TPTP Problem File: ITP081^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP081^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Irreducible problem prob_472__6626992_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_472__6626992_1 [Des21]
% Status : Theorem
% Rating : 0.60 v8.2.0, 0.46 v8.1.0, 0.45 v7.5.0
% Syntax : Number of formulae : 419 ( 146 unt; 68 typ; 0 def)
% Number of atoms : 1059 ( 603 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 4056 ( 203 ~; 21 |; 116 &;3214 @)
% ( 0 <=>; 502 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 8 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 242 ( 242 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 56 usr; 17 con; 0-7 aty)
% Number of variables : 1257 ( 23 ^;1157 !; 77 ?;1257 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:40:45.367
%------------------------------------------------------------------------------
% 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 (56)
thf(sy_c_Finite__Set_Ofinite_001tf__node,type,
finite_finite_node: set_node > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__val,type,
finite_finite_val: set_val > $o ).
thf(sy_c_Graph__path_Ograph__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_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_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____,type,
rs2: list_node ).
thf(sy_v_s,type,
s: val ).
% Relevant facts (350)
thf(fact_0_False,axiom,
r != phi_r ).
% False
thf(fact_1_ri__is__valid,axiom,
( i
= ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_r ) ) ).
% ri_is_valid
thf(fact_2_old_Oinvar,axiom,
! [G: g] : ( invar @ G ) ).
% old.invar
thf(fact_3_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_4_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_5_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_6__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_7_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_8_assms_I10_J,axiom,
sSA_CF1252180629de_val @ alpha_n @ defs @ phis @ g2 @ phi_r @ r ).
% assms(10)
thf(fact_9_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_10_ms_H__props_I1_J,axiom,
graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ g2 @ m @ ms2 @ i ).
% ms'_props(1)
thf(fact_11__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_12_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_13_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_14_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_15_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_16_old_Opath2__hd__in___092_060alpha_062n,axiom,
! [G: g,N: node,Ns: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) ) ) ).
% old.path2_hd_in_\<alpha>n
thf(fact_17_old_Opath2__hd__in__ns,axiom,
! [G: g,N: node,Ns: list_node,M: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( member_node @ N @ ( set_node2 @ Ns ) ) ) ).
% old.path2_hd_in_ns
thf(fact_18_old_Opath2__in___092_060alpha_062n,axiom,
! [G: g,N: node,Ns: list_node,M: node,L: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( member_node @ L @ ( set_node2 @ Ns ) )
=> ( member_node @ L @ ( set_node2 @ ( alpha_n @ G ) ) ) ) ) ).
% old.path2_in_\<alpha>n
thf(fact_19_old_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_20_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_21_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_22_ri__props_I2_J,axiom,
member_node @ i @ ( set_node2 @ ms ) ).
% ri_props(2)
thf(fact_23_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_24_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_25_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_26_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_27_rs_H__loopfree,axiom,
~ ( member_node @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ r ) @ ( set_node2 @ ( tl_node @ rs ) ) ) ).
% rs'_loopfree
thf(fact_28_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_29_ri__props_I4_J,axiom,
prefix_node @ ri @ rs2 ).
% ri_props(4)
thf(fact_30_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_31_ms_H__props_I3_J,axiom,
~ ( member_node @ i @ ( set_node2 @ ( butlast_node @ ms2 ) ) ) ).
% ms'_props(3)
thf(fact_32_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_33_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_34_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_35_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_36_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_37_list__e__eq__lel_I2_J,axiom,
! [L1: list_val,E: val,L2: list_val,E2: val] :
( ( ( append_val @ L1 @ ( cons_val @ E @ L2 ) )
= ( cons_val @ E2 @ nil_val ) )
= ( ( L1 = nil_val )
& ( E = E2 )
& ( L2 = nil_val ) ) ) ).
% list_e_eq_lel(2)
thf(fact_38_list__e__eq__lel_I2_J,axiom,
! [L1: list_list_node,E: list_node,L2: list_list_node,E2: list_node] :
( ( ( append_list_node @ L1 @ ( cons_list_node @ E @ L2 ) )
= ( cons_list_node @ E2 @ nil_list_node ) )
= ( ( L1 = nil_list_node )
& ( E = E2 )
& ( L2 = nil_list_node ) ) ) ).
% list_e_eq_lel(2)
thf(fact_39_list__e__eq__lel_I2_J,axiom,
! [L1: list_node,E: node,L2: list_node,E2: node] :
( ( ( append_node @ L1 @ ( cons_node @ E @ L2 ) )
= ( cons_node @ E2 @ nil_node ) )
= ( ( L1 = nil_node )
& ( E = E2 )
& ( L2 = nil_node ) ) ) ).
% list_e_eq_lel(2)
thf(fact_40_list__e__eq__lel_I1_J,axiom,
! [E2: val,L1: list_val,E: val,L2: list_val] :
( ( ( cons_val @ E2 @ nil_val )
= ( append_val @ L1 @ ( cons_val @ E @ L2 ) ) )
= ( ( L1 = nil_val )
& ( E = E2 )
& ( L2 = nil_val ) ) ) ).
% list_e_eq_lel(1)
thf(fact_41_list__e__eq__lel_I1_J,axiom,
! [E2: list_node,L1: list_list_node,E: list_node,L2: list_list_node] :
( ( ( cons_list_node @ E2 @ nil_list_node )
= ( append_list_node @ L1 @ ( cons_list_node @ E @ L2 ) ) )
= ( ( L1 = nil_list_node )
& ( E = E2 )
& ( L2 = nil_list_node ) ) ) ).
% list_e_eq_lel(1)
thf(fact_42_list__e__eq__lel_I1_J,axiom,
! [E2: node,L1: list_node,E: node,L2: list_node] :
( ( ( cons_node @ E2 @ nil_node )
= ( append_node @ L1 @ ( cons_node @ E @ L2 ) ) )
= ( ( L1 = nil_node )
& ( E = E2 )
& ( L2 = nil_node ) ) ) ).
% list_e_eq_lel(1)
thf(fact_43_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_44_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_45_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_46_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_47_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_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_list__ee__eq__leel_I2_J,axiom,
! [L1: list_val,E1: val,E22: val,L2: list_val,E12: val,E23: val] :
( ( ( append_val @ L1 @ ( cons_val @ E1 @ ( cons_val @ E22 @ L2 ) ) )
= ( cons_val @ E12 @ ( cons_val @ E23 @ nil_val ) ) )
= ( ( L1 = nil_val )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L2 = nil_val ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_56_list__ee__eq__leel_I2_J,axiom,
! [L1: list_list_node,E1: list_node,E22: list_node,L2: list_list_node,E12: list_node,E23: list_node] :
( ( ( append_list_node @ L1 @ ( cons_list_node @ E1 @ ( cons_list_node @ E22 @ L2 ) ) )
= ( cons_list_node @ E12 @ ( cons_list_node @ E23 @ nil_list_node ) ) )
= ( ( L1 = nil_list_node )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L2 = nil_list_node ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_57_list__ee__eq__leel_I2_J,axiom,
! [L1: list_node,E1: node,E22: node,L2: list_node,E12: node,E23: node] :
( ( ( append_node @ L1 @ ( cons_node @ E1 @ ( cons_node @ E22 @ L2 ) ) )
= ( cons_node @ E12 @ ( cons_node @ E23 @ nil_node ) ) )
= ( ( L1 = nil_node )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L2 = nil_node ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_58_list__ee__eq__leel_I1_J,axiom,
! [E12: val,E23: val,L1: list_val,E1: val,E22: val,L2: list_val] :
( ( ( cons_val @ E12 @ ( cons_val @ E23 @ nil_val ) )
= ( append_val @ L1 @ ( cons_val @ E1 @ ( cons_val @ E22 @ L2 ) ) ) )
= ( ( L1 = nil_val )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L2 = nil_val ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_59_list__ee__eq__leel_I1_J,axiom,
! [E12: list_node,E23: list_node,L1: list_list_node,E1: list_node,E22: list_node,L2: list_list_node] :
( ( ( cons_list_node @ E12 @ ( cons_list_node @ E23 @ nil_list_node ) )
= ( append_list_node @ L1 @ ( cons_list_node @ E1 @ ( cons_list_node @ E22 @ L2 ) ) ) )
= ( ( L1 = nil_list_node )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L2 = nil_list_node ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_60_list__ee__eq__leel_I1_J,axiom,
! [E12: node,E23: node,L1: list_node,E1: node,E22: node,L2: list_node] :
( ( ( cons_node @ E12 @ ( cons_node @ E23 @ nil_node ) )
= ( append_node @ L1 @ ( cons_node @ E1 @ ( cons_node @ E22 @ L2 ) ) ) )
= ( ( L1 = nil_node )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L2 = nil_node ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_61_ms_H__props_I2_J,axiom,
prefix_node @ ms2 @ ms ).
% ms'_props(2)
thf(fact_62_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_63_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_64_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_65_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_66_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_67_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_68_mem__Collect__eq,axiom,
! [A: val,P: val > $o] :
( ( member_val @ A @ ( collect_val @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A: node,P: node > $o] :
( ( member_node @ A @ ( collect_node @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A2: set_val] :
( ( collect_val
@ ^ [X3: val] : ( member_val @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A2: set_node] :
( ( collect_node
@ ^ [X3: node] : ( member_node @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_72_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_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_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_81_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_82_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_83_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_84_append__Nil2,axiom,
! [Xs: list_list_node] :
( ( append_list_node @ Xs @ nil_list_node )
= Xs ) ).
% append_Nil2
thf(fact_85_append__Nil2,axiom,
! [Xs: list_val] :
( ( append_val @ Xs @ nil_val )
= Xs ) ).
% append_Nil2
thf(fact_86_append__Nil2,axiom,
! [Xs: list_node] :
( ( append_node @ Xs @ nil_node )
= Xs ) ).
% append_Nil2
thf(fact_87_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_88_append__self__conv,axiom,
! [Xs: list_val,Ys: list_val] :
( ( ( append_val @ Xs @ Ys )
= Xs )
= ( Ys = nil_val ) ) ).
% append_self_conv
thf(fact_89_append__self__conv,axiom,
! [Xs: list_node,Ys: list_node] :
( ( ( append_node @ Xs @ Ys )
= Xs )
= ( Ys = nil_node ) ) ).
% append_self_conv
thf(fact_90_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_91_self__append__conv,axiom,
! [Xs: list_val,Ys: list_val] :
( ( Xs
= ( append_val @ Xs @ Ys ) )
= ( Ys = nil_val ) ) ).
% self_append_conv
thf(fact_92_self__append__conv,axiom,
! [Xs: list_node,Ys: list_node] :
( ( Xs
= ( append_node @ Xs @ Ys ) )
= ( Ys = nil_node ) ) ).
% self_append_conv
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_val,Ys: list_val] :
( ( ( append_val @ Xs @ Ys )
= Ys )
= ( Xs = nil_val ) ) ).
% append_self_conv2
thf(fact_95_append__self__conv2,axiom,
! [Xs: list_node,Ys: list_node] :
( ( ( append_node @ Xs @ Ys )
= Ys )
= ( Xs = nil_node ) ) ).
% append_self_conv2
thf(fact_96_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_97_self__append__conv2,axiom,
! [Ys: list_val,Xs: list_val] :
( ( Ys
= ( append_val @ Xs @ Ys ) )
= ( Xs = nil_val ) ) ).
% self_append_conv2
thf(fact_98_self__append__conv2,axiom,
! [Ys: list_node,Xs: list_node] :
( ( Ys
= ( append_node @ Xs @ Ys ) )
= ( Xs = nil_node ) ) ).
% self_append_conv2
thf(fact_99_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_100_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_101_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_102_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_103_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_104_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_105_empty__append__eq__id,axiom,
( ( append_list_node @ nil_list_node )
= ( ^ [X3: list_list_node] : X3 ) ) ).
% empty_append_eq_id
thf(fact_106_empty__append__eq__id,axiom,
( ( append_val @ nil_val )
= ( ^ [X3: list_val] : X3 ) ) ).
% empty_append_eq_id
thf(fact_107_empty__append__eq__id,axiom,
( ( append_node @ nil_node )
= ( ^ [X3: list_node] : X3 ) ) ).
% empty_append_eq_id
thf(fact_108_append_Oright__neutral,axiom,
! [A: list_list_node] :
( ( append_list_node @ A @ nil_list_node )
= A ) ).
% append.right_neutral
thf(fact_109_append_Oright__neutral,axiom,
! [A: list_val] :
( ( append_val @ A @ nil_val )
= A ) ).
% append.right_neutral
thf(fact_110_append_Oright__neutral,axiom,
! [A: list_node] :
( ( append_node @ A @ nil_node )
= A ) ).
% append.right_neutral
thf(fact_111_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_112_old_Opath2__prefix__ex,axiom,
! [G: g,N: node,Ns: list_node,M: node,M2: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( member_node @ M2 @ ( set_node2 @ Ns ) )
=> ~ ! [Ns3: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns3 @ M2 )
=> ( ( prefix_node @ Ns3 @ Ns )
=> ( member_node @ M2 @ ( set_node2 @ ( butlast_node @ Ns3 ) ) ) ) ) ) ) ).
% old.path2_prefix_ex
thf(fact_113_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 ) )
=> ~ ! [M3: node,Ns3: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns3 @ M3 )
=> ( ( P @ M3 )
=> ( ! [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ ( butlast_node @ Ns3 ) ) )
=> ~ ( P @ X2 ) )
=> ~ ( prefix_node @ Ns3 @ Ns ) ) ) ) ) ) ).
% old.path2_split_first_prop
thf(fact_114_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 ) )
=> ( ! [Ns4: list_node,N3: node,N4: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N4 @ Ns4 @ M )
=> ( ( P @ N4 @ Ns4 @ M )
=> ( ( member_node @ N3 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ N4 ) ) )
=> ( P @ N3 @ ( cons_node @ N3 @ Ns4 ) @ M ) ) ) )
=> ( P @ N @ Ns @ M ) ) ) ) ).
% old.path2_induct
thf(fact_115_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_116_old_Opath2__prefix,axiom,
! [G: g,N: node,Ns: list_node,M: node,Ns2: list_node,M2: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( prefix_node @ ( append_node @ Ns2 @ ( cons_node @ M2 @ nil_node ) ) @ Ns )
=> ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( append_node @ Ns2 @ ( cons_node @ M2 @ nil_node ) ) @ M2 ) ) ) ).
% old.path2_prefix
thf(fact_117_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 ) )
=> ( ! [Ns4: list_node,M3: node,M4: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns4 @ M3 )
=> ( ( P @ N @ Ns4 @ M3 )
=> ( ( member_node @ M3 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ M4 ) ) )
=> ( P @ N @ ( append_node @ Ns4 @ ( cons_node @ M4 @ nil_node ) ) @ M4 ) ) ) )
=> ( P @ N @ Ns @ M ) ) ) ) ).
% old.path2_rev_induct
thf(fact_118_old_Opath2__snoc,axiom,
! [G: g,N: node,Ns: list_node,M: node,M2: node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns @ M )
=> ( ( member_node @ M @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ M2 ) ) )
=> ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ ( append_node @ Ns @ ( cons_node @ M2 @ nil_node ) ) @ M2 ) ) ) ).
% old.path2_snoc
thf(fact_119_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_120_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_121_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_122_butlast__snoc,axiom,
! [Xs: list_val,X: val] :
( ( butlast_val @ ( append_val @ Xs @ ( cons_val @ X @ nil_val ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_123_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_124_butlast__snoc,axiom,
! [Xs: list_node,X: node] :
( ( butlast_node @ ( append_node @ Xs @ ( cons_node @ X @ nil_node ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_125__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_126__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_127_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_128_butlast_Osimps_I1_J,axiom,
( ( butlast_list_node @ nil_list_node )
= nil_list_node ) ).
% butlast.simps(1)
thf(fact_129_butlast_Osimps_I1_J,axiom,
( ( butlast_val @ nil_val )
= nil_val ) ).
% butlast.simps(1)
thf(fact_130_butlast_Osimps_I1_J,axiom,
( ( butlast_node @ nil_node )
= nil_node ) ).
% butlast.simps(1)
thf(fact_131_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_132_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_133_butlast__tl,axiom,
! [Xs: list_val] :
( ( butlast_val @ ( tl_val @ Xs ) )
= ( tl_val @ ( butlast_val @ Xs ) ) ) ).
% butlast_tl
thf(fact_134_butlast__tl,axiom,
! [Xs: list_node] :
( ( butlast_node @ ( tl_node @ Xs ) )
= ( tl_node @ ( butlast_node @ Xs ) ) ) ).
% butlast_tl
thf(fact_135_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_136_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_137_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_138_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_139_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_140_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_141_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_142_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_143_butlast__eq__cons__conv,axiom,
! [L: list_val,X: val,Xs: list_val] :
( ( ( butlast_val @ L )
= ( cons_val @ X @ Xs ) )
= ( ? [Xl: val] :
( L
= ( cons_val @ X @ ( append_val @ Xs @ ( cons_val @ Xl @ nil_val ) ) ) ) ) ) ).
% butlast_eq_cons_conv
thf(fact_144_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_145_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_146_butlast__eq__consE,axiom,
! [L: list_val,X: val,Xs: list_val] :
( ( ( butlast_val @ L )
= ( cons_val @ X @ Xs ) )
=> ~ ! [Xl2: val] :
( L
!= ( cons_val @ X @ ( append_val @ Xs @ ( cons_val @ Xl2 @ nil_val ) ) ) ) ) ).
% butlast_eq_consE
thf(fact_147_butlast__eq__consE,axiom,
! [L: list_list_node,X: list_node,Xs: list_list_node] :
( ( ( butlast_list_node @ L )
= ( cons_list_node @ X @ Xs ) )
=> ~ ! [Xl2: list_node] :
( L
!= ( cons_list_node @ X @ ( append_list_node @ Xs @ ( cons_list_node @ Xl2 @ nil_list_node ) ) ) ) ) ).
% butlast_eq_consE
thf(fact_148_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_149_list__tail__coinc,axiom,
! [N1: val,R1: list_val,N22: val,R2: list_val] :
( ( ( cons_val @ N1 @ R1 )
= ( cons_val @ N22 @ R2 ) )
=> ( ( N1 = N22 )
& ( R1 = R2 ) ) ) ).
% list_tail_coinc
thf(fact_150_list__tail__coinc,axiom,
! [N1: list_node,R1: list_list_node,N22: list_node,R2: list_list_node] :
( ( ( cons_list_node @ N1 @ R1 )
= ( cons_list_node @ N22 @ R2 ) )
=> ( ( N1 = N22 )
& ( R1 = R2 ) ) ) ).
% list_tail_coinc
thf(fact_151_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_152_not__Cons__self2,axiom,
! [X: val,Xs: list_val] :
( ( cons_val @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_153_not__Cons__self2,axiom,
! [X: list_node,Xs: list_list_node] :
( ( cons_list_node @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_154_not__Cons__self2,axiom,
! [X: node,Xs: list_node] :
( ( cons_node @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_155_append__eq__appendI,axiom,
! [Xs: list_val,Xs1: list_val,Zs: list_val,Ys: list_val,Us: list_val] :
( ( ( append_val @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_val @ Xs1 @ Us ) )
=> ( ( append_val @ Xs @ Ys )
= ( append_val @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_156_append__eq__appendI,axiom,
! [Xs: list_node,Xs1: list_node,Zs: list_node,Ys: list_node,Us: list_node] :
( ( ( append_node @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_node @ Xs1 @ Us ) )
=> ( ( append_node @ Xs @ Ys )
= ( append_node @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_157_append__eq__append__conv2,axiom,
! [Xs: list_val,Ys: list_val,Zs: list_val,Ts: list_val] :
( ( ( append_val @ Xs @ Ys )
= ( append_val @ Zs @ Ts ) )
= ( ? [Us2: list_val] :
( ( ( Xs
= ( append_val @ Zs @ Us2 ) )
& ( ( append_val @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_val @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_val @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_158_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 ) )
= ( ? [Us2: list_node] :
( ( ( Xs
= ( append_node @ Zs @ Us2 ) )
& ( ( append_node @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_node @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_node @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_159_strict__sorted_Oinduct,axiom,
! [P: list_val > $o,A0: list_val] :
( ( P @ nil_val )
=> ( ! [X4: val,Ys2: list_val] :
( ( P @ Ys2 )
=> ( P @ ( cons_val @ X4 @ Ys2 ) ) )
=> ( P @ A0 ) ) ) ).
% strict_sorted.induct
thf(fact_160_strict__sorted_Oinduct,axiom,
! [P: list_node > $o,A0: list_node] :
( ( P @ nil_node )
=> ( ! [X4: node,Ys2: list_node] :
( ( P @ Ys2 )
=> ( P @ ( cons_node @ X4 @ Ys2 ) ) )
=> ( P @ A0 ) ) ) ).
% strict_sorted.induct
thf(fact_161_mergesort__by__rel__merge__induct,axiom,
! [P: list_node > list_val > $o,R: node > val > $o,Xs: list_node,Ys: list_val] :
( ! [Xs2: list_node] : ( P @ Xs2 @ nil_val )
=> ( ! [X_1: list_val] : ( P @ nil_node @ X_1 )
=> ( ! [X4: node,Xs2: list_node,Y2: val,Ys2: list_val] :
( ( R @ X4 @ Y2 )
=> ( ( P @ Xs2 @ ( cons_val @ Y2 @ Ys2 ) )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ ( cons_val @ Y2 @ Ys2 ) ) ) )
=> ( ! [X4: node,Xs2: list_node,Y2: val,Ys2: list_val] :
( ~ ( R @ X4 @ Y2 )
=> ( ( P @ ( cons_node @ X4 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ ( cons_val @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_162_mergesort__by__rel__merge__induct,axiom,
! [P: list_val > list_val > $o,R: val > val > $o,Xs: list_val,Ys: list_val] :
( ! [Xs2: list_val] : ( P @ Xs2 @ nil_val )
=> ( ! [X_1: list_val] : ( P @ nil_val @ X_1 )
=> ( ! [X4: val,Xs2: list_val,Y2: val,Ys2: list_val] :
( ( R @ X4 @ Y2 )
=> ( ( P @ Xs2 @ ( cons_val @ Y2 @ Ys2 ) )
=> ( P @ ( cons_val @ X4 @ Xs2 ) @ ( cons_val @ Y2 @ Ys2 ) ) ) )
=> ( ! [X4: val,Xs2: list_val,Y2: val,Ys2: list_val] :
( ~ ( R @ X4 @ Y2 )
=> ( ( P @ ( cons_val @ X4 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_val @ X4 @ Xs2 ) @ ( cons_val @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_163_mergesort__by__rel__merge__induct,axiom,
! [P: list_list_node > list_val > $o,R: list_node > val > $o,Xs: list_list_node,Ys: list_val] :
( ! [Xs2: list_list_node] : ( P @ Xs2 @ nil_val )
=> ( ! [X_1: list_val] : ( P @ nil_list_node @ X_1 )
=> ( ! [X4: list_node,Xs2: list_list_node,Y2: val,Ys2: list_val] :
( ( R @ X4 @ Y2 )
=> ( ( P @ Xs2 @ ( cons_val @ Y2 @ Ys2 ) )
=> ( P @ ( cons_list_node @ X4 @ Xs2 ) @ ( cons_val @ Y2 @ Ys2 ) ) ) )
=> ( ! [X4: list_node,Xs2: list_list_node,Y2: val,Ys2: list_val] :
( ~ ( R @ X4 @ Y2 )
=> ( ( P @ ( cons_list_node @ X4 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_list_node @ X4 @ Xs2 ) @ ( cons_val @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_164_mergesort__by__rel__merge__induct,axiom,
! [P: list_node > list_list_node > $o,R: node > list_node > $o,Xs: list_node,Ys: list_list_node] :
( ! [Xs2: list_node] : ( P @ Xs2 @ nil_list_node )
=> ( ! [X_1: list_list_node] : ( P @ nil_node @ X_1 )
=> ( ! [X4: node,Xs2: list_node,Y2: list_node,Ys2: list_list_node] :
( ( R @ X4 @ Y2 )
=> ( ( P @ Xs2 @ ( cons_list_node @ Y2 @ Ys2 ) )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ ( cons_list_node @ Y2 @ Ys2 ) ) ) )
=> ( ! [X4: node,Xs2: list_node,Y2: list_node,Ys2: list_list_node] :
( ~ ( R @ X4 @ Y2 )
=> ( ( P @ ( cons_node @ X4 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ ( cons_list_node @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_165_mergesort__by__rel__merge__induct,axiom,
! [P: list_val > list_list_node > $o,R: val > list_node > $o,Xs: list_val,Ys: list_list_node] :
( ! [Xs2: list_val] : ( P @ Xs2 @ nil_list_node )
=> ( ! [X_1: list_list_node] : ( P @ nil_val @ X_1 )
=> ( ! [X4: val,Xs2: list_val,Y2: list_node,Ys2: list_list_node] :
( ( R @ X4 @ Y2 )
=> ( ( P @ Xs2 @ ( cons_list_node @ Y2 @ Ys2 ) )
=> ( P @ ( cons_val @ X4 @ Xs2 ) @ ( cons_list_node @ Y2 @ Ys2 ) ) ) )
=> ( ! [X4: val,Xs2: list_val,Y2: list_node,Ys2: list_list_node] :
( ~ ( R @ X4 @ Y2 )
=> ( ( P @ ( cons_val @ X4 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_val @ X4 @ Xs2 ) @ ( cons_list_node @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_166_mergesort__by__rel__merge__induct,axiom,
! [P: list_list_node > list_list_node > $o,R: list_node > list_node > $o,Xs: list_list_node,Ys: list_list_node] :
( ! [Xs2: list_list_node] : ( P @ Xs2 @ nil_list_node )
=> ( ! [X_1: list_list_node] : ( P @ nil_list_node @ X_1 )
=> ( ! [X4: list_node,Xs2: list_list_node,Y2: list_node,Ys2: list_list_node] :
( ( R @ X4 @ Y2 )
=> ( ( P @ Xs2 @ ( cons_list_node @ Y2 @ Ys2 ) )
=> ( P @ ( cons_list_node @ X4 @ Xs2 ) @ ( cons_list_node @ Y2 @ Ys2 ) ) ) )
=> ( ! [X4: list_node,Xs2: list_list_node,Y2: list_node,Ys2: list_list_node] :
( ~ ( R @ X4 @ Y2 )
=> ( ( P @ ( cons_list_node @ X4 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_list_node @ X4 @ Xs2 ) @ ( cons_list_node @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_167_mergesort__by__rel__merge__induct,axiom,
! [P: list_node > list_node > $o,R: 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,Y2: node,Ys2: list_node] :
( ( R @ X4 @ Y2 )
=> ( ( P @ Xs2 @ ( cons_node @ Y2 @ Ys2 ) )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ ( cons_node @ Y2 @ Ys2 ) ) ) )
=> ( ! [X4: node,Xs2: list_node,Y2: node,Ys2: list_node] :
( ~ ( R @ X4 @ Y2 )
=> ( ( P @ ( cons_node @ X4 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ ( cons_node @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_168_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] :
( ! [R3: node > node > $o,X4: node,Xs2: list_node,Y2: node,Ys2: list_node] :
( ( ( R3 @ X4 @ Y2 )
=> ( P @ R3 @ Xs2 @ ( cons_node @ Y2 @ Ys2 ) ) )
=> ( ( ~ ( R3 @ X4 @ Y2 )
=> ( P @ R3 @ ( cons_node @ X4 @ Xs2 ) @ Ys2 ) )
=> ( P @ R3 @ ( cons_node @ X4 @ Xs2 ) @ ( cons_node @ Y2 @ Ys2 ) ) ) )
=> ( ! [R3: node > node > $o,Xs2: list_node] : ( P @ R3 @ Xs2 @ nil_node )
=> ( ! [R3: node > node > $o,V: node,Va: list_node] : ( P @ R3 @ nil_node @ ( cons_node @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% mergesort_by_rel_merge.induct
thf(fact_169_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_170_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_171_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_172_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,Y2: node,Xs2: list_node] :
( ( P @ P2 @ ( cons_node @ Y2 @ Xs2 ) )
=> ( P @ P2 @ ( cons_node @ X4 @ ( cons_node @ Y2 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_173_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_174_remdups__adj_Oinduct,axiom,
! [P: list_node > $o,A0: list_node] :
( ( P @ nil_node )
=> ( ! [X4: node] : ( P @ ( cons_node @ X4 @ nil_node ) )
=> ( ! [X4: node,Y2: node,Xs2: list_node] :
( ( ( X4 = Y2 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) ) )
=> ( ( ( X4 != Y2 )
=> ( P @ ( cons_node @ Y2 @ Xs2 ) ) )
=> ( P @ ( cons_node @ X4 @ ( cons_node @ Y2 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_175_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,Ys2: list_node] :
( ( P @ P2 @ Ys2 )
=> ( P @ P2 @ ( cons_node @ X4 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_176_remdups__adj_Ocases,axiom,
! [X: list_node] :
( ( X != nil_node )
=> ( ! [X4: node] :
( X
!= ( cons_node @ X4 @ nil_node ) )
=> ~ ! [X4: node,Y2: node,Xs2: list_node] :
( X
!= ( cons_node @ X4 @ ( cons_node @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_177_list__2pre__induct,axiom,
! [P: list_node > list_node > $o,W1: list_node,W2: list_node] :
( ( P @ nil_node @ nil_node )
=> ( ! [E3: node,W12: list_node,W22: list_node] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons_node @ E3 @ W12 ) @ W22 ) )
=> ( ! [E3: node,W13: list_node,W23: list_node] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons_node @ E3 @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_178_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_179_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,Y2: node,Ys2: list_node] :
( ( P @ Xs2 @ ( cons_node @ Y2 @ Ys2 ) )
=> ( ( P @ ( cons_node @ X4 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ ( cons_node @ Y2 @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_180_min__list_Oinduct,axiom,
! [P: list_node > $o,A0: list_node] :
( ! [X4: node,Xs2: list_node] :
( ! [X212: node,X222: list_node] :
( ( Xs2
= ( cons_node @ X212 @ X222 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons_node @ X4 @ Xs2 ) ) )
=> ( ( P @ nil_node )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_181_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_182_induct__list012,axiom,
! [P: list_node > $o,Xs: list_node] :
( ( P @ nil_node )
=> ( ! [X4: node] : ( P @ ( cons_node @ X4 @ nil_node ) )
=> ( ! [X4: node,Y2: node,Zs2: list_node] :
( ( P @ Zs2 )
=> ( ( P @ ( cons_node @ Y2 @ Zs2 ) )
=> ( P @ ( cons_node @ X4 @ ( cons_node @ Y2 @ Zs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_183_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,Ys2: list_node] :
( ( P @ Ys2 @ Xs2 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_184_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 )
=> ( ! [Y2: node,Ys2: list_node] : ( P @ nil_node @ ( cons_node @ Y2 @ Ys2 ) )
=> ( ! [X4: node,Xs2: list_node,Y2: node,Ys2: list_node] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_node @ X4 @ Xs2 ) @ ( cons_node @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_185_neq__Nil__conv,axiom,
! [Xs: list_node] :
( ( Xs != nil_node )
= ( ? [Y3: node,Ys3: list_node] :
( Xs
= ( cons_node @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_186_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_187_list_Oexhaust,axiom,
! [Y: list_node] :
( ( Y != nil_node )
=> ~ ! [X213: node,X223: list_node] :
( Y
!= ( cons_node @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_188_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_189_list_OdiscI,axiom,
! [List: list_node,X21: node,X22: list_node] :
( ( List
= ( cons_node @ X21 @ X22 ) )
=> ( List != nil_node ) ) ).
% list.discI
thf(fact_190_neq__NilE,axiom,
! [L: list_node] :
( ( L != nil_node )
=> ~ ! [X4: node,Xs2: list_node] :
( L
!= ( cons_node @ X4 @ Xs2 ) ) ) ).
% neq_NilE
thf(fact_191_list_Odistinct_I1_J,axiom,
! [X21: node,X22: list_node] :
( nil_node
!= ( cons_node @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_192_list_Oset__cases,axiom,
! [E2: val,A: list_val] :
( ( member_val @ E2 @ ( set_val2 @ A ) )
=> ( ! [Z2: list_val] :
( A
!= ( cons_val @ E2 @ Z2 ) )
=> ~ ! [Z1: val,Z2: list_val] :
( ( A
= ( cons_val @ Z1 @ Z2 ) )
=> ~ ( member_val @ E2 @ ( set_val2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_193_list_Oset__cases,axiom,
! [E2: node,A: list_node] :
( ( member_node @ E2 @ ( set_node2 @ A ) )
=> ( ! [Z2: list_node] :
( A
!= ( cons_node @ E2 @ Z2 ) )
=> ~ ! [Z1: node,Z2: list_node] :
( ( A
= ( cons_node @ Z1 @ Z2 ) )
=> ~ ( member_node @ E2 @ ( set_node2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_194_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_195_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_196_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_197_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_198_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_199_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_200_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_201_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_202_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_203_eq__Nil__appendI,axiom,
! [Xs: list_node,Ys: list_node] :
( ( Xs = Ys )
=> ( Xs
= ( append_node @ nil_node @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_204_append__Nil,axiom,
! [Ys: list_node] :
( ( append_node @ nil_node @ Ys )
= Ys ) ).
% append_Nil
thf(fact_205_append_Oleft__neutral,axiom,
! [A: list_node] :
( ( append_node @ nil_node @ A )
= A ) ).
% append.left_neutral
thf(fact_206_list_Osel_I3_J,axiom,
! [X21: node,X22: list_node] :
( ( tl_node @ ( cons_node @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_207_list_Osel_I2_J,axiom,
( ( tl_node @ nil_node )
= nil_node ) ).
% list.sel(2)
thf(fact_208_list__append__eq__Cons__cases,axiom,
! [Ys: list_node,Zs: list_node,X: node,Xs: list_node] :
( ( ( append_node @ Ys @ Zs )
= ( cons_node @ X @ Xs ) )
=> ( ( ( Ys = nil_node )
=> ( Zs
!= ( cons_node @ X @ Xs ) ) )
=> ~ ! [Ys4: list_node] :
( ( Ys
= ( cons_node @ X @ Ys4 ) )
=> ( ( append_node @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_209_list__Cons__eq__append__cases,axiom,
! [X: node,Xs: list_node,Ys: list_node,Zs: list_node] :
( ( ( cons_node @ X @ Xs )
= ( append_node @ Ys @ Zs ) )
=> ( ( ( Ys = nil_node )
=> ( Zs
!= ( cons_node @ X @ Xs ) ) )
=> ~ ! [Ys4: list_node] :
( ( Ys
= ( cons_node @ X @ Ys4 ) )
=> ( ( append_node @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_210_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,Y2: node] : ( P @ ( cons_node @ X4 @ nil_node ) @ ( cons_node @ Y2 @ nil_node ) )
=> ( ! [X4: node,Xs2: list_node,Y2: node] :
( ( Xs2 != nil_node )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) @ ( cons_node @ Y2 @ nil_node ) ) )
=> ( ! [X4: node,Y2: node,Ys2: list_node] :
( ( Ys2 != nil_node )
=> ( P @ ( cons_node @ X4 @ nil_node ) @ ( append_node @ Ys2 @ ( cons_node @ Y2 @ nil_node ) ) ) )
=> ( ! [X4: node,Xs2: list_node,Y2: node,Ys2: list_node] :
( ( P @ Xs2 @ Ys2 )
=> ( ( Xs2 != nil_node )
=> ( ( Ys2 != nil_node )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) @ ( append_node @ Ys2 @ ( cons_node @ Y2 @ nil_node ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_211_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_212_append__eq__Cons__conv,axiom,
! [Ys: list_node,Zs: list_node,X: node,Xs: list_node] :
( ( ( append_node @ Ys @ Zs )
= ( cons_node @ X @ Xs ) )
= ( ( ( Ys = nil_node )
& ( Zs
= ( cons_node @ X @ Xs ) ) )
| ? [Ys5: list_node] :
( ( Ys
= ( cons_node @ X @ Ys5 ) )
& ( ( append_node @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_213_Cons__eq__append__conv,axiom,
! [X: node,Xs: list_node,Ys: list_node,Zs: list_node] :
( ( ( cons_node @ X @ Xs )
= ( append_node @ Ys @ Zs ) )
= ( ( ( Ys = nil_node )
& ( ( cons_node @ X @ Xs )
= Zs ) )
| ? [Ys5: list_node] :
( ( ( cons_node @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_node @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_214_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_215_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 )
=> ( ! [Y2: node,Ys2: list_node] : ( P @ nil_node @ ( append_node @ Ys2 @ ( cons_node @ Y2 @ nil_node ) ) )
=> ( ! [X4: node,Xs2: list_node,Y2: node,Ys2: list_node] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) @ ( append_node @ Ys2 @ ( cons_node @ Y2 @ nil_node ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_216_neq__Nil__revE,axiom,
! [L: list_node] :
( ( L != nil_node )
=> ~ ! [Ll: list_node,E3: node] :
( L
!= ( append_node @ Ll @ ( cons_node @ E3 @ nil_node ) ) ) ) ).
% neq_Nil_revE
thf(fact_217_rev__exhaust,axiom,
! [Xs: list_node] :
( ( Xs != nil_node )
=> ~ ! [Ys2: list_node,Y2: node] :
( Xs
!= ( append_node @ Ys2 @ ( cons_node @ Y2 @ nil_node ) ) ) ) ).
% rev_exhaust
thf(fact_218_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_219_split__list__first__prop__iff,axiom,
! [Xs: list_node,P: node > $o] :
( ( ? [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_node,X3: node] :
( ? [Zs3: list_node] :
( Xs
= ( append_node @ Ys3 @ ( cons_node @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: node] :
( ( member_node @ Y3 @ ( set_node2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_220_split__list__last__prop__iff,axiom,
! [Xs: list_node,P: node > $o] :
( ( ? [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_node,X3: node,Zs3: list_node] :
( ( Xs
= ( append_node @ Ys3 @ ( cons_node @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: node] :
( ( member_node @ Y3 @ ( set_node2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_221_in__set__conv__decomp__first,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
= ( ? [Ys3: list_val,Zs3: list_val] :
( ( Xs
= ( append_val @ Ys3 @ ( cons_val @ X @ Zs3 ) ) )
& ~ ( member_val @ X @ ( set_val2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_222_in__set__conv__decomp__first,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
= ( ? [Ys3: list_node,Zs3: list_node] :
( ( Xs
= ( append_node @ Ys3 @ ( cons_node @ X @ Zs3 ) ) )
& ~ ( member_node @ X @ ( set_node2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_223_in__set__conv__decomp__last,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
= ( ? [Ys3: list_val,Zs3: list_val] :
( ( Xs
= ( append_val @ Ys3 @ ( cons_val @ X @ Zs3 ) ) )
& ~ ( member_val @ X @ ( set_val2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_224_in__set__conv__decomp__last,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
= ( ? [Ys3: list_node,Zs3: list_node] :
( ( Xs
= ( append_node @ Ys3 @ ( cons_node @ X @ Zs3 ) ) )
& ~ ( member_node @ X @ ( set_node2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_225_split__list__first__propE,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys2: list_node,X4: node] :
( ? [Zs2: list_node] :
( Xs
= ( append_node @ Ys2 @ ( cons_node @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: node] :
( ( member_node @ Xa @ ( set_node2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_226_split__list__last__propE,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys2: list_node,X4: node,Zs2: list_node] :
( ( Xs
= ( append_node @ Ys2 @ ( cons_node @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: node] :
( ( member_node @ Xa @ ( set_node2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_227_split__list__first__prop,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys2: list_node,X4: node] :
( ? [Zs2: list_node] :
( Xs
= ( append_node @ Ys2 @ ( cons_node @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa: node] :
( ( member_node @ Xa @ ( set_node2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_228_split__list__last__prop,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys2: list_node,X4: node,Zs2: list_node] :
( ( Xs
= ( append_node @ Ys2 @ ( cons_node @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa: node] :
( ( member_node @ Xa @ ( set_node2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_229_in__set__list__format,axiom,
! [E2: val,L: list_val] :
( ( member_val @ E2 @ ( set_val2 @ L ) )
=> ~ ! [L12: list_val,L22: list_val] :
( L
!= ( append_val @ L12 @ ( cons_val @ E2 @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_230_in__set__list__format,axiom,
! [E2: node,L: list_node] :
( ( member_node @ E2 @ ( set_node2 @ L ) )
=> ~ ! [L12: list_node,L22: list_node] :
( L
!= ( append_node @ L12 @ ( cons_node @ E2 @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_231_in__set__conv__decomp,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
= ( ? [Ys3: list_val,Zs3: list_val] :
( Xs
= ( append_val @ Ys3 @ ( cons_val @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_232_in__set__conv__decomp,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
= ( ? [Ys3: list_node,Zs3: list_node] :
( Xs
= ( append_node @ Ys3 @ ( cons_node @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_233_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_234_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_235_split__list__propE,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys2: list_node,X4: node] :
( ? [Zs2: list_node] :
( Xs
= ( append_node @ Ys2 @ ( cons_node @ X4 @ Zs2 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_236_split__list__first,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ? [Ys2: list_val,Zs2: list_val] :
( ( Xs
= ( append_val @ Ys2 @ ( cons_val @ X @ Zs2 ) ) )
& ~ ( member_val @ X @ ( set_val2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_237_split__list__first,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ? [Ys2: list_node,Zs2: list_node] :
( ( Xs
= ( append_node @ Ys2 @ ( cons_node @ X @ Zs2 ) ) )
& ~ ( member_node @ X @ ( set_node2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_238_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_239_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_240_split__list__prop,axiom,
! [Xs: list_node,P: node > $o] :
( ? [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys2: list_node,X4: node] :
( ? [Zs2: list_node] :
( Xs
= ( append_node @ Ys2 @ ( cons_node @ X4 @ Zs2 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_241_split__list__last,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ? [Ys2: list_val,Zs2: list_val] :
( ( Xs
= ( append_val @ Ys2 @ ( cons_val @ X @ Zs2 ) ) )
& ~ ( member_val @ X @ ( set_val2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_242_split__list__last,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ? [Ys2: list_node,Zs2: list_node] :
( ( Xs
= ( append_node @ Ys2 @ ( cons_node @ X @ Zs2 ) ) )
& ~ ( member_node @ X @ ( set_node2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_243_split__list,axiom,
! [X: val,Xs: list_val] :
( ( member_val @ X @ ( set_val2 @ Xs ) )
=> ? [Ys2: list_val,Zs2: list_val] :
( Xs
= ( append_val @ Ys2 @ ( cons_val @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_244_split__list,axiom,
! [X: node,Xs: list_node] :
( ( member_node @ X @ ( set_node2 @ Xs ) )
=> ? [Ys2: list_node,Zs2: list_node] :
( Xs
= ( append_node @ Ys2 @ ( cons_node @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_245_tl__obtain__elem,axiom,
! [Xs: list_node] :
( ( Xs != nil_node )
=> ( ( ( tl_node @ Xs )
= nil_node )
=> ~ ! [E3: node] :
( Xs
!= ( cons_node @ E3 @ nil_node ) ) ) ) ).
% tl_obtain_elem
thf(fact_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_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 ) )
=> ~ ! [Ns3: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N @ Ns3 @ N )
=> ( ( member_node @ N2 @ ( set_node2 @ Ns3 ) )
=> ( ~ ( member_node @ N @ ( set_node2 @ ( tl_node @ ( butlast_node @ Ns3 ) ) ) )
=> ~ ( ord_less_eq_set_node @ ( set_node2 @ Ns3 ) @ ( set_node2 @ Ns ) ) ) ) ) ) ) ).
% old.path2_simple_loop
thf(fact_254_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_255_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_256_phiUses__finite,axiom,
! [N: node,G: g] :
( ( member_node @ N @ ( set_node2 @ ( alpha_n @ G ) ) )
=> ( finite_finite_val @ ( sSA_CF848637139eD_val @ alpha_n @ inEdges @ phis @ G @ N ) ) ) ).
% phiUses_finite
thf(fact_257_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,Ns3: list_node] :
( ( graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ G @ N3 @ Ns3 @ M )
=> ( ( P @ N3 )
=> ( ! [X2: node] :
( ( member_node @ X2 @ ( set_node2 @ ( tl_node @ Ns3 ) ) )
=> ~ ( P @ X2 ) )
=> ~ ( suffix_node @ Ns3 @ Ns ) ) ) ) ) ) ).
% old.path2_split_last_prop
thf(fact_258_defs__finite,axiom,
! [G: g,N: node] : ( finite_finite_val @ ( defs @ G @ N ) ) ).
% defs_finite
thf(fact_259_prefix__order_Oorder__refl,axiom,
! [X: list_node] : ( prefix_node @ X @ X ) ).
% prefix_order.order_refl
thf(fact_260_prefix__order_Odual__order_Orefl,axiom,
! [A: list_node] : ( prefix_node @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_261_suffix__order_Oorder__refl,axiom,
! [X: list_node] : ( suffix_node @ X @ X ) ).
% suffix_order.order_refl
thf(fact_262_suffix__order_Odual__order_Orefl,axiom,
! [A: list_node] : ( suffix_node @ A @ A ) ).
% suffix_order.dual_order.refl
thf(fact_263_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_264_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_265_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_266_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_267_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_268_List_Ofinite__set,axiom,
! [Xs: list_node] : ( finite_finite_node @ ( set_node2 @ Xs ) ) ).
% List.finite_set
thf(fact_269_List_Ofinite__set,axiom,
! [Xs: list_val] : ( finite_finite_val @ ( set_val2 @ Xs ) ) ).
% List.finite_set
thf(fact_270_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_271_prefix__code_I1_J,axiom,
! [Xs: list_node] : ( prefix_node @ nil_node @ Xs ) ).
% prefix_code(1)
thf(fact_272_prefix__Nil,axiom,
! [Xs: list_node] :
( ( prefix_node @ Xs @ nil_node )
= ( Xs = nil_node ) ) ).
% prefix_Nil
thf(fact_273_prefix__bot_Obot_Oextremum__unique,axiom,
! [A: list_node] :
( ( prefix_node @ A @ nil_node )
= ( A = nil_node ) ) ).
% prefix_bot.bot.extremum_unique
thf(fact_274_suffix__bot_Obot_Oextremum__unique,axiom,
! [A: list_node] :
( ( suffix_node @ A @ nil_node )
= ( A = nil_node ) ) ).
% suffix_bot.bot.extremum_unique
thf(fact_275_suffix__Nil,axiom,
! [Xs: list_node] :
( ( suffix_node @ Xs @ nil_node )
= ( Xs = nil_node ) ) ).
% suffix_Nil
thf(fact_276_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_277_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_278_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_279_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_280_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_281_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_282_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 ) ) ) )
=> ( ! [Ns4: list_node,N3: node] :
( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ Ns4 )
=> ( ( P @ Ns4 )
=> ( ( member_node @ N3 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ ( hd_node @ Ns4 ) ) ) )
=> ( P @ ( cons_node @ N3 @ Ns4 ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% old.path.inducts
thf(fact_283_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 ) ) )
=> ~ ! [Ns4: list_node,N3: node] :
( ( A
= ( cons_node @ N3 @ Ns4 ) )
=> ( ( graph_435229452_edgeD @ alpha_n @ invar @ inEdges @ G @ Ns4 )
=> ~ ( member_node @ N3 @ ( set_node2 @ ( graph_272749361_edgeD @ inEdges @ G @ ( hd_node @ Ns4 ) ) ) ) ) ) ) ) ).
% old.path.cases
thf(fact_284_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_285_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_286_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_287_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_288_hd__Cons__tl,axiom,
! [Xs: list_node] :
( ( Xs != nil_node )
=> ( ( cons_node @ ( hd_node @ Xs ) @ ( tl_node @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_289_list_Ocollapse,axiom,
! [List: list_node] :
( ( List != nil_node )
=> ( ( cons_node @ ( hd_node @ List ) @ ( tl_node @ List ) )
= List ) ) ).
% list.collapse
thf(fact_290_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_291_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_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_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_296_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_297_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_298_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_299_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_300_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_301_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_302_suffix__bot_Obot_Oextremum__uniqueI,axiom,
! [A: list_node] :
( ( suffix_node @ A @ nil_node )
=> ( A = nil_node ) ) ).
% suffix_bot.bot.extremum_uniqueI
thf(fact_303_suffix__bot_Obot_Oextremum,axiom,
! [A: list_node] : ( suffix_node @ nil_node @ A ) ).
% suffix_bot.bot.extremum
thf(fact_304_Nil__suffix,axiom,
! [Xs: list_node] : ( suffix_node @ nil_node @ Xs ) ).
% Nil_suffix
thf(fact_305_suffixE,axiom,
! [Xs: list_node,Ys: list_node] :
( ( suffix_node @ Xs @ Ys )
=> ~ ! [Zs2: list_node] :
( Ys
!= ( append_node @ Zs2 @ Xs ) ) ) ).
% suffixE
thf(fact_306_suffixI,axiom,
! [Ys: list_node,Zs: list_node,Xs: list_node] :
( ( Ys
= ( append_node @ Zs @ Xs ) )
=> ( suffix_node @ Xs @ Ys ) ) ).
% suffixI
thf(fact_307_Sublist_Osuffix__def,axiom,
( suffix_node
= ( ^ [Xs3: list_node,Ys3: list_node] :
? [Zs3: list_node] :
( Ys3
= ( append_node @ Zs3 @ Xs3 ) ) ) ) ).
% Sublist.suffix_def
thf(fact_308_suffix__append,axiom,
! [Xs: list_node,Ys: list_node,Zs: list_node] :
( ( suffix_node @ Xs @ ( append_node @ Ys @ Zs ) )
= ( ( suffix_node @ Xs @ Zs )
| ? [Xs5: list_node] :
( ( Xs
= ( append_node @ Xs5 @ Zs ) )
& ( suffix_node @ Xs5 @ Ys ) ) ) ) ).
% suffix_append
thf(fact_309_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_310_suffix__appendI,axiom,
! [Xs: list_node,Ys: list_node,Zs: list_node] :
( ( suffix_node @ Xs @ Ys )
=> ( suffix_node @ Xs @ ( append_node @ Zs @ Ys ) ) ) ).
% suffix_appendI
thf(fact_311_suffix__tl,axiom,
! [Xs: list_node] : ( suffix_node @ ( tl_node @ Xs ) @ Xs ) ).
% suffix_tl
thf(fact_312_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_313_suffix__order_Oeq__iff,axiom,
( ( ^ [Y4: list_node,Z: list_node] : Y4 = Z )
= ( ^ [X3: list_node,Y3: list_node] :
( ( suffix_node @ X3 @ Y3 )
& ( suffix_node @ Y3 @ X3 ) ) ) ) ).
% suffix_order.eq_iff
thf(fact_314_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_315_suffix__order_Oeq__refl,axiom,
! [X: list_node,Y: list_node] :
( ( X = Y )
=> ( suffix_node @ X @ Y ) ) ).
% suffix_order.eq_refl
thf(fact_316_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_317_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_318_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_319_suffix__order_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: list_node,Z: list_node] : Y4 = Z )
= ( ^ [A4: list_node,B3: list_node] :
( ( suffix_node @ A4 @ B3 )
& ( suffix_node @ B3 @ A4 ) ) ) ) ).
% suffix_order.order.eq_iff
thf(fact_320_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_321_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_322_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_323_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_324_suffix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: list_node,Z: list_node] : Y4 = Z )
= ( ^ [A4: list_node,B3: list_node] :
( ( suffix_node @ B3 @ A4 )
& ( suffix_node @ A4 @ B3 ) ) ) ) ).
% suffix_order.dual_order.eq_iff
thf(fact_325_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_326_list_Osel_I1_J,axiom,
! [X21: node,X22: list_node] :
( ( hd_node @ ( cons_node @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_327_finite__list,axiom,
! [A2: set_node] :
( ( finite_finite_node @ A2 )
=> ? [Xs2: list_node] :
( ( set_node2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_328_finite__list,axiom,
! [A2: set_val] :
( ( finite_finite_val @ A2 )
=> ? [Xs2: list_val] :
( ( set_val2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_329_subset__code_I1_J,axiom,
! [Xs: list_val,B4: set_val] :
( ( ord_less_eq_set_val @ ( set_val2 @ Xs ) @ B4 )
= ( ! [X3: val] :
( ( member_val @ X3 @ ( set_val2 @ Xs ) )
=> ( member_val @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_330_subset__code_I1_J,axiom,
! [Xs: list_node,B4: set_node] :
( ( ord_less_eq_set_node @ ( set_node2 @ Xs ) @ B4 )
= ( ! [X3: node] :
( ( member_node @ X3 @ ( set_node2 @ Xs ) )
=> ( member_node @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_331_set__mono__prefix,axiom,
! [Xs: list_node,Ys: list_node] :
( ( prefix_node @ Xs @ Ys )
=> ( ord_less_eq_set_node @ ( set_node2 @ Xs ) @ ( set_node2 @ Ys ) ) ) ).
% set_mono_prefix
thf(fact_332_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_333_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_334_hd__in__set,axiom,
! [Xs: list_val] :
( ( Xs != nil_val )
=> ( member_val @ ( hd_val @ Xs ) @ ( set_val2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_335_hd__in__set,axiom,
! [Xs: list_node] :
( ( Xs != nil_node )
=> ( member_node @ ( hd_node @ Xs ) @ ( set_node2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_336_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_337_longest__common__prefix,axiom,
! [Xs: list_node,Ys: list_node] :
? [Ps: list_node,Xs6: list_node,Ys4: list_node] :
( ( Xs
= ( append_node @ Ps @ Xs6 ) )
& ( Ys
= ( append_node @ Ps @ Ys4 ) )
& ( ( Xs6 = nil_node )
| ( Ys4 = nil_node )
| ( ( hd_node @ Xs6 )
!= ( hd_node @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_338_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_339_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,Y2: node,Ys2: list_node] :
( ( X4 != Y2 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) @ ( append_node @ Ys2 @ ( cons_node @ Y2 @ nil_node ) ) ) )
=> ( ! [X4: node,Xs2: list_node,Y2: node,Ys2: list_node] :
( ( X4 = Y2 )
=> ( ~ ( suffix_node @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_node @ Xs2 @ ( cons_node @ X4 @ nil_node ) ) @ ( append_node @ Ys2 @ ( cons_node @ Y2 @ nil_node ) ) ) ) ) )
=> ( P @ Ps2 @ Ls ) ) ) ) ) ).
% not_suffix_induct
thf(fact_340_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_341_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_342_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_343_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_344_list_Oexhaust__sel,axiom,
! [List: list_node] :
( ( List != nil_node )
=> ( List
= ( cons_node @ ( hd_node @ List ) @ ( tl_node @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_345_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_346_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_347_prefix__same__cases,axiom,
! [Xs_1: list_node,Ys: list_node,Xs_2: list_node] :
( ( prefix_node @ Xs_1 @ Ys )
=> ( ( prefix_node @ Xs_2 @ Ys )
=> ( ( prefix_node @ Xs_1 @ Xs_2 )
| ( prefix_node @ Xs_2 @ Xs_1 ) ) ) ) ).
% prefix_same_cases
thf(fact_348_prefix__order_Oeq__iff,axiom,
( ( ^ [Y4: list_node,Z: list_node] : Y4 = Z )
= ( ^ [X3: list_node,Y3: list_node] :
( ( prefix_node @ X3 @ Y3 )
& ( prefix_node @ Y3 @ X3 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_349_prefix__order_Oantisym,axiom,
! [X: list_node,Y: list_node] :
( ( prefix_node @ X @ Y )
=> ( ( prefix_node @ Y @ X )
=> ( X = Y ) ) ) ).
% prefix_order.antisym
% Conjectures (1)
thf(conj_0,conjecture,
graph_1012773594_edgeD @ alpha_n @ invar @ inEdges @ g2 @ n @ ( append_node @ ns @ ( tl_node @ ri ) ) @ ( sSA_CF551432799de_val @ alpha_n @ defs @ phis @ g2 @ phi_r ) ).
%------------------------------------------------------------------------------