TPTP Problem File: ITP067^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP067^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HeapImperative problem prob_172__5340680_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 : HeapImperative/prob_172__5340680_1 [Des21]
% Status : Theorem
% Rating : 0.60 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0
% Syntax : Number of formulae : 216 ( 63 unt; 47 typ; 0 def)
% Number of atoms : 532 ( 157 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 2064 ( 75 ~; 8 |; 42 &;1668 @)
% ( 0 <=>; 271 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 224 ( 224 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 43 usr; 14 con; 0-3 aty)
% Number of variables : 553 ( 40 ^; 506 !; 7 ?; 553 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:30:00.885
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $tType ).
thf(ty_n_t__Heap__OTree_Itf__a_J,type,
tree_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (43)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__a_J,type,
minus_1649712171iset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
zero_zero_multiset_a: multiset_a ).
thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oleft_001tf__a,type,
heapIm1140443833left_a: tree_a > tree_a ).
thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oright_001tf__a,type,
heapIm1257206334ight_a: tree_a > tree_a ).
thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OsiftDown_001tf__a,type,
heapIm1091024090Down_a: tree_a > tree_a ).
thf(sy_c_Heap_OTree_OE_001tf__a,type,
e_a: tree_a ).
thf(sy_c_Heap_OTree_OT_001tf__a,type,
t_a: a > tree_a > tree_a > tree_a ).
thf(sy_c_Heap_OTree_Opred__Tree_001tf__a,type,
pred_Tree_a: ( a > $o ) > tree_a > $o ).
thf(sy_c_Heap_OTree_Oset__Tree_001tf__a,type,
set_Tree_a: tree_a > set_a ).
thf(sy_c_Heap_Oin__tree_001tf__a,type,
in_tree_a: a > tree_a > $o ).
thf(sy_c_Heap_Ois__heap_001tf__a,type,
is_heap_a: tree_a > $o ).
thf(sy_c_Heap_Omultiset_001tf__a,type,
multiset_a2: tree_a > multiset_a ).
thf(sy_c_Heap_Oval_001tf__a,type,
val_a: tree_a > a ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001tf__a,type,
lattic146396397_Max_a: set_a > a ).
thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
add_mset_a: a > multiset_a > multiset_a ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mtf__a_J_J,type,
ord_less_eq_o_o_a: ( $o > $o > a ) > ( $o > $o > a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mtf__a_J,type,
ord_less_eq_o_a: ( $o > a ) > ( $o > a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Heap__OTree_Itf__a_J_M_Eo_J,type,
ord_less_eq_Tree_a_o: ( tree_a > $o ) > ( tree_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_Itf__a_J,type,
ord_le1199012836iset_a: multiset_a > multiset_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mtf__a_J,type,
order_Greatest_o_a: ( ( $o > a ) > $o ) > $o > a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
order_Greatest_a: ( a > $o ) > a ).
thf(sy_c_Set_OBall_001tf__a,type,
ball_a: set_a > ( a > $o ) > $o ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_l1____,type,
l1: tree_a ).
thf(sy_v_l2____,type,
l2: tree_a ).
thf(sy_v_r1____,type,
r1: tree_a ).
thf(sy_v_r2____,type,
r2: tree_a ).
thf(sy_v_t,type,
t: tree_a ).
thf(sy_v_v,type,
v: a ).
thf(sy_v_v1____,type,
v1: a ).
thf(sy_v_v2____,type,
v2: a ).
thf(sy_v_v_H____,type,
v3: a ).
% Relevant facts (168)
thf(fact_0_False,axiom,
~ ( ( v = v3 )
| ( v = v1 )
| ( v = v2 ) ) ).
% False
thf(fact_1__092_060open_062in__tree_Av_A_IsiftDown_At_J_092_060close_062,axiom,
in_tree_a @ v @ ( heapIm1091024090Down_a @ t ) ).
% \<open>in_tree v (siftDown t)\<close>
thf(fact_2_True,axiom,
ord_less_eq_a @ v2 @ v1 ).
% True
thf(fact_3__C5__1_Oprems_C,axiom,
in_tree_a @ v @ ( heapIm1091024090Down_a @ ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ).
% "5_1.prems"
thf(fact_4_Tree_Oinject,axiom,
! [X21: a,X22: tree_a,X23: tree_a,Y21: a,Y22: tree_a,Y23: tree_a] :
( ( ( t_a @ X21 @ X22 @ X23 )
= ( t_a @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% Tree.inject
thf(fact_5_in__tree_Osimps_I2_J,axiom,
! [V: a,V2: a,L: tree_a,R: tree_a] :
( ( in_tree_a @ V @ ( t_a @ V2 @ L @ R ) )
= ( ( V = V2 )
| ( in_tree_a @ V @ L )
| ( in_tree_a @ V @ R ) ) ) ).
% in_tree.simps(2)
thf(fact_6__C5__1_Ohyps_C_I2_J,axiom,
( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
=> ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ v3 )
=> ( ( in_tree_a @ v @ ( heapIm1091024090Down_a @ ( t_a @ v3 @ ( heapIm1140443833left_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) )
=> ( in_tree_a @ v @ ( t_a @ v3 @ ( heapIm1140443833left_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) ) ) ) ).
% "5_1.hyps"(2)
thf(fact_7__C5__1_Ohyps_C_I1_J,axiom,
( ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
=> ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ v3 )
=> ( ( in_tree_a @ v @ ( heapIm1091024090Down_a @ ( t_a @ v3 @ ( heapIm1140443833left_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) )
=> ( in_tree_a @ v @ ( t_a @ v3 @ ( heapIm1140443833left_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ) ) ) ).
% "5_1.hyps"(1)
thf(fact_8_left_Osimps,axiom,
! [V: a,L: tree_a,R: tree_a] :
( ( heapIm1140443833left_a @ ( t_a @ V @ L @ R ) )
= L ) ).
% left.simps
thf(fact_9_right_Osimps,axiom,
! [V: a,L: tree_a,R: tree_a] :
( ( heapIm1257206334ight_a @ ( t_a @ V @ L @ R ) )
= R ) ).
% right.simps
thf(fact_10_siftDown_Ocases,axiom,
! [X: tree_a] :
( ( X != e_a )
=> ( ! [V3: a] :
( X
!= ( t_a @ V3 @ e_a @ e_a ) )
=> ( ! [V3: a,Va: a,Vb: tree_a,Vc: tree_a] :
( X
!= ( t_a @ V3 @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
=> ( ! [V3: a,Va: a,Vb: tree_a,Vc: tree_a] :
( X
!= ( t_a @ V3 @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
=> ~ ! [V3: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
( X
!= ( t_a @ V3 @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ).
% siftDown.cases
thf(fact_11_Tree_Opred__inject_I2_J,axiom,
! [P: a > $o,A: a,Aa: tree_a,Ab: tree_a] :
( ( pred_Tree_a @ P @ ( t_a @ A @ Aa @ Ab ) )
= ( ( P @ A )
& ( pred_Tree_a @ P @ Aa )
& ( pred_Tree_a @ P @ Ab ) ) ) ).
% Tree.pred_inject(2)
thf(fact_12_in__tree_Osimps_I1_J,axiom,
! [V: a] :
~ ( in_tree_a @ V @ e_a ) ).
% in_tree.simps(1)
thf(fact_13_Tree_Oset__intros_I3_J,axiom,
! [Ya: a,X23: tree_a,X21: a,X22: tree_a] :
( ( member_a @ Ya @ ( set_Tree_a @ X23 ) )
=> ( member_a @ Ya @ ( set_Tree_a @ ( t_a @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(3)
thf(fact_14_Tree_Opred__inject_I1_J,axiom,
! [P: a > $o] : ( pred_Tree_a @ P @ e_a ) ).
% Tree.pred_inject(1)
thf(fact_15_siftDown_Osimps_I6_J,axiom,
! [Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a,V: a] :
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(6)
thf(fact_16_siftDown_Osimps_I5_J,axiom,
! [Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a,V: a] :
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(5)
thf(fact_17_siftDown_Osimps_I4_J,axiom,
! [Va2: a,Vb2: tree_a,Vc2: tree_a,V: a] :
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ e_a @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ).
% siftDown.simps(4)
thf(fact_18_siftDown_Osimps_I3_J,axiom,
! [Va2: a,Vb2: tree_a,Vc2: tree_a,V: a] :
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
= ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
= ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) @ e_a ) ) ) ) ).
% siftDown.simps(3)
thf(fact_19_siftDown_Osimps_I1_J,axiom,
( ( heapIm1091024090Down_a @ e_a )
= e_a ) ).
% siftDown.simps(1)
thf(fact_20_Tree_Opred__cong,axiom,
! [X: tree_a,Ya: tree_a,P: a > $o,Pa: a > $o] :
( ( X = Ya )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( set_Tree_a @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( pred_Tree_a @ P @ X )
= ( pred_Tree_a @ Pa @ Ya ) ) ) ) ).
% Tree.pred_cong
thf(fact_21_Tree_Opred__mono__strong,axiom,
! [P: a > $o,X: tree_a,Pa: a > $o] :
( ( pred_Tree_a @ P @ X )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( set_Tree_a @ X ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( pred_Tree_a @ Pa @ X ) ) ) ).
% Tree.pred_mono_strong
thf(fact_22_siftDown__in__tree,axiom,
! [T: tree_a] :
( ( T != e_a )
=> ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ T ) ) @ T ) ) ).
% siftDown_in_tree
thf(fact_23_siftDown__Node,axiom,
! [T: tree_a,V: a,L: tree_a,R: tree_a] :
( ( T
= ( t_a @ V @ L @ R ) )
=> ? [L2: tree_a,V4: a,R2: tree_a] :
( ( ( heapIm1091024090Down_a @ T )
= ( t_a @ V4 @ L2 @ R2 ) )
& ( ord_less_eq_a @ V @ V4 ) ) ) ).
% siftDown_Node
thf(fact_24_is__heap_Ocases,axiom,
! [X: tree_a] :
( ( X != e_a )
=> ( ! [V3: a] :
( X
!= ( t_a @ V3 @ e_a @ e_a ) )
=> ( ! [V3: a,Va: a,Vb: tree_a,Vc: tree_a] :
( X
!= ( t_a @ V3 @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
=> ( ! [V3: a,Va: a,Vb: tree_a,Vc: tree_a] :
( X
!= ( t_a @ V3 @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
=> ~ ! [V3: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
( X
!= ( t_a @ V3 @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ).
% is_heap.cases
thf(fact_25_Tree_Oexhaust,axiom,
! [Y: tree_a] :
( ( Y != e_a )
=> ~ ! [X212: a,X222: tree_a,X232: tree_a] :
( Y
!= ( t_a @ X212 @ X222 @ X232 ) ) ) ).
% Tree.exhaust
thf(fact_26_Tree_Oinduct,axiom,
! [P: tree_a > $o,Tree: tree_a] :
( ( P @ e_a )
=> ( ! [X1: a,X2: tree_a,X3: tree_a] :
( ( P @ X2 )
=> ( ( P @ X3 )
=> ( P @ ( t_a @ X1 @ X2 @ X3 ) ) ) )
=> ( P @ Tree ) ) ) ).
% Tree.induct
thf(fact_27_Tree_Odistinct_I1_J,axiom,
! [X21: a,X22: tree_a,X23: tree_a] :
( e_a
!= ( t_a @ X21 @ X22 @ X23 ) ) ).
% Tree.distinct(1)
thf(fact_28_siftDown_Osimps_I2_J,axiom,
! [V: a] :
( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ e_a ) )
= ( t_a @ V @ e_a @ e_a ) ) ).
% siftDown.simps(2)
thf(fact_29_val_Osimps,axiom,
! [V: a,Uu: tree_a,Uv: tree_a] :
( ( val_a @ ( t_a @ V @ Uu @ Uv ) )
= V ) ).
% val.simps
thf(fact_30_Tree_Oset__cases,axiom,
! [E: a,A: tree_a] :
( ( member_a @ E @ ( set_Tree_a @ A ) )
=> ( ! [Z2: tree_a,Z3: tree_a] :
( A
!= ( t_a @ E @ Z2 @ Z3 ) )
=> ( ! [Z1: a,Z2: tree_a] :
( ? [Z3: tree_a] :
( A
= ( t_a @ Z1 @ Z2 @ Z3 ) )
=> ~ ( member_a @ E @ ( set_Tree_a @ Z2 ) ) )
=> ~ ! [Z1: a,Z2: tree_a,Z3: tree_a] :
( ( A
= ( t_a @ Z1 @ Z2 @ Z3 ) )
=> ~ ( member_a @ E @ ( set_Tree_a @ Z3 ) ) ) ) ) ) ).
% Tree.set_cases
thf(fact_31_Tree_Oset__intros_I1_J,axiom,
! [X21: a,X22: tree_a,X23: tree_a] : ( member_a @ X21 @ ( set_Tree_a @ ( t_a @ X21 @ X22 @ X23 ) ) ) ).
% Tree.set_intros(1)
thf(fact_32_Tree_Oset__intros_I2_J,axiom,
! [Y: a,X22: tree_a,X21: a,X23: tree_a] :
( ( member_a @ Y @ ( set_Tree_a @ X22 ) )
=> ( member_a @ Y @ ( set_Tree_a @ ( t_a @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(2)
thf(fact_33_order__refl,axiom,
! [X: $o > a] : ( ord_less_eq_o_a @ X @ X ) ).
% order_refl
thf(fact_34_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_35_is__heap_Osimps_I4_J,axiom,
! [V: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
= ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
& ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ).
% is_heap.simps(4)
thf(fact_36_is__heap_Osimps_I3_J,axiom,
! [V: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( ( is_heap_a @ ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
& ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ).
% is_heap.simps(3)
thf(fact_37_is__heap__max,axiom,
! [V: a,T: tree_a] :
( ( in_tree_a @ V @ T )
=> ( ( is_heap_a @ T )
=> ( ord_less_eq_a @ V @ ( val_a @ T ) ) ) ) ).
% is_heap_max
thf(fact_38_is__heap_Osimps_I6_J,axiom,
! [V: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( ( is_heap_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
& ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
& ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
& ( is_heap_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ).
% is_heap.simps(6)
thf(fact_39_is__heap_Osimps_I5_J,axiom,
! [V: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
& ( is_heap_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
& ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
& ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ).
% is_heap.simps(5)
thf(fact_40_le__funD,axiom,
! [F: $o > a,G: $o > a,X: $o] :
( ( ord_less_eq_o_a @ F @ G )
=> ( ord_less_eq_a @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_41_le__funE,axiom,
! [F: $o > a,G: $o > a,X: $o] :
( ( ord_less_eq_o_a @ F @ G )
=> ( ord_less_eq_a @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_42_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_44_le__funI,axiom,
! [F: $o > a,G: $o > a] :
( ! [X5: $o] : ( ord_less_eq_a @ ( F @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq_o_a @ F @ G ) ) ).
% le_funI
thf(fact_45_le__fun__def,axiom,
( ord_less_eq_o_a
= ( ^ [F2: $o > a,G2: $o > a] :
! [X4: $o] : ( ord_less_eq_a @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_46_Tree_Opred__mono,axiom,
! [P: a > $o,Pa: a > $o] :
( ( ord_less_eq_a_o @ P @ Pa )
=> ( ord_less_eq_Tree_a_o @ ( pred_Tree_a @ P ) @ ( pred_Tree_a @ Pa ) ) ) ).
% Tree.pred_mono
thf(fact_47_is__heap_Osimps_I1_J,axiom,
is_heap_a @ e_a ).
% is_heap.simps(1)
thf(fact_48_is__heap_Osimps_I2_J,axiom,
! [V: a] : ( is_heap_a @ ( t_a @ V @ e_a @ e_a ) ) ).
% is_heap.simps(2)
thf(fact_49_dual__order_Oantisym,axiom,
! [B: $o > a,A: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( ord_less_eq_o_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_50_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_51_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: $o > a,Z4: $o > a] : Y2 = Z4 )
= ( ^ [A3: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A3 )
& ( ord_less_eq_o_a @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_52_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: a,Z4: a] : Y2 = Z4 )
= ( ^ [A3: a,B2: a] :
( ( ord_less_eq_a @ B2 @ A3 )
& ( ord_less_eq_a @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_53_dual__order_Otrans,axiom,
! [B: $o > a,A: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( ord_less_eq_o_a @ C @ B )
=> ( ord_less_eq_o_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_54_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_55_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: a,B3: a] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_56_dual__order_Orefl,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% dual_order.refl
thf(fact_57_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_58_order__trans,axiom,
! [X: $o > a,Y: $o > a,Z5: $o > a] :
( ( ord_less_eq_o_a @ X @ Y )
=> ( ( ord_less_eq_o_a @ Y @ Z5 )
=> ( ord_less_eq_o_a @ X @ Z5 ) ) ) ).
% order_trans
thf(fact_59_order__trans,axiom,
! [X: a,Y: a,Z5: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z5 )
=> ( ord_less_eq_a @ X @ Z5 ) ) ) ).
% order_trans
thf(fact_60_order__class_Oorder_Oantisym,axiom,
! [A: $o > a,B: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_61_order__class_Oorder_Oantisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_62_ord__le__eq__trans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_63_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_64_ord__eq__le__trans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( A = B )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_65_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_66_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: $o > a,Z4: $o > a] : Y2 = Z4 )
= ( ^ [A3: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ A3 @ B2 )
& ( ord_less_eq_o_a @ B2 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_67_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: a,Z4: a] : Y2 = Z4 )
= ( ^ [A3: a,B2: a] :
( ( ord_less_eq_a @ A3 @ B2 )
& ( ord_less_eq_a @ B2 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_68_antisym__conv,axiom,
! [Y: $o > a,X: $o > a] :
( ( ord_less_eq_o_a @ Y @ X )
=> ( ( ord_less_eq_o_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_69_antisym__conv,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_70_le__cases3,axiom,
! [X: a,Y: a,Z5: a] :
( ( ( ord_less_eq_a @ X @ Y )
=> ~ ( ord_less_eq_a @ Y @ Z5 ) )
=> ( ( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_eq_a @ X @ Z5 ) )
=> ( ( ( ord_less_eq_a @ X @ Z5 )
=> ~ ( ord_less_eq_a @ Z5 @ Y ) )
=> ( ( ( ord_less_eq_a @ Z5 @ Y )
=> ~ ( ord_less_eq_a @ Y @ X ) )
=> ( ( ( ord_less_eq_a @ Y @ Z5 )
=> ~ ( ord_less_eq_a @ Z5 @ X ) )
=> ~ ( ( ord_less_eq_a @ Z5 @ X )
=> ~ ( ord_less_eq_a @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_71_order_Otrans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% order.trans
thf(fact_72_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_73_le__cases,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% le_cases
thf(fact_74_eq__refl,axiom,
! [X: $o > a,Y: $o > a] :
( ( X = Y )
=> ( ord_less_eq_o_a @ X @ Y ) ) ).
% eq_refl
thf(fact_75_eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% eq_refl
thf(fact_76_linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_eq_a @ Y @ X ) ) ).
% linear
thf(fact_77_antisym,axiom,
! [X: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X @ Y )
=> ( ( ord_less_eq_o_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_78_antisym,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_79_eq__iff,axiom,
( ( ^ [Y2: $o > a,Z4: $o > a] : Y2 = Z4 )
= ( ^ [X4: $o > a,Y3: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y3 )
& ( ord_less_eq_o_a @ Y3 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_80_eq__iff,axiom,
( ( ^ [Y2: a,Z4: a] : Y2 = Z4 )
= ( ^ [X4: a,Y3: a] :
( ( ord_less_eq_a @ X4 @ Y3 )
& ( ord_less_eq_a @ Y3 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_81_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: a,Y4: a] :
( ( ord_less_eq_a @ X5 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_82_ord__le__eq__subst,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_83_ord__le__eq__subst,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_84_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: a,Y4: a] :
( ( ord_less_eq_a @ X5 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_85_ord__eq__le__subst,axiom,
! [A: $o > a,F: a > $o > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X5: a,Y4: a] :
( ( ord_less_eq_a @ X5 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_86_ord__eq__le__subst,axiom,
! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X5: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_87_ord__eq__le__subst,axiom,
! [A: $o > a,F: ( $o > a ) > $o > a,B: $o > a,C: $o > a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X5: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_88_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X5: a,Y4: a] :
( ( ord_less_eq_a @ X5 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_89_order__subst2,axiom,
! [A: a,B: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
=> ( ! [X5: a,Y4: a] :
( ( ord_less_eq_a @ X5 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_90_order__subst2,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X5: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_91_order__subst2,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
=> ( ! [X5: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_92_order__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X5: a,Y4: a] :
( ( ord_less_eq_a @ X5 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_93_order__subst1,axiom,
! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X5: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_94_order__subst1,axiom,
! [A: $o > a,F: a > $o > a,B: a,C: a] :
( ( ord_less_eq_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X5: a,Y4: a] :
( ( ord_less_eq_a @ X5 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_95_order__subst1,axiom,
! [A: $o > a,F: ( $o > a ) > $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X5: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y4 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_96_order__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X5: a,Y4: a] :
( ( ord_less_eq_a @ X5 @ Y4 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_97_Greatest__equality,axiom,
! [P: ( $o > a ) > $o,X: $o > a] :
( ( P @ X )
=> ( ! [Y4: $o > a] :
( ( P @ Y4 )
=> ( ord_less_eq_o_a @ Y4 @ X ) )
=> ( ( order_Greatest_o_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_98_Greatest__equality,axiom,
! [P: a > $o,X: a] :
( ( P @ X )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X ) )
=> ( ( order_Greatest_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_99_GreatestI2__order,axiom,
! [P: ( $o > a ) > $o,X: $o > a,Q: ( $o > a ) > $o] :
( ( P @ X )
=> ( ! [Y4: $o > a] :
( ( P @ Y4 )
=> ( ord_less_eq_o_a @ Y4 @ X ) )
=> ( ! [X5: $o > a] :
( ( P @ X5 )
=> ( ! [Y5: $o > a] :
( ( P @ Y5 )
=> ( ord_less_eq_o_a @ Y5 @ X5 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( order_Greatest_o_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_100_GreatestI2__order,axiom,
! [P: a > $o,X: a,Q: a > $o] :
( ( P @ X )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( ord_less_eq_a @ Y4 @ X ) )
=> ( ! [X5: a] :
( ( P @ X5 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ Y5 @ X5 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_101_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_a
= ( ^ [X6: $o > $o > a,Y6: $o > $o > a] :
( ( ord_less_eq_o_a @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_a @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_102_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_a
= ( ^ [X6: $o > a,Y6: $o > a] :
( ( ord_less_eq_a @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_a @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_103_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_104_Tree_Osimps_I14_J,axiom,
( ( set_Tree_a @ e_a )
= bot_bot_set_a ) ).
% Tree.simps(14)
thf(fact_105_Tree_Opred__set,axiom,
( pred_Tree_a
= ( ^ [P2: a > $o,X4: tree_a] :
! [Y3: a] :
( ( member_a @ Y3 @ ( set_Tree_a @ X4 ) )
=> ( P2 @ Y3 ) ) ) ) ).
% Tree.pred_set
thf(fact_106_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_107_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_108_subsetI,axiom,
! [A2: set_a,B4: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( member_a @ X5 @ B4 ) )
=> ( ord_less_eq_set_a @ A2 @ B4 ) ) ).
% subsetI
thf(fact_109_in__mono,axiom,
! [A2: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_110_subsetD,axiom,
! [A2: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_111_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( member_a @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_112_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A5 )
=> ( member_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_113_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_114_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_115_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_116_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_117_Ball__def,axiom,
( ball_a
= ( ^ [A5: set_a,P2: a > $o] :
! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( P2 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_118_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X5: a] :
~ ( member_a @ X5 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_119_ball__reg,axiom,
! [R3: set_a,P: a > $o,Q: a > $o] :
( ! [X5: a] :
( ( member_a @ X5 @ R3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ! [X5: a] :
( ( member_a @ X5 @ R3 )
=> ( P @ X5 ) )
=> ! [X7: a] :
( ( member_a @ X7 @ R3 )
=> ( Q @ X7 ) ) ) ) ).
% ball_reg
thf(fact_120_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_121_heap__top__geq,axiom,
! [A: a,T: tree_a] :
( ( member_a @ A @ ( set_mset_a @ ( multiset_a2 @ T ) ) )
=> ( ( is_heap_a @ T )
=> ( ord_less_eq_a @ A @ ( val_a @ T ) ) ) ) ).
% heap_top_geq
thf(fact_122_heap__top__max,axiom,
! [T: tree_a] :
( ( T != e_a )
=> ( ( is_heap_a @ T )
=> ( ( val_a @ T )
= ( lattic146396397_Max_a @ ( set_mset_a @ ( multiset_a2 @ T ) ) ) ) ) ) ).
% heap_top_max
thf(fact_123_multiset_Osimps_I1_J,axiom,
( ( multiset_a2 @ e_a )
= zero_zero_multiset_a ) ).
% multiset.simps(1)
thf(fact_124_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X5: a,Y4: a] :
( ( member_a @ X5 @ A2 )
=> ( ( member_a @ Y4 @ A2 )
=> ( X5 = Y4 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_125_Diff__iff,axiom,
! [C: a,A2: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B4 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_126_DiffI,axiom,
! [C: a,A2: set_a,B4: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_127_DiffD2,axiom,
! [C: a,A2: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B4 ) )
=> ~ ( member_a @ C @ B4 ) ) ).
% DiffD2
thf(fact_128_DiffD1,axiom,
! [C: a,A2: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B4 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_129_DiffE,axiom,
! [C: a,A2: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B4 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B4 ) ) ) ).
% DiffE
thf(fact_130_set__mset__eq__empty__iff,axiom,
! [M: multiset_a] :
( ( ( set_mset_a @ M )
= bot_bot_set_a )
= ( M = zero_zero_multiset_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_131_set__mset__empty,axiom,
( ( set_mset_a @ zero_zero_multiset_a )
= bot_bot_set_a ) ).
% set_mset_empty
thf(fact_132_multiset__induct__max,axiom,
! [P: multiset_a > $o,M: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X5: a,M2: multiset_a] :
( ( P @ M2 )
=> ( ! [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ M2 ) )
=> ( ord_less_eq_a @ Xa @ X5 ) )
=> ( P @ ( add_mset_a @ X5 @ M2 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_max
thf(fact_133_multiset__induct__min,axiom,
! [P: multiset_a > $o,M: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X5: a,M2: multiset_a] :
( ( P @ M2 )
=> ( ! [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ M2 ) )
=> ( ord_less_eq_a @ X5 @ Xa ) )
=> ( P @ ( add_mset_a @ X5 @ M2 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_min
thf(fact_134_at__most__one__mset__mset__diff,axiom,
! [A: a,M: multiset_a] :
( ~ ( member_a @ A @ ( set_mset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( minus_minus_set_a @ ( set_mset_a @ M ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_135_multiset_Osimps_I2_J,axiom,
! [V: a,L: tree_a,R: tree_a] :
( ( multiset_a2 @ ( t_a @ V @ L @ R ) )
= ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ ( multiset_a2 @ L ) @ ( add_mset_a @ V @ zero_zero_multiset_a ) ) @ ( multiset_a2 @ R ) ) ) ).
% multiset.simps(2)
thf(fact_136_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_137_insertCI,axiom,
! [A: a,B4: set_a,B: a] :
( ( ~ ( member_a @ A @ B4 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_138_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_139_insert__subset,axiom,
! [X: a,A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B4 )
= ( ( member_a @ X @ B4 )
& ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ).
% insert_subset
thf(fact_140_insert__Diff1,axiom,
! [X: a,B4: set_a,A2: set_a] :
( ( member_a @ X @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B4 )
= ( minus_minus_set_a @ A2 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_141_Diff__insert0,axiom,
! [X: a,A2: set_a,B4: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B4 ) )
= ( minus_minus_set_a @ A2 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_142_insert__Diff__if,axiom,
! [X: a,B4: set_a,A2: set_a] :
( ( ( member_a @ X @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B4 )
= ( minus_minus_set_a @ A2 @ B4 ) ) )
& ( ~ ( member_a @ X @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B4 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_143_verit__sum__simplify,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
= A ) ).
% verit_sum_simplify
thf(fact_144_add__right__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( ord_le1199012836iset_a @ A @ B )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% add_right_mono
thf(fact_145_add__left__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( ord_le1199012836iset_a @ A @ B )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% add_left_mono
thf(fact_146_add__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
( ( ord_le1199012836iset_a @ A @ B )
=> ( ( ord_le1199012836iset_a @ C @ D )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% add_mono
thf(fact_147_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
( ( ( ord_le1199012836iset_a @ I @ J )
& ( ord_le1199012836iset_a @ K @ L ) )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_148_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
( ( ( I = J )
& ( ord_le1199012836iset_a @ K @ L ) )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_149_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: multiset_a,J: multiset_a,K: multiset_a,L: multiset_a] :
( ( ( ord_le1199012836iset_a @ I @ J )
& ( K = L ) )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_150_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B6: set_a] :
( ( A2
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_151_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B4: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B4 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A2 = B4 ) )
& ( ( A != B )
=> ? [C2: set_a] :
( ( A2
= ( insert_a @ B @ C2 ) )
& ~ ( member_a @ B @ C2 )
& ( B4
= ( insert_a @ A @ C2 ) )
& ~ ( member_a @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_152_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_153_insert__ident,axiom,
! [X: a,A2: set_a,B4: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B4 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B4 ) )
= ( A2 = B4 ) ) ) ) ).
% insert_ident
thf(fact_154_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B6: set_a] :
( ( A2
= ( insert_a @ X @ B6 ) )
=> ( member_a @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_155_insertI2,axiom,
! [A: a,B4: set_a,B: a] :
( ( member_a @ A @ B4 )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_156_insertI1,axiom,
! [A: a,B4: set_a] : ( member_a @ A @ ( insert_a @ A @ B4 ) ) ).
% insertI1
thf(fact_157_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_158_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_159_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_160_insert__subsetI,axiom,
! [X: a,A2: set_a,X8: set_a] :
( ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ X8 @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X8 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_161_subset__insert,axiom,
! [X: a,A2: set_a,B4: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B4 ) )
= ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ).
% subset_insert
thf(fact_162_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_163_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_164_subset__Diff__insert,axiom,
! [A2: set_a,B4: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B4 @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B4 @ C3 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_165_subset__insert__iff,axiom,
! [A2: set_a,X: a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B4 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B4 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_166_set__mset__single,axiom,
! [B: a] :
( ( set_mset_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( insert_a @ B @ bot_bot_set_a ) ) ).
% set_mset_single
thf(fact_167_Max__singleton,axiom,
! [X: a] :
( ( lattic146396397_Max_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% Max_singleton
% Conjectures (1)
thf(conj_0,conjecture,
in_tree_a @ v @ ( t_a @ v3 @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ).
%------------------------------------------------------------------------------