TPTP Problem File: ITP082^2.p
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%------------------------------------------------------------------------------
% File : ITP082^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Irreducible problem prob_559__6628256_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Irreducible/prob_559__6628256_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 315 ( 104 unt; 51 typ; 0 def)
% Number of atoms : 779 ( 365 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 5240 ( 161 ~; 16 |; 66 &;4591 @)
% ( 0 <=>; 406 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 10 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 232 ( 232 >; 0 *; 0 +; 0 <<)
% Number of symbols : 48 ( 47 usr; 16 con; 0-12 aty)
% Number of variables : 1147 ( 16 ^;1034 !; 60 ?;1147 :)
% ( 37 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:27:26.217
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_edgeD,type,
edgeD: $tType ).
thf(ty_tf_node,type,
node: $tType ).
thf(ty_tf_val,type,
val: $tType ).
thf(ty_tf_g,type,
g: $tType ).
% Explicit typings (43)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_c_Graph__path_Ograph__path__base_Opath,type,
graph_1146196722h_path:
!>[G: $tType,Node: $tType,EdgeD: $tType] : ( ( G > ( list @ Node ) ) > ( G > $o ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > G > ( list @ Node ) > $o ) ).
thf(sy_c_Graph__path_Ograph__path__base_Opath2,type,
graph_1661282752_path2:
!>[G: $tType,Node: $tType,EdgeD: $tType] : ( ( G > ( list @ Node ) ) > ( G > $o ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > G > Node > ( list @ Node ) > Node > $o ) ).
thf(sy_c_Graph__path_Ograph__path__base_Opredecessors,type,
graph_1201503639essors:
!>[G: $tType,Node: $tType,EdgeD: $tType] : ( ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > G > Node > ( list @ Node ) ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Minimality_Ograph__path_OpathsConverge,type,
graph_pathsConverge:
!>[G: $tType,Node: $tType,EdgeD: $tType] : ( ( G > ( list @ Node ) ) > ( G > $o ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > G > Node > ( list @ Node ) > Node > ( list @ Node ) > Node > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__base_OphiUses,type,
sSA_CFG_SSA_phiUses:
!>[G: $tType,Node: $tType,EdgeD: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( list @ ( product_prod @ Node @ EdgeD ) ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Node > ( set @ Val ) ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_OdefNode,type,
sSA_CF1081484811efNode:
!>[G: $tType,Node: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Val > Node ) ).
thf(sy_c_SSA__CFG_OCFG__SSA__wf__base_OphiArg,type,
sSA_CF1165125185phiArg:
!>[G: $tType,Node: $tType,Val: $tType] : ( ( G > ( list @ Node ) ) > ( G > Node > ( set @ Val ) ) > ( G > ( product_prod @ Node @ Val ) > ( option @ ( list @ Val ) ) ) > G > Val > Val > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Sublist_Oprefix,type,
prefix:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_Sublist_Osuffix,type,
suffix:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $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 @ ( product_prod @ node @ 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 > ( product_prod @ node @ val ) > ( option @ ( list @ val ) ) ).
thf(sy_v_pred_092_060_094sub_062_092_060phi_062_092_060_094sub_062r____,type,
pred_phi_r: node ).
thf(sy_v_r,type,
r: val ).
thf(sy_v_ri____,type,
ri: list @ node ).
thf(sy_v_rs_H____,type,
rs: list @ node ).
thf(sy_v_rs_H__rest____,type,
rs_rest: list @ node ).
thf(sy_v_rs____,type,
rs2: list @ node ).
thf(sy_v_s,type,
s: val ).
thf(sy_v_tmp____,type,
tmp: list @ node ).
% Relevant facts (255)
thf(fact_0_m__i__differ_I2_J,axiom,
m != i ).
% m_i_differ(2)
thf(fact_1_rs_H__rest__def,axiom,
( rs
= ( append @ node @ tmp @ ( cons @ node @ i @ rs_rest ) ) ) ).
% rs'_rest_def
thf(fact_2_old_Oinvar,axiom,
! [G2: g] : ( invar @ G2 ) ).
% old.invar
thf(fact_3_old_Opath2__split_I2_J,axiom,
! [G2: g,N: node,Ns: list @ node,N2: node,Ns2: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ Ns @ ( cons @ node @ N2 @ Ns2 ) ) @ M )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N2 @ ( cons @ node @ N2 @ Ns2 ) @ M ) ) ).
% old.path2_split(2)
thf(fact_4_False,axiom,
r != phi_r ).
% False
thf(fact_5_rs_H__props_I1_J,axiom,
graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ r ) @ rs @ pred_phi_r ).
% rs'_props(1)
thf(fact_6_ms_H__props_I1_J,axiom,
graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ m @ ms2 @ i ).
% ms'_props(1)
thf(fact_7_rs_H__rest__prop,axiom,
( rs
= ( append @ node @ ri @ rs_rest ) ) ).
% rs'_rest_prop
thf(fact_8_rs_H__props_I3_J,axiom,
member @ val @ r @ ( sSA_CFG_SSA_phiUses @ g @ node @ edgeD @ val @ alpha_n @ inEdges @ phis @ g2 @ pred_phi_r ) ).
% rs'_props(3)
thf(fact_9_old_Opath2__not__Nil,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( Ns
!= ( nil @ node ) ) ) ).
% old.path2_not_Nil
thf(fact_10_old_Opath2__not__Nil2,axiom,
! [G2: g,N: node,M: node] :
~ ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( nil @ node ) @ M ) ).
% old.path2_not_Nil2
thf(fact_11_old_Opath2__hd__in___092_060alpha_062n,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ).
% old.path2_hd_in_\<alpha>n
thf(fact_12_old_Opath2__hd__in__ns,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( member @ node @ N @ ( set2 @ node @ Ns ) ) ) ).
% old.path2_hd_in_ns
thf(fact_13_old_Opath2__in___092_060alpha_062n,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,L: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( member @ node @ L @ ( set2 @ node @ Ns ) )
=> ( member @ node @ L @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ) ).
% old.path2_in_\<alpha>n
thf(fact_14_old_Opath2__last__in__ns,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( member @ node @ M @ ( set2 @ node @ Ns ) ) ) ).
% old.path2_last_in_ns
thf(fact_15_old_Opath2__tl__in___092_060alpha_062n,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( member @ node @ M @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ).
% old.path2_tl_in_\<alpha>n
thf(fact_16_old_Opath2__forget__hd,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( hd @ node @ Ns ) @ Ns @ M ) ) ).
% old.path2_forget_hd
thf(fact_17_old_Opath2__hd,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( N
= ( hd @ node @ Ns ) ) ) ).
% old.path2_hd
thf(fact_18__092_060open_062ri_A_061_Atmp_A_064_A_091i_093_092_060close_062,axiom,
( ri
= ( append @ node @ tmp @ ( cons @ node @ i @ ( nil @ node ) ) ) ) ).
% \<open>ri = tmp @ [i]\<close>
thf(fact_19__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062rs_H__rest_O_Ars_H_A_061_Ari_A_064_Ars_H__rest_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Rs_rest: list @ node] :
( rs
!= ( append @ node @ ri @ Rs_rest ) ) ).
% \<open>\<And>thesis. (\<And>rs'_rest. rs' = ri @ rs'_rest \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_20_ri__props_I2_J,axiom,
member @ node @ i @ ( set2 @ node @ ms ) ).
% ri_props(2)
thf(fact_21_m__i__differ_I1_J,axiom,
( i
!= ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r ) ) ).
% m_i_differ(1)
thf(fact_22__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062tmp_O_Ari_A_061_Atmp_A_064_A_091i_093_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Tmp: list @ node] :
( ri
!= ( append @ node @ Tmp @ ( cons @ node @ i @ ( nil @ node ) ) ) ) ).
% \<open>\<And>thesis. (\<And>tmp. ri = tmp @ [i] \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_23_ri__props_I1_J,axiom,
graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ r ) @ ri @ i ).
% ri_props(1)
thf(fact_24_old_Opath2__split_I1_J,axiom,
! [G2: g,N: node,Ns: list @ node,N2: node,Ns2: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ Ns @ ( cons @ node @ N2 @ Ns2 ) ) @ M )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ Ns @ ( cons @ node @ N2 @ ( nil @ node ) ) ) @ N2 ) ) ).
% old.path2_split(1)
thf(fact_25__092_060open_062defNode_Ag_A_092_060phi_062_092_060_094sub_062r_A_092_060noteq_062_AdefNode_Ag_Ar_092_060close_062,axiom,
( ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r )
!= ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ r ) ) ).
% \<open>defNode g \<phi>\<^sub>r \<noteq> defNode g r\<close>
thf(fact_26__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062i_A_061_AdefNode_Ag_A_092_060phi_062_092_060_094sub_062r_A_092_060Longrightarrow_062_Athesis_059_A_092_060lbrakk_062i_A_092_060noteq_062_AdefNode_Ag_A_092_060phi_062_092_060_094sub_062r_059_Am_A_061_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_059_A_092_060lbrakk_062i_A_092_060noteq_062_AdefNode_Ag_A_092_060phi_062_092_060_094sub_062r_059_Am_A_092_060noteq_062_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
( ( i
!= ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r ) )
=> ( ( ( i
!= ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r ) )
=> ( m != i ) )
=> ~ ( ( i
!= ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r ) )
=> ( m = i ) ) ) ) ).
% \<open>\<And>thesis. \<lbrakk>i = defNode g \<phi>\<^sub>r \<Longrightarrow> thesis; \<lbrakk>i \<noteq> defNode g \<phi>\<^sub>r; m = i\<rbrakk> \<Longrightarrow> thesis; \<lbrakk>i \<noteq> defNode g \<phi>\<^sub>r; m \<noteq> i\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_27_old_Oempty__path2,axiom,
! [N: node,G2: g] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( invar @ G2 )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( cons @ node @ N @ ( nil @ node ) ) @ N ) ) ) ).
% old.empty_path2
thf(fact_28_ms_H__props_I3_J,axiom,
~ ( member @ node @ i @ ( set2 @ node @ ( butlast @ node @ ms2 ) ) ) ).
% ms'_props(3)
thf(fact_29_assms_I10_J,axiom,
sSA_CF1165125185phiArg @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r @ r ).
% assms(10)
thf(fact_30_rs_H__props_I4_J,axiom,
member @ node @ pred_phi_r @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ g2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r ) ) ) ).
% rs'_props(4)
thf(fact_31_assms_I7_J,axiom,
graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ n @ ns @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ r ) ).
% assms(7)
thf(fact_32_rs__props_I1_J,axiom,
graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ r ) @ rs2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r ) ).
% rs_props(1)
thf(fact_33_rs__def,axiom,
( rs2
= ( append @ node @ rs @ ( cons @ node @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ phi_r ) @ ( nil @ node ) ) ) ) ).
% rs_def
thf(fact_34_assms_I8_J,axiom,
graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ m @ ms @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ s ) ).
% assms(8)
thf(fact_35_rs_H__loopfree,axiom,
~ ( member @ node @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ r ) @ ( set2 @ node @ ( tl @ node @ rs ) ) ) ).
% rs'_loopfree
thf(fact_36_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_37_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_38_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E2: A,L1: list @ A,E12: A,E22: A,L2: list @ A] :
( ( ( cons @ A @ E1 @ ( cons @ A @ E2 @ ( nil @ A ) ) )
= ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E22 @ L2 ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_39_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L1: list @ A,E12: A,E22: A,L2: list @ A,E1: A,E2: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E22 @ L2 ) ) )
= ( cons @ A @ E1 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_40_list__se__match_I1_J,axiom,
! [A: $tType,L1: list @ A,L2: list @ A,A2: A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L2 )
= ( cons @ A @ A2 @ ( nil @ A ) ) )
= ( ( L1
= ( cons @ A @ A2 @ ( nil @ A ) ) )
& ( L2
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_41_ri__props_I3_J,axiom,
! [X2: node] :
( ( member @ node @ X2 @ ( set2 @ node @ ( butlast @ node @ ri ) ) )
=> ~ ( member @ node @ X2 @ ( set2 @ node @ ms ) ) ) ).
% ri_props(3)
thf(fact_42_old_Oelem__set__implies__elem__tl__app__cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ ( append @ A @ Ys @ ( cons @ A @ Y @ Xs ) ) ) ) ) ) ).
% old.elem_set_implies_elem_tl_app_cons
thf(fact_43_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_44_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G2: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G2 @ X4 ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_49_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_50_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_51_append_Oassoc,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C: list @ A] :
( ( append @ A @ ( append @ A @ A2 @ B2 ) @ C )
= ( append @ A @ A2 @ ( append @ A @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_52_old_Opredecessor__is__node,axiom,
! [N: node,G2: g,N2: node] :
( ( member @ node @ N @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N2 ) ) )
=> ( ( invar @ G2 )
=> ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ) ).
% old.predecessor_is_node
thf(fact_53_old_Osuccessor__is__node,axiom,
! [N: node,G2: g,N2: node] :
( ( member @ node @ N @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N2 ) ) )
=> ( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( invar @ G2 )
=> ( member @ node @ N2 @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ) ) ).
% old.successor_is_node
thf(fact_54_old_Osuccessor__in___092_060alpha_062n,axiom,
! [G2: g,N: node] :
( ( ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N )
!= ( nil @ node ) )
=> ( ( invar @ G2 )
=> ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ) ).
% old.successor_in_\<alpha>n
thf(fact_55_old_Opath2__app_H,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,Ms: list @ node,L: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ M @ Ms @ L )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ ( butlast @ node @ Ns ) @ Ms ) @ L ) ) ) ).
% old.path2_app'
thf(fact_56_old_Opath2__app,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,Ms: list @ node,L: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ M @ Ms @ L )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ Ns @ ( tl @ node @ Ms ) ) @ L ) ) ) ).
% old.path2_app
thf(fact_57_append_Oright__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ A2 @ ( nil @ A ) )
= A2 ) ).
% append.right_neutral
thf(fact_58_empty__append__eq__id,axiom,
! [A: $tType] :
( ( append @ A @ ( nil @ A ) )
= ( ^ [X3: list @ A] : X3 ) ) ).
% empty_append_eq_id
thf(fact_59_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_60_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_61_self__append__conv2,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( append @ A @ Xs @ Ys ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_62_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_63_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_64_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_65_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_66_old_Opath2__split__ex_H,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,X: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( member @ node @ X @ ( set2 @ node @ Ns ) )
=> ~ ! [Ns_1: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns_1 @ X )
=> ! [Ns_2: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ X @ Ns_2 @ M )
=> ( Ns
!= ( append @ node @ ( butlast @ node @ Ns_1 ) @ Ns_2 ) ) ) ) ) ) ).
% old.path2_split_ex'
thf(fact_67_old_Opath2__induct,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,P: node > ( list @ node ) > node > $o] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( ( invar @ G2 )
=> ( P @ M @ ( cons @ node @ M @ ( nil @ node ) ) @ M ) )
=> ( ! [Ns3: list @ node,N3: node,N4: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N4 @ Ns3 @ M )
=> ( ( P @ N4 @ Ns3 @ M )
=> ( ( member @ node @ N3 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N4 ) ) )
=> ( P @ N3 @ ( cons @ node @ N3 @ Ns3 ) @ M ) ) ) )
=> ( P @ N @ Ns @ M ) ) ) ) ).
% old.path2_induct
thf(fact_68_old_Opath2__split__ex,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,X: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( member @ node @ X @ ( set2 @ node @ Ns ) )
=> ~ ! [Ns_1: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns_1 @ X )
=> ! [Ns_2: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ 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_69_old_Opath2__rev__induct,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,P: node > ( list @ node ) > node > $o] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( P @ N @ ( cons @ node @ N @ ( nil @ node ) ) @ N ) )
=> ( ! [Ns3: list @ node,M2: node,M3: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns3 @ M2 )
=> ( ( P @ N @ Ns3 @ M2 )
=> ( ( member @ node @ M2 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ M3 ) ) )
=> ( P @ N @ ( append @ node @ Ns3 @ ( cons @ node @ M3 @ ( nil @ node ) ) ) @ M3 ) ) ) )
=> ( P @ N @ Ns @ M ) ) ) ) ).
% old.path2_rev_induct
thf(fact_70_old_Opath2__snoc,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,M4: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( member @ node @ M @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ M4 ) ) )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ Ns @ ( cons @ node @ M4 @ ( nil @ node ) ) ) @ M4 ) ) ) ).
% old.path2_snoc
thf(fact_71_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L1: list @ A,E: A,L2: list @ A,E3: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E @ L2 ) )
= ( cons @ A @ E3 @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E = E3 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_72_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E3: A,L1: list @ A,E: A,L2: list @ A] :
( ( ( cons @ A @ E3 @ ( nil @ A ) )
= ( append @ A @ L1 @ ( cons @ A @ E @ L2 ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E = E3 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_73_list__se__match_I4_J,axiom,
! [A: $tType,L2: list @ A,A2: A,L1: list @ A] :
( ( L2
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A2 @ ( nil @ A ) )
= ( append @ A @ L1 @ L2 ) )
= ( ( L1
= ( nil @ A ) )
& ( L2
= ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_74_list__se__match_I3_J,axiom,
! [A: $tType,L1: list @ A,A2: A,L2: list @ A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A2 @ ( nil @ A ) )
= ( append @ A @ L1 @ L2 ) )
= ( ( L1
= ( cons @ A @ A2 @ ( nil @ A ) ) )
& ( L2
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_75_list__se__match_I2_J,axiom,
! [A: $tType,L2: list @ A,L1: list @ A,A2: A] :
( ( L2
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L2 )
= ( cons @ A @ A2 @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( L2
= ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_76_tl__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( tl @ A @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_77_old_Opath2__cases,axiom,
! [G2: g,N: node,Ns: list @ node,M: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( ( Ns
= ( cons @ node @ N @ ( nil @ node ) ) )
=> ( M != N ) )
=> ~ ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( hd @ node @ ( tl @ node @ Ns ) ) @ ( tl @ node @ Ns ) @ M )
=> ~ ( member @ node @ N @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ ( hd @ node @ ( tl @ node @ Ns ) ) ) ) ) ) ) ) ).
% old.path2_cases
thf(fact_78_rs__props_I3_J,axiom,
~ ( member @ node @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ r ) @ ( set2 @ node @ ( tl @ node @ rs2 ) ) ) ).
% rs_props(3)
thf(fact_79_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_80_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_81_hd__Cons__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_82_in__hd__or__tl__conv,axiom,
! [A: $tType,L: list @ A,X: A] :
( ( L
!= ( nil @ A ) )
=> ( ( ( X
= ( hd @ A @ L ) )
| ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ L ) ) ) )
= ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_83_ri__props_I4_J,axiom,
prefix @ node @ ri @ rs2 ).
% ri_props(4)
thf(fact_84_ms_H__props_I2_J,axiom,
prefix @ node @ ms2 @ ms ).
% ms'_props(2)
thf(fact_85_old_OCons__path2,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,N2: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( member @ node @ N2 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ N ) ) )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N2 @ ( cons @ node @ N2 @ Ns ) @ M ) ) ) ).
% old.Cons_path2
thf(fact_86_butlast__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% butlast_tl
thf(fact_87_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_88_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_89_in__set__butlastD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_90_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_91_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_92_tl__obtain__elem,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( tl @ A @ Xs )
= ( nil @ A ) )
=> ~ ! [E4: A] :
( Xs
!= ( cons @ A @ E4 @ ( nil @ A ) ) ) ) ) ).
% tl_obtain_elem
thf(fact_93_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X3: A] :
( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_94_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X3: A] :
( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_95_butlast__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_96_list_Oset__sel_I2_J,axiom,
! [A: $tType,A2: list @ A,X: A] :
( ( A2
!= ( nil @ A ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ A2 ) ) )
=> ( member @ A @ X @ ( set2 @ A @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_97_in__set__butlast__appendI,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
| ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ Ys ) ) ) )
=> ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ ( append @ A @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_98_list_Oexpand,axiom,
! [A: $tType,List: list @ A,List2: list @ A] :
( ( ( List
= ( nil @ A ) )
= ( List2
= ( nil @ A ) ) )
=> ( ( ( List
!= ( nil @ A ) )
=> ( ( List2
!= ( nil @ A ) )
=> ( ( ( hd @ A @ List )
= ( hd @ A @ List2 ) )
& ( ( tl @ A @ List )
= ( tl @ A @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_99_not__hd__in__tl,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( X
!= ( hd @ A @ Xs ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% not_hd_in_tl
thf(fact_100_butlast__eq__cons__conv,axiom,
! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X @ Xs ) )
= ( ? [Xl: A] :
( L
= ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl @ ( nil @ A ) ) ) ) ) ) ) ).
% butlast_eq_cons_conv
thf(fact_101_butlast__eq__consE,axiom,
! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X @ Xs ) )
=> ~ ! [Xl2: A] :
( L
!= ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl2 @ ( nil @ A ) ) ) ) ) ) ).
% butlast_eq_consE
thf(fact_102_list_Oexhaust__sel,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_103_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_104_list__tail__coinc,axiom,
! [A: $tType,N1: A,R1: list @ A,N22: A,R2: list @ A] :
( ( ( cons @ A @ N1 @ R1 )
= ( cons @ A @ N22 @ R2 ) )
=> ( ( N1 = N22 )
& ( R1 = R2 ) ) ) ).
% list_tail_coinc
thf(fact_105_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us ) )
& ( ( append @ A @ Us @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us )
= Zs )
& ( Ys
= ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_106_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us2: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us2 ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_107_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_108_neq__NilE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [X4: A,Xs2: list @ A] :
( L
!= ( cons @ A @ X4 @ Xs2 ) ) ) ).
% neq_NilE
thf(fact_109_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_110_revg_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [A4: A,As: list @ A,B3: list @ A] :
( ( P @ As @ ( cons @ A @ A4 @ B3 ) )
=> ( P @ ( cons @ A @ A4 @ As ) @ B3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% revg.induct
thf(fact_111_zipf_Oinduct,axiom,
! [A: $tType,C2: $tType,B: $tType,P: ( A > B > C2 ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B > C2,A1: list @ A,A22: list @ B] :
( ! [F2: A > B > C2] : ( P @ F2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [F2: A > B > C2,A4: A,As: list @ A,B3: B,Bs: list @ B] :
( ( P @ F2 @ As @ Bs )
=> ( P @ F2 @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B3 @ Bs ) ) )
=> ( ! [A4: A > B > C2,V: A,Va: list @ A] : ( P @ A4 @ ( cons @ A @ V @ Va ) @ ( nil @ B ) )
=> ( ! [A4: A > B > C2,V: B,Va: list @ B] : ( P @ A4 @ ( nil @ A ) @ ( cons @ B @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% zipf.induct
thf(fact_112_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_113_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X1: A,X23: list @ A] :
( ( P @ X23 )
=> ( P @ ( cons @ A @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_114_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y2: A,Ys2: list @ A] :
( Xs
= ( cons @ A @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_115_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X4: A,Xs2: list @ A] : ( P @ ( cons @ A @ X4 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys3 ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_116_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X4: A,Xs2: list @ A,Ys3: list @ A] :
( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons @ A @ X4 @ Xs2 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_117_induct__list012,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X4: A] : ( P @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Y3: A,Zs2: list @ A] :
( ( P @ Zs2 )
=> ( ( P @ ( cons @ A @ Y3 @ Zs2 ) )
=> ( P @ ( cons @ A @ X4 @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_118_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A] :
( ! [X4: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X4 @ Xs2 ) )
=> ( X
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_119_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ! [X4: A,Xs2: list @ A] :
( ! [X213: A,X223: list @ A] :
( ( Xs2
= ( cons @ A @ X213 @ X223 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons @ A @ X4 @ Xs2 ) ) )
=> ( ( P @ ( nil @ A ) )
=> ( P @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_120_shuffles_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ A ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
( ( P @ Xs2 @ ( cons @ A @ Y3 @ Ys3 ) )
=> ( ( P @ ( cons @ A @ X4 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_121_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W2: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [E4: A,W12: list @ A,W22: list @ B] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons @ A @ E4 @ W12 ) @ W22 ) )
=> ( ! [E4: B,W13: list @ A,W23: list @ B] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons @ B @ E4 @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_122_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X4: A] :
( X
!= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ~ ! [X4: A,Y3: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X4 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_123_sorted__wrt_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
=> ( ! [P2: A > A > $o,X4: A,Ys3: list @ A] :
( ( P @ P2 @ Ys3 )
=> ( P @ P2 @ ( cons @ A @ X4 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_124_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X4: A] : ( P @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Y3: A,Xs2: list @ A] :
( ( ( X4 = Y3 )
=> ( P @ ( cons @ A @ X4 @ Xs2 ) ) )
=> ( ( ( X4 != Y3 )
=> ( P @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X4 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_125_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X4: A] : ( P @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X1: A,X23: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_126_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
( ! [F2: A > B,X4: A] : ( P @ F2 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [F2: A > B,X4: A,Y3: A,Zs2: list @ A] :
( ( P @ F2 @ ( cons @ A @ Y3 @ Zs2 ) )
=> ( P @ F2 @ ( cons @ A @ X4 @ ( cons @ A @ Y3 @ Zs2 ) ) ) )
=> ( ! [A4: A > B] : ( P @ A4 @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_127_successively_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
=> ( ! [P2: A > A > $o,X4: A] : ( P @ P2 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [P2: A > A > $o,X4: A,Y3: A,Xs2: list @ A] :
( ( P @ P2 @ ( cons @ A @ Y3 @ Xs2 ) )
=> ( P @ P2 @ ( cons @ A @ X4 @ ( cons @ A @ Y3 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_128_list__all__zip_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B > $o,A1: list @ A,A22: list @ B] :
( ! [P2: A > B > $o] : ( P @ P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [P2: A > B > $o,A4: A,As: list @ A,B3: B,Bs: list @ B] :
( ( P @ P2 @ As @ Bs )
=> ( P @ P2 @ ( cons @ A @ A4 @ As ) @ ( cons @ B @ B3 @ Bs ) ) )
=> ( ! [P2: A > B > $o,V: A,Va: list @ A] : ( P @ P2 @ ( cons @ A @ V @ Va ) @ ( nil @ B ) )
=> ( ! [P2: A > B > $o,V: B,Va: list @ B] : ( P @ P2 @ ( nil @ A ) @ ( cons @ B @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).
% list_all_zip.induct
thf(fact_129_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X4: A] : ( P @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_130_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F2: A > B,X_1: list @ B] : ( P @ F2 @ ( nil @ A ) @ X_1 )
=> ( ! [F2: A > B,A4: A,As: list @ A,Bs: list @ B] :
( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A4 ) @ Bs ) )
=> ( P @ F2 @ ( cons @ A @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_131_mergesort__by__rel__merge_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A,A22: list @ A] :
( ! [R: A > A > $o,X4: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
( ( ( R @ X4 @ Y3 )
=> ( P @ R @ Xs2 @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ( ~ ( R @ X4 @ Y3 )
=> ( P @ R @ ( cons @ A @ X4 @ Xs2 ) @ Ys3 ) )
=> ( P @ R @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) )
=> ( ! [R: A > A > $o,Xs2: list @ A] : ( P @ R @ Xs2 @ ( nil @ A ) )
=> ( ! [R: A > A > $o,V: A,Va: list @ A] : ( P @ R @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% mergesort_by_rel_merge.induct
thf(fact_132_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R3: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ B ) )
=> ( ! [X_1: list @ B] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X4: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
( ( R3 @ X4 @ Y3 )
=> ( ( P @ Xs2 @ ( cons @ B @ Y3 @ Ys3 ) )
=> ( P @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
( ~ ( R3 @ X4 @ Y3 )
=> ( ( P @ ( cons @ A @ X4 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_133_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X4: A,Ys3: list @ A] :
( ( P @ Ys3 )
=> ( P @ ( cons @ A @ X4 @ Ys3 ) ) )
=> ( P @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_134_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X4: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_135_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y: A,X22: list @ A,X21: A] :
( ( member @ A @ Y @ ( set2 @ A @ X22 ) )
=> ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_136_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] : ( member @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_137_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_138_list_Oset__cases,axiom,
! [A: $tType,E3: A,A2: list @ A] :
( ( member @ A @ E3 @ ( set2 @ A @ A2 ) )
=> ( ! [Z2: list @ A] :
( A2
!= ( cons @ A @ E3 @ Z2 ) )
=> ~ ! [Z1: A,Z2: list @ A] :
( ( A2
= ( cons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E3 @ ( set2 @ A @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_139_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_140_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_141_list__match__lel__lel,axiom,
! [A: $tType,C1: list @ A,Qs: A,C22: list @ A,C12: list @ A,Qs2: A,C23: list @ A] :
( ( ( append @ A @ C1 @ ( cons @ A @ Qs @ C22 ) )
= ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C23 ) ) )
=> ( ! [C21: list @ A] :
( ( C1
= ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C21 ) ) )
=> ( C23
!= ( append @ A @ C21 @ ( cons @ A @ Qs @ C22 ) ) ) )
=> ( ( ( C12 = C1 )
=> ( ( Qs2 = Qs )
=> ( C23 != C22 ) ) )
=> ~ ! [C212: list @ A] :
( ( C12
= ( append @ A @ C1 @ ( cons @ A @ Qs @ C212 ) ) )
=> ( C22
!= ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C23 ) ) ) ) ) ) ) ).
% list_match_lel_lel
thf(fact_142_append_Oleft__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ ( nil @ A ) @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_143_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_144_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_145_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_146_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X4: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_147_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys3: list @ A,Y3: A] :
( Xs
!= ( append @ A @ Ys3 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_148_neq__Nil__revE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [Ll: list @ A,E4: A] :
( L
!= ( append @ A @ Ll @ ( cons @ A @ E4 @ ( nil @ A ) ) ) ) ) ).
% neq_Nil_revE
thf(fact_149_rev__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X4: A,Xs2: list @ A] : ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_150_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
= ( ? [Xs3: list @ A,X3: A] :
( L
= ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_151_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys4: list @ A] :
( ( ( cons @ A @ X @ Ys4 )
= Ys )
& ( Xs
= ( append @ A @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_152_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
& ( ( append @ A @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_153_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X4: A] : ( P @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_154_rev__nonempty__induct2_H,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ! [X4: A,Y3: B] : ( P @ ( cons @ A @ X4 @ ( nil @ A ) ) @ ( cons @ B @ Y3 @ ( nil @ B ) ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: B] :
( ( Xs2
!= ( nil @ A ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) )
=> ( ! [X4: A,Y3: B,Ys3: list @ B] :
( ( Ys3
!= ( nil @ B ) )
=> ( P @ ( cons @ A @ X4 @ ( nil @ A ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
( ( P @ Xs2 @ Ys3 )
=> ( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys3
!= ( nil @ B ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y3 @ ( nil @ B ) ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_155_list__Cons__eq__append__cases,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys5 ) )
=> ( ( append @ A @ Ys5 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_156_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys5 ) )
=> ( ( append @ A @ Ys5 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_157_split__list,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys3: list @ A,Zs2: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_158_split__list__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys3: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_159_split__list__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys3: list @ A,X4: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs2 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_160_xy__in__set__cases,axiom,
! [A: $tType,X: A,L: list @ A,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ L ) )
=> ( ( member @ A @ Y @ ( set2 @ A @ L ) )
=> ( ( ( X = Y )
=> ! [L12: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ Y @ L22 ) ) ) )
=> ( ( ( X != Y )
=> ! [L12: list @ A,L22: list @ A,L3: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ X @ ( append @ A @ L22 @ ( cons @ A @ Y @ L3 ) ) ) ) ) )
=> ~ ( ( X != Y )
=> ! [L12: list @ A,L22: list @ A,L3: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ Y @ ( append @ A @ L22 @ ( cons @ A @ X @ L3 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_161_split__list__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys3: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_162_split__list__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys3: list @ A,X4: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs2 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_163_append__Cons__eq__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Xs4: list @ A,Ys6: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ( ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) )
= ( append @ A @ Xs4 @ ( cons @ A @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_164_in__set__conv__decomp,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys2: list @ A,Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_165_in__set__list__format,axiom,
! [A: $tType,E3: A,L: list @ A] :
( ( member @ A @ E3 @ ( set2 @ A @ L ) )
=> ~ ! [L12: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ E3 @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_166_split__list__last__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys3: list @ A,X4: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_167_split__list__first__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys3: list @ A,X4: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_168_split__list__last__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys3: list @ A,X4: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_169_split__list__first__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys3: list @ A,X4: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_170_in__set__conv__decomp__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys2: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_171_in__set__conv__decomp__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys2: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_172_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list @ A,X3: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_173_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list @ A,X3: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_174_hd__in__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ Xs ) ) ) ).
% hd_in_set
thf(fact_175_list_Oset__sel_I1_J,axiom,
! [A: $tType,A2: list @ A] :
( ( A2
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ A2 ) @ ( set2 @ A @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_176_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ps: list @ A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs
= ( append @ A @ Ps @ Xs5 ) )
& ( Ys
= ( append @ A @ Ps @ Ys5 ) )
& ( ( Xs5
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( hd @ A @ Xs5 )
!= ( hd @ A @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_177_hd__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Ys ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ) ).
% hd_append
thf(fact_178__092_060open_062old_OpathsConverge_Ag_Am_Ams_H_An_A_Ins_A_064_Atl_Ari_J_Ai_092_060close_062,axiom,
graph_pathsConverge @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ m @ ms2 @ n @ ( append @ node @ ns @ ( tl @ node @ ri ) ) @ i ).
% \<open>old.pathsConverge g m ms' n (ns @ tl ri) i\<close>
thf(fact_179__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_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ ( sSA_CF1081484811efNode @ g @ node @ val @ alpha_n @ defs @ phis @ g2 @ r ) @ Ri @ I )
=> ( ( member @ node @ I @ ( set2 @ node @ ms ) )
=> ( ! [X2: node] :
( ( member @ node @ X2 @ ( set2 @ node @ ( butlast @ node @ Ri ) ) )
=> ~ ( member @ node @ X2 @ ( set2 @ node @ 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_180__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_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ m @ Ms2 @ i )
=> ( ( prefix @ node @ Ms2 @ ms )
=> ( member @ node @ i @ ( set2 @ node @ ( 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_181_old_Opath2__simple__loop,axiom,
! [G2: g,N: node,Ns: list @ node,N2: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ N )
=> ( ( member @ node @ N2 @ ( set2 @ node @ Ns ) )
=> ~ ! [Ns4: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns4 @ N )
=> ( ( member @ node @ N2 @ ( set2 @ node @ Ns4 ) )
=> ( ~ ( member @ node @ N @ ( set2 @ node @ ( tl @ node @ ( butlast @ node @ Ns4 ) ) ) )
=> ~ ( ord_less_eq @ ( set @ node ) @ ( set2 @ node @ Ns4 ) @ ( set2 @ node @ Ns ) ) ) ) ) ) ) ).
% old.path2_simple_loop
thf(fact_182_old_Opath_Ocases,axiom,
! [G2: g,A2: list @ node] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ A2 )
=> ( ! [N4: node] :
( ( A2
= ( cons @ node @ N4 @ ( nil @ node ) ) )
=> ( ( member @ node @ N4 @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ~ ( invar @ G2 ) ) )
=> ~ ! [Ns3: list @ node,N3: node] :
( ( A2
= ( cons @ node @ N3 @ Ns3 ) )
=> ( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ Ns3 )
=> ~ ( member @ node @ N3 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ ( hd @ node @ Ns3 ) ) ) ) ) ) ) ) ).
% old.path.cases
thf(fact_183_old_Opath_Oinducts,axiom,
! [G2: g,X: list @ node,P: ( list @ node ) > $o] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ X )
=> ( ! [N4: node] :
( ( member @ node @ N4 @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( invar @ G2 )
=> ( P @ ( cons @ node @ N4 @ ( nil @ node ) ) ) ) )
=> ( ! [Ns3: list @ node,N3: node] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ Ns3 )
=> ( ( P @ Ns3 )
=> ( ( member @ node @ N3 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ ( hd @ node @ Ns3 ) ) ) )
=> ( P @ ( cons @ node @ N3 @ Ns3 ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% old.path.inducts
thf(fact_184_old_Opath_Osimps,axiom,
! [G2: g,A2: list @ node] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ A2 )
= ( ? [N5: node] :
( ( A2
= ( cons @ node @ N5 @ ( nil @ node ) ) )
& ( member @ node @ N5 @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
& ( invar @ G2 ) )
| ? [Ns5: list @ node,N6: node] :
( ( A2
= ( cons @ node @ N6 @ Ns5 ) )
& ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ Ns5 )
& ( member @ node @ N6 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ ( hd @ node @ Ns5 ) ) ) ) ) ) ) ).
% old.path.simps
thf(fact_185_old_Opath__snoc,axiom,
! [G2: g,Ns: list @ node,N: node,M: node] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( append @ node @ Ns @ ( cons @ node @ N @ ( nil @ node ) ) ) )
=> ( ( member @ node @ N @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ M ) ) )
=> ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( append @ node @ Ns @ ( cons @ node @ N @ ( cons @ node @ M @ ( nil @ node ) ) ) ) ) ) ) ).
% old.path_snoc
thf(fact_186_ri__rs_H__prefix,axiom,
prefix @ node @ ri @ rs ).
% ri_rs'_prefix
thf(fact_187_old_Opath__not__Nil,axiom,
! [G2: g,Ns: list @ node] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ Ns )
=> ( Ns
!= ( nil @ node ) ) ) ).
% old.path_not_Nil
thf(fact_188_old_Opath__hd,axiom,
! [G2: g,N: node,Ns: list @ node] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( cons @ node @ N @ Ns ) )
=> ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( cons @ node @ N @ ( nil @ node ) ) ) ) ).
% old.path_hd
thf(fact_189_old_Opath__split_I2_J,axiom,
! [G2: g,Ns: list @ node,M: node,Ns2: list @ node] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( append @ node @ Ns @ ( cons @ node @ M @ Ns2 ) ) )
=> ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( cons @ node @ M @ Ns2 ) ) ) ).
% old.path_split(2)
thf(fact_190_old_Oempty__path,axiom,
! [N: node,G2: g] :
( ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) )
=> ( ( invar @ G2 )
=> ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( cons @ node @ N @ ( nil @ node ) ) ) ) ) ).
% old.empty_path
thf(fact_191_old_Opath__split_I1_J,axiom,
! [G2: g,Ns: list @ node,M: node,Ns2: list @ node] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( append @ node @ Ns @ ( cons @ node @ M @ Ns2 ) ) )
=> ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( append @ node @ Ns @ ( cons @ node @ M @ ( nil @ node ) ) ) ) ) ).
% old.path_split(1)
thf(fact_192_old_Opath2__split__first__prop,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,P: node > $o] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ? [X2: node] :
( ( member @ node @ X2 @ ( set2 @ node @ Ns ) )
& ( P @ X2 ) )
=> ~ ! [M2: node,Ns4: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns4 @ M2 )
=> ( ( P @ M2 )
=> ( ! [X2: node] :
( ( member @ node @ X2 @ ( set2 @ node @ ( butlast @ node @ Ns4 ) ) )
=> ~ ( P @ X2 ) )
=> ~ ( prefix @ node @ Ns4 @ Ns ) ) ) ) ) ) ).
% old.path2_split_first_prop
thf(fact_193_old_Opath2__prefix__ex,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,M4: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( member @ node @ M4 @ ( set2 @ node @ Ns ) )
=> ~ ! [Ns4: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns4 @ M4 )
=> ( ( prefix @ node @ Ns4 @ Ns )
=> ( member @ node @ M4 @ ( set2 @ node @ ( butlast @ node @ Ns4 ) ) ) ) ) ) ) ).
% old.path2_prefix_ex
thf(fact_194_old_OCons__path,axiom,
! [G2: g,Ns: list @ node,N2: node] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ Ns )
=> ( ( member @ node @ N2 @ ( set2 @ node @ ( graph_1201503639essors @ g @ node @ edgeD @ inEdges @ G2 @ ( hd @ node @ Ns ) ) ) )
=> ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( cons @ node @ N2 @ Ns ) ) ) ) ).
% old.Cons_path
thf(fact_195_old_Opath2__prefix,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,Ns2: list @ node,M4: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( prefix @ node @ ( append @ node @ Ns2 @ ( cons @ node @ M4 @ ( nil @ node ) ) ) @ Ns )
=> ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ Ns2 @ ( cons @ node @ M4 @ ( nil @ node ) ) ) @ M4 ) ) ) ).
% old.path2_prefix
thf(fact_196_old_Opath__invar,axiom,
! [G2: g,Ns: list @ node] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ Ns )
=> ( invar @ G2 ) ) ).
% old.path_invar
thf(fact_197_old_Opath__in___092_060alpha_062n,axiom,
! [G2: g,Ns: list @ node,N: node] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ Ns )
=> ( ( member @ node @ N @ ( set2 @ node @ Ns ) )
=> ( member @ node @ N @ ( set2 @ node @ ( alpha_n @ G2 ) ) ) ) ) ).
% old.path_in_\<alpha>n
thf(fact_198_old_Opath__by__tail,axiom,
! [G2: g,N: node,N2: node,Ns: list @ node,Ms: list @ node] :
( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( cons @ node @ N @ ( cons @ node @ N2 @ Ns ) ) )
=> ( ( ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( cons @ node @ N2 @ Ns ) )
=> ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( cons @ node @ N2 @ Ms ) ) )
=> ( graph_1146196722h_path @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ ( cons @ node @ N @ ( cons @ node @ N2 @ Ms ) ) ) ) ) ).
% old.path_by_tail
thf(fact_199_subset__Collect__conv,axiom,
! [A: $tType,S: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( collect @ A @ P ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( P @ X3 ) ) ) ) ).
% subset_Collect_conv
thf(fact_200_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ( C = D )
=> ( ord_less_eq @ A @ A2 @ D ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_201_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B4 )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_202_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_203_tl__subset,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( tl @ A @ Xs ) ) @ A3 ) ) ) ).
% tl_subset
thf(fact_204_butlast__subset,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( butlast @ A @ Xs ) ) @ A3 ) ) ) ).
% butlast_subset
thf(fact_205_prefix__snoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Y: A] :
( ( prefix @ A @ Xs @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
| ( prefix @ A @ Xs @ Ys ) ) ) ).
% prefix_snoc
thf(fact_206_old_Opath2__split__first__last,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,X: node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ( member @ node @ X @ ( set2 @ node @ 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_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ ( append @ node @ Ns_1 @ ( cons @ node @ X @ ( nil @ node ) ) ) @ X )
=> ( ~ ( member @ node @ X @ ( set2 @ node @ Ns_1 ) )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ X @ Ns_3 @ X )
=> ( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ X @ ( cons @ node @ X @ Ns_2 ) @ M )
=> ( member @ node @ X @ ( set2 @ node @ Ns_2 ) ) ) ) ) ) ) ) ) ) ) ).
% old.path2_split_first_last
thf(fact_207_same__prefix__nil,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( prefix @ A @ ( append @ A @ Xs @ Ys ) @ Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% same_prefix_nil
thf(fact_208_prefix__order_Odual__order_Orefl,axiom,
! [A: $tType,A2: list @ A] : ( prefix @ A @ A2 @ A2 ) ).
% prefix_order.dual_order.refl
thf(fact_209_prefix__order_Oorder__refl,axiom,
! [A: $tType,X: list @ A] : ( prefix @ A @ X @ X ) ).
% prefix_order.order_refl
thf(fact_210_suffix__order_Odual__order_Orefl,axiom,
! [A: $tType,A2: list @ A] : ( suffix @ A @ A2 @ A2 ) ).
% suffix_order.dual_order.refl
thf(fact_211_suffix__order_Oorder__refl,axiom,
! [A: $tType,X: list @ A] : ( suffix @ A @ X @ X ) ).
% suffix_order.order_refl
thf(fact_212_Cons__prefix__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( prefix @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( X = Y )
& ( prefix @ A @ Xs @ Ys ) ) ) ).
% Cons_prefix_Cons
thf(fact_213_prefix__code_I1_J,axiom,
! [A: $tType,Xs: list @ A] : ( prefix @ A @ ( nil @ A ) @ Xs ) ).
% prefix_code(1)
thf(fact_214_prefix__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( prefix @ A @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% prefix_Nil
thf(fact_215_prefix__bot_Obot_Oextremum__unique,axiom,
! [A: $tType,A2: list @ A] :
( ( prefix @ A @ A2 @ ( nil @ A ) )
= ( A2
= ( nil @ A ) ) ) ).
% prefix_bot.bot.extremum_unique
thf(fact_216_suffix__bot_Obot_Oextremum__unique,axiom,
! [A: $tType,A2: list @ A] :
( ( suffix @ A @ A2 @ ( nil @ A ) )
= ( A2
= ( nil @ A ) ) ) ).
% suffix_bot.bot.extremum_unique
thf(fact_217_suffix__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( suffix @ A @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% suffix_Nil
thf(fact_218_same__prefix__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( prefix @ A @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) )
= ( prefix @ A @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_219_same__suffix__suffix,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( suffix @ A @ ( append @ A @ Ys @ Xs ) @ ( append @ A @ Zs @ Xs ) )
= ( suffix @ A @ Ys @ Zs ) ) ).
% same_suffix_suffix
thf(fact_220_old_Opath2__split__last__prop,axiom,
! [G2: g,N: node,Ns: list @ node,M: node,P: node > $o] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N @ Ns @ M )
=> ( ? [X2: node] :
( ( member @ node @ X2 @ ( set2 @ node @ Ns ) )
& ( P @ X2 ) )
=> ~ ! [N3: node,Ns4: list @ node] :
( ( graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ G2 @ N3 @ Ns4 @ M )
=> ( ( P @ N3 )
=> ( ! [X2: node] :
( ( member @ node @ X2 @ ( set2 @ node @ ( tl @ node @ Ns4 ) ) )
=> ~ ( P @ X2 ) )
=> ~ ( suffix @ node @ Ns4 @ Ns ) ) ) ) ) ) ).
% old.path2_split_last_prop
thf(fact_221_same__suffix__nil,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( suffix @ A @ ( append @ A @ Ys @ Xs ) @ Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% same_suffix_nil
thf(fact_222_snoc__suffix__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( suffix @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( X = Y )
& ( suffix @ A @ Xs @ Ys ) ) ) ).
% snoc_suffix_snoc
thf(fact_223_suffix__snoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Y: A] :
( ( suffix @ A @ Xs @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs
= ( nil @ A ) )
| ? [Zs3: list @ A] :
( ( Xs
= ( append @ A @ Zs3 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
& ( suffix @ A @ Zs3 @ Ys ) ) ) ) ).
% suffix_snoc
thf(fact_224_suffix__tl,axiom,
! [A: $tType,Xs: list @ A] : ( suffix @ A @ ( tl @ A @ Xs ) @ Xs ) ).
% suffix_tl
thf(fact_225_suffix__Cons,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] :
( ( suffix @ A @ Xs @ ( cons @ A @ Y @ Ys ) )
= ( ( Xs
= ( cons @ A @ Y @ Ys ) )
| ( suffix @ A @ Xs @ Ys ) ) ) ).
% suffix_Cons
thf(fact_226_suffix__ConsD,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( suffix @ A @ ( cons @ A @ X @ Xs ) @ Ys )
=> ( suffix @ A @ Xs @ Ys ) ) ).
% suffix_ConsD
thf(fact_227_suffix__ConsI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Y: A] :
( ( suffix @ A @ Xs @ Ys )
=> ( suffix @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% suffix_ConsI
thf(fact_228_suffix__ConsD2,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( suffix @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
=> ( suffix @ A @ Xs @ Ys ) ) ).
% suffix_ConsD2
thf(fact_229_suffix__bot_Obot_Oextremum__uniqueI,axiom,
! [A: $tType,A2: list @ A] :
( ( suffix @ A @ A2 @ ( nil @ A ) )
=> ( A2
= ( nil @ A ) ) ) ).
% suffix_bot.bot.extremum_uniqueI
thf(fact_230_suffix__bot_Obot_Oextremum,axiom,
! [A: $tType,A2: list @ A] : ( suffix @ A @ ( nil @ A ) @ A2 ) ).
% suffix_bot.bot.extremum
thf(fact_231_Nil__suffix,axiom,
! [A: $tType,Xs: list @ A] : ( suffix @ A @ ( nil @ A ) @ Xs ) ).
% Nil_suffix
thf(fact_232_set__mono__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( suffix @ A @ Xs @ Ys )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_mono_suffix
thf(fact_233_not__suffix__cases,axiom,
! [A: $tType,Ps2: list @ A,Ls: list @ A] :
( ~ ( suffix @ A @ Ps2 @ Ls )
=> ( ( ( Ps2
!= ( nil @ A ) )
=> ( Ls
!= ( nil @ A ) ) )
=> ( ! [A4: A,As: list @ A] :
( ( Ps2
= ( append @ A @ As @ ( cons @ A @ A4 @ ( nil @ A ) ) ) )
=> ! [X4: A,Xs2: list @ A] :
( ( Ls
= ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) )
=> ( ( X4 = A4 )
=> ( suffix @ A @ As @ Xs2 ) ) ) )
=> ~ ! [A4: A] :
( ? [As: list @ A] :
( Ps2
= ( append @ A @ As @ ( cons @ A @ A4 @ ( nil @ A ) ) ) )
=> ! [X4: A] :
( ? [Xs2: list @ A] :
( Ls
= ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) )
=> ( X4 = A4 ) ) ) ) ) ) ).
% not_suffix_cases
thf(fact_234_not__suffix__induct,axiom,
! [A: $tType,Ps2: list @ A,Ls: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ~ ( suffix @ A @ Ps2 @ Ls )
=> ( ! [X4: A,Xs2: list @ A] : ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) @ ( nil @ A ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
( ( X4 != Y3 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) @ ( append @ A @ Ys3 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
( ( X4 = Y3 )
=> ( ~ ( suffix @ A @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) @ ( append @ A @ Ys3 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ) )
=> ( P @ Ps2 @ Ls ) ) ) ) ) ).
% not_suffix_induct
thf(fact_235_suffix__order_Odual__order_Oantisym,axiom,
! [A: $tType,B2: list @ A,A2: list @ A] :
( ( suffix @ A @ B2 @ A2 )
=> ( ( suffix @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% suffix_order.dual_order.antisym
thf(fact_236_suffix__order_Odual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z: list @ A] : Y4 = Z )
= ( ^ [A5: list @ A,B5: list @ A] :
( ( suffix @ A @ B5 @ A5 )
& ( suffix @ A @ A5 @ B5 ) ) ) ) ).
% suffix_order.dual_order.eq_iff
thf(fact_237_suffix__order_Odual__order_Otrans,axiom,
! [A: $tType,B2: list @ A,A2: list @ A,C: list @ A] :
( ( suffix @ A @ B2 @ A2 )
=> ( ( suffix @ A @ C @ B2 )
=> ( suffix @ A @ C @ A2 ) ) ) ).
% suffix_order.dual_order.trans
thf(fact_238_suffix__order_Oord__le__eq__trans,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C: list @ A] :
( ( suffix @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( suffix @ A @ A2 @ C ) ) ) ).
% suffix_order.ord_le_eq_trans
thf(fact_239_suffix__order_Oord__eq__le__trans,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C: list @ A] :
( ( A2 = B2 )
=> ( ( suffix @ A @ B2 @ C )
=> ( suffix @ A @ A2 @ C ) ) ) ).
% suffix_order.ord_eq_le_trans
thf(fact_240_suffix__order_Oorder_Oantisym,axiom,
! [A: $tType,A2: list @ A,B2: list @ A] :
( ( suffix @ A @ A2 @ B2 )
=> ( ( suffix @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% suffix_order.order.antisym
thf(fact_241_suffix__order_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z: list @ A] : Y4 = Z )
= ( ^ [A5: list @ A,B5: list @ A] :
( ( suffix @ A @ A5 @ B5 )
& ( suffix @ A @ B5 @ A5 ) ) ) ) ).
% suffix_order.order.eq_iff
thf(fact_242_suffix__order_Oantisym__conv,axiom,
! [A: $tType,Y: list @ A,X: list @ A] :
( ( suffix @ A @ Y @ X )
=> ( ( suffix @ A @ X @ Y )
= ( X = Y ) ) ) ).
% suffix_order.antisym_conv
thf(fact_243_suffix__order_Oorder__trans,axiom,
! [A: $tType,X: list @ A,Y: list @ A,Z3: list @ A] :
( ( suffix @ A @ X @ Y )
=> ( ( suffix @ A @ Y @ Z3 )
=> ( suffix @ A @ X @ Z3 ) ) ) ).
% suffix_order.order_trans
thf(fact_244_suffix__order_Oorder_Otrans,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C: list @ A] :
( ( suffix @ A @ A2 @ B2 )
=> ( ( suffix @ A @ B2 @ C )
=> ( suffix @ A @ A2 @ C ) ) ) ).
% suffix_order.order.trans
thf(fact_245_suffix__order_Oeq__refl,axiom,
! [A: $tType,X: list @ A,Y: list @ A] :
( ( X = Y )
=> ( suffix @ A @ X @ Y ) ) ).
% suffix_order.eq_refl
thf(fact_246_suffix__order_Oantisym,axiom,
! [A: $tType,X: list @ A,Y: list @ A] :
( ( suffix @ A @ X @ Y )
=> ( ( suffix @ A @ Y @ X )
=> ( X = Y ) ) ) ).
% suffix_order.antisym
thf(fact_247_suffix__order_Oeq__iff,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z: list @ A] : Y4 = Z )
= ( ^ [X3: list @ A,Y2: list @ A] :
( ( suffix @ A @ X3 @ Y2 )
& ( suffix @ A @ Y2 @ X3 ) ) ) ) ).
% suffix_order.eq_iff
thf(fact_248_suffix__same__cases,axiom,
! [A: $tType,Xs_1: list @ A,Ys: list @ A,Xs_2: list @ A] :
( ( suffix @ A @ Xs_1 @ Ys )
=> ( ( suffix @ A @ Xs_2 @ Ys )
=> ( ( suffix @ A @ Xs_1 @ Xs_2 )
| ( suffix @ A @ Xs_2 @ Xs_1 ) ) ) ) ).
% suffix_same_cases
thf(fact_249_suffixE,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( suffix @ A @ Xs @ Ys )
=> ~ ! [Zs2: list @ A] :
( Ys
!= ( append @ A @ Zs2 @ Xs ) ) ) ).
% suffixE
thf(fact_250_suffixI,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,Xs: list @ A] :
( ( Ys
= ( append @ A @ Zs @ Xs ) )
=> ( suffix @ A @ Xs @ Ys ) ) ).
% suffixI
thf(fact_251_Sublist_Osuffix__def,axiom,
! [A: $tType] :
( ( suffix @ A )
= ( ^ [Xs3: list @ A,Ys2: list @ A] :
? [Zs3: list @ A] :
( Ys2
= ( append @ A @ Zs3 @ Xs3 ) ) ) ) ).
% Sublist.suffix_def
thf(fact_252_suffix__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( suffix @ A @ Xs @ ( append @ A @ Ys @ Zs ) )
= ( ( suffix @ A @ Xs @ Zs )
| ? [Xs6: list @ A] :
( ( Xs
= ( append @ A @ Xs6 @ Zs ) )
& ( suffix @ A @ Xs6 @ Ys ) ) ) ) ).
% suffix_append
thf(fact_253_suffix__appendD,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( suffix @ A @ ( append @ A @ Zs @ Xs ) @ Ys )
=> ( suffix @ A @ Xs @ Ys ) ) ).
% suffix_appendD
thf(fact_254_suffix__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( suffix @ A @ Xs @ Ys )
=> ( suffix @ A @ Xs @ ( append @ A @ Zs @ Ys ) ) ) ).
% suffix_appendI
% Subclasses (2)
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( type @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ord @ A ) ) ).
% Type constructors (4)
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 )
=> ( ord @ ( A6 > A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_1,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_2,axiom,
ord @ $o ).
% Free types (2)
thf(tfree_0,hypothesis,
linorder @ node ).
thf(tfree_1,hypothesis,
linorder @ val ).
% Conjectures (1)
thf(conj_0,conjecture,
graph_1661282752_path2 @ g @ node @ edgeD @ alpha_n @ invar @ inEdges @ g2 @ i @ ( cons @ node @ i @ rs_rest ) @ pred_phi_r ).
%------------------------------------------------------------------------------