TPTP Problem File: ITP082^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP082^1 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Irreducible problem prob_559__6628256_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_559__6628256_1 [Des21]
% Status : Theorem
% Rating : 0.50 v9.0.0, 0.60 v8.2.0, 0.38 v8.1.0, 0.45 v7.5.0
% Syntax : Number of formulae : 423 ( 153 unt; 70 typ; 0 def)
% Number of atoms : 1084 ( 638 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 3833 ( 215 ~; 24 |; 113 &;3023 @)
% ( 0 <=>; 458 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 229 ( 229 >; 0 *; 0 +; 0 <<)
% Number of symbols : 59 ( 58 usr; 19 con; 0-9 aty)
% Number of variables : 1148 ( 19 ^;1052 !; 77 ?;1148 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:42:19.440
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
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__Set__Oset_It__List__Olist_Itf__node_J_J,type,
set_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 (58)
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_Opath_001tf__g_001tf__node_001tf__edgeD,type,
graph_435229452_edgeD: ( g > list_node ) > ( g > $o ) > ( g > node > list_P561207620_edgeD ) > g > list_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__List__Olist_Itf__node_J,type,
append_list_node: list_list_node > list_list_node > list_list_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_Obutlast_001t__List__Olist_Itf__node_J,type,
butlast_list_node: list_list_node > list_list_node ).
thf(sy_c_List_Obutlast_001tf__node,type,
butlast_node: list_node > list_node ).
thf(sy_c_List_Obutlast_001tf__val,type,
butlast_val: list_val > list_val ).
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_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_Itf__node_J,type,
nil_list_node: list_list_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__List__Olist_Itf__node_J,type,
hd_list_node: list_list_node > list_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__List__Olist_Itf__node_J,type,
set_list_node2: list_list_node > set_list_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__List__Olist_Itf__node_J,type,
tl_list_node: list_list_node > list_list_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__path_OpathsConverge_001tf__g_001tf__node_001tf__edgeD,type,
graph_2009891965_edgeD: ( g > list_node ) > ( g > $o ) > ( g > node > list_P561207620_edgeD ) > g > node > list_node > node > list_node > node > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__node_J,type,
ord_less_eq_set_node: set_node > set_node > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__val_J,type,
ord_less_eq_set_val: set_val > set_val > $o ).
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_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_Oprefix_001tf__node,type,
prefix_node: list_node > list_node > $o ).
thf(sy_c_Sublist_Osuffix_001tf__node,type,
suffix_node: list_node > list_node > $o ).
thf(sy_c_member_001t__List__Olist_Itf__node_J,type,
member_list_node: list_node > set_list_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__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_i____,type,
i: node ).
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_ms_H____,type,
ms2: 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_ri____,type,
ri: list_node ).
thf(sy_v_rs_H____,type,
rs: list_node ).
thf(sy_v_rs_H__rest____,type,
rs_rest: list_node ).
thf(sy_v_rs____,type,
rs2: list_node ).
thf(sy_v_s,type,
s: val ).
thf(sy_v_tmp____,type,
tmp: list_node ).
% Relevant facts (352)
thf(fact_0_m__i__differ_I2_J,axiom,
m != i ).
% m_i_differ(2)
thf(fact_1_rs_H__rest__def,axiom,
( rs
= ( append_node @ tmp @ ( cons_node @ i @ rs_rest ) ) ) ).
% rs'_rest_def
thf(fact_2_old_Oinvar,axiom,
! [G: g] : ( invar @ G ) ).
% old.invar
thf(fact_3_old_Opath2__split_I2_J,axiom,
! [G: g,N: node,Ns: list_node,N2: node,Ns2: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( append_node @ Ns @ ( cons_node @ N2 @ Ns2 ) ) @ M )
=> ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N2 @ ( cons_node @ N2 @ Ns2 ) @ M ) ) ).
% old.path2_split(2)
thf(fact_4_False,axiom,
r != phi_r ).
% False
thf(fact_5_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_6_ms_H__props_I1_J,axiom,
graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ g2 @ m @ ms2 @ i ).
% ms'_props(1)
thf(fact_7_rs_H__rest__prop,axiom,
( rs
= ( append_node @ ri @ rs_rest ) ) ).
% rs'_rest_prop
thf(fact_8_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_9_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_10_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_11_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_12_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_13_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_14_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_15_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_16_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_17_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_18__092_060open_062ri_A_061_Atmp_A_064_A_091i_093_092_060close_062,axiom,
( ri
= ( append_node @ tmp @ ( cons_node @ i @ nil_node ) ) ) ).
% \<open>ri = tmp @ [i]\<close>
thf(fact_19__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062rs_H__rest_O_Ars_H_A_061_Ari_A_064_Ars_H__rest_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Rs_rest: list_node] :
( rs
!= ( append_node @ ri @ Rs_rest ) ) ).
% \<open>\<And>thesis. (\<And>rs'_rest. rs' = ri @ rs'_rest \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_20_ri__props_I2_J,axiom,
member_node @ i @ ( set_node2 @ ms ) ).
% ri_props(2)
thf(fact_21_m__i__differ_I1_J,axiom,
( i
!= ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_r ) ) ).
% m_i_differ(1)
thf(fact_22__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062tmp_O_Ari_A_061_Atmp_A_064_A_091i_093_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Tmp: list_node] :
( ri
!= ( append_node @ Tmp @ ( cons_node @ i @ nil_node ) ) ) ).
% \<open>\<And>thesis. (\<And>tmp. ri = tmp @ [i] \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_23_ri__props_I1_J,axiom,
graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ g2 @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ r ) @ ri @ i ).
% ri_props(1)
thf(fact_24_old_Opath2__split_I1_J,axiom,
! [G: g,N: node,Ns: list_node,N2: node,Ns2: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( append_node @ Ns @ ( cons_node @ N2 @ Ns2 ) ) @ 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_25__092_060open_062defNode_Ag_A_092_060phi_062_092_060_094sub_062r_A_092_060noteq_062_AdefNode_Ag_Ar_092_060close_062,axiom,
( ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_r )
!= ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ r ) ) ).
% \<open>defNode g \<phi>\<^sub>r \<noteq> defNode g r\<close>
thf(fact_26__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062i_A_061_AdefNode_Ag_A_092_060phi_062_092_060_094sub_062r_A_092_060Longrightarrow_062_Athesis_059_A_092_060lbrakk_062i_A_092_060noteq_062_AdefNode_Ag_A_092_060phi_062_092_060_094sub_062r_059_Am_A_061_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_059_A_092_060lbrakk_062i_A_092_060noteq_062_AdefNode_Ag_A_092_060phi_062_092_060_094sub_062r_059_Am_A_092_060noteq_062_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
( ( i
!= ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_r ) )
=> ( ( ( i
!= ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_r ) )
=> ( m != i ) )
=> ~ ( ( i
!= ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_r ) )
=> ( m = i ) ) ) ) ).
% \<open>\<And>thesis. \<lbrakk>i = defNode g \<phi>\<^sub>r \<Longrightarrow> thesis; \<lbrakk>i \<noteq> defNode g \<phi>\<^sub>r; m = i\<rbrakk> \<Longrightarrow> thesis; \<lbrakk>i \<noteq> defNode g \<phi>\<^sub>r; m \<noteq> i\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_27_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_28_ms_H__props_I3_J,axiom,
~ ( member_node @ i @ ( set_node2 @ ( butlast_node @ ms2 ) ) ) ).
% ms'_props(3)
thf(fact_29_assms_I10_J,axiom,
sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ g2 @ phi_r @ r ).
% assms(10)
thf(fact_30_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_31_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_32_rs__props_I1_J,axiom,
graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ g2 @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ r ) @ rs2 @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_r ) ).
% rs_props(1)
thf(fact_33_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_34_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_35_rs_H__loopfree,axiom,
~ ( member_node @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ r ) @ ( set_node2 @ ( tl_node @ rs ) ) ) ).
% rs'_loopfree
thf(fact_36_hd__append2,axiom,
! [Xs: list_val,Ys: list_val] :
( ( Xs != nil_val )
=> ( ( hd_val @ ( append_val @ Xs @ Ys ) )
= ( hd_val @ Xs ) ) ) ).
% hd_append2
thf(fact_37_hd__append2,axiom,
! [Xs: list_list_node,Ys: list_list_node] :
( ( Xs != nil_list_node )
=> ( ( hd_list_node @ ( append_list_node @ Xs @ Ys ) )
= ( hd_list_node @ Xs ) ) ) ).
% hd_append2
thf(fact_38_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_39_append1__eq__conv,axiom,
! [Xs: list_val,X: val,Ys: list_val,Y: val] :
( ( ( append_val @ Xs @ ( cons_val @ X @ nil_val ) )
= ( append_val @ Ys @ ( cons_val @ Y @ nil_val ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_40_append1__eq__conv,axiom,
! [Xs: list_list_node,X: list_node,Ys: list_list_node,Y: list_node] :
( ( ( append_list_node @ Xs @ ( cons_list_node @ X @ nil_list_node ) )
= ( append_list_node @ Ys @ ( cons_list_node @ Y @ nil_list_node ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_41_append1__eq__conv,axiom,
! [Xs: list_node,X: node,Ys: list_node,Y: node] :
( ( ( append_node @ Xs @ ( cons_node @ X @ nil_node ) )
= ( append_node @ Ys @ ( cons_node @ Y @ nil_node ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_42_list__ee__eq__leel_I1_J,axiom,
! [E1: val,E2: val,L1: list_val,E12: val,E22: val,L2: list_val] :
( ( ( cons_val @ E1 @ ( cons_val @ E2 @ nil_val ) )
= ( append_val @ L1 @ ( cons_val @ E12 @ ( cons_val @ E22 @ L2 ) ) ) )
= ( ( L1 = nil_val )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2 = nil_val ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_43_list__ee__eq__leel_I1_J,axiom,
! [E1: list_node,E2: list_node,L1: list_list_node,E12: list_node,E22: list_node,L2: list_list_node] :
( ( ( cons_list_node @ E1 @ ( cons_list_node @ E2 @ nil_list_node ) )
= ( append_list_node @ L1 @ ( cons_list_node @ E12 @ ( cons_list_node @ E22 @ L2 ) ) ) )
= ( ( L1 = nil_list_node )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2 = nil_list_node ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_44_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_45_list__ee__eq__leel_I2_J,axiom,
! [L1: list_val,E12: val,E22: val,L2: list_val,E1: val,E2: val] :
( ( ( append_val @ L1 @ ( cons_val @ E12 @ ( cons_val @ E22 @ L2 ) ) )
= ( cons_val @ E1 @ ( cons_val @ E2 @ nil_val ) ) )
= ( ( L1 = nil_val )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2 = nil_val ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_46_list__ee__eq__leel_I2_J,axiom,
! [L1: list_list_node,E12: list_node,E22: list_node,L2: list_list_node,E1: list_node,E2: list_node] :
( ( ( append_list_node @ L1 @ ( cons_list_node @ E12 @ ( cons_list_node @ E22 @ L2 ) ) )
= ( cons_list_node @ E1 @ ( cons_list_node @ E2 @ nil_list_node ) ) )
= ( ( L1 = nil_list_node )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2 = nil_list_node ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_47_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_48_list__se__match_I1_J,axiom,
! [L1: list_val,L2: list_val,A: val] :
( ( L1 != nil_val )
=> ( ( ( append_val @ L1 @ L2 )
= ( cons_val @ A @ nil_val ) )
= ( ( L1
= ( cons_val @ A @ nil_val ) )
& ( L2 = nil_val ) ) ) ) ).
% list_se_match(1)
thf(fact_49_list__se__match_I1_J,axiom,
! [L1: list_list_node,L2: list_list_node,A: list_node] :
( ( L1 != nil_list_node )
=> ( ( ( append_list_node @ L1 @ L2 )
= ( cons_list_node @ A @ nil_list_node ) )
= ( ( L1
= ( cons_list_node @ A @ nil_list_node ) )
& ( L2 = nil_list_node ) ) ) ) ).
% list_se_match(1)
thf(fact_50_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_51_ri__props_I3_J,axiom,
! [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ ( butlast_node @ ri ) ) )
=> ~ ( member_node @ X2 @ ( set_node2 @ ms ) ) ) ).
% ri_props(3)
thf(fact_52_old_Oelem__set__implies__elem__tl__app__cons,axiom,
! [X: list_node,Xs: list_list_node,Ys: list_list_node,Y: list_node] :
( ( member_list_node @ X @ ( set_list_node2 @ Xs ) )
=> ( member_list_node @ X @ ( set_list_node2 @ ( tl_list_node @ ( append_list_node @ Ys @ ( cons_list_node @ Y @ Xs ) ) ) ) ) ) ).
% old.elem_set_implies_elem_tl_app_cons
thf(fact_53_old_Oelem__set__implies__elem__tl__app__cons,axiom,
! [X: val,Xs: list_val,Ys: list_val,Y: val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ( member_val @ X @ ( set_val2 @ ( tl_val @ ( append_val @ Ys @ ( cons_val @ Y @ Xs ) ) ) ) ) ) ).
% old.elem_set_implies_elem_tl_app_cons
thf(fact_54_old_Oelem__set__implies__elem__tl__app__cons,axiom,
! [X: node,Xs: list_node,Ys: list_node,Y: node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ( member_node @ X @ ( set_node2 @ ( tl_node @ ( append_node @ Ys @ ( cons_node @ Y @ Xs ) ) ) ) ) ) ).
% old.elem_set_implies_elem_tl_app_cons
thf(fact_55_list_Oinject,axiom,
! [X21: val,X22: list_val,Y21: val,Y22: list_val] :
( ( ( cons_val @ X21 @ X22 )
= ( cons_val @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_56_list_Oinject,axiom,
! [X21: list_node,X22: list_list_node,Y21: list_node,Y22: list_list_node] :
( ( ( cons_list_node @ X21 @ X22 )
= ( cons_list_node @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_57_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_58_same__append__eq,axiom,
! [Xs: list_val,Ys: list_val,Zs: list_val] :
( ( ( append_val @ Xs @ Ys )
= ( append_val @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_59_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_60_mem__Collect__eq,axiom,
! [A: val,P: val > $o] :
( ( member_val @ A @ ( collect_val @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: node,P: node > $o] :
( ( member_node @ A @ ( collect_node @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_val] :
( ( collect_val
@ ^ [X3: val] : ( member_val @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_Collect__mem__eq,axiom,
! [A2: set_node] :
( ( collect_node
@ ^ [X3: node] : ( member_node @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_64_Collect__cong,axiom,
! [P: node > $o,Q: node > $o] :
( ! [X4: node] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_node @ P )
= ( collect_node @ Q ) ) ) ).
% Collect_cong
thf(fact_65_Collect__cong,axiom,
! [P: val > $o,Q: val > $o] :
( ! [X4: val] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_val @ P )
= ( collect_val @ Q ) ) ) ).
% Collect_cong
thf(fact_66_append__same__eq,axiom,
! [Ys: list_val,Xs: list_val,Zs: list_val] :
( ( ( append_val @ Ys @ Xs )
= ( append_val @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_67_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_68_append__assoc,axiom,
! [Xs: list_val,Ys: list_val,Zs: list_val] :
( ( append_val @ ( append_val @ Xs @ Ys ) @ Zs )
= ( append_val @ Xs @ ( append_val @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_69_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_70_append_Oassoc,axiom,
! [A: list_val,B: list_val,C: list_val] :
( ( append_val @ ( append_val @ A @ B ) @ C )
= ( append_val @ A @ ( append_val @ B @ C ) ) ) ).
% append.assoc
thf(fact_71_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_72_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_73_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_74_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_75_old_Opath2__app_H,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 @ ( butlast_node @ Ns ) @ Ms ) @ L ) ) ) ).
% old.path2_app'
thf(fact_76_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_77_append_Oright__neutral,axiom,
! [A: list_val] :
( ( append_val @ A @ nil_val )
= A ) ).
% append.right_neutral
thf(fact_78_append_Oright__neutral,axiom,
! [A: list_list_node] :
( ( append_list_node @ A @ nil_list_node )
= A ) ).
% append.right_neutral
thf(fact_79_append_Oright__neutral,axiom,
! [A: list_node] :
( ( append_node @ A @ nil_node )
= A ) ).
% append.right_neutral
thf(fact_80_empty__append__eq__id,axiom,
( ( append_val @ nil_val )
= ( ^ [X3: list_val] : X3 ) ) ).
% empty_append_eq_id
thf(fact_81_empty__append__eq__id,axiom,
( ( append_list_node @ nil_list_node )
= ( ^ [X3: list_list_node] : X3 ) ) ).
% empty_append_eq_id
thf(fact_82_empty__append__eq__id,axiom,
( ( append_node @ nil_node )
= ( ^ [X3: list_node] : X3 ) ) ).
% empty_append_eq_id
thf(fact_83_append__is__Nil__conv,axiom,
! [Xs: list_val,Ys: list_val] :
( ( ( append_val @ Xs @ Ys )
= nil_val )
= ( ( Xs = nil_val )
& ( Ys = nil_val ) ) ) ).
% append_is_Nil_conv
thf(fact_84_append__is__Nil__conv,axiom,
! [Xs: list_list_node,Ys: list_list_node] :
( ( ( append_list_node @ Xs @ Ys )
= nil_list_node )
= ( ( Xs = nil_list_node )
& ( Ys = nil_list_node ) ) ) ).
% append_is_Nil_conv
thf(fact_85_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_86_Nil__is__append__conv,axiom,
! [Xs: list_val,Ys: list_val] :
( ( nil_val
= ( append_val @ Xs @ Ys ) )
= ( ( Xs = nil_val )
& ( Ys = nil_val ) ) ) ).
% Nil_is_append_conv
thf(fact_87_Nil__is__append__conv,axiom,
! [Xs: list_list_node,Ys: list_list_node] :
( ( nil_list_node
= ( append_list_node @ Xs @ Ys ) )
= ( ( Xs = nil_list_node )
& ( Ys = nil_list_node ) ) ) ).
% Nil_is_append_conv
thf(fact_88_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_89_self__append__conv2,axiom,
! [Ys: list_val,Xs: list_val] :
( ( Ys
= ( append_val @ Xs @ Ys ) )
= ( Xs = nil_val ) ) ).
% self_append_conv2
thf(fact_90_self__append__conv2,axiom,
! [Ys: list_list_node,Xs: list_list_node] :
( ( Ys
= ( append_list_node @ Xs @ Ys ) )
= ( Xs = nil_list_node ) ) ).
% self_append_conv2
thf(fact_91_self__append__conv2,axiom,
! [Ys: list_node,Xs: list_node] :
( ( Ys
= ( append_node @ Xs @ Ys ) )
= ( Xs = nil_node ) ) ).
% self_append_conv2
thf(fact_92_append__self__conv2,axiom,
! [Xs: list_val,Ys: list_val] :
( ( ( append_val @ Xs @ Ys )
= Ys )
= ( Xs = nil_val ) ) ).
% append_self_conv2
thf(fact_93_append__self__conv2,axiom,
! [Xs: list_list_node,Ys: list_list_node] :
( ( ( append_list_node @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_node ) ) ).
% append_self_conv2
thf(fact_94_append__self__conv2,axiom,
! [Xs: list_node,Ys: list_node] :
( ( ( append_node @ Xs @ Ys )
= Ys )
= ( Xs = nil_node ) ) ).
% append_self_conv2
thf(fact_95_self__append__conv,axiom,
! [Xs: list_val,Ys: list_val] :
( ( Xs
= ( append_val @ Xs @ Ys ) )
= ( Ys = nil_val ) ) ).
% self_append_conv
thf(fact_96_self__append__conv,axiom,
! [Xs: list_list_node,Ys: list_list_node] :
( ( Xs
= ( append_list_node @ Xs @ Ys ) )
= ( Ys = nil_list_node ) ) ).
% self_append_conv
thf(fact_97_self__append__conv,axiom,
! [Xs: list_node,Ys: list_node] :
( ( Xs
= ( append_node @ Xs @ Ys ) )
= ( Ys = nil_node ) ) ).
% self_append_conv
thf(fact_98_append__self__conv,axiom,
! [Xs: list_val,Ys: list_val] :
( ( ( append_val @ Xs @ Ys )
= Xs )
= ( Ys = nil_val ) ) ).
% append_self_conv
thf(fact_99_append__self__conv,axiom,
! [Xs: list_list_node,Ys: list_list_node] :
( ( ( append_list_node @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_node ) ) ).
% append_self_conv
thf(fact_100_append__self__conv,axiom,
! [Xs: list_node,Ys: list_node] :
( ( ( append_node @ Xs @ Ys )
= Xs )
= ( Ys = nil_node ) ) ).
% append_self_conv
thf(fact_101_append__Nil2,axiom,
! [Xs: list_val] :
( ( append_val @ Xs @ nil_val )
= Xs ) ).
% append_Nil2
thf(fact_102_append__Nil2,axiom,
! [Xs: list_list_node] :
( ( append_list_node @ Xs @ nil_list_node )
= Xs ) ).
% append_Nil2
thf(fact_103_append__Nil2,axiom,
! [Xs: list_node] :
( ( append_node @ Xs @ nil_node )
= Xs ) ).
% append_Nil2
thf(fact_104_old_Opath2__split__ex_H,axiom,
! [G: g,N: node,Ns: list_node,M: node,X: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( member_node @ X @ ( set_node2 @ Ns ) )
=> ~ ! [Ns_1: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns_1 @ X )
=> ! [Ns_2: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ X @ Ns_2 @ M )
=> ( Ns
!= ( append_node @ ( butlast_node @ Ns_1 ) @ Ns_2 ) ) ) ) ) ) ).
% old.path2_split_ex'
thf(fact_105_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 ) )
=> ( ! [Ns3: list_node,N3: node,N4: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N4 @ Ns3 @ M )
=> ( ( P @ N4 @ Ns3 @ M )
=> ( ( member_node @ N3 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ N4 ) ) )
=> ( P @ N3 @ ( cons_node @ N3 @ Ns3 ) @ M ) ) ) )
=> ( P @ N @ Ns @ M ) ) ) ) ).
% old.path2_induct
thf(fact_106_old_Opath2__split__ex,axiom,
! [G: g,N: node,Ns: list_node,M: node,X: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( member_node @ X @ ( set_node2 @ Ns ) )
=> ~ ! [Ns_1: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns_1 @ X )
=> ! [Ns_2: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ X @ Ns_2 @ M )
=> ( ( Ns
= ( append_node @ Ns_1 @ ( tl_node @ Ns_2 ) ) )
=> ( Ns
!= ( append_node @ ( butlast_node @ Ns_1 ) @ Ns_2 ) ) ) ) ) ) ) ).
% old.path2_split_ex
thf(fact_107_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 ) )
=> ( ! [Ns3: list_node,M2: node,M3: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns3 @ M2 )
=> ( ( P @ N @ Ns3 @ M2 )
=> ( ( member_node @ M2 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ M3 ) ) )
=> ( P @ N @ ( append_node @ Ns3 @ ( cons_node @ M3 @ nil_node ) ) @ M3 ) ) ) )
=> ( P @ N @ Ns @ M ) ) ) ) ).
% old.path2_rev_induct
thf(fact_108_old_Opath2__snoc,axiom,
! [G: g,N: node,Ns: list_node,M: node,M4: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( member_node @ M @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ M4 ) ) )
=> ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( append_node @ Ns @ ( cons_node @ M4 @ nil_node ) ) @ M4 ) ) ) ).
% old.path2_snoc
thf(fact_109_list__e__eq__lel_I2_J,axiom,
! [L1: list_val,E: val,L2: list_val,E3: val] :
( ( ( append_val @ L1 @ ( cons_val @ E @ L2 ) )
= ( cons_val @ E3 @ nil_val ) )
= ( ( L1 = nil_val )
& ( E = E3 )
& ( L2 = nil_val ) ) ) ).
% list_e_eq_lel(2)
thf(fact_110_list__e__eq__lel_I2_J,axiom,
! [L1: list_list_node,E: list_node,L2: list_list_node,E3: list_node] :
( ( ( append_list_node @ L1 @ ( cons_list_node @ E @ L2 ) )
= ( cons_list_node @ E3 @ nil_list_node ) )
= ( ( L1 = nil_list_node )
& ( E = E3 )
& ( L2 = nil_list_node ) ) ) ).
% list_e_eq_lel(2)
thf(fact_111_list__e__eq__lel_I2_J,axiom,
! [L1: list_node,E: node,L2: list_node,E3: node] :
( ( ( append_node @ L1 @ ( cons_node @ E @ L2 ) )
= ( cons_node @ E3 @ nil_node ) )
= ( ( L1 = nil_node )
& ( E = E3 )
& ( L2 = nil_node ) ) ) ).
% list_e_eq_lel(2)
thf(fact_112_list__e__eq__lel_I1_J,axiom,
! [E3: val,L1: list_val,E: val,L2: list_val] :
( ( ( cons_val @ E3 @ nil_val )
= ( append_val @ L1 @ ( cons_val @ E @ L2 ) ) )
= ( ( L1 = nil_val )
& ( E = E3 )
& ( L2 = nil_val ) ) ) ).
% list_e_eq_lel(1)
thf(fact_113_list__e__eq__lel_I1_J,axiom,
! [E3: list_node,L1: list_list_node,E: list_node,L2: list_list_node] :
( ( ( cons_list_node @ E3 @ nil_list_node )
= ( append_list_node @ L1 @ ( cons_list_node @ E @ L2 ) ) )
= ( ( L1 = nil_list_node )
& ( E = E3 )
& ( L2 = nil_list_node ) ) ) ).
% list_e_eq_lel(1)
thf(fact_114_list__e__eq__lel_I1_J,axiom,
! [E3: node,L1: list_node,E: node,L2: list_node] :
( ( ( cons_node @ E3 @ nil_node )
= ( append_node @ L1 @ ( cons_node @ E @ L2 ) ) )
= ( ( L1 = nil_node )
& ( E = E3 )
& ( L2 = nil_node ) ) ) ).
% list_e_eq_lel(1)
thf(fact_115_list__se__match_I4_J,axiom,
! [L2: list_val,A: val,L1: list_val] :
( ( L2 != nil_val )
=> ( ( ( cons_val @ A @ nil_val )
= ( append_val @ L1 @ L2 ) )
= ( ( L1 = nil_val )
& ( L2
= ( cons_val @ A @ nil_val ) ) ) ) ) ).
% list_se_match(4)
thf(fact_116_list__se__match_I4_J,axiom,
! [L2: list_list_node,A: list_node,L1: list_list_node] :
( ( L2 != nil_list_node )
=> ( ( ( cons_list_node @ A @ nil_list_node )
= ( append_list_node @ L1 @ L2 ) )
= ( ( L1 = nil_list_node )
& ( L2
= ( cons_list_node @ A @ nil_list_node ) ) ) ) ) ).
% list_se_match(4)
thf(fact_117_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_118_list__se__match_I3_J,axiom,
! [L1: list_val,A: val,L2: list_val] :
( ( L1 != nil_val )
=> ( ( ( cons_val @ A @ nil_val )
= ( append_val @ L1 @ L2 ) )
= ( ( L1
= ( cons_val @ A @ nil_val ) )
& ( L2 = nil_val ) ) ) ) ).
% list_se_match(3)
thf(fact_119_list__se__match_I3_J,axiom,
! [L1: list_list_node,A: list_node,L2: list_list_node] :
( ( L1 != nil_list_node )
=> ( ( ( cons_list_node @ A @ nil_list_node )
= ( append_list_node @ L1 @ L2 ) )
= ( ( L1
= ( cons_list_node @ A @ nil_list_node ) )
& ( L2 = nil_list_node ) ) ) ) ).
% list_se_match(3)
thf(fact_120_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_121_list__se__match_I2_J,axiom,
! [L2: list_val,L1: list_val,A: val] :
( ( L2 != nil_val )
=> ( ( ( append_val @ L1 @ L2 )
= ( cons_val @ A @ nil_val ) )
= ( ( L1 = nil_val )
& ( L2
= ( cons_val @ A @ nil_val ) ) ) ) ) ).
% list_se_match(2)
thf(fact_122_list__se__match_I2_J,axiom,
! [L2: list_list_node,L1: list_list_node,A: list_node] :
( ( L2 != nil_list_node )
=> ( ( ( append_list_node @ L1 @ L2 )
= ( cons_list_node @ A @ nil_list_node ) )
= ( ( L1 = nil_list_node )
& ( L2
= ( cons_list_node @ A @ nil_list_node ) ) ) ) ) ).
% list_se_match(2)
thf(fact_123_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_124_tl__append2,axiom,
! [Xs: list_val,Ys: list_val] :
( ( Xs != nil_val )
=> ( ( tl_val @ ( append_val @ Xs @ Ys ) )
= ( append_val @ ( tl_val @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_125_tl__append2,axiom,
! [Xs: list_list_node,Ys: list_list_node] :
( ( Xs != nil_list_node )
=> ( ( tl_list_node @ ( append_list_node @ Xs @ Ys ) )
= ( append_list_node @ ( tl_list_node @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_126_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_127_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_128_rs__props_I3_J,axiom,
~ ( member_node @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ r ) @ ( set_node2 @ ( tl_node @ rs2 ) ) ) ).
% rs_props(3)
thf(fact_129_butlast__snoc,axiom,
! [Xs: list_val,X: val] :
( ( butlast_val @ ( append_val @ Xs @ ( cons_val @ X @ nil_val ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_130_butlast__snoc,axiom,
! [Xs: list_list_node,X: list_node] :
( ( butlast_list_node @ ( append_list_node @ Xs @ ( cons_list_node @ X @ nil_list_node ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_131_butlast__snoc,axiom,
! [Xs: list_node,X: node] :
( ( butlast_node @ ( append_node @ Xs @ ( cons_node @ X @ nil_node ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_132_list_Ocollapse,axiom,
! [List: list_val] :
( ( List != nil_val )
=> ( ( cons_val @ ( hd_val @ List ) @ ( tl_val @ List ) )
= List ) ) ).
% list.collapse
thf(fact_133_list_Ocollapse,axiom,
! [List: list_list_node] :
( ( List != nil_list_node )
=> ( ( cons_list_node @ ( hd_list_node @ List ) @ ( tl_list_node @ List ) )
= List ) ) ).
% list.collapse
thf(fact_134_list_Ocollapse,axiom,
! [List: list_node] :
( ( List != nil_node )
=> ( ( cons_node @ ( hd_node @ List ) @ ( tl_node @ List ) )
= List ) ) ).
% list.collapse
thf(fact_135_hd__Cons__tl,axiom,
! [Xs: list_val] :
( ( Xs != nil_val )
=> ( ( cons_val @ ( hd_val @ Xs ) @ ( tl_val @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_136_hd__Cons__tl,axiom,
! [Xs: list_list_node] :
( ( Xs != nil_list_node )
=> ( ( cons_list_node @ ( hd_list_node @ Xs ) @ ( tl_list_node @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_137_hd__Cons__tl,axiom,
! [Xs: list_node] :
( ( Xs != nil_node )
=> ( ( cons_node @ ( hd_node @ Xs ) @ ( tl_node @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_138_in__hd__or__tl__conv,axiom,
! [L: list_list_node,X: list_node] :
( ( L != nil_list_node )
=> ( ( ( X
= ( hd_list_node @ L ) )
| ( member_list_node @ X @ ( set_list_node2 @ ( tl_list_node @ L ) ) ) )
= ( member_list_node @ X @ ( set_list_node2 @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_139_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_140_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_141_ri__props_I4_J,axiom,
prefix_node @ ri @ rs2 ).
% ri_props(4)
thf(fact_142_ms_H__props_I2_J,axiom,
prefix_node @ ms2 @ ms ).
% ms'_props(2)
thf(fact_143_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_144_butlast__tl,axiom,
! [Xs: list_val] :
( ( butlast_val @ ( tl_val @ Xs ) )
= ( tl_val @ ( butlast_val @ Xs ) ) ) ).
% butlast_tl
thf(fact_145_butlast__tl,axiom,
! [Xs: list_node] :
( ( butlast_node @ ( tl_node @ Xs ) )
= ( tl_node @ ( butlast_node @ Xs ) ) ) ).
% butlast_tl
thf(fact_146_butlast_Osimps_I1_J,axiom,
( ( butlast_val @ nil_val )
= nil_val ) ).
% butlast.simps(1)
thf(fact_147_butlast_Osimps_I1_J,axiom,
( ( butlast_list_node @ nil_list_node )
= nil_list_node ) ).
% butlast.simps(1)
thf(fact_148_butlast_Osimps_I1_J,axiom,
( ( butlast_node @ nil_node )
= nil_node ) ).
% butlast.simps(1)
thf(fact_149_list_Osel_I3_J,axiom,
! [X21: val,X22: list_val] :
( ( tl_val @ ( cons_val @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_150_list_Osel_I3_J,axiom,
! [X21: list_node,X22: list_list_node] :
( ( tl_list_node @ ( cons_list_node @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_151_list_Osel_I3_J,axiom,
! [X21: node,X22: list_node] :
( ( tl_node @ ( cons_node @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_152_in__set__butlastD,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ ( butlast_val @ Xs ) ) )
=> ( member_val @ X @ ( set_val2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_153_in__set__butlastD,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ ( butlast_node @ Xs ) ) )
=> ( member_node @ X @ ( set_node2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_154_list_Osel_I2_J,axiom,
( ( tl_val @ nil_val )
= nil_val ) ).
% list.sel(2)
thf(fact_155_list_Osel_I2_J,axiom,
( ( tl_list_node @ nil_list_node )
= nil_list_node ) ).
% list.sel(2)
thf(fact_156_list_Osel_I2_J,axiom,
( ( tl_node @ nil_node )
= nil_node ) ).
% list.sel(2)
thf(fact_157_butlast_Osimps_I2_J,axiom,
! [Xs: list_val,X: val] :
( ( ( Xs = nil_val )
=> ( ( butlast_val @ ( cons_val @ X @ Xs ) )
= nil_val ) )
& ( ( Xs != nil_val )
=> ( ( butlast_val @ ( cons_val @ X @ Xs ) )
= ( cons_val @ X @ ( butlast_val @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_158_butlast_Osimps_I2_J,axiom,
! [Xs: list_list_node,X: list_node] :
( ( ( Xs = nil_list_node )
=> ( ( butlast_list_node @ ( cons_list_node @ X @ Xs ) )
= nil_list_node ) )
& ( ( Xs != nil_list_node )
=> ( ( butlast_list_node @ ( cons_list_node @ X @ Xs ) )
= ( cons_list_node @ X @ ( butlast_list_node @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_159_butlast_Osimps_I2_J,axiom,
! [Xs: list_node,X: node] :
( ( ( Xs = nil_node )
=> ( ( butlast_node @ ( cons_node @ X @ Xs ) )
= nil_node ) )
& ( ( Xs != nil_node )
=> ( ( butlast_node @ ( cons_node @ X @ Xs ) )
= ( cons_node @ X @ ( butlast_node @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_160_tl__obtain__elem,axiom,
! [Xs: list_val] :
( ( Xs != nil_val )
=> ( ( ( tl_val @ Xs )
= nil_val )
=> ~ ! [E4: val] :
( Xs
!= ( cons_val @ E4 @ nil_val ) ) ) ) ).
% tl_obtain_elem
thf(fact_161_tl__obtain__elem,axiom,
! [Xs: list_list_node] :
( ( Xs != nil_list_node )
=> ( ( ( tl_list_node @ Xs )
= nil_list_node )
=> ~ ! [E4: list_node] :
( Xs
!= ( cons_list_node @ E4 @ nil_list_node ) ) ) ) ).
% tl_obtain_elem
thf(fact_162_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_163_tl__Nil,axiom,
! [Xs: list_val] :
( ( ( tl_val @ Xs )
= nil_val )
= ( ( Xs = nil_val )
| ? [X3: val] :
( Xs
= ( cons_val @ X3 @ nil_val ) ) ) ) ).
% tl_Nil
thf(fact_164_tl__Nil,axiom,
! [Xs: list_list_node] :
( ( ( tl_list_node @ Xs )
= nil_list_node )
= ( ( Xs = nil_list_node )
| ? [X3: list_node] :
( Xs
= ( cons_list_node @ X3 @ nil_list_node ) ) ) ) ).
% tl_Nil
thf(fact_165_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_166_Nil__tl,axiom,
! [Xs: list_val] :
( ( nil_val
= ( tl_val @ Xs ) )
= ( ( Xs = nil_val )
| ? [X3: val] :
( Xs
= ( cons_val @ X3 @ nil_val ) ) ) ) ).
% Nil_tl
thf(fact_167_Nil__tl,axiom,
! [Xs: list_list_node] :
( ( nil_list_node
= ( tl_list_node @ Xs ) )
= ( ( Xs = nil_list_node )
| ? [X3: list_node] :
( Xs
= ( cons_list_node @ X3 @ nil_list_node ) ) ) ) ).
% Nil_tl
thf(fact_168_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_169_butlast__append,axiom,
! [Ys: list_val,Xs: list_val] :
( ( ( Ys = nil_val )
=> ( ( butlast_val @ ( append_val @ Xs @ Ys ) )
= ( butlast_val @ Xs ) ) )
& ( ( Ys != nil_val )
=> ( ( butlast_val @ ( append_val @ Xs @ Ys ) )
= ( append_val @ Xs @ ( butlast_val @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_170_butlast__append,axiom,
! [Ys: list_list_node,Xs: list_list_node] :
( ( ( Ys = nil_list_node )
=> ( ( butlast_list_node @ ( append_list_node @ Xs @ Ys ) )
= ( butlast_list_node @ Xs ) ) )
& ( ( Ys != nil_list_node )
=> ( ( butlast_list_node @ ( append_list_node @ Xs @ Ys ) )
= ( append_list_node @ Xs @ ( butlast_list_node @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_171_butlast__append,axiom,
! [Ys: list_node,Xs: list_node] :
( ( ( Ys = nil_node )
=> ( ( butlast_node @ ( append_node @ Xs @ Ys ) )
= ( butlast_node @ Xs ) ) )
& ( ( Ys != nil_node )
=> ( ( butlast_node @ ( append_node @ Xs @ Ys ) )
= ( append_node @ Xs @ ( butlast_node @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_172_list_Oset__sel_I2_J,axiom,
! [A: list_list_node,X: list_node] :
( ( A != nil_list_node )
=> ( ( member_list_node @ X @ ( set_list_node2 @ ( tl_list_node @ A ) ) )
=> ( member_list_node @ X @ ( set_list_node2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_173_list_Oset__sel_I2_J,axiom,
! [A: list_val,X: val] :
( ( A != nil_val )
=> ( ( member_val @ X @ ( set_val2 @ ( tl_val @ A ) ) )
=> ( member_val @ X @ ( set_val2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_174_list_Oset__sel_I2_J,axiom,
! [A: list_node,X: node] :
( ( A != nil_node )
=> ( ( member_node @ X @ ( set_node2 @ ( tl_node @ A ) ) )
=> ( member_node @ X @ ( set_node2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_175_in__set__butlast__appendI,axiom,
! [X: val,Xs: list_val,Ys: list_val] :
( ( ( member_val @ X @ ( set_val2 @ ( butlast_val @ Xs ) ) )
| ( member_val @ X @ ( set_val2 @ ( butlast_val @ Ys ) ) ) )
=> ( member_val @ X @ ( set_val2 @ ( butlast_val @ ( append_val @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_176_in__set__butlast__appendI,axiom,
! [X: node,Xs: list_node,Ys: list_node] :
( ( ( member_node @ X @ ( set_node2 @ ( butlast_node @ Xs ) ) )
| ( member_node @ X @ ( set_node2 @ ( butlast_node @ Ys ) ) ) )
=> ( member_node @ X @ ( set_node2 @ ( butlast_node @ ( append_node @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_177_list_Oexpand,axiom,
! [List: list_val,List2: list_val] :
( ( ( List = nil_val )
= ( List2 = nil_val ) )
=> ( ( ( List != nil_val )
=> ( ( List2 != nil_val )
=> ( ( ( hd_val @ List )
= ( hd_val @ List2 ) )
& ( ( tl_val @ List )
= ( tl_val @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_178_list_Oexpand,axiom,
! [List: list_list_node,List2: list_list_node] :
( ( ( List = nil_list_node )
= ( List2 = nil_list_node ) )
=> ( ( ( List != nil_list_node )
=> ( ( List2 != nil_list_node )
=> ( ( ( hd_list_node @ List )
= ( hd_list_node @ List2 ) )
& ( ( tl_list_node @ List )
= ( tl_list_node @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_179_list_Oexpand,axiom,
! [List: list_node,List2: list_node] :
( ( ( List = nil_node )
= ( List2 = nil_node ) )
=> ( ( ( List != nil_node )
=> ( ( List2 != nil_node )
=> ( ( ( hd_node @ List )
= ( hd_node @ List2 ) )
& ( ( tl_node @ List )
= ( tl_node @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_180_not__hd__in__tl,axiom,
! [X: val,Xs: list_val] :
( ( X
!= ( hd_val @ Xs ) )
=> ( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ( member_val @ X @ ( set_val2 @ ( tl_val @ Xs ) ) ) ) ) ).
% not_hd_in_tl
thf(fact_181_not__hd__in__tl,axiom,
! [X: node,Xs: list_node] :
( ( X
!= ( hd_node @ Xs ) )
=> ( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ( member_node @ X @ ( set_node2 @ ( tl_node @ Xs ) ) ) ) ) ).
% not_hd_in_tl
thf(fact_182_butlast__eq__cons__conv,axiom,
! [L: list_list_node,X: list_node,Xs: list_list_node] :
( ( ( butlast_list_node @ L )
= ( cons_list_node @ X @ Xs ) )
= ( ? [Xl: list_node] :
( L
= ( cons_list_node @ X @ ( append_list_node @ Xs @ ( cons_list_node @ Xl @ nil_list_node ) ) ) ) ) ) ).
% butlast_eq_cons_conv
thf(fact_183_butlast__eq__cons__conv,axiom,
! [L: list_node,X: node,Xs: list_node] :
( ( ( butlast_node @ L )
= ( cons_node @ X @ Xs ) )
= ( ? [Xl: node] :
( L
= ( cons_node @ X @ ( append_node @ Xs @ ( cons_node @ Xl @ nil_node ) ) ) ) ) ) ).
% butlast_eq_cons_conv
thf(fact_184_butlast__eq__consE,axiom,
! [L: list_node,X: node,Xs: list_node] :
( ( ( butlast_node @ L )
= ( cons_node @ X @ Xs ) )
=> ~ ! [Xl2: node] :
( L
!= ( cons_node @ X @ ( append_node @ Xs @ ( cons_node @ Xl2 @ nil_node ) ) ) ) ) ).
% butlast_eq_consE
thf(fact_185_list_Oexhaust__sel,axiom,
! [List: list_node] :
( ( List != nil_node )
=> ( List
= ( cons_node @ ( hd_node @ List ) @ ( tl_node @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_186_not__Cons__self2,axiom,
! [X: node,Xs: list_node] :
( ( cons_node @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_187_list__tail__coinc,axiom,
! [N1: node,R1: list_node,N22: node,R2: list_node] :
( ( ( cons_node @ N1 @ R1 )
= ( cons_node @ N22 @ R2 ) )
=> ( ( N1 = N22 )
& ( R1 = R2 ) ) ) ).
% list_tail_coinc
thf(fact_188_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_189_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_190_list_Odistinct_I1_J,axiom,
! [X21: node,X22: list_node] :
( nil_node
!= ( cons_node @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_191_neq__NilE,axiom,
! [L: list_node] :
( ( L != nil_node )
=> ~ ! [X4: node,Xs2: list_node] :
( L
!= ( cons_node @ X4 @ Xs2 ) ) ) ).
% neq_NilE
thf(fact_192_list_OdiscI,axiom,
! [List: list_node,X21: node,X22: list_node] :
( ( List
= ( cons_node @ X21 @ X22 ) )
=> ( List != nil_node ) ) ).
% list.discI
thf(fact_193_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_194_list_Oexhaust,axiom,
! [Y: list_node] :
( ( Y != nil_node )
=> ~ ! [X212: node,X222: list_node] :
( Y
!= ( cons_node @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_195_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_196_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_197_list__induct2_H,axiom,
! [P: list_node > list_node > $o,Xs: list_node,Ys: list_node] :
( ( P @ nil_node @ nil_node )
=> ( ! [X4: node,Xs2: list_node] : ( P @ ( cons_node @ X4 @ Xs2 ) @ nil_node )
=> ( ! [Y3: node,Ys3: list_node] : ( P @ nil_node @ ( cons_node @ Y3 @ Ys3 ) )
=> ( ! [X4: node,Xs2: list_node,Y3: node,Ys3: list_node] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ ( cons_node @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_198_splice_Oinduct,axiom,
! [P: list_node > list_node > $o,A0: list_node,A1: list_node] :
( ! [X_1: list_node] : ( P @ nil_node @ X_1 )
=> ( ! [X4: node,Xs2: list_node,Ys3: list_node] :
( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_199_induct__list012,axiom,
! [P: list_node > $o,Xs: list_node] :
( ( P @ nil_node )
=> ( ! [X4: node] : ( P @ ( cons_node @ X4 @ nil_node ) )
=> ( ! [X4: node,Y3: node,Zs2: list_node] :
( ( P @ Zs2 )
=> ( ( P @ ( cons_node @ Y3 @ Zs2 ) )
=> ( P @ ( cons_node @ X4 @ ( cons_node @ Y3 @ Zs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_200_min__list_Ocases,axiom,
! [X: list_node] :
( ! [X4: node,Xs2: list_node] :
( X
!= ( cons_node @ X4 @ Xs2 ) )
=> ( X = nil_node ) ) ).
% min_list.cases
thf(fact_201_min__list_Oinduct,axiom,
! [P: list_node > $o,A0: list_node] :
( ! [X4: node,Xs2: list_node] :
( ! [X213: node,X223: list_node] :
( ( Xs2
= ( cons_node @ X213 @ X223 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons_node @ X4 @ Xs2 ) ) )
=> ( ( P @ nil_node )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_202_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 )
=> ( ! [X4: node,Xs2: list_node,Y3: node,Ys3: list_node] :
( ( P @ Xs2 @ ( cons_node @ Y3 @ Ys3 ) )
=> ( ( P @ ( cons_node @ X4 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ ( cons_node @ Y3 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_203_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_204_remdups__adj_Ocases,axiom,
! [X: list_node] :
( ( X != nil_node )
=> ( ! [X4: node] :
( X
!= ( cons_node @ X4 @ nil_node ) )
=> ~ ! [X4: node,Y3: node,Xs2: list_node] :
( X
!= ( cons_node @ X4 @ ( cons_node @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_205_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,X4: node,Ys3: list_node] :
( ( P @ P2 @ Ys3 )
=> ( P @ P2 @ ( cons_node @ X4 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_206_remdups__adj_Oinduct,axiom,
! [P: list_node > $o,A0: list_node] :
( ( P @ nil_node )
=> ( ! [X4: node] : ( P @ ( cons_node @ X4 @ nil_node ) )
=> ( ! [X4: node,Y3: node,Xs2: list_node] :
( ( ( X4 = Y3 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) ) )
=> ( ( ( X4 != Y3 )
=> ( P @ ( cons_node @ Y3 @ Xs2 ) ) )
=> ( P @ ( cons_node @ X4 @ ( cons_node @ Y3 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_207_list__induct__first2,axiom,
! [P: list_node > $o,Xs: list_node] :
( ( P @ nil_node )
=> ( ! [X4: node] : ( P @ ( cons_node @ X4 @ 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_208_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,X4: node] : ( P @ P2 @ ( cons_node @ X4 @ nil_node ) )
=> ( ! [P2: node > node > $o,X4: node,Y3: node,Xs2: list_node] :
( ( P @ P2 @ ( cons_node @ Y3 @ Xs2 ) )
=> ( P @ P2 @ ( cons_node @ X4 @ ( cons_node @ Y3 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_209_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,V: node,Va: list_node] : ( P @ P2 @ ( cons_node @ V @ Va ) @ nil_node )
=> ( ! [P2: node > node > $o,V: node,Va: list_node] : ( P @ P2 @ nil_node @ ( cons_node @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% list_all_zip.induct
thf(fact_210_list__nonempty__induct,axiom,
! [Xs: list_node,P: list_node > $o] :
( ( Xs != nil_node )
=> ( ! [X4: node] : ( P @ ( cons_node @ X4 @ nil_node ) )
=> ( ! [X4: node,Xs2: list_node] :
( ( Xs2 != nil_node )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_211_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_212_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,X4: node,Xs2: list_node,Y3: node,Ys3: list_node] :
( ( ( R @ X4 @ Y3 )
=> ( P @ R @ Xs2 @ ( cons_node @ Y3 @ Ys3 ) ) )
=> ( ( ~ ( R @ X4 @ Y3 )
=> ( P @ R @ ( cons_node @ X4 @ Xs2 ) @ Ys3 ) )
=> ( P @ R @ ( cons_node @ X4 @ Xs2 ) @ ( cons_node @ Y3 @ Ys3 ) ) ) )
=> ( ! [R: node > node > $o,Xs2: list_node] : ( P @ R @ Xs2 @ nil_node )
=> ( ! [R: node > node > $o,V: node,Va: list_node] : ( P @ R @ nil_node @ ( cons_node @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% mergesort_by_rel_merge.induct
thf(fact_213_mergesort__by__rel__merge__induct,axiom,
! [P: list_node > list_node > $o,R3: 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 )
=> ( ! [X4: node,Xs2: list_node,Y3: node,Ys3: list_node] :
( ( R3 @ X4 @ Y3 )
=> ( ( P @ Xs2 @ ( cons_node @ Y3 @ Ys3 ) )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ ( cons_node @ Y3 @ Ys3 ) ) ) )
=> ( ! [X4: node,Xs2: list_node,Y3: node,Ys3: list_node] :
( ~ ( R3 @ X4 @ Y3 )
=> ( ( P @ ( cons_node @ X4 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ ( cons_node @ Y3 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_214_strict__sorted_Oinduct,axiom,
! [P: list_node > $o,A0: list_node] :
( ( P @ nil_node )
=> ( ! [X4: node,Ys3: list_node] :
( ( P @ Ys3 )
=> ( P @ ( cons_node @ X4 @ Ys3 ) ) )
=> ( P @ A0 ) ) ) ).
% strict_sorted.induct
thf(fact_215_transpose_Ocases,axiom,
! [X: list_list_node] :
( ( X != nil_list_node )
=> ( ! [Xss: list_list_node] :
( X
!= ( cons_list_node @ nil_node @ Xss ) )
=> ~ ! [X4: node,Xs2: list_node,Xss: list_list_node] :
( X
!= ( cons_list_node @ ( cons_node @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_216_list_Oset__intros_I2_J,axiom,
! [Y: val,X22: list_val,X21: val] :
( ( member_val @ Y @ ( set_val2 @ X22 ) )
=> ( member_val @ Y @ ( set_val2 @ ( cons_val @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_217_list_Oset__intros_I2_J,axiom,
! [Y: node,X22: list_node,X21: node] :
( ( member_node @ Y @ ( set_node2 @ X22 ) )
=> ( member_node @ Y @ ( set_node2 @ ( cons_node @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_218_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_219_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_220_set__ConsD,axiom,
! [Y: val,X: val,Xs: list_val] :
( ( member_val @ Y @ ( set_val2 @ ( cons_val @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_val @ Y @ ( set_val2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_221_set__ConsD,axiom,
! [Y: node,X: node,Xs: list_node] :
( ( member_node @ Y @ ( set_node2 @ ( cons_node @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_node @ Y @ ( set_node2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_222_list_Oset__cases,axiom,
! [E3: val,A: list_val] :
( ( member_val @ E3 @ ( set_val2 @ A ) )
=> ( ! [Z2: list_val] :
( A
!= ( cons_val @ E3 @ Z2 ) )
=> ~ ! [Z1: val,Z2: list_val] :
( ( A
= ( cons_val @ Z1 @ Z2 ) )
=> ~ ( member_val @ E3 @ ( set_val2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_223_list_Oset__cases,axiom,
! [E3: node,A: list_node] :
( ( member_node @ E3 @ ( set_node2 @ A ) )
=> ( ! [Z2: list_node] :
( A
!= ( cons_node @ E3 @ Z2 ) )
=> ~ ! [Z1: node,Z2: list_node] :
( ( A
= ( cons_node @ Z1 @ Z2 ) )
=> ~ ( member_node @ E3 @ ( set_node2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_224_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_225_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_226_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_227_append_Oleft__neutral,axiom,
! [A: list_node] :
( ( append_node @ nil_node @ A )
= A ) ).
% append.left_neutral
thf(fact_228_append__Nil,axiom,
! [Ys: list_node] :
( ( append_node @ nil_node @ Ys )
= Ys ) ).
% append_Nil
thf(fact_229_eq__Nil__appendI,axiom,
! [Xs: list_node,Ys: list_node] :
( ( Xs = Ys )
=> ( Xs
= ( append_node @ nil_node @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_230_list_Osel_I1_J,axiom,
! [X21: node,X22: list_node] :
( ( hd_node @ ( cons_node @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_231_rev__induct,axiom,
! [P: list_node > $o,Xs: list_node] :
( ( P @ nil_node )
=> ( ! [X4: node,Xs2: list_node] :
( ( P @ Xs2 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_232_rev__exhaust,axiom,
! [Xs: list_node] :
( ( Xs != nil_node )
=> ~ ! [Ys3: list_node,Y3: node] :
( Xs
!= ( append_node @ Ys3 @ ( cons_node @ Y3 @ nil_node ) ) ) ) ).
% rev_exhaust
thf(fact_233_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_234_rev__induct2_H,axiom,
! [P: list_node > list_node > $o,Xs: list_node,Ys: list_node] :
( ( P @ nil_node @ nil_node )
=> ( ! [X4: node,Xs2: list_node] : ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) @ nil_node )
=> ( ! [Y3: node,Ys3: list_node] : ( P @ nil_node @ ( append_node @ Ys3 @ ( cons_node @ Y3 @ nil_node ) ) )
=> ( ! [X4: node,Xs2: list_node,Y3: node,Ys3: list_node] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) @ ( append_node @ Ys3 @ ( cons_node @ Y3 @ nil_node ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_235_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_236_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 ) )
| ? [Ys4: list_node] :
( ( ( cons_node @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_node @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_237_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 ) ) )
| ? [Ys4: list_node] :
( ( Ys
= ( cons_node @ X @ Ys4 ) )
& ( ( append_node @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_238_rev__nonempty__induct,axiom,
! [Xs: list_node,P: list_node > $o] :
( ( Xs != nil_node )
=> ( ! [X4: node] : ( P @ ( cons_node @ X4 @ nil_node ) )
=> ( ! [X4: node,Xs2: list_node] :
( ( Xs2 != nil_node )
=> ( ( P @ Xs2 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_239_rev__nonempty__induct2_H,axiom,
! [Xs: list_node,Ys: list_node,P: list_node > list_node > $o] :
( ( Xs != nil_node )
=> ( ( Ys != nil_node )
=> ( ! [X4: node,Y3: node] : ( P @ ( cons_node @ X4 @ nil_node ) @ ( cons_node @ Y3 @ nil_node ) )
=> ( ! [X4: node,Xs2: list_node,Y3: node] :
( ( Xs2 != nil_node )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) @ ( cons_node @ Y3 @ nil_node ) ) )
=> ( ! [X4: node,Y3: node,Ys3: list_node] :
( ( Ys3 != nil_node )
=> ( P @ ( cons_node @ X4 @ nil_node ) @ ( append_node @ Ys3 @ ( cons_node @ Y3 @ nil_node ) ) ) )
=> ( ! [X4: node,Xs2: list_node,Y3: node,Ys3: list_node] :
( ( P @ Xs2 @ Ys3 )
=> ( ( Xs2 != nil_node )
=> ( ( Ys3 != nil_node )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) @ ( append_node @ Ys3 @ ( cons_node @ Y3 @ nil_node ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_240_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 ) ) )
=> ~ ! [Ys5: list_node] :
( ( Ys
= ( cons_node @ X @ Ys5 ) )
=> ( ( append_node @ Ys5 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_241_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 ) ) )
=> ~ ! [Ys5: list_node] :
( ( Ys
= ( cons_node @ X @ Ys5 ) )
=> ( ( append_node @ Ys5 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_242_split__list,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ? [Ys3: list_val,Zs2: list_val] :
( Xs
= ( append_val @ Ys3 @ ( cons_val @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_243_split__list,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ? [Ys3: list_node,Zs2: list_node] :
( Xs
= ( append_node @ Ys3 @ ( cons_node @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_244_split__list__last,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ? [Ys3: list_val,Zs2: list_val] :
( ( Xs
= ( append_val @ Ys3 @ ( cons_val @ X @ Zs2 ) ) )
& ~ ( member_val @ X @ ( set_val2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_245_split__list__last,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ? [Ys3: list_node,Zs2: list_node] :
( ( Xs
= ( append_node @ Ys3 @ ( cons_node @ X @ Zs2 ) ) )
& ~ ( member_node @ X @ ( set_node2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_246_split__list__prop,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys3: list_node,X4: node] :
( ? [Zs2: list_node] :
( Xs
= ( append_node @ Ys3 @ ( cons_node @ X4 @ Zs2 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_247_xy__in__set__cases,axiom,
! [X: val,L: list_val,Y: val] :
( ( member_val @ X @ ( set_val2 @ L ) )
=> ( ( member_val @ Y @ ( set_val2 @ L ) )
=> ( ( ( X = Y )
=> ! [L12: list_val,L22: list_val] :
( L
!= ( append_val @ L12 @ ( cons_val @ Y @ L22 ) ) ) )
=> ( ( ( X != Y )
=> ! [L12: list_val,L22: list_val,L3: list_val] :
( L
!= ( append_val @ L12 @ ( cons_val @ X @ ( append_val @ L22 @ ( cons_val @ Y @ L3 ) ) ) ) ) )
=> ~ ( ( X != Y )
=> ! [L12: list_val,L22: list_val,L3: list_val] :
( L
!= ( append_val @ L12 @ ( cons_val @ Y @ ( append_val @ L22 @ ( cons_val @ X @ L3 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_248_xy__in__set__cases,axiom,
! [X: node,L: list_node,Y: node] :
( ( member_node @ X @ ( set_node2 @ L ) )
=> ( ( member_node @ Y @ ( set_node2 @ L ) )
=> ( ( ( X = Y )
=> ! [L12: list_node,L22: list_node] :
( L
!= ( append_node @ L12 @ ( cons_node @ Y @ L22 ) ) ) )
=> ( ( ( X != Y )
=> ! [L12: list_node,L22: list_node,L3: list_node] :
( L
!= ( append_node @ L12 @ ( cons_node @ X @ ( append_node @ L22 @ ( cons_node @ Y @ L3 ) ) ) ) ) )
=> ~ ( ( X != Y )
=> ! [L12: list_node,L22: list_node,L3: list_node] :
( L
!= ( append_node @ L12 @ ( cons_node @ Y @ ( append_node @ L22 @ ( cons_node @ X @ L3 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_249_split__list__first,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ? [Ys3: list_val,Zs2: list_val] :
( ( Xs
= ( append_val @ Ys3 @ ( cons_val @ X @ Zs2 ) ) )
& ~ ( member_val @ X @ ( set_val2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_250_split__list__first,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ? [Ys3: list_node,Zs2: list_node] :
( ( Xs
= ( append_node @ Ys3 @ ( cons_node @ X @ Zs2 ) ) )
& ~ ( member_node @ X @ ( set_node2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_251_split__list__propE,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys3: list_node,X4: node] :
( ? [Zs2: list_node] :
( Xs
= ( append_node @ Ys3 @ ( cons_node @ X4 @ Zs2 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_252_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_253_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_254_in__set__conv__decomp,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
= ( ? [Ys2: list_val,Zs3: list_val] :
( Xs
= ( append_val @ Ys2 @ ( cons_val @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_255_in__set__conv__decomp,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
= ( ? [Ys2: list_node,Zs3: list_node] :
( Xs
= ( append_node @ Ys2 @ ( cons_node @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_256_in__set__list__format,axiom,
! [E3: val,L: list_val] :
( ( member_val @ E3 @ ( set_val2 @ L ) )
=> ~ ! [L12: list_val,L22: list_val] :
( L
!= ( append_val @ L12 @ ( cons_val @ E3 @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_257_in__set__list__format,axiom,
! [E3: node,L: list_node] :
( ( member_node @ E3 @ ( set_node2 @ L ) )
=> ~ ! [L12: list_node,L22: list_node] :
( L
!= ( append_node @ L12 @ ( cons_node @ E3 @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_258_split__list__last__prop,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys3: list_node,X4: node,Zs2: list_node] :
( ( Xs
= ( append_node @ Ys3 @ ( cons_node @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa: node] :
( ( member_node @ Xa @ ( set_node2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_259_split__list__first__prop,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys3: list_node,X4: node] :
( ? [Zs2: list_node] :
( Xs
= ( append_node @ Ys3 @ ( cons_node @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa: node] :
( ( member_node @ Xa @ ( set_node2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_260_split__list__last__propE,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys3: list_node,X4: node,Zs2: list_node] :
( ( Xs
= ( append_node @ Ys3 @ ( cons_node @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: node] :
( ( member_node @ Xa @ ( set_node2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_261_split__list__first__propE,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys3: list_node,X4: node] :
( ? [Zs2: list_node] :
( Xs
= ( append_node @ Ys3 @ ( cons_node @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: node] :
( ( member_node @ Xa @ ( set_node2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_262_in__set__conv__decomp__last,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
= ( ? [Ys2: list_val,Zs3: list_val] :
( ( Xs
= ( append_val @ Ys2 @ ( cons_val @ X @ Zs3 ) ) )
& ~ ( member_val @ X @ ( set_val2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_263_in__set__conv__decomp__last,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
= ( ? [Ys2: list_node,Zs3: list_node] :
( ( Xs
= ( append_node @ Ys2 @ ( cons_node @ X @ Zs3 ) ) )
& ~ ( member_node @ X @ ( set_node2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_264_in__set__conv__decomp__first,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
= ( ? [Ys2: list_val,Zs3: list_val] :
( ( Xs
= ( append_val @ Ys2 @ ( cons_val @ X @ Zs3 ) ) )
& ~ ( member_val @ X @ ( set_val2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_265_in__set__conv__decomp__first,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
= ( ? [Ys2: list_node,Zs3: list_node] :
( ( Xs
= ( append_node @ Ys2 @ ( cons_node @ X @ Zs3 ) ) )
& ~ ( member_node @ X @ ( set_node2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_266_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,Zs3: list_node] :
( ( Xs
= ( append_node @ Ys2 @ ( cons_node @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y2: node] :
( ( member_node @ Y2 @ ( set_node2 @ Zs3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_267_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] :
( ? [Zs3: list_node] :
( Xs
= ( append_node @ Ys2 @ ( cons_node @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y2: node] :
( ( member_node @ Y2 @ ( set_node2 @ Ys2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_268_hd__in__set,axiom,
! [Xs: list_val] :
( ( Xs != nil_val )
=> ( member_val @ ( hd_val @ Xs ) @ ( set_val2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_269_hd__in__set,axiom,
! [Xs: list_node] :
( ( Xs != nil_node )
=> ( member_node @ ( hd_node @ Xs ) @ ( set_node2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_270_list_Oset__sel_I1_J,axiom,
! [A: list_val] :
( ( A != nil_val )
=> ( member_val @ ( hd_val @ A ) @ ( set_val2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_271_list_Oset__sel_I1_J,axiom,
! [A: list_node] :
( ( A != nil_node )
=> ( member_node @ ( hd_node @ A ) @ ( set_node2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_272_longest__common__prefix,axiom,
! [Xs: list_node,Ys: list_node] :
? [Ps: list_node,Xs5: list_node,Ys5: list_node] :
( ( Xs
= ( append_node @ Ps @ Xs5 ) )
& ( Ys
= ( append_node @ Ps @ Ys5 ) )
& ( ( Xs5 = nil_node )
| ( Ys5 = nil_node )
| ( ( hd_node @ Xs5 )
!= ( hd_node @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_273_hd__append,axiom,
! [Xs: list_node,Ys: list_node] :
( ( ( Xs = nil_node )
=> ( ( hd_node @ ( append_node @ Xs @ Ys ) )
= ( hd_node @ Ys ) ) )
& ( ( Xs != nil_node )
=> ( ( hd_node @ ( append_node @ Xs @ Ys ) )
= ( hd_node @ Xs ) ) ) ) ).
% hd_append
thf(fact_274__092_060open_062old_OpathsConverge_Ag_Am_Ams_H_An_A_Ins_A_064_Atl_Ari_J_Ai_092_060close_062,axiom,
graph_2009891965_edgeD @ alpha_n @ invar @ inEdges @ g2 @ m @ ms2 @ n @ ( append_node @ ns @ ( tl_node @ ri ) ) @ i ).
% \<open>old.pathsConverge g m ms' n (ns @ tl ri) i\<close>
thf(fact_275__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062i_Ari_O_A_092_060lbrakk_062g_A_092_060turnstile_062_AdefNode_Ag_Ar_Nri_092_060rightarrow_062i_059_Ai_A_092_060in_062_Aset_Ams_059_A_092_060forall_062n_092_060in_062set_A_Ibutlast_Ari_J_O_An_A_092_060notin_062_Aset_Ams_059_Aprefix_Ari_Ars_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [I: node,Ri: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ g2 @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ r ) @ Ri @ I )
=> ( ( member_node @ I @ ( set_node2 @ ms ) )
=> ( ! [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ ( butlast_node @ Ri ) ) )
=> ~ ( member_node @ X2 @ ( set_node2 @ ms ) ) )
=> ~ ( prefix_node @ Ri @ rs2 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>i ri. \<lbrakk>g \<turnstile> defNode g r-ri\<rightarrow>i; i \<in> set ms; \<forall>n\<in>set (butlast ri). n \<notin> set ms; prefix ri rs\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_276__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062ms_H_O_A_092_060lbrakk_062g_A_092_060turnstile_062_Am_Nms_H_092_060rightarrow_062i_059_Aprefix_Ams_H_Ams_059_Ai_A_092_060notin_062_Aset_A_Ibutlast_Ams_H_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Ms2: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ g2 @ m @ Ms2 @ i )
=> ( ( prefix_node @ Ms2 @ ms )
=> ( member_node @ i @ ( set_node2 @ ( butlast_node @ Ms2 ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>ms'. \<lbrakk>g \<turnstile> m-ms'\<rightarrow>i; prefix ms' ms; i \<notin> set (butlast ms')\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_277_old_Opath2__simple__loop,axiom,
! [G: g,N: node,Ns: list_node,N2: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ N )
=> ( ( member_node @ N2 @ ( set_node2 @ Ns ) )
=> ~ ! [Ns4: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns4 @ N )
=> ( ( member_node @ N2 @ ( set_node2 @ Ns4 ) )
=> ( ~ ( member_node @ N @ ( set_node2 @ ( tl_node @ ( butlast_node @ Ns4 ) ) ) )
=> ~ ( ord_less_eq_set_node @ ( set_node2 @ Ns4 ) @ ( set_node2 @ Ns ) ) ) ) ) ) ) ).
% old.path2_simple_loop
thf(fact_278_old_Opath_Ocases,axiom,
! [G: g,A: list_node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ A )
=> ( ! [N4: node] :
( ( A
= ( cons_node @ N4 @ nil_node ) )
=> ( ( member_node @ N4 @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ~ ( invar @ G ) ) )
=> ~ ! [Ns3: list_node,N3: node] :
( ( A
= ( cons_node @ N3 @ Ns3 ) )
=> ( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ Ns3 )
=> ~ ( member_node @ N3 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ ( hd_node @ Ns3 ) ) ) ) ) ) ) ) ).
% old.path.cases
thf(fact_279_old_Opath_Oinducts,axiom,
! [G: g,X: list_node,P: list_node > $o] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ X )
=> ( ! [N4: node] :
( ( member_node @ N4 @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( invar @ G )
=> ( P @ ( cons_node @ N4 @ nil_node ) ) ) )
=> ( ! [Ns3: list_node,N3: node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ Ns3 )
=> ( ( P @ Ns3 )
=> ( ( member_node @ N3 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ ( hd_node @ Ns3 ) ) ) )
=> ( P @ ( cons_node @ N3 @ Ns3 ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% old.path.inducts
thf(fact_280_old_Opath_Osimps,axiom,
! [G: g,A: list_node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ A )
= ( ? [N5: node] :
( ( A
= ( cons_node @ N5 @ nil_node ) )
& ( member_node @ N5 @ ( set_node2 @ ( alpha_n @ G ) ) )
& ( invar @ G ) )
| ? [Ns5: list_node,N6: node] :
( ( A
= ( cons_node @ N6 @ Ns5 ) )
& ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ Ns5 )
& ( member_node @ N6 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ ( hd_node @ Ns5 ) ) ) ) ) ) ) ).
% old.path.simps
thf(fact_281_old_Opath__snoc,axiom,
! [G: g,Ns: list_node,N: node,M: node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( append_node @ Ns @ ( cons_node @ N @ nil_node ) ) )
=> ( ( member_node @ N @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ M ) ) )
=> ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( append_node @ Ns @ ( cons_node @ N @ ( cons_node @ M @ nil_node ) ) ) ) ) ) ).
% old.path_snoc
thf(fact_282_ri__rs_H__prefix,axiom,
prefix_node @ ri @ rs ).
% ri_rs'_prefix
thf(fact_283_old_Opath__not__Nil,axiom,
! [G: g,Ns: list_node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ Ns )
=> ( Ns != nil_node ) ) ).
% old.path_not_Nil
thf(fact_284_old_Opath__hd,axiom,
! [G: g,N: node,Ns: list_node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( cons_node @ N @ Ns ) )
=> ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( cons_node @ N @ nil_node ) ) ) ).
% old.path_hd
thf(fact_285_old_Opath__split_I2_J,axiom,
! [G: g,Ns: list_node,M: node,Ns2: list_node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( append_node @ Ns @ ( cons_node @ M @ Ns2 ) ) )
=> ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( cons_node @ M @ Ns2 ) ) ) ).
% old.path_split(2)
thf(fact_286_old_Oempty__path,axiom,
! [N: node,G: g] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( ( invar @ G )
=> ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( cons_node @ N @ nil_node ) ) ) ) ).
% old.empty_path
thf(fact_287_old_Opath__split_I1_J,axiom,
! [G: g,Ns: list_node,M: node,Ns2: list_node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( append_node @ Ns @ ( cons_node @ M @ Ns2 ) ) )
=> ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( append_node @ Ns @ ( cons_node @ M @ nil_node ) ) ) ) ).
% old.path_split(1)
thf(fact_288_old_Opath2__split__first__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 )
=> ( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Ns ) )
& ( P @ X2 ) )
=> ~ ! [M2: node,Ns4: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns4 @ M2 )
=> ( ( P @ M2 )
=> ( ! [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ ( butlast_node @ Ns4 ) ) )
=> ~ ( P @ X2 ) )
=> ~ ( prefix_node @ Ns4 @ Ns ) ) ) ) ) ) ).
% old.path2_split_first_prop
thf(fact_289_old_Opath2__prefix__ex,axiom,
! [G: g,N: node,Ns: list_node,M: node,M4: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( member_node @ M4 @ ( set_node2 @ Ns ) )
=> ~ ! [Ns4: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns4 @ M4 )
=> ( ( prefix_node @ Ns4 @ Ns )
=> ( member_node @ M4 @ ( set_node2 @ ( butlast_node @ Ns4 ) ) ) ) ) ) ) ).
% old.path2_prefix_ex
thf(fact_290_old_OCons__path,axiom,
! [G: g,Ns: list_node,N2: node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ Ns )
=> ( ( member_node @ N2 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ ( hd_node @ Ns ) ) ) )
=> ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( cons_node @ N2 @ Ns ) ) ) ) ).
% old.Cons_path
thf(fact_291_old_Opath2__prefix,axiom,
! [G: g,N: node,Ns: list_node,M: node,Ns2: list_node,M4: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( prefix_node @ ( append_node @ Ns2 @ ( cons_node @ M4 @ nil_node ) ) @ Ns )
=> ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( append_node @ Ns2 @ ( cons_node @ M4 @ nil_node ) ) @ M4 ) ) ) ).
% old.path2_prefix
thf(fact_292_old_Opath__invar,axiom,
! [G: g,Ns: list_node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ Ns )
=> ( invar @ G ) ) ).
% old.path_invar
thf(fact_293_old_Opath__in___092_060alpha_062n,axiom,
! [G: g,Ns: list_node,N: node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ Ns )
=> ( ( member_node @ N @ ( set_node2 @ Ns ) )
=> ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) ) ) ) ).
% old.path_in_\<alpha>n
thf(fact_294_old_Opath__by__tail,axiom,
! [G: g,N: node,N2: node,Ns: list_node,Ms: list_node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( cons_node @ N @ ( cons_node @ N2 @ Ns ) ) )
=> ( ( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( cons_node @ N2 @ Ns ) )
=> ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( cons_node @ N2 @ Ms ) ) )
=> ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ ( cons_node @ N @ ( cons_node @ N2 @ Ms ) ) ) ) ) ).
% old.path_by_tail
thf(fact_295_subset__Collect__conv,axiom,
! [S: set_node,P: node > $o] :
( ( ord_less_eq_set_node @ S @ ( collect_node @ P ) )
= ( ! [X3: node] :
( ( member_node @ X3 @ S )
=> ( P @ X3 ) ) ) ) ).
% subset_Collect_conv
thf(fact_296_ord__eq__le__eq__trans,axiom,
! [A: set_node,B: set_node,C: set_node,D: set_node] :
( ( A = B )
=> ( ( ord_less_eq_set_node @ B @ C )
=> ( ( C = D )
=> ( ord_less_eq_set_node @ A @ D ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_297_subset__code_I1_J,axiom,
! [Xs: list_val,B3: set_val] :
( ( ord_less_eq_set_val @ ( set_val2 @ Xs ) @ B3 )
= ( ! [X3: val] :
( ( member_val @ X3 @ ( set_val2 @ Xs ) )
=> ( member_val @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_298_subset__code_I1_J,axiom,
! [Xs: list_node,B3: set_node] :
( ( ord_less_eq_set_node @ ( set_node2 @ Xs ) @ B3 )
= ( ! [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ Xs ) )
=> ( member_node @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_299_set__subset__Cons,axiom,
! [Xs: list_node,X: node] : ( ord_less_eq_set_node @ ( set_node2 @ Xs ) @ ( set_node2 @ ( cons_node @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_300_tl__subset,axiom,
! [Xs: list_node,A2: set_node] :
( ( Xs != nil_node )
=> ( ( ord_less_eq_set_node @ ( set_node2 @ Xs ) @ A2 )
=> ( ord_less_eq_set_node @ ( set_node2 @ ( tl_node @ Xs ) ) @ A2 ) ) ) ).
% tl_subset
thf(fact_301_butlast__subset,axiom,
! [Xs: list_node,A2: set_node] :
( ( Xs != nil_node )
=> ( ( ord_less_eq_set_node @ ( set_node2 @ Xs ) @ A2 )
=> ( ord_less_eq_set_node @ ( set_node2 @ ( butlast_node @ Xs ) ) @ A2 ) ) ) ).
% butlast_subset
thf(fact_302_prefix__snoc,axiom,
! [Xs: list_node,Ys: list_node,Y: node] :
( ( prefix_node @ Xs @ ( append_node @ Ys @ ( cons_node @ Y @ nil_node ) ) )
= ( ( Xs
= ( append_node @ Ys @ ( cons_node @ Y @ nil_node ) ) )
| ( prefix_node @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_303_old_Opath2__split__first__last,axiom,
! [G: g,N: node,Ns: list_node,M: node,X: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( member_node @ X @ ( set_node2 @ Ns ) )
=> ~ ! [Ns_1: list_node,Ns_3: list_node,Ns_2: list_node] :
( ( Ns
= ( append_node @ Ns_1 @ ( append_node @ Ns_3 @ Ns_2 ) ) )
=> ( ( prefix_node @ ( append_node @ Ns_1 @ ( cons_node @ X @ nil_node ) ) @ Ns )
=> ( ( suffix_node @ ( cons_node @ X @ Ns_2 ) @ Ns )
=> ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( append_node @ Ns_1 @ ( cons_node @ X @ nil_node ) ) @ X )
=> ( ~ ( member_node @ X @ ( set_node2 @ Ns_1 ) )
=> ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ X @ Ns_3 @ X )
=> ( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ X @ ( cons_node @ X @ Ns_2 ) @ M )
=> ( member_node @ X @ ( set_node2 @ Ns_2 ) ) ) ) ) ) ) ) ) ) ) ).
% old.path2_split_first_last
thf(fact_304_same__prefix__nil,axiom,
! [Xs: list_node,Ys: list_node] :
( ( prefix_node @ ( append_node @ Xs @ Ys ) @ Xs )
= ( Ys = nil_node ) ) ).
% same_prefix_nil
thf(fact_305_prefix__order_Odual__order_Orefl,axiom,
! [A: list_node] : ( prefix_node @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_306_prefix__order_Oorder__refl,axiom,
! [X: list_node] : ( prefix_node @ X @ X ) ).
% prefix_order.order_refl
thf(fact_307_suffix__order_Odual__order_Orefl,axiom,
! [A: list_node] : ( suffix_node @ A @ A ) ).
% suffix_order.dual_order.refl
thf(fact_308_suffix__order_Oorder__refl,axiom,
! [X: list_node] : ( suffix_node @ X @ X ) ).
% suffix_order.order_refl
thf(fact_309_Cons__prefix__Cons,axiom,
! [X: node,Xs: list_node,Y: node,Ys: list_node] :
( ( prefix_node @ ( cons_node @ X @ Xs ) @ ( cons_node @ Y @ Ys ) )
= ( ( X = Y )
& ( prefix_node @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_310_prefix__code_I1_J,axiom,
! [Xs: list_node] : ( prefix_node @ nil_node @ Xs ) ).
% prefix_code(1)
thf(fact_311_prefix__Nil,axiom,
! [Xs: list_node] :
( ( prefix_node @ Xs @ nil_node )
= ( Xs = nil_node ) ) ).
% prefix_Nil
thf(fact_312_prefix__bot_Obot_Oextremum__unique,axiom,
! [A: list_node] :
( ( prefix_node @ A @ nil_node )
= ( A = nil_node ) ) ).
% prefix_bot.bot.extremum_unique
thf(fact_313_suffix__bot_Obot_Oextremum__unique,axiom,
! [A: list_node] :
( ( suffix_node @ A @ nil_node )
= ( A = nil_node ) ) ).
% suffix_bot.bot.extremum_unique
thf(fact_314_suffix__Nil,axiom,
! [Xs: list_node] :
( ( suffix_node @ Xs @ nil_node )
= ( Xs = nil_node ) ) ).
% suffix_Nil
thf(fact_315_same__prefix__prefix,axiom,
! [Xs: list_node,Ys: list_node,Zs: list_node] :
( ( prefix_node @ ( append_node @ Xs @ Ys ) @ ( append_node @ Xs @ Zs ) )
= ( prefix_node @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_316_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_317_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 )
=> ( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Ns ) )
& ( P @ X2 ) )
=> ~ ! [N3: node,Ns4: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N3 @ Ns4 @ M )
=> ( ( P @ N3 )
=> ( ! [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ ( tl_node @ Ns4 ) ) )
=> ~ ( P @ X2 ) )
=> ~ ( suffix_node @ Ns4 @ Ns ) ) ) ) ) ) ).
% old.path2_split_last_prop
thf(fact_318_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_319_snoc__suffix__snoc,axiom,
! [Xs: list_node,X: node,Ys: list_node,Y: node] :
( ( suffix_node @ ( append_node @ Xs @ ( cons_node @ X @ nil_node ) ) @ ( append_node @ Ys @ ( cons_node @ Y @ nil_node ) ) )
= ( ( X = Y )
& ( suffix_node @ Xs @ Ys ) ) ) ).
% snoc_suffix_snoc
thf(fact_320_suffix__snoc,axiom,
! [Xs: list_node,Ys: list_node,Y: node] :
( ( suffix_node @ Xs @ ( append_node @ Ys @ ( cons_node @ Y @ nil_node ) ) )
= ( ( Xs = nil_node )
| ? [Zs3: list_node] :
( ( Xs
= ( append_node @ Zs3 @ ( cons_node @ Y @ nil_node ) ) )
& ( suffix_node @ Zs3 @ Ys ) ) ) ) ).
% suffix_snoc
thf(fact_321_suffix__tl,axiom,
! [Xs: list_node] : ( suffix_node @ ( tl_node @ Xs ) @ Xs ) ).
% suffix_tl
thf(fact_322_suffix__Cons,axiom,
! [Xs: list_node,Y: node,Ys: list_node] :
( ( suffix_node @ Xs @ ( cons_node @ Y @ Ys ) )
= ( ( Xs
= ( cons_node @ Y @ Ys ) )
| ( suffix_node @ Xs @ Ys ) ) ) ).
% suffix_Cons
thf(fact_323_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_324_suffix__ConsI,axiom,
! [Xs: list_node,Ys: list_node,Y: node] :
( ( suffix_node @ Xs @ Ys )
=> ( suffix_node @ Xs @ ( cons_node @ Y @ Ys ) ) ) ).
% suffix_ConsI
thf(fact_325_suffix__ConsD2,axiom,
! [X: node,Xs: list_node,Y: node,Ys: list_node] :
( ( suffix_node @ ( cons_node @ X @ Xs ) @ ( cons_node @ Y @ Ys ) )
=> ( suffix_node @ Xs @ Ys ) ) ).
% suffix_ConsD2
thf(fact_326_suffix__bot_Obot_Oextremum__uniqueI,axiom,
! [A: list_node] :
( ( suffix_node @ A @ nil_node )
=> ( A = nil_node ) ) ).
% suffix_bot.bot.extremum_uniqueI
thf(fact_327_suffix__bot_Obot_Oextremum,axiom,
! [A: list_node] : ( suffix_node @ nil_node @ A ) ).
% suffix_bot.bot.extremum
thf(fact_328_Nil__suffix,axiom,
! [Xs: list_node] : ( suffix_node @ nil_node @ Xs ) ).
% Nil_suffix
thf(fact_329_set__mono__suffix,axiom,
! [Xs: list_node,Ys: list_node] :
( ( suffix_node @ Xs @ Ys )
=> ( ord_less_eq_set_node @ ( set_node2 @ Xs ) @ ( set_node2 @ Ys ) ) ) ).
% set_mono_suffix
thf(fact_330_not__suffix__cases,axiom,
! [Ps2: list_node,Ls: list_node] :
( ~ ( suffix_node @ Ps2 @ Ls )
=> ( ( ( Ps2 != nil_node )
=> ( Ls != nil_node ) )
=> ( ! [A3: node,As: list_node] :
( ( Ps2
= ( append_node @ As @ ( cons_node @ A3 @ nil_node ) ) )
=> ! [X4: node,Xs2: list_node] :
( ( Ls
= ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) )
=> ( ( X4 = A3 )
=> ( suffix_node @ As @ Xs2 ) ) ) )
=> ~ ! [A3: node] :
( ? [As: list_node] :
( Ps2
= ( append_node @ As @ ( cons_node @ A3 @ nil_node ) ) )
=> ! [X4: node] :
( ? [Xs2: list_node] :
( Ls
= ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) )
=> ( X4 = A3 ) ) ) ) ) ) ).
% not_suffix_cases
thf(fact_331_not__suffix__induct,axiom,
! [Ps2: list_node,Ls: list_node,P: list_node > list_node > $o] :
( ~ ( suffix_node @ Ps2 @ Ls )
=> ( ! [X4: node,Xs2: list_node] : ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) @ nil_node )
=> ( ! [X4: node,Xs2: list_node,Y3: node,Ys3: list_node] :
( ( X4 != Y3 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) @ ( append_node @ Ys3 @ ( cons_node @ Y3 @ nil_node ) ) ) )
=> ( ! [X4: node,Xs2: list_node,Y3: node,Ys3: list_node] :
( ( X4 = Y3 )
=> ( ~ ( suffix_node @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) @ ( append_node @ Ys3 @ ( cons_node @ Y3 @ nil_node ) ) ) ) ) )
=> ( P @ Ps2 @ Ls ) ) ) ) ) ).
% not_suffix_induct
thf(fact_332_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_333_suffix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: list_node,Z: list_node] : ( Y4 = Z ) )
= ( ^ [A4: list_node,B4: list_node] :
( ( suffix_node @ B4 @ A4 )
& ( suffix_node @ A4 @ B4 ) ) ) ) ).
% suffix_order.dual_order.eq_iff
thf(fact_334_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_335_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_336_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_337_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_338_suffix__order_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: list_node,Z: list_node] : ( Y4 = Z ) )
= ( ^ [A4: list_node,B4: list_node] :
( ( suffix_node @ A4 @ B4 )
& ( suffix_node @ B4 @ A4 ) ) ) ) ).
% suffix_order.order.eq_iff
thf(fact_339_suffix__order_Oantisym__conv,axiom,
! [Y: list_node,X: list_node] :
( ( suffix_node @ Y @ X )
=> ( ( suffix_node @ X @ Y )
= ( X = Y ) ) ) ).
% suffix_order.antisym_conv
thf(fact_340_suffix__order_Oorder__trans,axiom,
! [X: list_node,Y: list_node,Z3: list_node] :
( ( suffix_node @ X @ Y )
=> ( ( suffix_node @ Y @ Z3 )
=> ( suffix_node @ X @ Z3 ) ) ) ).
% suffix_order.order_trans
thf(fact_341_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_342_suffix__order_Oeq__refl,axiom,
! [X: list_node,Y: list_node] :
( ( X = Y )
=> ( suffix_node @ X @ Y ) ) ).
% suffix_order.eq_refl
thf(fact_343_suffix__order_Oantisym,axiom,
! [X: list_node,Y: list_node] :
( ( suffix_node @ X @ Y )
=> ( ( suffix_node @ Y @ X )
=> ( X = Y ) ) ) ).
% suffix_order.antisym
thf(fact_344_suffix__order_Oeq__iff,axiom,
( ( ^ [Y4: list_node,Z: list_node] : ( Y4 = Z ) )
= ( ^ [X3: list_node,Y2: list_node] :
( ( suffix_node @ X3 @ Y2 )
& ( suffix_node @ Y2 @ X3 ) ) ) ) ).
% suffix_order.eq_iff
thf(fact_345_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_346_suffixE,axiom,
! [Xs: list_node,Ys: list_node] :
( ( suffix_node @ Xs @ Ys )
=> ~ ! [Zs2: list_node] :
( Ys
!= ( append_node @ Zs2 @ Xs ) ) ) ).
% suffixE
thf(fact_347_suffixI,axiom,
! [Ys: list_node,Zs: list_node,Xs: list_node] :
( ( Ys
= ( append_node @ Zs @ Xs ) )
=> ( suffix_node @ Xs @ Ys ) ) ).
% suffixI
thf(fact_348_Sublist_Osuffix__def,axiom,
( suffix_node
= ( ^ [Xs3: list_node,Ys2: list_node] :
? [Zs3: list_node] :
( Ys2
= ( append_node @ Zs3 @ Xs3 ) ) ) ) ).
% Sublist.suffix_def
thf(fact_349_suffix__append,axiom,
! [Xs: list_node,Ys: list_node,Zs: list_node] :
( ( suffix_node @ Xs @ ( append_node @ Ys @ Zs ) )
= ( ( suffix_node @ Xs @ Zs )
| ? [Xs6: list_node] :
( ( Xs
= ( append_node @ Xs6 @ Zs ) )
& ( suffix_node @ Xs6 @ Ys ) ) ) ) ).
% suffix_append
thf(fact_350_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_351_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
% Conjectures (1)
thf(conj_0,conjecture,
graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ g2 @ i @ ( cons_node @ i @ rs_rest ) @ pred_phi_r ).
%------------------------------------------------------------------------------