thf(fact_128_old_OEntry__loop,axiom,
! [G: g,Ns: list_node] :
( ( invar @ G )
=> ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ ( entry @ G ) @ Ns @ ( entry @ G ) )
=> ( Ns
= ( cons_node @ ( entry @ G ) @ nil_node ) ) ) ) ).
% old.Entry_loop
thf(fact_129_old_OEntryPath__triv,axiom,
! [G: g,N: node] : ( graph_1994935542_edgeD @ alpha_n @ invar @ inEdges @ entry @ G @ ( cons_node @ N @ nil_node ) ) ).
% old.EntryPath_triv
thf(fact_130_append1__eq__conv,axiom,
! [Xs: list_P738500740D_node,X: produc1453890942D_node,Ys: list_P738500740D_node,Y3: produc1453890942D_node] :
( ( ( append2096883353D_node @ Xs @ ( cons_P1018517044D_node @ X @ nil_Pr1769730692D_node ) )
= ( append2096883353D_node @ Ys @ ( cons_P1018517044D_node @ Y3 @ nil_Pr1769730692D_node ) ) )
= ( ( Xs = Ys )
& ( X = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_131_append1__eq__conv,axiom,
! [Xs: list_node,X: node,Ys: list_node,Y3: node] :
( ( ( append_node @ Xs @ ( cons_node @ X @ nil_node ) )
= ( append_node @ Ys @ ( cons_node @ Y3 @ nil_node ) ) )
= ( ( Xs = Ys )
& ( X = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_132_list__ee__eq__leel_I1_J,axiom,
! [E1: produc1453890942D_node,E2: produc1453890942D_node,L1: list_P738500740D_node,E12: produc1453890942D_node,E22: produc1453890942D_node,L2: list_P738500740D_node] :
( ( ( cons_P1018517044D_node @ E1 @ ( cons_P1018517044D_node @ E2 @ nil_Pr1769730692D_node ) )
= ( append2096883353D_node @ L1 @ ( cons_P1018517044D_node @ E12 @ ( cons_P1018517044D_node @ E22 @ L2 ) ) ) )
= ( ( L1 = nil_Pr1769730692D_node )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2 = nil_Pr1769730692D_node ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_133_list__ee__eq__leel_I1_J,axiom,
! [E1: node,E2: node,L1: list_node,E12: node,E22: node,L2: list_node] :
( ( ( cons_node @ E1 @ ( cons_node @ E2 @ nil_node ) )
= ( append_node @ L1 @ ( cons_node @ E12 @ ( cons_node @ E22 @ L2 ) ) ) )
= ( ( L1 = nil_node )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2 = nil_node ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_134_list__ee__eq__leel_I2_J,axiom,
! [L1: list_P738500740D_node,E12: produc1453890942D_node,E22: produc1453890942D_node,L2: list_P738500740D_node,E1: produc1453890942D_node,E2: produc1453890942D_node] :
( ( ( append2096883353D_node @ L1 @ ( cons_P1018517044D_node @ E12 @ ( cons_P1018517044D_node @ E22 @ L2 ) ) )
= ( cons_P1018517044D_node @ E1 @ ( cons_P1018517044D_node @ E2 @ nil_Pr1769730692D_node ) ) )
= ( ( L1 = nil_Pr1769730692D_node )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2 = nil_Pr1769730692D_node ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_135_list__ee__eq__leel_I2_J,axiom,
! [L1: list_node,E12: node,E22: node,L2: list_node,E1: node,E2: node] :
( ( ( append_node @ L1 @ ( cons_node @ E12 @ ( cons_node @ E22 @ L2 ) ) )
= ( cons_node @ E1 @ ( cons_node @ E2 @ nil_node ) ) )
= ( ( L1 = nil_node )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2 = nil_node ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_136_list__se__match_I1_J,axiom,
! [L1: list_P738500740D_node,L2: list_P738500740D_node,A: produc1453890942D_node] :
( ( L1 != nil_Pr1769730692D_node )
=> ( ( ( append2096883353D_node @ L1 @ L2 )
= ( cons_P1018517044D_node @ A @ nil_Pr1769730692D_node ) )
= ( ( L1
= ( cons_P1018517044D_node @ A @ nil_Pr1769730692D_node ) )
& ( L2 = nil_Pr1769730692D_node ) ) ) ) ).
% list_se_match(1)
thf(fact_137_list__se__match_I1_J,axiom,
! [L1: list_node,L2: list_node,A: node] :
( ( L1 != nil_node )
=> ( ( ( append_node @ L1 @ L2 )
= ( cons_node @ A @ nil_node ) )
= ( ( L1
= ( cons_node @ A @ nil_node ) )
& ( L2 = nil_node ) ) ) ) ).
% list_se_match(1)
thf(fact_138_list__se__match_I2_J,axiom,
! [L2: list_P738500740D_node,L1: list_P738500740D_node,A: produc1453890942D_node] :
( ( L2 != nil_Pr1769730692D_node )
=> ( ( ( append2096883353D_node @ L1 @ L2 )
= ( cons_P1018517044D_node @ A @ nil_Pr1769730692D_node ) )
= ( ( L1 = nil_Pr1769730692D_node )
& ( L2
= ( cons_P1018517044D_node @ A @ nil_Pr1769730692D_node ) ) ) ) ) ).
% list_se_match(2)
thf(fact_139_list__se__match_I2_J,axiom,
! [L2: list_node,L1: list_node,A: node] :
( ( L2 != nil_node )
=> ( ( ( append_node @ L1 @ L2 )
= ( cons_node @ A @ nil_node ) )
= ( ( L1 = nil_node )
& ( L2
= ( cons_node @ A @ nil_node ) ) ) ) ) ).
% list_se_match(2)
thf(fact_140_list__se__match_I3_J,axiom,
! [L1: list_P738500740D_node,A: produc1453890942D_node,L2: list_P738500740D_node] :
( ( L1 != nil_Pr1769730692D_node )
=> ( ( ( cons_P1018517044D_node @ A @ nil_Pr1769730692D_node )
= ( append2096883353D_node @ L1 @ L2 ) )
= ( ( L1
= ( cons_P1018517044D_node @ A @ nil_Pr1769730692D_node ) )
& ( L2 = nil_Pr1769730692D_node ) ) ) ) ).
% list_se_match(3)
thf(fact_141_list__se__match_I3_J,axiom,
! [L1: list_node,A: node,L2: list_node] :
( ( L1 != nil_node )
=> ( ( ( cons_node @ A @ nil_node )
= ( append_node @ L1 @ L2 ) )
= ( ( L1
= ( cons_node @ A @ nil_node ) )
& ( L2 = nil_node ) ) ) ) ).
% list_se_match(3)
thf(fact_142_list__se__match_I4_J,axiom,
! [L2: list_P738500740D_node,A: produc1453890942D_node,L1: list_P738500740D_node] :
( ( L2 != nil_Pr1769730692D_node )
=> ( ( ( cons_P1018517044D_node @ A @ nil_Pr1769730692D_node )
= ( append2096883353D_node @ L1 @ L2 ) )
= ( ( L1 = nil_Pr1769730692D_node )
& ( L2
= ( cons_P1018517044D_node @ A @ nil_Pr1769730692D_node ) ) ) ) ) ).
% list_se_match(4)
thf(fact_143_list__se__match_I4_J,axiom,
! [L2: list_node,A: node,L1: list_node] :
( ( L2 != nil_node )
=> ( ( ( cons_node @ A @ nil_node )
= ( append_node @ L1 @ L2 ) )
= ( ( L1 = nil_node )
& ( L2
= ( cons_node @ A @ nil_node ) ) ) ) ) ).
% list_se_match(4)
thf(fact_144_list__e__eq__lel_I1_J,axiom,
! [E: produc1453890942D_node,L1: list_P738500740D_node,E3: produc1453890942D_node,L2: list_P738500740D_node] :
( ( ( cons_P1018517044D_node @ E @ nil_Pr1769730692D_node )
= ( append2096883353D_node @ L1 @ ( cons_P1018517044D_node @ E3 @ L2 ) ) )
= ( ( L1 = nil_Pr1769730692D_node )
& ( E3 = E )
& ( L2 = nil_Pr1769730692D_node ) ) ) ).
% list_e_eq_lel(1)
thf(fact_145_list__e__eq__lel_I1_J,axiom,
! [E: node,L1: list_node,E3: node,L2: list_node] :
( ( ( cons_node @ E @ nil_node )
= ( append_node @ L1 @ ( cons_node @ E3 @ L2 ) ) )
= ( ( L1 = nil_node )
& ( E3 = E )
& ( L2 = nil_node ) ) ) ).
% list_e_eq_lel(1)
thf(fact_146_list__e__eq__lel_I2_J,axiom,
! [L1: list_P738500740D_node,E3: produc1453890942D_node,L2: list_P738500740D_node,E: produc1453890942D_node] :
( ( ( append2096883353D_node @ L1 @ ( cons_P1018517044D_node @ E3 @ L2 ) )
= ( cons_P1018517044D_node @ E @ nil_Pr1769730692D_node ) )
= ( ( L1 = nil_Pr1769730692D_node )
& ( E3 = E )
& ( L2 = nil_Pr1769730692D_node ) ) ) ).
% list_e_eq_lel(2)
thf(fact_147_list__e__eq__lel_I2_J,axiom,
! [L1: list_node,E3: node,L2: list_node,E: node] :
( ( ( append_node @ L1 @ ( cons_node @ E3 @ L2 ) )
= ( cons_node @ E @ nil_node ) )
= ( ( L1 = nil_node )
& ( E3 = E )
& ( L2 = nil_node ) ) ) ).
% list_e_eq_lel(2)
thf(fact_148_snoc__suffix__snoc,axiom,
! [Xs: list_P738500740D_node,X: produc1453890942D_node,Ys: list_P738500740D_node,Y3: produc1453890942D_node] :
( ( suffix1143830554D_node @ ( append2096883353D_node @ Xs @ ( cons_P1018517044D_node @ X @ nil_Pr1769730692D_node ) ) @ ( append2096883353D_node @ Ys @ ( cons_P1018517044D_node @ Y3 @ nil_Pr1769730692D_node ) ) )
= ( ( X = Y3 )
& ( suffix1143830554D_node @ Xs @ Ys ) ) ) ).
% snoc_suffix_snoc
thf(fact_149_snoc__suffix__snoc,axiom,
! [Xs: list_node,X: node,Ys: list_node,Y3: node] :
( ( suffix_node @ ( append_node @ Xs @ ( cons_node @ X @ nil_node ) ) @ ( append_node @ Ys @ ( cons_node @ Y3 @ nil_node ) ) )
= ( ( X = Y3 )
& ( suffix_node @ Xs @ Ys ) ) ) ).
% snoc_suffix_snoc
thf(fact_150_suffix__snoc,axiom,
! [Xs: list_P738500740D_node,Ys: list_P738500740D_node,Y3: produc1453890942D_node] :
( ( suffix1143830554D_node @ Xs @ ( append2096883353D_node @ Ys @ ( cons_P1018517044D_node @ Y3 @ nil_Pr1769730692D_node ) ) )
= ( ( Xs = nil_Pr1769730692D_node )
| ? [Zs2: list_P738500740D_node] :
( ( Xs
= ( append2096883353D_node @ Zs2 @ ( cons_P1018517044D_node @ Y3 @ nil_Pr1769730692D_node ) ) )
& ( suffix1143830554D_node @ Zs2 @ Ys ) ) ) ) ).
% suffix_snoc
thf(fact_151_suffix__snoc,axiom,
! [Xs: list_node,Ys: list_node,Y3: node] :
( ( suffix_node @ Xs @ ( append_node @ Ys @ ( cons_node @ Y3 @ nil_node ) ) )
= ( ( Xs = nil_node )
| ? [Zs2: list_node] :
( ( Xs
= ( append_node @ Zs2 @ ( cons_node @ Y3 @ nil_node ) ) )
& ( suffix_node @ Zs2 @ Ys ) ) ) ) ).
% suffix_snoc
thf(fact_152_list_Ocollapse,axiom,
! [List: list_P738500740D_node] :
( ( List != nil_Pr1769730692D_node )
=> ( ( cons_P1018517044D_node @ ( hd_Pro1395892457D_node @ List ) @ ( tl_Pro1633633005D_node @ List ) )
= List ) ) ).
% list.collapse
thf(fact_153_list_Ocollapse,axiom,
! [List: list_node] :
( ( List != nil_node )
=> ( ( cons_node @ ( hd_node @ List ) @ ( tl_node @ List ) )
= List ) ) ).
% list.collapse
thf(fact_154_hd__Cons__tl,axiom,
! [Xs: list_P738500740D_node] :
( ( Xs != nil_Pr1769730692D_node )
=> ( ( cons_P1018517044D_node @ ( hd_Pro1395892457D_node @ Xs ) @ ( tl_Pro1633633005D_node @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_155_hd__Cons__tl,axiom,
! [Xs: list_node] :
( ( Xs != nil_node )
=> ( ( cons_node @ ( hd_node @ Xs ) @ ( tl_node @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_156_rs__def,axiom,
( rs2
= ( append_node @ rs @ ( cons_node @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_r ) @ nil_node ) ) ) ).
% rs_def
thf(fact_157_old_Oempty__path2,axiom,
! [N: node,G: g] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( invar @ G )
=> ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( cons_node @ N @ nil_node ) @ N ) ) ) ).
% old.empty_path2
thf(fact_158_old_OCons__path2,axiom,
! [G: g,N: node,Ns: list_node,M: node,N2: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( member_node @ N2 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ N ) ) )
=> ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N2 @ ( cons_node @ N2 @ Ns ) @ M ) ) ) ).
% old.Cons_path2
thf(fact_159_list_Odistinct_I1_J,axiom,
! [X21: produc1453890942D_node,X22: list_P738500740D_node] :
( nil_Pr1769730692D_node
!= ( cons_P1018517044D_node @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_160_list_Odistinct_I1_J,axiom,
! [X21: node,X22: list_node] :
( nil_node
!= ( cons_node @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_161_neq__NilE,axiom,
! [L: list_P738500740D_node] :
( ( L != nil_Pr1769730692D_node )
=> ~ ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node] :
( L
!= ( cons_P1018517044D_node @ X2 @ Xs2 ) ) ) ).
% neq_NilE
thf(fact_162_neq__NilE,axiom,
! [L: list_node] :
( ( L != nil_node )
=> ~ ! [X2: node,Xs2: list_node] :
( L
!= ( cons_node @ X2 @ Xs2 ) ) ) ).
% neq_NilE
thf(fact_163_list_OdiscI,axiom,
! [List: list_P738500740D_node,X21: produc1453890942D_node,X22: list_P738500740D_node] :
( ( List
= ( cons_P1018517044D_node @ X21 @ X22 ) )
=> ( List != nil_Pr1769730692D_node ) ) ).
% list.discI
thf(fact_164_list_OdiscI,axiom,
! [List: list_node,X21: node,X22: list_node] :
( ( List
= ( cons_node @ X21 @ X22 ) )
=> ( List != nil_node ) ) ).
% list.discI
thf(fact_165_revg_Oinduct,axiom,
! [P: list_P738500740D_node > list_P738500740D_node > $o,A0: list_P738500740D_node,A1: list_P738500740D_node] :
( ! [X_1: list_P738500740D_node] : ( P @ nil_Pr1769730692D_node @ X_1 )
=> ( ! [A3: produc1453890942D_node,As: list_P738500740D_node,B2: list_P738500740D_node] :
( ( P @ As @ ( cons_P1018517044D_node @ A3 @ B2 ) )
=> ( P @ ( cons_P1018517044D_node @ A3 @ As ) @ B2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% revg.induct
thf(fact_166_revg_Oinduct,axiom,
! [P: list_node > list_node > $o,A0: list_node,A1: list_node] :
( ! [X_1: list_node] : ( P @ nil_node @ X_1 )
=> ( ! [A3: node,As: list_node,B2: list_node] :
( ( P @ As @ ( cons_node @ A3 @ B2 ) )
=> ( P @ ( cons_node @ A3 @ As ) @ B2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% revg.induct
thf(fact_167_list_Oexhaust,axiom,
! [Y3: list_P738500740D_node] :
( ( Y3 != nil_Pr1769730692D_node )
=> ~ ! [X212: produc1453890942D_node,X222: list_P738500740D_node] :
( Y3
!= ( cons_P1018517044D_node @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_168_list_Oexhaust,axiom,
! [Y3: list_node] :
( ( Y3 != nil_node )
=> ~ ! [X212: node,X222: list_node] :
( Y3
!= ( cons_node @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_169_list_Oinducts,axiom,
! [P: list_P738500740D_node > $o,List: list_P738500740D_node] :
( ( P @ nil_Pr1769730692D_node )
=> ( ! [X1: produc1453890942D_node,X23: list_P738500740D_node] :
( ( P @ X23 )
=> ( P @ ( cons_P1018517044D_node @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_170_list_Oinducts,axiom,
! [P: list_node > $o,List: list_node] :
( ( P @ nil_node )
=> ( ! [X1: node,X23: list_node] :
( ( P @ X23 )
=> ( P @ ( cons_node @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_171_neq__Nil__conv,axiom,
! [Xs: list_P738500740D_node] :
( ( Xs != nil_Pr1769730692D_node )
= ( ? [Y2: produc1453890942D_node,Ys2: list_P738500740D_node] :
( Xs
= ( cons_P1018517044D_node @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_172_neq__Nil__conv,axiom,
! [Xs: list_node] :
( ( Xs != nil_node )
= ( ? [Y2: node,Ys2: list_node] :
( Xs
= ( cons_node @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_173_list__induct2_H,axiom,
! [P: list_P738500740D_node > list_P738500740D_node > $o,Xs: list_P738500740D_node,Ys: list_P738500740D_node] :
( ( P @ nil_Pr1769730692D_node @ nil_Pr1769730692D_node )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node] : ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ nil_Pr1769730692D_node )
=> ( ! [Y4: produc1453890942D_node,Ys3: list_P738500740D_node] : ( P @ nil_Pr1769730692D_node @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_174_list__induct2_H,axiom,
! [P: list_P738500740D_node > list_node > $o,Xs: list_P738500740D_node,Ys: list_node] :
( ( P @ nil_Pr1769730692D_node @ nil_node )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node] : ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ nil_node )
=> ( ! [Y4: node,Ys3: list_node] : ( P @ nil_Pr1769730692D_node @ ( cons_node @ Y4 @ Ys3 ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: node,Ys3: list_node] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ ( cons_node @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_175_list__induct2_H,axiom,
! [P: list_node > list_P738500740D_node > $o,Xs: list_node,Ys: list_P738500740D_node] :
( ( P @ nil_node @ nil_Pr1769730692D_node )
=> ( ! [X2: node,Xs2: list_node] : ( P @ ( cons_node @ X2 @ Xs2 ) @ nil_Pr1769730692D_node )
=> ( ! [Y4: produc1453890942D_node,Ys3: list_P738500740D_node] : ( P @ nil_node @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) )
=> ( ! [X2: node,Xs2: list_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_node @ X2 @ Xs2 ) @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_176_list__induct2_H,axiom,
! [P: list_node > list_node > $o,Xs: list_node,Ys: list_node] :
( ( P @ nil_node @ nil_node )
=> ( ! [X2: node,Xs2: list_node] : ( P @ ( cons_node @ X2 @ Xs2 ) @ nil_node )
=> ( ! [Y4: node,Ys3: list_node] : ( P @ nil_node @ ( cons_node @ Y4 @ Ys3 ) )
=> ( ! [X2: node,Xs2: list_node,Y4: node,Ys3: list_node] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_node @ X2 @ Xs2 ) @ ( cons_node @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_177_splice_Oinduct,axiom,
! [P: list_P738500740D_node > list_P738500740D_node > $o,A0: list_P738500740D_node,A1: list_P738500740D_node] :
( ! [X_1: list_P738500740D_node] : ( P @ nil_Pr1769730692D_node @ X_1 )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Ys3: list_P738500740D_node] :
( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_178_splice_Oinduct,axiom,
! [P: list_node > list_node > $o,A0: list_node,A1: list_node] :
( ! [X_1: list_node] : ( P @ nil_node @ X_1 )
=> ( ! [X2: node,Xs2: list_node,Ys3: list_node] :
( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons_node @ X2 @ Xs2 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_179_induct__list012,axiom,
! [P: list_P738500740D_node > $o,Xs: list_P738500740D_node] :
( ( P @ nil_Pr1769730692D_node )
=> ( ! [X2: produc1453890942D_node] : ( P @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) )
=> ( ! [X2: produc1453890942D_node,Y4: produc1453890942D_node,Zs3: list_P738500740D_node] :
( ( P @ Zs3 )
=> ( ( P @ ( cons_P1018517044D_node @ Y4 @ Zs3 ) )
=> ( P @ ( cons_P1018517044D_node @ X2 @ ( cons_P1018517044D_node @ Y4 @ Zs3 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_180_induct__list012,axiom,
! [P: list_node > $o,Xs: list_node] :
( ( P @ nil_node )
=> ( ! [X2: node] : ( P @ ( cons_node @ X2 @ nil_node ) )
=> ( ! [X2: node,Y4: node,Zs3: list_node] :
( ( P @ Zs3 )
=> ( ( P @ ( cons_node @ Y4 @ Zs3 ) )
=> ( P @ ( cons_node @ X2 @ ( cons_node @ Y4 @ Zs3 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_181_min__list_Ocases,axiom,
! [X: list_node] :
( ! [X2: node,Xs2: list_node] :
( X
!= ( cons_node @ X2 @ Xs2 ) )
=> ( X = nil_node ) ) ).
% min_list.cases
thf(fact_182_suffix__Cons,axiom,
! [Xs: list_node,Y3: node,Ys: list_node] :
( ( suffix_node @ Xs @ ( cons_node @ Y3 @ Ys ) )
= ( ( Xs
= ( cons_node @ Y3 @ Ys ) )
| ( suffix_node @ Xs @ Ys ) ) ) ).
% suffix_Cons
thf(fact_183_Cons__eq__appendI,axiom,
! [X: node,Xs1: list_node,Ys: list_node,Xs: list_node,Zs: list_node] :
( ( ( cons_node @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_node @ Xs1 @ Zs ) )
=> ( ( cons_node @ X @ Xs )
= ( append_node @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_184_min__list_Oinduct,axiom,
! [P: list_node > $o,A0: list_node] :
( ! [X2: node,Xs2: list_node] :
( ! [X213: node,X223: list_node] :
( ( Xs2
= ( cons_node @ X213 @ X223 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons_node @ X2 @ Xs2 ) ) )
=> ( ( P @ nil_node )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_185_shuffles_Oinduct,axiom,
! [P: list_P738500740D_node > list_P738500740D_node > $o,A0: list_P738500740D_node,A1: list_P738500740D_node] :
( ! [X_1: list_P738500740D_node] : ( P @ nil_Pr1769730692D_node @ X_1 )
=> ( ! [Xs2: list_P738500740D_node] : ( P @ Xs2 @ nil_Pr1769730692D_node )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( P @ Xs2 @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) )
=> ( ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_186_shuffles_Oinduct,axiom,
! [P: list_node > list_node > $o,A0: list_node,A1: list_node] :
( ! [X_1: list_node] : ( P @ nil_node @ X_1 )
=> ( ! [Xs2: list_node] : ( P @ Xs2 @ nil_node )
=> ( ! [X2: node,Xs2: list_node,Y4: node,Ys3: list_node] :
( ( P @ Xs2 @ ( cons_node @ Y4 @ Ys3 ) )
=> ( ( P @ ( cons_node @ X2 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons_node @ X2 @ Xs2 ) @ ( cons_node @ Y4 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_187_transpose_Ocases,axiom,
! [X: list_l1129649930D_node] :
( ( X != nil_li1626782346D_node )
=> ( ! [Xss: list_l1129649930D_node] :
( X
!= ( cons_l1288865338D_node @ nil_Pr1769730692D_node @ Xss ) )
=> ~ ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Xss: list_l1129649930D_node] :
( X
!= ( cons_l1288865338D_node @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_188_transpose_Ocases,axiom,
! [X: list_list_node] :
( ( X != nil_list_node )
=> ( ! [Xss: list_list_node] :
( X
!= ( cons_list_node @ nil_node @ Xss ) )
=> ~ ! [X2: node,Xs2: list_node,Xss: list_list_node] :
( X
!= ( cons_list_node @ ( cons_node @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_189_suffix__ConsD,axiom,
! [X: node,Xs: list_node,Ys: list_node] :
( ( suffix_node @ ( cons_node @ X @ Xs ) @ Ys )
=> ( suffix_node @ Xs @ Ys ) ) ).
% suffix_ConsD
thf(fact_190_suffix__ConsI,axiom,
! [Xs: list_node,Ys: list_node,Y3: node] :
( ( suffix_node @ Xs @ Ys )
=> ( suffix_node @ Xs @ ( cons_node @ Y3 @ Ys ) ) ) ).
% suffix_ConsI
thf(fact_191_list__2pre__induct,axiom,
! [P: list_P738500740D_node > list_P738500740D_node > $o,W1: list_P738500740D_node,W2: list_P738500740D_node] :
( ( P @ nil_Pr1769730692D_node @ nil_Pr1769730692D_node )
=> ( ! [E4: produc1453890942D_node,W12: list_P738500740D_node,W22: list_P738500740D_node] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons_P1018517044D_node @ E4 @ W12 ) @ W22 ) )
=> ( ! [E4: produc1453890942D_node,W13: list_P738500740D_node,W23: list_P738500740D_node] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons_P1018517044D_node @ E4 @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_192_list__2pre__induct,axiom,
! [P: list_P738500740D_node > list_node > $o,W1: list_P738500740D_node,W2: list_node] :
( ( P @ nil_Pr1769730692D_node @ nil_node )
=> ( ! [E4: produc1453890942D_node,W12: list_P738500740D_node,W22: list_node] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons_P1018517044D_node @ E4 @ W12 ) @ W22 ) )
=> ( ! [E4: node,W13: list_P738500740D_node,W23: list_node] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons_node @ E4 @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_193_list__2pre__induct,axiom,
! [P: list_node > list_P738500740D_node > $o,W1: list_node,W2: list_P738500740D_node] :
( ( P @ nil_node @ nil_Pr1769730692D_node )
=> ( ! [E4: node,W12: list_node,W22: list_P738500740D_node] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons_node @ E4 @ W12 ) @ W22 ) )
=> ( ! [E4: produc1453890942D_node,W13: list_node,W23: list_P738500740D_node] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons_P1018517044D_node @ E4 @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_194_list__2pre__induct,axiom,
! [P: list_node > list_node > $o,W1: list_node,W2: list_node] :
( ( P @ nil_node @ nil_node )
=> ( ! [E4: node,W12: list_node,W22: list_node] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons_node @ E4 @ W12 ) @ W22 ) )
=> ( ! [E4: node,W13: list_node,W23: list_node] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons_node @ E4 @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_195_suffix__ConsD2,axiom,
! [X: node,Xs: list_node,Y3: node,Ys: list_node] :
( ( suffix_node @ ( cons_node @ X @ Xs ) @ ( cons_node @ Y3 @ Ys ) )
=> ( suffix_node @ Xs @ Ys ) ) ).
% suffix_ConsD2
thf(fact_196_remdups__adj_Ocases,axiom,
! [X: list_P738500740D_node] :
( ( X != nil_Pr1769730692D_node )
=> ( ! [X2: produc1453890942D_node] :
( X
!= ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) )
=> ~ ! [X2: produc1453890942D_node,Y4: produc1453890942D_node,Xs2: list_P738500740D_node] :
( X
!= ( cons_P1018517044D_node @ X2 @ ( cons_P1018517044D_node @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_197_remdups__adj_Ocases,axiom,
! [X: list_node] :
( ( X != nil_node )
=> ( ! [X2: node] :
( X
!= ( cons_node @ X2 @ nil_node ) )
=> ~ ! [X2: node,Y4: node,Xs2: list_node] :
( X
!= ( cons_node @ X2 @ ( cons_node @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_198_sorted__wrt_Oinduct,axiom,
! [P: ( produc1453890942D_node > produc1453890942D_node > $o ) > list_P738500740D_node > $o,A0: produc1453890942D_node > produc1453890942D_node > $o,A1: list_P738500740D_node] :
( ! [P2: produc1453890942D_node > produc1453890942D_node > $o] : ( P @ P2 @ nil_Pr1769730692D_node )
=> ( ! [P2: produc1453890942D_node > produc1453890942D_node > $o,X2: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( P @ P2 @ Ys3 )
=> ( P @ P2 @ ( cons_P1018517044D_node @ X2 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_199_sorted__wrt_Oinduct,axiom,
! [P: ( node > node > $o ) > list_node > $o,A0: node > node > $o,A1: list_node] :
( ! [P2: node > node > $o] : ( P @ P2 @ nil_node )
=> ( ! [P2: node > node > $o,X2: node,Ys3: list_node] :
( ( P @ P2 @ Ys3 )
=> ( P @ P2 @ ( cons_node @ X2 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_200_append__Cons,axiom,
! [X: node,Xs: list_node,Ys: list_node] :
( ( append_node @ ( cons_node @ X @ Xs ) @ Ys )
= ( cons_node @ X @ ( append_node @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_201_remdups__adj_Oinduct,axiom,
! [P: list_P738500740D_node > $o,A0: list_P738500740D_node] :
( ( P @ nil_Pr1769730692D_node )
=> ( ! [X2: produc1453890942D_node] : ( P @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) )
=> ( ! [X2: produc1453890942D_node,Y4: produc1453890942D_node,Xs2: list_P738500740D_node] :
( ( ( X2 = Y4 )
=> ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y4 )
=> ( P @ ( cons_P1018517044D_node @ Y4 @ Xs2 ) ) )
=> ( P @ ( cons_P1018517044D_node @ X2 @ ( cons_P1018517044D_node @ Y4 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_202_remdups__adj_Oinduct,axiom,
! [P: list_node > $o,A0: list_node] :
( ( P @ nil_node )
=> ( ! [X2: node] : ( P @ ( cons_node @ X2 @ nil_node ) )
=> ( ! [X2: node,Y4: node,Xs2: list_node] :
( ( ( X2 = Y4 )
=> ( P @ ( cons_node @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y4 )
=> ( P @ ( cons_node @ Y4 @ Xs2 ) ) )
=> ( P @ ( cons_node @ X2 @ ( cons_node @ Y4 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_203_list__induct__first2,axiom,
! [P: list_P738500740D_node > $o,Xs: list_P738500740D_node] :
( ( P @ nil_Pr1769730692D_node )
=> ( ! [X2: produc1453890942D_node] : ( P @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) )
=> ( ! [X1: produc1453890942D_node,X23: produc1453890942D_node,Xs2: list_P738500740D_node] :
( ( P @ Xs2 )
=> ( P @ ( cons_P1018517044D_node @ X1 @ ( cons_P1018517044D_node @ X23 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_204_list__induct__first2,axiom,
! [P: list_node > $o,Xs: list_node] :
( ( P @ nil_node )
=> ( ! [X2: node] : ( P @ ( cons_node @ X2 @ nil_node ) )
=> ( ! [X1: node,X23: node,Xs2: list_node] :
( ( P @ Xs2 )
=> ( P @ ( cons_node @ X1 @ ( cons_node @ X23 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_205_list__match__lel__lel,axiom,
! [C1: list_node,Qs: node,C2: list_node,C12: list_node,Qs2: node,C22: list_node] :
( ( ( append_node @ C1 @ ( cons_node @ Qs @ C2 ) )
= ( append_node @ C12 @ ( cons_node @ Qs2 @ C22 ) ) )
=> ( ! [C21: list_node] :
( ( C1
= ( append_node @ C12 @ ( cons_node @ Qs2 @ C21 ) ) )
=> ( C22
!= ( append_node @ C21 @ ( cons_node @ Qs @ C2 ) ) ) )
=> ( ( ( C12 = C1 )
=> ( ( Qs2 = Qs )
=> ( C22 != C2 ) ) )
=> ~ ! [C212: list_node] :
( ( C12
= ( append_node @ C1 @ ( cons_node @ Qs @ C212 ) ) )
=> ( C2
!= ( append_node @ C212 @ ( cons_node @ Qs2 @ C22 ) ) ) ) ) ) ) ).
% list_match_lel_lel
thf(fact_206_successively_Oinduct,axiom,
! [P: ( produc1453890942D_node > produc1453890942D_node > $o ) > list_P738500740D_node > $o,A0: produc1453890942D_node > produc1453890942D_node > $o,A1: list_P738500740D_node] :
( ! [P2: produc1453890942D_node > produc1453890942D_node > $o] : ( P @ P2 @ nil_Pr1769730692D_node )
=> ( ! [P2: produc1453890942D_node > produc1453890942D_node > $o,X2: produc1453890942D_node] : ( P @ P2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) )
=> ( ! [P2: produc1453890942D_node > produc1453890942D_node > $o,X2: produc1453890942D_node,Y4: produc1453890942D_node,Xs2: list_P738500740D_node] :
( ( P @ P2 @ ( cons_P1018517044D_node @ Y4 @ Xs2 ) )
=> ( P @ P2 @ ( cons_P1018517044D_node @ X2 @ ( cons_P1018517044D_node @ Y4 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_207_successively_Oinduct,axiom,
! [P: ( node > node > $o ) > list_node > $o,A0: node > node > $o,A1: list_node] :
( ! [P2: node > node > $o] : ( P @ P2 @ nil_node )
=> ( ! [P2: node > node > $o,X2: node] : ( P @ P2 @ ( cons_node @ X2 @ nil_node ) )
=> ( ! [P2: node > node > $o,X2: node,Y4: node,Xs2: list_node] :
( ( P @ P2 @ ( cons_node @ Y4 @ Xs2 ) )
=> ( P @ P2 @ ( cons_node @ X2 @ ( cons_node @ Y4 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_208_list__all__zip_Oinduct,axiom,
! [P: ( produc1453890942D_node > produc1453890942D_node > $o ) > list_P738500740D_node > list_P738500740D_node > $o,A0: produc1453890942D_node > produc1453890942D_node > $o,A1: list_P738500740D_node,A22: list_P738500740D_node] :
( ! [P2: produc1453890942D_node > produc1453890942D_node > $o] : ( P @ P2 @ nil_Pr1769730692D_node @ nil_Pr1769730692D_node )
=> ( ! [P2: produc1453890942D_node > produc1453890942D_node > $o,A3: produc1453890942D_node,As: list_P738500740D_node,B2: produc1453890942D_node,Bs: list_P738500740D_node] :
( ( P @ P2 @ As @ Bs )
=> ( P @ P2 @ ( cons_P1018517044D_node @ A3 @ As ) @ ( cons_P1018517044D_node @ B2 @ Bs ) ) )
=> ( ! [P2: produc1453890942D_node > produc1453890942D_node > $o,V2: produc1453890942D_node,Va: list_P738500740D_node] : ( P @ P2 @ ( cons_P1018517044D_node @ V2 @ Va ) @ nil_Pr1769730692D_node )
=> ( ! [P2: produc1453890942D_node > produc1453890942D_node > $o,V2: produc1453890942D_node,Va: list_P738500740D_node] : ( P @ P2 @ nil_Pr1769730692D_node @ ( cons_P1018517044D_node @ V2 @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% list_all_zip.induct
thf(fact_209_list__all__zip_Oinduct,axiom,
! [P: ( produc1453890942D_node > node > $o ) > list_P738500740D_node > list_node > $o,A0: produc1453890942D_node > node > $o,A1: list_P738500740D_node,A22: list_node] :
( ! [P2: produc1453890942D_node > node > $o] : ( P @ P2 @ nil_Pr1769730692D_node @ nil_node )
=> ( ! [P2: produc1453890942D_node > node > $o,A3: produc1453890942D_node,As: list_P738500740D_node,B2: node,Bs: list_node] :
( ( P @ P2 @ As @ Bs )
=> ( P @ P2 @ ( cons_P1018517044D_node @ A3 @ As ) @ ( cons_node @ B2 @ Bs ) ) )
=> ( ! [P2: produc1453890942D_node > node > $o,V2: produc1453890942D_node,Va: list_P738500740D_node] : ( P @ P2 @ ( cons_P1018517044D_node @ V2 @ Va ) @ nil_node )
=> ( ! [P2: produc1453890942D_node > node > $o,V2: node,Va: list_node] : ( P @ P2 @ nil_Pr1769730692D_node @ ( cons_node @ V2 @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% list_all_zip.induct
thf(fact_210_list__all__zip_Oinduct,axiom,
! [P: ( node > produc1453890942D_node > $o ) > list_node > list_P738500740D_node > $o,A0: node > produc1453890942D_node > $o,A1: list_node,A22: list_P738500740D_node] :
( ! [P2: node > produc1453890942D_node > $o] : ( P @ P2 @ nil_node @ nil_Pr1769730692D_node )
=> ( ! [P2: node > produc1453890942D_node > $o,A3: node,As: list_node,B2: produc1453890942D_node,Bs: list_P738500740D_node] :
( ( P @ P2 @ As @ Bs )
=> ( P @ P2 @ ( cons_node @ A3 @ As ) @ ( cons_P1018517044D_node @ B2 @ Bs ) ) )
=> ( ! [P2: node > produc1453890942D_node > $o,V2: node,Va: list_node] : ( P @ P2 @ ( cons_node @ V2 @ Va ) @ nil_Pr1769730692D_node )
=> ( ! [P2: node > produc1453890942D_node > $o,V2: produc1453890942D_node,Va: list_P738500740D_node] : ( P @ P2 @ nil_node @ ( cons_P1018517044D_node @ V2 @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% list_all_zip.induct
thf(fact_211_list__all__zip_Oinduct,axiom,
! [P: ( node > node > $o ) > list_node > list_node > $o,A0: node > node > $o,A1: list_node,A22: list_node] :
( ! [P2: node > node > $o] : ( P @ P2 @ nil_node @ nil_node )
=> ( ! [P2: node > node > $o,A3: node,As: list_node,B2: node,Bs: list_node] :
( ( P @ P2 @ As @ Bs )
=> ( P @ P2 @ ( cons_node @ A3 @ As ) @ ( cons_node @ B2 @ Bs ) ) )
=> ( ! [P2: node > node > $o,V2: node,Va: list_node] : ( P @ P2 @ ( cons_node @ V2 @ Va ) @ nil_node )
=> ( ! [P2: node > node > $o,V2: node,Va: list_node] : ( P @ P2 @ nil_node @ ( cons_node @ V2 @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% list_all_zip.induct
thf(fact_212_list__nonempty__induct,axiom,
! [Xs: list_P738500740D_node,P: list_P738500740D_node > $o] :
( ( Xs != nil_Pr1769730692D_node )
=> ( ! [X2: produc1453890942D_node] : ( P @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node] :
( ( Xs2 != nil_Pr1769730692D_node )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_213_list__nonempty__induct,axiom,
! [Xs: list_node,P: list_node > $o] :
( ( Xs != nil_node )
=> ( ! [X2: node] : ( P @ ( cons_node @ X2 @ nil_node ) )
=> ( ! [X2: node,Xs2: list_node] :
( ( Xs2 != nil_node )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_node @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_214_map__tailrec__rev_Oinduct,axiom,
! [P: ( produc1453890942D_node > node ) > list_P738500740D_node > list_node > $o,A0: produc1453890942D_node > node,A1: list_P738500740D_node,A22: list_node] :
( ! [F: produc1453890942D_node > node,X_1: list_node] : ( P @ F @ nil_Pr1769730692D_node @ X_1 )
=> ( ! [F: produc1453890942D_node > node,A3: produc1453890942D_node,As: list_P738500740D_node,Bs: list_node] :
( ( P @ F @ As @ ( cons_node @ ( F @ A3 ) @ Bs ) )
=> ( P @ F @ ( cons_P1018517044D_node @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_215_map__tailrec__rev_Oinduct,axiom,
! [P: ( node > node ) > list_node > list_node > $o,A0: node > node,A1: list_node,A22: list_node] :
( ! [F: node > node,X_1: list_node] : ( P @ F @ nil_node @ X_1 )
=> ( ! [F: node > node,A3: node,As: list_node,Bs: list_node] :
( ( P @ F @ As @ ( cons_node @ ( F @ A3 ) @ Bs ) )
=> ( P @ F @ ( cons_node @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_216_mergesort__by__rel__merge_Oinduct,axiom,
! [P: ( produc1453890942D_node > produc1453890942D_node > $o ) > list_P738500740D_node > list_P738500740D_node > $o,A0: produc1453890942D_node > produc1453890942D_node > $o,A1: list_P738500740D_node,A22: list_P738500740D_node] :
( ! [R: produc1453890942D_node > produc1453890942D_node > $o,X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( ( R @ X2 @ Y4 )
=> ( P @ R @ Xs2 @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) ) )
=> ( ( ~ ( R @ X2 @ Y4 )
=> ( P @ R @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ Ys3 ) )
=> ( P @ R @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) ) ) )
=> ( ! [R: produc1453890942D_node > produc1453890942D_node > $o,Xs2: list_P738500740D_node] : ( P @ R @ Xs2 @ nil_Pr1769730692D_node )
=> ( ! [R: produc1453890942D_node > produc1453890942D_node > $o,V2: produc1453890942D_node,Va: list_P738500740D_node] : ( P @ R @ nil_Pr1769730692D_node @ ( cons_P1018517044D_node @ V2 @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% mergesort_by_rel_merge.induct
thf(fact_217_mergesort__by__rel__merge_Oinduct,axiom,
! [P: ( node > node > $o ) > list_node > list_node > $o,A0: node > node > $o,A1: list_node,A22: list_node] :
( ! [R: node > node > $o,X2: node,Xs2: list_node,Y4: node,Ys3: list_node] :
( ( ( R @ X2 @ Y4 )
=> ( P @ R @ Xs2 @ ( cons_node @ Y4 @ Ys3 ) ) )
=> ( ( ~ ( R @ X2 @ Y4 )
=> ( P @ R @ ( cons_node @ X2 @ Xs2 ) @ Ys3 ) )
=> ( P @ R @ ( cons_node @ X2 @ Xs2 ) @ ( cons_node @ Y4 @ Ys3 ) ) ) )
=> ( ! [R: node > node > $o,Xs2: list_node] : ( P @ R @ Xs2 @ nil_node )
=> ( ! [R: node > node > $o,V2: node,Va: list_node] : ( P @ R @ nil_node @ ( cons_node @ V2 @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% mergesort_by_rel_merge.induct
thf(fact_218_mergesort__by__rel__merge__induct,axiom,
! [P: list_P738500740D_node > list_P738500740D_node > $o,R2: produc1453890942D_node > produc1453890942D_node > $o,Xs: list_P738500740D_node,Ys: list_P738500740D_node] :
( ! [Xs2: list_P738500740D_node] : ( P @ Xs2 @ nil_Pr1769730692D_node )
=> ( ! [X_1: list_P738500740D_node] : ( P @ nil_Pr1769730692D_node @ X_1 )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( R2 @ X2 @ Y4 )
=> ( ( P @ Xs2 @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) )
=> ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) ) ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ~ ( R2 @ X2 @ Y4 )
=> ( ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_219_mergesort__by__rel__merge__induct,axiom,
! [P: list_node > list_P738500740D_node > $o,R2: node > produc1453890942D_node > $o,Xs: list_node,Ys: list_P738500740D_node] :
( ! [Xs2: list_node] : ( P @ Xs2 @ nil_Pr1769730692D_node )
=> ( ! [X_1: list_P738500740D_node] : ( P @ nil_node @ X_1 )
=> ( ! [X2: node,Xs2: list_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( R2 @ X2 @ Y4 )
=> ( ( P @ Xs2 @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) )
=> ( P @ ( cons_node @ X2 @ Xs2 ) @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) ) ) )
=> ( ! [X2: node,Xs2: list_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ~ ( R2 @ X2 @ Y4 )
=> ( ( P @ ( cons_node @ X2 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons_node @ X2 @ Xs2 ) @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_220_mergesort__by__rel__merge__induct,axiom,
! [P: list_P738500740D_node > list_node > $o,R2: produc1453890942D_node > node > $o,Xs: list_P738500740D_node,Ys: list_node] :
( ! [Xs2: list_P738500740D_node] : ( P @ Xs2 @ nil_node )
=> ( ! [X_1: list_node] : ( P @ nil_Pr1769730692D_node @ X_1 )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: node,Ys3: list_node] :
( ( R2 @ X2 @ Y4 )
=> ( ( P @ Xs2 @ ( cons_node @ Y4 @ Ys3 ) )
=> ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ ( cons_node @ Y4 @ Ys3 ) ) ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: node,Ys3: list_node] :
( ~ ( R2 @ X2 @ Y4 )
=> ( ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ ( cons_node @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_221_mergesort__by__rel__merge__induct,axiom,
! [P: list_node > list_node > $o,R2: node > node > $o,Xs: list_node,Ys: list_node] :
( ! [Xs2: list_node] : ( P @ Xs2 @ nil_node )
=> ( ! [X_1: list_node] : ( P @ nil_node @ X_1 )
=> ( ! [X2: node,Xs2: list_node,Y4: node,Ys3: list_node] :
( ( R2 @ X2 @ Y4 )
=> ( ( P @ Xs2 @ ( cons_node @ Y4 @ Ys3 ) )
=> ( P @ ( cons_node @ X2 @ Xs2 ) @ ( cons_node @ Y4 @ Ys3 ) ) ) )
=> ( ! [X2: node,Xs2: list_node,Y4: node,Ys3: list_node] :
( ~ ( R2 @ X2 @ Y4 )
=> ( ( P @ ( cons_node @ X2 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons_node @ X2 @ Xs2 ) @ ( cons_node @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_222_longest__common__prefix_Oinduct,axiom,
! [P: list_P738500740D_node > list_P738500740D_node > $o,A0: list_P738500740D_node,A1: list_P738500740D_node] :
( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( ( X2 = Y4 )
=> ( P @ Xs2 @ Ys3 ) )
=> ( P @ ( cons_P1018517044D_node @ X2 @ Xs2 ) @ ( cons_P1018517044D_node @ Y4 @ Ys3 ) ) )
=> ( ! [X_1: list_P738500740D_node] : ( P @ nil_Pr1769730692D_node @ X_1 )
=> ( ! [Uu: list_P738500740D_node] : ( P @ Uu @ nil_Pr1769730692D_node )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_223_longest__common__prefix_Oinduct,axiom,
! [P: list_node > list_node > $o,A0: list_node,A1: list_node] :
( ! [X2: node,Xs2: list_node,Y4: node,Ys3: list_node] :
( ( ( X2 = Y4 )
=> ( P @ Xs2 @ Ys3 ) )
=> ( P @ ( cons_node @ X2 @ Xs2 ) @ ( cons_node @ Y4 @ Ys3 ) ) )
=> ( ! [X_1: list_node] : ( P @ nil_node @ X_1 )
=> ( ! [Uu: list_node] : ( P @ Uu @ nil_node )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_224_strict__sorted_Oinduct,axiom,
! [P: list_node > $o,A0: list_node] :
( ( P @ nil_node )
=> ( ! [X2: node,Ys3: list_node] :
( ( P @ Ys3 )
=> ( P @ ( cons_node @ X2 @ Ys3 ) ) )
=> ( P @ A0 ) ) ) ).
% strict_sorted.induct
thf(fact_225_list_Osel_I1_J,axiom,
! [X21: node,X22: list_node] :
( ( hd_node @ ( cons_node @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_226_list_Osel_I3_J,axiom,
! [X21: node,X22: list_node] :
( ( tl_node @ ( cons_node @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_227_list__tail__coinc,axiom,
! [N1: node,R1: list_node,N22: node,R22: list_node] :
( ( ( cons_node @ N1 @ R1 )
= ( cons_node @ N22 @ R22 ) )
=> ( ( N1 = N22 )
& ( R1 = R22 ) ) ) ).
% list_tail_coinc
thf(fact_228_not__Cons__self2,axiom,
! [X: node,Xs: list_node] :
( ( cons_node @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_229_distinct__length__2__or__more,axiom,
! [A: node,B: node,Xs: list_node] :
( ( distinct_node @ ( cons_node @ A @ ( cons_node @ B @ Xs ) ) )
= ( ( A != B )
& ( distinct_node @ ( cons_node @ A @ Xs ) )
& ( distinct_node @ ( cons_node @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_230_list_Oset__cases,axiom,
! [E: val,A: list_val] :
( ( member_val @ E @ ( set_val2 @ A ) )
=> ( ! [Z2: list_val] :
( A
!= ( cons_val @ E @ Z2 ) )
=> ~ ! [Z1: val,Z2: list_val] :
( ( A
= ( cons_val @ Z1 @ Z2 ) )
=> ~ ( member_val @ E @ ( set_val2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_231_list_Oset__cases,axiom,
! [E: node,A: list_node] :
( ( member_node @ E @ ( set_node2 @ A ) )
=> ( ! [Z2: list_node] :
( A
!= ( cons_node @ E @ Z2 ) )
=> ~ ! [Z1: node,Z2: list_node] :
( ( A
= ( cons_node @ Z1 @ Z2 ) )
=> ~ ( member_node @ E @ ( set_node2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_232_set__ConsD,axiom,
! [Y3: val,X: val,Xs: list_val] :
( ( member_val @ Y3 @ ( set_val2 @ ( cons_val @ X @ Xs ) ) )
=> ( ( Y3 = X )
| ( member_val @ Y3 @ ( set_val2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_233_set__ConsD,axiom,
! [Y3: node,X: node,Xs: list_node] :
( ( member_node @ Y3 @ ( set_node2 @ ( cons_node @ X @ Xs ) ) )
=> ( ( Y3 = X )
| ( member_node @ Y3 @ ( set_node2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_234_list_Oset__intros_I1_J,axiom,
! [X21: val,X22: list_val] : ( member_val @ X21 @ ( set_val2 @ ( cons_val @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_235_list_Oset__intros_I1_J,axiom,
! [X21: node,X22: list_node] : ( member_node @ X21 @ ( set_node2 @ ( cons_node @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_236_list_Oset__intros_I2_J,axiom,
! [Y3: val,X22: list_val,X21: val] :
( ( member_val @ Y3 @ ( set_val2 @ X22 ) )
=> ( member_val @ Y3 @ ( set_val2 @ ( cons_val @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_237_list_Oset__intros_I2_J,axiom,
! [Y3: node,X22: list_node,X21: node] :
( ( member_node @ Y3 @ ( set_node2 @ X22 ) )
=> ( member_node @ Y3 @ ( set_node2 @ ( cons_node @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_238_list__append__eq__Cons__cases,axiom,
! [Ys: list_P738500740D_node,Zs: list_P738500740D_node,X: produc1453890942D_node,Xs: list_P738500740D_node] :
( ( ( append2096883353D_node @ Ys @ Zs )
= ( cons_P1018517044D_node @ X @ Xs ) )
=> ( ( ( Ys = nil_Pr1769730692D_node )
=> ( Zs
!= ( cons_P1018517044D_node @ X @ Xs ) ) )
=> ~ ! [Ys4: list_P738500740D_node] :
( ( Ys
= ( cons_P1018517044D_node @ X @ Ys4 ) )
=> ( ( append2096883353D_node @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_239_list__append__eq__Cons__cases,axiom,
! [Ys: list_node,Zs: list_node,X: node,Xs: list_node] :
( ( ( append_node @ Ys @ Zs )
= ( cons_node @ X @ Xs ) )
=> ( ( ( Ys = nil_node )
=> ( Zs
!= ( cons_node @ X @ Xs ) ) )
=> ~ ! [Ys4: list_node] :
( ( Ys
= ( cons_node @ X @ Ys4 ) )
=> ( ( append_node @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_240_list__Cons__eq__append__cases,axiom,
! [X: produc1453890942D_node,Xs: list_P738500740D_node,Ys: list_P738500740D_node,Zs: list_P738500740D_node] :
( ( ( cons_P1018517044D_node @ X @ Xs )
= ( append2096883353D_node @ Ys @ Zs ) )
=> ( ( ( Ys = nil_Pr1769730692D_node )
=> ( Zs
!= ( cons_P1018517044D_node @ X @ Xs ) ) )
=> ~ ! [Ys4: list_P738500740D_node] :
( ( Ys
= ( cons_P1018517044D_node @ X @ Ys4 ) )
=> ( ( append2096883353D_node @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_241_list__Cons__eq__append__cases,axiom,
! [X: node,Xs: list_node,Ys: list_node,Zs: list_node] :
( ( ( cons_node @ X @ Xs )
= ( append_node @ Ys @ Zs ) )
=> ( ( ( Ys = nil_node )
=> ( Zs
!= ( cons_node @ X @ Xs ) ) )
=> ~ ! [Ys4: list_node] :
( ( Ys
= ( cons_node @ X @ Ys4 ) )
=> ( ( append_node @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_242_rev__nonempty__induct2_H,axiom,
! [Xs: list_P738500740D_node,Ys: list_P738500740D_node,P: list_P738500740D_node > list_P738500740D_node > $o] :
( ( Xs != nil_Pr1769730692D_node )
=> ( ( Ys != nil_Pr1769730692D_node )
=> ( ! [X2: produc1453890942D_node,Y4: produc1453890942D_node] : ( P @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: produc1453890942D_node] :
( ( Xs2 != nil_Pr1769730692D_node )
=> ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) )
=> ( ! [X2: produc1453890942D_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( Ys3 != nil_Pr1769730692D_node )
=> ( P @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) @ ( append2096883353D_node @ Ys3 @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( P @ Xs2 @ Ys3 )
=> ( ( Xs2 != nil_Pr1769730692D_node )
=> ( ( Ys3 != nil_Pr1769730692D_node )
=> ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) @ ( append2096883353D_node @ Ys3 @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_243_rev__nonempty__induct2_H,axiom,
! [Xs: list_P738500740D_node,Ys: list_node,P: list_P738500740D_node > list_node > $o] :
( ( Xs != nil_Pr1769730692D_node )
=> ( ( Ys != nil_node )
=> ( ! [X2: produc1453890942D_node,Y4: node] : ( P @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) @ ( cons_node @ Y4 @ nil_node ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: node] :
( ( Xs2 != nil_Pr1769730692D_node )
=> ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) @ ( cons_node @ Y4 @ nil_node ) ) )
=> ( ! [X2: produc1453890942D_node,Y4: node,Ys3: list_node] :
( ( Ys3 != nil_node )
=> ( P @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) @ ( append_node @ Ys3 @ ( cons_node @ Y4 @ nil_node ) ) ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: node,Ys3: list_node] :
( ( P @ Xs2 @ Ys3 )
=> ( ( Xs2 != nil_Pr1769730692D_node )
=> ( ( Ys3 != nil_node )
=> ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) @ ( append_node @ Ys3 @ ( cons_node @ Y4 @ nil_node ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_244_rev__nonempty__induct2_H,axiom,
! [Xs: list_node,Ys: list_P738500740D_node,P: list_node > list_P738500740D_node > $o] :
( ( Xs != nil_node )
=> ( ( Ys != nil_Pr1769730692D_node )
=> ( ! [X2: node,Y4: produc1453890942D_node] : ( P @ ( cons_node @ X2 @ nil_node ) @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) )
=> ( ! [X2: node,Xs2: list_node,Y4: produc1453890942D_node] :
( ( Xs2 != nil_node )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) )
=> ( ! [X2: node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( Ys3 != nil_Pr1769730692D_node )
=> ( P @ ( cons_node @ X2 @ nil_node ) @ ( append2096883353D_node @ Ys3 @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) ) )
=> ( ! [X2: node,Xs2: list_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( P @ Xs2 @ Ys3 )
=> ( ( Xs2 != nil_node )
=> ( ( Ys3 != nil_Pr1769730692D_node )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) @ ( append2096883353D_node @ Ys3 @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_245_rev__nonempty__induct2_H,axiom,
! [Xs: list_node,Ys: list_node,P: list_node > list_node > $o] :
( ( Xs != nil_node )
=> ( ( Ys != nil_node )
=> ( ! [X2: node,Y4: node] : ( P @ ( cons_node @ X2 @ nil_node ) @ ( cons_node @ Y4 @ nil_node ) )
=> ( ! [X2: node,Xs2: list_node,Y4: node] :
( ( Xs2 != nil_node )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) @ ( cons_node @ Y4 @ nil_node ) ) )
=> ( ! [X2: node,Y4: node,Ys3: list_node] :
( ( Ys3 != nil_node )
=> ( P @ ( cons_node @ X2 @ nil_node ) @ ( append_node @ Ys3 @ ( cons_node @ Y4 @ nil_node ) ) ) )
=> ( ! [X2: node,Xs2: list_node,Y4: node,Ys3: list_node] :
( ( P @ Xs2 @ Ys3 )
=> ( ( Xs2 != nil_node )
=> ( ( Ys3 != nil_node )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) @ ( append_node @ Ys3 @ ( cons_node @ Y4 @ nil_node ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_246_rev__nonempty__induct,axiom,
! [Xs: list_P738500740D_node,P: list_P738500740D_node > $o] :
( ( Xs != nil_Pr1769730692D_node )
=> ( ! [X2: produc1453890942D_node] : ( P @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node] :
( ( Xs2 != nil_Pr1769730692D_node )
=> ( ( P @ Xs2 )
=> ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_247_rev__nonempty__induct,axiom,
! [Xs: list_node,P: list_node > $o] :
( ( Xs != nil_node )
=> ( ! [X2: node] : ( P @ ( cons_node @ X2 @ nil_node ) )
=> ( ! [X2: node,Xs2: list_node] :
( ( Xs2 != nil_node )
=> ( ( P @ Xs2 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_248_append__eq__Cons__conv,axiom,
! [Ys: list_P738500740D_node,Zs: list_P738500740D_node,X: produc1453890942D_node,Xs: list_P738500740D_node] :
( ( ( append2096883353D_node @ Ys @ Zs )
= ( cons_P1018517044D_node @ X @ Xs ) )
= ( ( ( Ys = nil_Pr1769730692D_node )
& ( Zs
= ( cons_P1018517044D_node @ X @ Xs ) ) )
| ? [Ys5: list_P738500740D_node] :
( ( Ys
= ( cons_P1018517044D_node @ X @ Ys5 ) )
& ( ( append2096883353D_node @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_249_append__eq__Cons__conv,axiom,
! [Ys: list_node,Zs: list_node,X: node,Xs: list_node] :
( ( ( append_node @ Ys @ Zs )
= ( cons_node @ X @ Xs ) )
= ( ( ( Ys = nil_node )
& ( Zs
= ( cons_node @ X @ Xs ) ) )
| ? [Ys5: list_node] :
( ( Ys
= ( cons_node @ X @ Ys5 ) )
& ( ( append_node @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_250_Cons__eq__append__conv,axiom,
! [X: produc1453890942D_node,Xs: list_P738500740D_node,Ys: list_P738500740D_node,Zs: list_P738500740D_node] :
( ( ( cons_P1018517044D_node @ X @ Xs )
= ( append2096883353D_node @ Ys @ Zs ) )
= ( ( ( Ys = nil_Pr1769730692D_node )
& ( ( cons_P1018517044D_node @ X @ Xs )
= Zs ) )
| ? [Ys5: list_P738500740D_node] :
( ( ( cons_P1018517044D_node @ X @ Ys5 )
= Ys )
& ( Xs
= ( append2096883353D_node @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_251_Cons__eq__append__conv,axiom,
! [X: node,Xs: list_node,Ys: list_node,Zs: list_node] :
( ( ( cons_node @ X @ Xs )
= ( append_node @ Ys @ Zs ) )
= ( ( ( Ys = nil_node )
& ( ( cons_node @ X @ Xs )
= Zs ) )
| ? [Ys5: list_node] :
( ( ( cons_node @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_node @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_252_neq__Nil__rev__conv,axiom,
! [L: list_P738500740D_node] :
( ( L != nil_Pr1769730692D_node )
= ( ? [Xs3: list_P738500740D_node,X3: produc1453890942D_node] :
( L
= ( append2096883353D_node @ Xs3 @ ( cons_P1018517044D_node @ X3 @ nil_Pr1769730692D_node ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_253_neq__Nil__rev__conv,axiom,
! [L: list_node] :
( ( L != nil_node )
= ( ? [Xs3: list_node,X3: node] :
( L
= ( append_node @ Xs3 @ ( cons_node @ X3 @ nil_node ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_254_rev__induct2_H,axiom,
! [P: list_P738500740D_node > list_P738500740D_node > $o,Xs: list_P738500740D_node,Ys: list_P738500740D_node] :
( ( P @ nil_Pr1769730692D_node @ nil_Pr1769730692D_node )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node] : ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) @ nil_Pr1769730692D_node )
=> ( ! [Y4: produc1453890942D_node,Ys3: list_P738500740D_node] : ( P @ nil_Pr1769730692D_node @ ( append2096883353D_node @ Ys3 @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) @ ( append2096883353D_node @ Ys3 @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_255_rev__induct2_H,axiom,
! [P: list_P738500740D_node > list_node > $o,Xs: list_P738500740D_node,Ys: list_node] :
( ( P @ nil_Pr1769730692D_node @ nil_node )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node] : ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) @ nil_node )
=> ( ! [Y4: node,Ys3: list_node] : ( P @ nil_Pr1769730692D_node @ ( append_node @ Ys3 @ ( cons_node @ Y4 @ nil_node ) ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: node,Ys3: list_node] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) @ ( append_node @ Ys3 @ ( cons_node @ Y4 @ nil_node ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_256_rev__induct2_H,axiom,
! [P: list_node > list_P738500740D_node > $o,Xs: list_node,Ys: list_P738500740D_node] :
( ( P @ nil_node @ nil_Pr1769730692D_node )
=> ( ! [X2: node,Xs2: list_node] : ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) @ nil_Pr1769730692D_node )
=> ( ! [Y4: produc1453890942D_node,Ys3: list_P738500740D_node] : ( P @ nil_node @ ( append2096883353D_node @ Ys3 @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) )
=> ( ! [X2: node,Xs2: list_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) @ ( append2096883353D_node @ Ys3 @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_257_rev__induct2_H,axiom,
! [P: list_node > list_node > $o,Xs: list_node,Ys: list_node] :
( ( P @ nil_node @ nil_node )
=> ( ! [X2: node,Xs2: list_node] : ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) @ nil_node )
=> ( ! [Y4: node,Ys3: list_node] : ( P @ nil_node @ ( append_node @ Ys3 @ ( cons_node @ Y4 @ nil_node ) ) )
=> ( ! [X2: node,Xs2: list_node,Y4: node,Ys3: list_node] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) @ ( append_node @ Ys3 @ ( cons_node @ Y4 @ nil_node ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_258_neq__Nil__revE,axiom,
! [L: list_P738500740D_node] :
( ( L != nil_Pr1769730692D_node )
=> ~ ! [Ll: list_P738500740D_node,E4: produc1453890942D_node] :
( L
!= ( append2096883353D_node @ Ll @ ( cons_P1018517044D_node @ E4 @ nil_Pr1769730692D_node ) ) ) ) ).
% neq_Nil_revE
thf(fact_259_neq__Nil__revE,axiom,
! [L: list_node] :
( ( L != nil_node )
=> ~ ! [Ll: list_node,E4: node] :
( L
!= ( append_node @ Ll @ ( cons_node @ E4 @ nil_node ) ) ) ) ).
% neq_Nil_revE
thf(fact_260_rev__exhaust,axiom,
! [Xs: list_P738500740D_node] :
( ( Xs != nil_Pr1769730692D_node )
=> ~ ! [Ys3: list_P738500740D_node,Y4: produc1453890942D_node] :
( Xs
!= ( append2096883353D_node @ Ys3 @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) ) ) ).
% rev_exhaust
thf(fact_261_rev__exhaust,axiom,
! [Xs: list_node] :
( ( Xs != nil_node )
=> ~ ! [Ys3: list_node,Y4: node] :
( Xs
!= ( append_node @ Ys3 @ ( cons_node @ Y4 @ nil_node ) ) ) ) ).
% rev_exhaust
thf(fact_262_rev__induct,axiom,
! [P: list_P738500740D_node > $o,Xs: list_P738500740D_node] :
( ( P @ nil_Pr1769730692D_node )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node] :
( ( P @ Xs2 )
=> ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_263_rev__induct,axiom,
! [P: list_node > $o,Xs: list_node] :
( ( P @ nil_node )
=> ( ! [X2: node,Xs2: list_node] :
( ( P @ Xs2 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_264_split__list,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ? [Ys3: list_val,Zs3: list_val] :
( Xs
= ( append_val @ Ys3 @ ( cons_val @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_265_split__list,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ? [Ys3: list_node,Zs3: list_node] :
( Xs
= ( append_node @ Ys3 @ ( cons_node @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_266_split__list__last,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ? [Ys3: list_val,Zs3: list_val] :
( ( Xs
= ( append_val @ Ys3 @ ( cons_val @ X @ Zs3 ) ) )
& ~ ( member_val @ X @ ( set_val2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_267_split__list__last,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ? [Ys3: list_node,Zs3: list_node] :
( ( Xs
= ( append_node @ Ys3 @ ( cons_node @ X @ Zs3 ) ) )
& ~ ( member_node @ X @ ( set_node2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_268_split__list__prop,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X4: node] :
( ( member_node @ X4 @ ( set_node2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_node,X2: node] :
( ? [Zs3: list_node] :
( Xs
= ( append_node @ Ys3 @ ( cons_node @ X2 @ Zs3 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_269_xy__in__set__cases,axiom,
! [X: val,L: list_val,Y3: val] :
( ( member_val @ X @ ( set_val2 @ L ) )
=> ( ( member_val @ Y3 @ ( set_val2 @ L ) )
=> ( ( ( X = Y3 )
=> ! [L12: list_val,L22: list_val] :
( L
!= ( append_val @ L12 @ ( cons_val @ Y3 @ L22 ) ) ) )
=> ( ( ( X != Y3 )
=> ! [L12: list_val,L22: list_val,L3: list_val] :
( L
!= ( append_val @ L12 @ ( cons_val @ X @ ( append_val @ L22 @ ( cons_val @ Y3 @ L3 ) ) ) ) ) )
=> ~ ( ( X != Y3 )
=> ! [L12: list_val,L22: list_val,L3: list_val] :
( L
!= ( append_val @ L12 @ ( cons_val @ Y3 @ ( append_val @ L22 @ ( cons_val @ X @ L3 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_270_xy__in__set__cases,axiom,
! [X: node,L: list_node,Y3: node] :
( ( member_node @ X @ ( set_node2 @ L ) )
=> ( ( member_node @ Y3 @ ( set_node2 @ L ) )
=> ( ( ( X = Y3 )
=> ! [L12: list_node,L22: list_node] :
( L
!= ( append_node @ L12 @ ( cons_node @ Y3 @ L22 ) ) ) )
=> ( ( ( X != Y3 )
=> ! [L12: list_node,L22: list_node,L3: list_node] :
( L
!= ( append_node @ L12 @ ( cons_node @ X @ ( append_node @ L22 @ ( cons_node @ Y3 @ L3 ) ) ) ) ) )
=> ~ ( ( X != Y3 )
=> ! [L12: list_node,L22: list_node,L3: list_node] :
( L
!= ( append_node @ L12 @ ( cons_node @ Y3 @ ( append_node @ L22 @ ( cons_node @ X @ L3 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_271_split__list__first,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ? [Ys3: list_val,Zs3: list_val] :
( ( Xs
= ( append_val @ Ys3 @ ( cons_val @ X @ Zs3 ) ) )
& ~ ( member_val @ X @ ( set_val2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_272_split__list__first,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ? [Ys3: list_node,Zs3: list_node] :
( ( Xs
= ( append_node @ Ys3 @ ( cons_node @ X @ Zs3 ) ) )
& ~ ( member_node @ X @ ( set_node2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_273_split__list__propE,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X4: node] :
( ( member_node @ X4 @ ( set_node2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_node,X2: node] :
( ? [Zs3: list_node] :
( Xs
= ( append_node @ Ys3 @ ( cons_node @ X2 @ Zs3 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_274_append__Cons__eq__iff,axiom,
! [X: val,Xs: list_val,Ys: list_val,Xs4: list_val,Ys6: list_val] :
( ~ ( member_val @ X @ ( set_val2 @ Xs ) )
=> ( ~ ( member_val @ X @ ( set_val2 @ Ys ) )
=> ( ( ( append_val @ Xs @ ( cons_val @ X @ Ys ) )
= ( append_val @ Xs4 @ ( cons_val @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_275_append__Cons__eq__iff,axiom,
! [X: node,Xs: list_node,Ys: list_node,Xs4: list_node,Ys6: list_node] :
( ~ ( member_node @ X @ ( set_node2 @ Xs ) )
=> ( ~ ( member_node @ X @ ( set_node2 @ Ys ) )
=> ( ( ( append_node @ Xs @ ( cons_node @ X @ Ys ) )
= ( append_node @ Xs4 @ ( cons_node @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_276_in__set__conv__decomp,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
= ( ? [Ys2: list_val,Zs2: list_val] :
( Xs
= ( append_val @ Ys2 @ ( cons_val @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_277_in__set__conv__decomp,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
= ( ? [Ys2: list_node,Zs2: list_node] :
( Xs
= ( append_node @ Ys2 @ ( cons_node @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_278_in__set__list__format,axiom,
! [E: val,L: list_val] :
( ( member_val @ E @ ( set_val2 @ L ) )
=> ~ ! [L12: list_val,L22: list_val] :
( L
!= ( append_val @ L12 @ ( cons_val @ E @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_279_in__set__list__format,axiom,
! [E: node,L: list_node] :
( ( member_node @ E @ ( set_node2 @ L ) )
=> ~ ! [L12: list_node,L22: list_node] :
( L
!= ( append_node @ L12 @ ( cons_node @ E @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_280_split__list__last__prop,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X4: node] :
( ( member_node @ X4 @ ( set_node2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_node,X2: node,Zs3: list_node] :
( ( Xs
= ( append_node @ Ys3 @ ( cons_node @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: node] :
( ( member_node @ Xa @ ( set_node2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_281_split__list__first__prop,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X4: node] :
( ( member_node @ X4 @ ( set_node2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_node,X2: node] :
( ? [Zs3: list_node] :
( Xs
= ( append_node @ Ys3 @ ( cons_node @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: node] :
( ( member_node @ Xa @ ( set_node2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_282_split__list__last__propE,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X4: node] :
( ( member_node @ X4 @ ( set_node2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_node,X2: node,Zs3: list_node] :
( ( Xs
= ( append_node @ Ys3 @ ( cons_node @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: node] :
( ( member_node @ Xa @ ( set_node2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_283_split__list__first__propE,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X4: node] :
( ( member_node @ X4 @ ( set_node2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_node,X2: node] :
( ? [Zs3: list_node] :
( Xs
= ( append_node @ Ys3 @ ( cons_node @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: node] :
( ( member_node @ Xa @ ( set_node2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_284_in__set__conv__decomp__last,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
= ( ? [Ys2: list_val,Zs2: list_val] :
( ( Xs
= ( append_val @ Ys2 @ ( cons_val @ X @ Zs2 ) ) )
& ~ ( member_val @ X @ ( set_val2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_285_in__set__conv__decomp__last,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
= ( ? [Ys2: list_node,Zs2: list_node] :
( ( Xs
= ( append_node @ Ys2 @ ( cons_node @ X @ Zs2 ) ) )
& ~ ( member_node @ X @ ( set_node2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_286_in__set__conv__decomp__first,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
= ( ? [Ys2: list_val,Zs2: list_val] :
( ( Xs
= ( append_val @ Ys2 @ ( cons_val @ X @ Zs2 ) ) )
& ~ ( member_val @ X @ ( set_val2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_287_in__set__conv__decomp__first,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
= ( ? [Ys2: list_node,Zs2: list_node] :
( ( Xs
= ( append_node @ Ys2 @ ( cons_node @ X @ Zs2 ) ) )
& ~ ( member_node @ X @ ( set_node2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_288_split__list__last__prop__iff,axiom,
! [Xs: list_node,P: node > $o] :
( ( ? [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list_node,X3: node,Zs2: list_node] :
( ( Xs
= ( append_node @ Ys2 @ ( cons_node @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y2: node] :
( ( member_node @ Y2 @ ( set_node2 @ Zs2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_289_split__list__first__prop__iff,axiom,
! [Xs: list_node,P: node > $o] :
( ( ? [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list_node,X3: node] :
( ? [Zs2: list_node] :
( Xs
= ( append_node @ Ys2 @ ( cons_node @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y2: node] :
( ( member_node @ Y2 @ ( set_node2 @ Ys2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_290_distinct__singleton,axiom,
! [X: produc1453890942D_node] : ( distin1098566007D_node @ ( cons_P1018517044D_node @ X @ nil_Pr1769730692D_node ) ) ).
% distinct_singleton
thf(fact_291_distinct__singleton,axiom,
! [X: node] : ( distinct_node @ ( cons_node @ X @ nil_node ) ) ).
% distinct_singleton
thf(fact_292_distinct_Osimps_I2_J,axiom,
! [X: val,Xs: list_val] :
( ( distinct_val @ ( cons_val @ X @ Xs ) )
= ( ~ ( member_val @ X @ ( set_val2 @ Xs ) )
& ( distinct_val @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_293_distinct_Osimps_I2_J,axiom,
! [X: node,Xs: list_node] :
( ( distinct_node @ ( cons_node @ X @ Xs ) )
= ( ~ ( member_node @ X @ ( set_node2 @ Xs ) )
& ( distinct_node @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_294_distinct__match,axiom,
! [Al: list_node,E: node,Bl: list_node,Al2: list_node,Bl2: list_node] :
( ( distinct_node @ ( append_node @ Al @ ( cons_node @ E @ Bl ) ) )
=> ( ( ( append_node @ Al @ ( cons_node @ E @ Bl ) )
= ( append_node @ Al2 @ ( cons_node @ E @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( Bl = Bl2 ) ) ) ) ).
% distinct_match
thf(fact_295_Nil__tl,axiom,
! [Xs: list_P738500740D_node] :
( ( nil_Pr1769730692D_node
= ( tl_Pro1633633005D_node @ Xs ) )
= ( ( Xs = nil_Pr1769730692D_node )
| ? [X3: produc1453890942D_node] :
( Xs
= ( cons_P1018517044D_node @ X3 @ nil_Pr1769730692D_node ) ) ) ) ).
% Nil_tl
thf(fact_296_Nil__tl,axiom,
! [Xs: list_node] :
( ( nil_node
= ( tl_node @ Xs ) )
= ( ( Xs = nil_node )
| ? [X3: node] :
( Xs
= ( cons_node @ X3 @ nil_node ) ) ) ) ).
% Nil_tl
thf(fact_297_tl__Nil,axiom,
! [Xs: list_P738500740D_node] :
( ( ( tl_Pro1633633005D_node @ Xs )
= nil_Pr1769730692D_node )
= ( ( Xs = nil_Pr1769730692D_node )
| ? [X3: produc1453890942D_node] :
( Xs
= ( cons_P1018517044D_node @ X3 @ nil_Pr1769730692D_node ) ) ) ) ).
% tl_Nil
thf(fact_298_tl__Nil,axiom,
! [Xs: list_node] :
( ( ( tl_node @ Xs )
= nil_node )
= ( ( Xs = nil_node )
| ? [X3: node] :
( Xs
= ( cons_node @ X3 @ nil_node ) ) ) ) ).
% tl_Nil
thf(fact_299_tl__obtain__elem,axiom,
! [Xs: list_P738500740D_node] :
( ( Xs != nil_Pr1769730692D_node )
=> ( ( ( tl_Pro1633633005D_node @ Xs )
= nil_Pr1769730692D_node )
=> ~ ! [E4: produc1453890942D_node] :
( Xs
!= ( cons_P1018517044D_node @ E4 @ nil_Pr1769730692D_node ) ) ) ) ).
% tl_obtain_elem
thf(fact_300_tl__obtain__elem,axiom,
! [Xs: list_node] :
( ( Xs != nil_node )
=> ( ( ( tl_node @ Xs )
= nil_node )
=> ~ ! [E4: node] :
( Xs
!= ( cons_node @ E4 @ nil_node ) ) ) ) ).
% tl_obtain_elem
thf(fact_301_not__distinct__decomp,axiom,
! [Ws: list_P738500740D_node] :
( ~ ( distin1098566007D_node @ Ws )
=> ? [Xs2: list_P738500740D_node,Ys3: list_P738500740D_node,Zs3: list_P738500740D_node,Y4: produc1453890942D_node] :
( Ws
= ( append2096883353D_node @ Xs2 @ ( append2096883353D_node @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) @ ( append2096883353D_node @ Ys3 @ ( append2096883353D_node @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_302_not__distinct__decomp,axiom,
! [Ws: list_node] :
( ~ ( distinct_node @ Ws )
=> ? [Xs2: list_node,Ys3: list_node,Zs3: list_node,Y4: node] :
( Ws
= ( append_node @ Xs2 @ ( append_node @ ( cons_node @ Y4 @ nil_node ) @ ( append_node @ Ys3 @ ( append_node @ ( cons_node @ Y4 @ nil_node ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_303_not__distinct__conv__prefix,axiom,
! [As2: list_val] :
( ( ~ ( distinct_val @ As2 ) )
= ( ? [Xs3: list_val,Y2: val,Ys2: list_val] :
( ( member_val @ Y2 @ ( set_val2 @ Xs3 ) )
& ( distinct_val @ Xs3 )
& ( As2
= ( append_val @ Xs3 @ ( cons_val @ Y2 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_304_not__distinct__conv__prefix,axiom,
! [As2: list_node] :
( ( ~ ( distinct_node @ As2 ) )
= ( ? [Xs3: list_node,Y2: node,Ys2: list_node] :
( ( member_node @ Y2 @ ( set_node2 @ Xs3 ) )
& ( distinct_node @ Xs3 )
& ( As2
= ( append_node @ Xs3 @ ( cons_node @ Y2 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_305_not__suffix__induct,axiom,
! [Ps: list_P738500740D_node,Ls: list_P738500740D_node,P: list_P738500740D_node > list_P738500740D_node > $o] :
( ~ ( suffix1143830554D_node @ Ps @ Ls )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node] : ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) @ nil_Pr1769730692D_node )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( X2 != Y4 )
=> ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) @ ( append2096883353D_node @ Ys3 @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) ) )
=> ( ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node,Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( X2 = Y4 )
=> ( ~ ( suffix1143830554D_node @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) @ ( append2096883353D_node @ Ys3 @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_suffix_induct
thf(fact_306_not__suffix__induct,axiom,
! [Ps: list_node,Ls: list_node,P: list_node > list_node > $o] :
( ~ ( suffix_node @ Ps @ Ls )
=> ( ! [X2: node,Xs2: list_node] : ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) @ nil_node )
=> ( ! [X2: node,Xs2: list_node,Y4: node,Ys3: list_node] :
( ( X2 != Y4 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) @ ( append_node @ Ys3 @ ( cons_node @ Y4 @ nil_node ) ) ) )
=> ( ! [X2: node,Xs2: list_node,Y4: node,Ys3: list_node] :
( ( X2 = Y4 )
=> ( ~ ( suffix_node @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) @ ( append_node @ Ys3 @ ( cons_node @ Y4 @ nil_node ) ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_suffix_induct
thf(fact_307_not__suffix__cases,axiom,
! [Ps: list_P738500740D_node,Ls: list_P738500740D_node] :
( ~ ( suffix1143830554D_node @ Ps @ Ls )
=> ( ( ( Ps != nil_Pr1769730692D_node )
=> ( Ls != nil_Pr1769730692D_node ) )
=> ( ! [A3: produc1453890942D_node,As: list_P738500740D_node] :
( ( Ps
= ( append2096883353D_node @ As @ ( cons_P1018517044D_node @ A3 @ nil_Pr1769730692D_node ) ) )
=> ! [X2: produc1453890942D_node,Xs2: list_P738500740D_node] :
( ( Ls
= ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) )
=> ( ( X2 = A3 )
=> ( suffix1143830554D_node @ As @ Xs2 ) ) ) )
=> ~ ! [A3: produc1453890942D_node] :
( ? [As: list_P738500740D_node] :
( Ps
= ( append2096883353D_node @ As @ ( cons_P1018517044D_node @ A3 @ nil_Pr1769730692D_node ) ) )
=> ! [X2: produc1453890942D_node] :
( ? [Xs2: list_P738500740D_node] :
( Ls
= ( append2096883353D_node @ Xs2 @ ( cons_P1018517044D_node @ X2 @ nil_Pr1769730692D_node ) ) )
=> ( X2 = A3 ) ) ) ) ) ) ).
% not_suffix_cases
thf(fact_308_not__suffix__cases,axiom,
! [Ps: list_node,Ls: list_node] :
( ~ ( suffix_node @ Ps @ Ls )
=> ( ( ( Ps != nil_node )
=> ( Ls != nil_node ) )
=> ( ! [A3: node,As: list_node] :
( ( Ps
= ( append_node @ As @ ( cons_node @ A3 @ nil_node ) ) )
=> ! [X2: node,Xs2: list_node] :
( ( Ls
= ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) )
=> ( ( X2 = A3 )
=> ( suffix_node @ As @ Xs2 ) ) ) )
=> ~ ! [A3: node] :
( ? [As: list_node] :
( Ps
= ( append_node @ As @ ( cons_node @ A3 @ nil_node ) ) )
=> ! [X2: node] :
( ? [Xs2: list_node] :
( Ls
= ( append_node @ Xs2 @ ( cons_node @ X2 @ nil_node ) ) )
=> ( X2 = A3 ) ) ) ) ) ) ).
% not_suffix_cases
thf(fact_309_list_Oexhaust__sel,axiom,
! [List: list_P738500740D_node] :
( ( List != nil_Pr1769730692D_node )
=> ( List
= ( cons_P1018517044D_node @ ( hd_Pro1395892457D_node @ List ) @ ( tl_Pro1633633005D_node @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_310_list_Oexhaust__sel,axiom,
! [List: list_node] :
( ( List != nil_node )
=> ( List
= ( cons_node @ ( hd_node @ List ) @ ( tl_node @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_311_not__distinct__split__distinct,axiom,
! [Xs: list_val] :
( ~ ( distinct_val @ Xs )
=> ~ ! [Y4: val,Ys3: list_val] :
( ( distinct_val @ Ys3 )
=> ( ( member_val @ Y4 @ ( set_val2 @ Ys3 ) )
=> ! [Zs3: list_val] :
( Xs
!= ( append_val @ Ys3 @ ( append_val @ ( cons_val @ Y4 @ nil_val ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_312_not__distinct__split__distinct,axiom,
! [Xs: list_P738500740D_node] :
( ~ ( distin1098566007D_node @ Xs )
=> ~ ! [Y4: produc1453890942D_node,Ys3: list_P738500740D_node] :
( ( distin1098566007D_node @ Ys3 )
=> ( ( member1797643303D_node @ Y4 @ ( set_Pr1238794387D_node @ Ys3 ) )
=> ! [Zs3: list_P738500740D_node] :
( Xs
!= ( append2096883353D_node @ Ys3 @ ( append2096883353D_node @ ( cons_P1018517044D_node @ Y4 @ nil_Pr1769730692D_node ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_313_not__distinct__split__distinct,axiom,
! [Xs: list_node] :
( ~ ( distinct_node @ Xs )
=> ~ ! [Y4: node,Ys3: list_node] :
( ( distinct_node @ Ys3 )
=> ( ( member_node @ Y4 @ ( set_node2 @ Ys3 ) )
=> ! [Zs3: list_node] :
( Xs
!= ( append_node @ Ys3 @ ( append_node @ ( cons_node @ Y4 @ nil_node ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_314_append__eq__append__conv2,axiom,
! [Xs: list_node,Ys: list_node,Zs: list_node,Ts: list_node] :
( ( ( append_node @ Xs @ Ys )
= ( append_node @ Zs @ Ts ) )
= ( ? [Us: list_node] :
( ( ( Xs
= ( append_node @ Zs @ Us ) )
& ( ( append_node @ Us @ Ys )
= Ts ) )
| ( ( ( append_node @ Xs @ Us )
= Zs )
& ( Ys
= ( append_node @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_315_append__eq__appendI,axiom,
! [Xs: list_node,Xs1: list_node,Zs: list_node,Ys: list_node,Us2: list_node] :
( ( ( append_node @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_node @ Xs1 @ Us2 ) )
=> ( ( append_node @ Xs @ Ys )
= ( append_node @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_316_suffix__order_Odual__order_Oantisym,axiom,
! [B: list_node,A: list_node] :
( ( suffix_node @ B @ A )
=> ( ( suffix_node @ A @ B )
=> ( A = B ) ) ) ).
% suffix_order.dual_order.antisym
thf(fact_317_suffix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y5: list_node,Z: list_node] : Y5 = Z )
= ( ^ [A4: list_node,B3: list_node] :
( ( suffix_node @ B3 @ A4 )
& ( suffix_node @ A4 @ B3 ) ) ) ) ).
% suffix_order.dual_order.eq_iff
thf(fact_318_suffix__order_Odual__order_Otrans,axiom,
! [B: list_node,A: list_node,C: list_node] :
( ( suffix_node @ B @ A )
=> ( ( suffix_node @ C @ B )
=> ( suffix_node @ C @ A ) ) ) ).
% suffix_order.dual_order.trans
thf(fact_319_suffix__order_Oord__le__eq__trans,axiom,
! [A: list_node,B: list_node,C: list_node] :
( ( suffix_node @ A @ B )
=> ( ( B = C )
=> ( suffix_node @ A @ C ) ) ) ).
% suffix_order.ord_le_eq_trans
thf(fact_320_suffix__order_Oord__eq__le__trans,axiom,
! [A: list_node,B: list_node,C: list_node] :
( ( A = B )
=> ( ( suffix_node @ B @ C )
=> ( suffix_node @ A @ C ) ) ) ).
% suffix_order.ord_eq_le_trans
thf(fact_321_suffix__order_Oorder_Oantisym,axiom,
! [A: list_node,B: list_node] :
( ( suffix_node @ A @ B )
=> ( ( suffix_node @ B @ A )
=> ( A = B ) ) ) ).
% suffix_order.order.antisym
thf(fact_322_suffix__order_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: list_node,Z: list_node] : Y5 = Z )
= ( ^ [A4: list_node,B3: list_node] :
( ( suffix_node @ A4 @ B3 )
& ( suffix_node @ B3 @ A4 ) ) ) ) ).
% suffix_order.order.eq_iff
thf(fact_323_suffix__order_Oantisym__conv,axiom,
! [Y3: list_node,X: list_node] :
( ( suffix_node @ Y3 @ X )
=> ( ( suffix_node @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% suffix_order.antisym_conv
thf(fact_324_suffix__order_Oorder__trans,axiom,
! [X: list_node,Y3: list_node,Z3: list_node] :
( ( suffix_node @ X @ Y3 )
=> ( ( suffix_node @ Y3 @ Z3 )
=> ( suffix_node @ X @ Z3 ) ) ) ).
% suffix_order.order_trans
thf(fact_325_suffix__order_Oorder_Otrans,axiom,
! [A: list_node,B: list_node,C: list_node] :
( ( suffix_node @ A @ B )
=> ( ( suffix_node @ B @ C )
=> ( suffix_node @ A @ C ) ) ) ).
% suffix_order.order.trans
thf(fact_326_suffix__order_Oeq__refl,axiom,
! [X: list_node,Y3: list_node] :
( ( X = Y3 )
=> ( suffix_node @ X @ Y3 ) ) ).
% suffix_order.eq_refl
thf(fact_327_suffix__order_Oantisym,axiom,
! [X: list_node,Y3: list_node] :
( ( suffix_node @ X @ Y3 )
=> ( ( suffix_node @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% suffix_order.antisym
thf(fact_328_suffix__order_Oeq__iff,axiom,
( ( ^ [Y5: list_node,Z: list_node] : Y5 = Z )
= ( ^ [X3: list_node,Y2: list_node] :
( ( suffix_node @ X3 @ Y2 )
& ( suffix_node @ Y2 @ X3 ) ) ) ) ).
% suffix_order.eq_iff
thf(fact_329_suffix__same__cases,axiom,
! [Xs_1: list_node,Ys: list_node,Xs_2: list_node] :
( ( suffix_node @ Xs_1 @ Ys )
=> ( ( suffix_node @ Xs_2 @ Ys )
=> ( ( suffix_node @ Xs_1 @ Xs_2 )
| ( suffix_node @ Xs_2 @ Xs_1 ) ) ) ) ).
% suffix_same_cases
thf(fact_330_finite__list,axiom,
! [A2: set_node] :
( ( finite_finite_node @ A2 )
=> ? [Xs2: list_node] :
( ( set_node2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_331_finite__list,axiom,
! [A2: set_val] :
( ( finite_finite_val @ A2 )
=> ? [Xs2: list_val] :
( ( set_val2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_332_append_Oleft__neutral,axiom,
! [A: list_node] :
( ( append_node @ nil_node @ A )
= A ) ).
% append.left_neutral
thf(fact_333_append_Oleft__neutral,axiom,
! [A: list_P738500740D_node] :
( ( append2096883353D_node @ nil_Pr1769730692D_node @ A )
= A ) ).
% append.left_neutral
thf(fact_334_append__Nil,axiom,
! [Ys: list_node] :
( ( append_node @ nil_node @ Ys )
= Ys ) ).
% append_Nil
thf(fact_335_append__Nil,axiom,
! [Ys: list_P738500740D_node] :
( ( append2096883353D_node @ nil_Pr1769730692D_node @ Ys )
= Ys ) ).
% append_Nil
thf(fact_336_eq__Nil__appendI,axiom,
! [Xs: list_node,Ys: list_node] :
( ( Xs = Ys )
=> ( Xs
= ( append_node @ nil_node @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_337_eq__Nil__appendI,axiom,
! [Xs: list_P738500740D_node,Ys: list_P738500740D_node] :
( ( Xs = Ys )
=> ( Xs
= ( append2096883353D_node @ nil_Pr1769730692D_node @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_338_distinct_Osimps_I1_J,axiom,
distinct_node @ nil_node ).
% distinct.simps(1)
thf(fact_339_distinct_Osimps_I1_J,axiom,
distin1098566007D_node @ nil_Pr1769730692D_node ).
% distinct.simps(1)
thf(fact_340_suffix__bot_Obot_Oextremum__uniqueI,axiom,
! [A: list_P738500740D_node] :
( ( suffix1143830554D_node @ A @ nil_Pr1769730692D_node )
=> ( A = nil_Pr1769730692D_node ) ) ).
% suffix_bot.bot.extremum_uniqueI
thf(fact_341_suffix__bot_Obot_Oextremum__uniqueI,axiom,
! [A: list_node] :
( ( suffix_node @ A @ nil_node )
=> ( A = nil_node ) ) ).
% suffix_bot.bot.extremum_uniqueI
thf(fact_342_suffix__bot_Obot_Oextremum,axiom,
! [A: list_P738500740D_node] : ( suffix1143830554D_node @ nil_Pr1769730692D_node @ A ) ).
% suffix_bot.bot.extremum
thf(fact_343_suffix__bot_Obot_Oextremum,axiom,
! [A: list_node] : ( suffix_node @ nil_node @ A ) ).
% suffix_bot.bot.extremum
thf(fact_344_Nil__suffix,axiom,
! [Xs: list_P738500740D_node] : ( suffix1143830554D_node @ nil_Pr1769730692D_node @ Xs ) ).
% Nil_suffix
thf(fact_345_Nil__suffix,axiom,
! [Xs: list_node] : ( suffix_node @ nil_node @ Xs ) ).
% Nil_suffix
thf(fact_346_list_Osel_I2_J,axiom,
( ( tl_node @ nil_node )
= nil_node ) ).
% list.sel(2)
thf(fact_347_list_Osel_I2_J,axiom,
( ( tl_Pro1633633005D_node @ nil_Pr1769730692D_node )
= nil_Pr1769730692D_node ) ).
% list.sel(2)
thf(fact_348_suffix__appendI,axiom,
! [Xs: list_node,Ys: list_node,Zs: list_node] :
( ( suffix_node @ Xs @ Ys )
=> ( suffix_node @ Xs @ ( append_node @ Zs @ Ys ) ) ) ).
% suffix_appendI
thf(fact_349_suffix__appendD,axiom,
! [Zs: list_node,Xs: list_node,Ys: list_node] :
( ( suffix_node @ ( append_node @ Zs @ Xs ) @ Ys )
=> ( suffix_node @ Xs @ Ys ) ) ).
% suffix_appendD
thf(fact_350_old_OEntry__unreachable,axiom,
! [G: g] :
( ( invar @ G )
=> ( ( graph_1947481694_edgeD @ inEdges @ G @ ( entry @ G ) )
= nil_Pr1769730692D_node ) ) ).
% old.Entry_unreachable
% Conjectures (1)
thf(conj_0,conjecture,
~ ( member_node @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ r ) @ ( set_node2 @ ( tl_node @ rs ) ) ) ).
%------------------------------------------------------------------------------