TPTP Problem File: SLH0415^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Knights_Tour/0000_KnightsTour/prob_00592_023894__5772498_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1417 ( 667 unt; 132 typ; 0 def)
% Number of atoms : 3297 (1338 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 10076 ( 427 ~; 59 |; 211 &;8123 @)
% ( 0 <=>;1256 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 344 ( 344 >; 0 *; 0 +; 0 <<)
% Number of symbols : 115 ( 112 usr; 21 con; 0-3 aty)
% Number of variables : 3480 ( 248 ^;3113 !; 119 ?;3480 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:00:36.315
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
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% Explicit typings (112)
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thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
produc2261658324281137661nt_int: set_Pr958786334691620121nt_int > list_P5707943133018811711nt_int > produc2007852851243229709nt_int ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
insert5033312907999012233nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
is_sin8895854488172861613nt_int: set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
the_elem_int: set_int > int ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
the_el8326832613380209454nt_int: set_Pr958786334691620121nt_int > product_prod_int_int ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
accp_P5348991460554856342nt_int: ( produc2007852851243229709nt_int > produc2007852851243229709nt_int > $o ) > produc2007852851243229709nt_int > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
member7279096912039735102um_num: product_prod_num_num > set_Pr8218934625190621173um_num > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
member8566619992076573584nt_int: produc1219242969750017639nt_int > set_Pr2560585780119916871nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
member6077610525077772982nt_int: produc2007852851243229709nt_int > set_Pr2166573435379693421nt_int > $o ).
thf(sy_v_k____,type,
k: nat ).
thf(sy_v_m,type,
m: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_ps,type,
ps: list_P5707943133018811711nt_int ).
thf(sy_v_ps_092_060_094sub_062r____,type,
ps_r: list_P5707943133018811711nt_int ).
thf(sy_v_ps_H____,type,
ps2: list_P5707943133018811711nt_int ).
thf(sy_v_s_092_060_094sub_062j____,type,
s_j: product_prod_int_int ).
thf(sy_v_s_092_060_094sub_062k____,type,
s_k: product_prod_int_int ).
% Relevant facts (1273)
thf(fact_0__092_060open_062s_092_060_094sub_062j_A_092_060noteq_062_As_092_060_094sub_062k_092_060close_062,axiom,
s_j != s_k ).
% \<open>s\<^sub>j \<noteq> s\<^sub>k\<close>
thf(fact_1__092_060open_062s_092_060_094sub_062j_A_061_A_I3_M_A2_J_092_060close_062,axiom,
( s_j
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% \<open>s\<^sub>j = (3, 2)\<close>
thf(fact_2__092_060open_062_I1_M_A1_J_A_092_060in_062_Aboard_An_Am_092_060close_062,axiom,
member5262025264175285858nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( board @ n @ m ) ).
% \<open>(1, 1) \<in> board n m\<close>
thf(fact_3_ps_092_060_094sub_062r__prems_I2_J,axiom,
( ( hd_Pro282112905867057956nt_int @ ps_r )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% ps\<^sub>r_prems(2)
thf(fact_4_assms_I1_J,axiom,
knights_circuit @ ( board @ n @ m ) @ ps ).
% assms(1)
thf(fact_5__092_060open_062s_092_060_094sub_062j_A_092_060in_062_Aboard_An_Am_092_060close_062,axiom,
member5262025264175285858nt_int @ s_j @ ( board @ n @ m ) ).
% \<open>s\<^sub>j \<in> board n m\<close>
thf(fact_6__092_060open_062s_092_060_094sub_062k_A_092_060in_062_Aboard_An_Am_092_060close_062,axiom,
member5262025264175285858nt_int @ s_k @ ( board @ n @ m ) ).
% \<open>s\<^sub>k \<in> board n m\<close>
thf(fact_7__092_060open_062s_092_060_094sub_062j_A_061_A_I2_M_A3_J_A_092_060or_062_As_092_060_094sub_062j_A_061_A_I3_M_A2_J_092_060close_062,axiom,
( ( s_j
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
| ( s_j
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% \<open>s\<^sub>j = (2, 3) \<or> s\<^sub>j = (3, 2)\<close>
thf(fact_8__092_060open_062s_092_060_094sub_062k_A_061_A_I2_M_A3_J_A_092_060or_062_As_092_060_094sub_062k_A_061_A_I3_M_A2_J_092_060close_062,axiom,
( ( s_k
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
| ( s_k
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% \<open>s\<^sub>k = (2, 3) \<or> s\<^sub>k = (3, 2)\<close>
thf(fact_9__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062ps_092_060_094sub_062r_O_A_092_060lbrakk_062knights__circuit_A_Iboard_An_Am_J_Aps_092_060_094sub_062r_059_Ahd_Aps_092_060_094sub_062r_A_061_A_I1_M_A1_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Ps_r: list_P5707943133018811711nt_int] :
( ( knights_circuit @ ( board @ n @ m ) @ Ps_r )
=> ( ( hd_Pro282112905867057956nt_int @ Ps_r )
!= ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% \<open>\<And>thesis. (\<And>ps\<^sub>r. \<lbrakk>knights_circuit (board n m) ps\<^sub>r; hd ps\<^sub>r = (1, 1)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_10__092_060open_062_I1_M_A2_J_A_092_060in_062_Aboard_An_Am_092_060close_062,axiom,
member5262025264175285858nt_int @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( board @ n @ m ) ).
% \<open>(1, 2) \<in> board n m\<close>
thf(fact_11__092_060open_062_I1_M_A3_J_A_092_060in_062_Aboard_An_Am_092_060close_062,axiom,
member5262025264175285858nt_int @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) @ ( board @ n @ m ) ).
% \<open>(1, 3) \<in> board n m\<close>
thf(fact_12_ps_092_060_094sub_062r__prems_I1_J,axiom,
knights_circuit @ ( board @ n @ m ) @ ps_r ).
% ps\<^sub>r_prems(1)
thf(fact_13__092_060open_062s_092_060_094sub_062k_A_061_A_I3_M_A2_J_A_092_060or_062_As_092_060_094sub_062j_A_061_A_I3_M_A2_J_092_060close_062,axiom,
( ( s_k
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
| ( s_j
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% \<open>s\<^sub>k = (3, 2) \<or> s\<^sub>j = (3, 2)\<close>
thf(fact_14__092_060open_062knights__circuit_A_Iboard_An_Am_J_A_I_I1_M_A1_J_A_D_Arev_A_Is_092_060_094sub_062j_A_D_Aps_H_A_064_A_091s_092_060_094sub_062k_093_J_J_092_060close_062,axiom,
knights_circuit @ ( board @ n @ m ) @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( rev_Pr2923690841345412895nt_int @ ( cons_P3334398858971670639nt_int @ s_j @ ( append7030698103840186580nt_int @ ps2 @ ( cons_P3334398858971670639nt_int @ s_k @ nil_Pr2300489316682597567nt_int ) ) ) ) ) ).
% \<open>knights_circuit (board n m) ((1, 1) # rev (s\<^sub>j # ps' @ [s\<^sub>k]))\<close>
thf(fact_15_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_16_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_17_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_18_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_19_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_20_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_21_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_22_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_23_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_24_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_25_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X2: num] :
( Y
!= ( bit0 @ X2 ) )
=> ~ ! [X3: num] :
( Y
!= ( bit1 @ X3 ) ) ) ) ).
% num.exhaust
thf(fact_26_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_27_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_28_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_29_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_30__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_092_060_094sub_062j_Aps_H_As_092_060_094sub_062k_O_Aps_092_060_094sub_062r_A_061_A_I1_M_A1_J_A_D_As_092_060_094sub_062j_A_D_Aps_H_A_064_A_091s_092_060_094sub_062k_093_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [S_j: product_prod_int_int,Ps: list_P5707943133018811711nt_int,S_k: product_prod_int_int] :
( ps_r
!= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( cons_P3334398858971670639nt_int @ S_j @ ( append7030698103840186580nt_int @ Ps @ ( cons_P3334398858971670639nt_int @ S_k @ nil_Pr2300489316682597567nt_int ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>s\<^sub>j ps' s\<^sub>k. ps\<^sub>r = (1, 1) # s\<^sub>j # ps' @ [s\<^sub>k] \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_31_rev__simp,axiom,
( ( rev_Pr2923690841345412895nt_int @ ( cons_P3334398858971670639nt_int @ s_j @ ( append7030698103840186580nt_int @ ps2 @ ( cons_P3334398858971670639nt_int @ s_k @ nil_Pr2300489316682597567nt_int ) ) ) )
= ( cons_P3334398858971670639nt_int @ s_k @ ( append7030698103840186580nt_int @ ( rev_Pr2923690841345412895nt_int @ ps2 ) @ ( cons_P3334398858971670639nt_int @ s_j @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% rev_simp
thf(fact_32__092_060open_062ps_092_060_094sub_062r_A_061_A_I1_M_A1_J_A_D_As_092_060_094sub_062j_A_D_Aps_H_A_064_A_091s_092_060_094sub_062k_093_092_060close_062,axiom,
( ps_r
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( cons_P3334398858971670639nt_int @ s_j @ ( append7030698103840186580nt_int @ ps2 @ ( cons_P3334398858971670639nt_int @ s_k @ nil_Pr2300489316682597567nt_int ) ) ) ) ) ).
% \<open>ps\<^sub>r = (1, 1) # s\<^sub>j # ps' @ [s\<^sub>k]\<close>
thf(fact_33_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_34_knights__circuit__rotate1,axiom,
! [B: set_Pr958786334691620121nt_int,S_i: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
( ( knights_circuit @ B @ ( cons_P3334398858971670639nt_int @ S_i @ Ps2 ) )
=> ( knights_circuit @ B @ ( append7030698103840186580nt_int @ Ps2 @ ( cons_P3334398858971670639nt_int @ S_i @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% knights_circuit_rotate1
thf(fact_35_knights__circuit__rev,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int] :
( ( knights_circuit @ B @ Ps2 )
=> ( knights_circuit @ B @ ( rev_Pr2923690841345412895nt_int @ Ps2 ) ) ) ).
% knights_circuit_rev
thf(fact_36_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_37_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_38_vs,axiom,
valid_step @ ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( rev_Pr2923690841345412895nt_int @ ( cons_P3334398858971670639nt_int @ s_j @ ( append7030698103840186580nt_int @ ps2 @ ( cons_P3334398858971670639nt_int @ s_k @ nil_Pr2300489316682597567nt_int ) ) ) ) ) ) @ ( hd_Pro282112905867057956nt_int @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( rev_Pr2923690841345412895nt_int @ ( cons_P3334398858971670639nt_int @ s_j @ ( append7030698103840186580nt_int @ ps2 @ ( cons_P3334398858971670639nt_int @ s_k @ nil_Pr2300489316682597567nt_int ) ) ) ) ) ) ).
% vs
thf(fact_39__092_060open_062knights__path_A_Iboard_An_Am_J_A_I_I1_M_A1_J_A_D_Arev_A_Is_092_060_094sub_062j_A_D_Aps_H_A_064_A_091s_092_060_094sub_062k_093_J_J_092_060close_062,axiom,
knights_path @ ( board @ n @ m ) @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( rev_Pr2923690841345412895nt_int @ ( cons_P3334398858971670639nt_int @ s_j @ ( append7030698103840186580nt_int @ ps2 @ ( cons_P3334398858971670639nt_int @ s_k @ nil_Pr2300489316682597567nt_int ) ) ) ) ) ).
% \<open>knights_path (board n m) ((1, 1) # rev (s\<^sub>j # ps' @ [s\<^sub>k]))\<close>
thf(fact_40__092_060open_062valid__step_A_Ilast_Aps_092_060_094sub_062r_J_A_I1_M_A1_J_092_060close_062,axiom,
valid_step @ ( last_P3305686521732843992nt_int @ ps_r ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% \<open>valid_step (last ps\<^sub>r) (1, 1)\<close>
thf(fact_41_last__snoc,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) )
= X ) ).
% last_snoc
thf(fact_42_last__snoc,axiom,
! [Xs: list_int,X: int] :
( ( last_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
= X ) ).
% last_snoc
thf(fact_43_rev__eq__Cons__iff,axiom,
! [Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int] :
( ( ( rev_Pr2923690841345412895nt_int @ Xs )
= ( cons_P3334398858971670639nt_int @ Y @ Ys ) )
= ( Xs
= ( append7030698103840186580nt_int @ ( rev_Pr2923690841345412895nt_int @ Ys ) @ ( cons_P3334398858971670639nt_int @ Y @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_44_rev__eq__Cons__iff,axiom,
! [Xs: list_int,Y: int,Ys: list_int] :
( ( ( rev_int @ Xs )
= ( cons_int @ Y @ Ys ) )
= ( Xs
= ( append_int @ ( rev_int @ Ys ) @ ( cons_int @ Y @ nil_int ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_45_vs__s_092_060_094sub_062k,axiom,
valid_step @ s_k @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% vs_s\<^sub>k
thf(fact_46_mem__Collect__eq,axiom,
! [A: product_prod_int_int,P: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( collec213857154873943460nt_int
@ ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_48_vs__s_092_060_094sub_062j,axiom,
valid_step @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ s_j ).
% vs_s\<^sub>j
thf(fact_49__092_060open_062valid__step_A_I1_M_A1_J_As_092_060_094sub_062k_092_060close_062,axiom,
valid_step @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ s_k ).
% \<open>valid_step (1, 1) s\<^sub>k\<close>
thf(fact_50_last__appendR,axiom,
! [Ys: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
( ( Ys != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
= ( last_P3305686521732843992nt_int @ Ys ) ) ) ).
% last_appendR
thf(fact_51_last__appendL,axiom,
! [Ys: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
( ( Ys = nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) ) ).
% last_appendL
thf(fact_52_hd__append2,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( hd_Pro282112905867057956nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
= ( hd_Pro282112905867057956nt_int @ Xs ) ) ) ).
% hd_append2
thf(fact_53_list_Oinject,axiom,
! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int,Y21: product_prod_int_int,Y22: list_P5707943133018811711nt_int] :
( ( ( cons_P3334398858971670639nt_int @ X21 @ X22 )
= ( cons_P3334398858971670639nt_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_54_list_Oinject,axiom,
! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X22 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_55_same__append__eq,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Ys )
= ( append7030698103840186580nt_int @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_56_append__same__eq,axiom,
! [Ys: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Ys @ Xs )
= ( append7030698103840186580nt_int @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_57_append__assoc,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) @ Zs )
= ( append7030698103840186580nt_int @ Xs @ ( append7030698103840186580nt_int @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_58_append_Oassoc,axiom,
! [A: list_P5707943133018811711nt_int,B: list_P5707943133018811711nt_int,C: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ ( append7030698103840186580nt_int @ A @ B ) @ C )
= ( append7030698103840186580nt_int @ A @ ( append7030698103840186580nt_int @ B @ C ) ) ) ).
% append.assoc
thf(fact_59_rev__is__rev__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( ( rev_Pr2923690841345412895nt_int @ Xs )
= ( rev_Pr2923690841345412895nt_int @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_60_rev__rev__ident,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( rev_Pr2923690841345412895nt_int @ ( rev_Pr2923690841345412895nt_int @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_61_kp,axiom,
knights_path @ ( board @ n @ m ) @ ps_r ).
% kp
thf(fact_62_append__is__Nil__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Ys )
= nil_Pr2300489316682597567nt_int )
= ( ( Xs = nil_Pr2300489316682597567nt_int )
& ( Ys = nil_Pr2300489316682597567nt_int ) ) ) ).
% append_is_Nil_conv
thf(fact_63_Nil__is__append__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( nil_Pr2300489316682597567nt_int
= ( append7030698103840186580nt_int @ Xs @ Ys ) )
= ( ( Xs = nil_Pr2300489316682597567nt_int )
& ( Ys = nil_Pr2300489316682597567nt_int ) ) ) ).
% Nil_is_append_conv
thf(fact_64_self__append__conv2,axiom,
! [Y: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
( ( Y
= ( append7030698103840186580nt_int @ Xs @ Y ) )
= ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% self_append_conv2
thf(fact_65_append__self__conv2,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% append_self_conv2
thf(fact_66_self__append__conv,axiom,
! [Y: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( Y
= ( append7030698103840186580nt_int @ Y @ Ys ) )
= ( Ys = nil_Pr2300489316682597567nt_int ) ) ).
% self_append_conv
thf(fact_67_append__self__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr2300489316682597567nt_int ) ) ).
% append_self_conv
thf(fact_68_append__Nil2,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ Xs @ nil_Pr2300489316682597567nt_int )
= Xs ) ).
% append_Nil2
thf(fact_69_append_Oright__neutral,axiom,
! [A: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ A @ nil_Pr2300489316682597567nt_int )
= A ) ).
% append.right_neutral
thf(fact_70__092_060open_062knights__path_A_Iboard_An_Am_J_Aps_092_060close_062,axiom,
knights_path @ ( board @ n @ m ) @ ps ).
% \<open>knights_path (board n m) ps\<close>
thf(fact_71_rev__is__Nil__conv,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( ( rev_Pr2923690841345412895nt_int @ Xs )
= nil_Pr2300489316682597567nt_int )
= ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% rev_is_Nil_conv
thf(fact_72_Nil__is__rev__conv,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( nil_Pr2300489316682597567nt_int
= ( rev_Pr2923690841345412895nt_int @ Xs ) )
= ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% Nil_is_rev_conv
thf(fact_73_rev__append,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( rev_Pr2923690841345412895nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
= ( append7030698103840186580nt_int @ ( rev_Pr2923690841345412895nt_int @ Ys ) @ ( rev_Pr2923690841345412895nt_int @ Xs ) ) ) ).
% rev_append
thf(fact_74_append1__eq__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
( ( ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) )
= ( append7030698103840186580nt_int @ Ys @ ( cons_P3334398858971670639nt_int @ Y @ nil_Pr2300489316682597567nt_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_75_append1__eq__conv,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int] :
( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
= ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_76_singleton__rev__conv,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int )
= ( rev_Pr2923690841345412895nt_int @ Xs ) )
= ( ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_77_singleton__rev__conv,axiom,
! [X: int,Xs: list_int] :
( ( ( cons_int @ X @ nil_int )
= ( rev_int @ Xs ) )
= ( ( cons_int @ X @ nil_int )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_78_rev__singleton__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ( rev_Pr2923690841345412895nt_int @ Xs )
= ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) )
= ( Xs
= ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) ) ).
% rev_singleton_conv
thf(fact_79_rev__singleton__conv,axiom,
! [Xs: list_int,X: int] :
( ( ( rev_int @ Xs )
= ( cons_int @ X @ nil_int ) )
= ( Xs
= ( cons_int @ X @ nil_int ) ) ) ).
% rev_singleton_conv
thf(fact_80_valid__step__non__transitive,axiom,
! [S_i: product_prod_int_int,S_j2: product_prod_int_int,S_k2: product_prod_int_int] :
( ( valid_step @ S_i @ S_j2 )
=> ( ( valid_step @ S_j2 @ S_k2 )
=> ~ ( valid_step @ S_i @ S_k2 ) ) ) ).
% valid_step_non_transitive
thf(fact_81_knights__path__board__unique,axiom,
! [B_1: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int,B_2: set_Pr958786334691620121nt_int] :
( ( knights_path @ B_1 @ Ps2 )
=> ( ( knights_path @ B_2 @ Ps2 )
=> ( B_1 = B_2 ) ) ) ).
% knights_path_board_unique
thf(fact_82_valid__step__rev,axiom,
! [S_i: product_prod_int_int,S_j2: product_prod_int_int] :
( ( valid_step @ S_i @ S_j2 )
=> ( valid_step @ S_j2 @ S_i ) ) ).
% valid_step_rev
thf(fact_83_valid__step__neq,axiom,
! [S_i: product_prod_int_int,S_j2: product_prod_int_int] :
( ( valid_step @ S_i @ S_j2 )
=> ( S_i != S_j2 ) ) ).
% valid_step_neq
thf(fact_84_knights__path__non__nil,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int] :
( ( knights_path @ B @ Ps2 )
=> ( Ps2 != nil_Pr2300489316682597567nt_int ) ) ).
% knights_path_non_nil
thf(fact_85_knights__path__rev,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int] :
( ( knights_path @ B @ Ps2 )
=> ( knights_path @ B @ ( rev_Pr2923690841345412895nt_int @ Ps2 ) ) ) ).
% knights_path_rev
thf(fact_86_knights__circuit__def,axiom,
( knights_circuit
= ( ^ [B2: set_Pr958786334691620121nt_int,Ps3: list_P5707943133018811711nt_int] :
( ( knights_path @ B2 @ Ps3 )
& ( valid_step @ ( last_P3305686521732843992nt_int @ Ps3 ) @ ( hd_Pro282112905867057956nt_int @ Ps3 ) ) ) ) ) ).
% knights_circuit_def
thf(fact_87_not__Cons__self2,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( cons_P3334398858971670639nt_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_88_not__Cons__self2,axiom,
! [X: int,Xs: list_int] :
( ( cons_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_89_append__eq__append__conv2,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int,Ts: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Ys )
= ( append7030698103840186580nt_int @ Zs @ Ts ) )
= ( ? [Us: list_P5707943133018811711nt_int] :
( ( ( Xs
= ( append7030698103840186580nt_int @ Zs @ Us ) )
& ( ( append7030698103840186580nt_int @ Us @ Ys )
= Ts ) )
| ( ( ( append7030698103840186580nt_int @ Xs @ Us )
= Zs )
& ( Ys
= ( append7030698103840186580nt_int @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_90_append__eq__appendI,axiom,
! [Xs: list_P5707943133018811711nt_int,Xs1: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Us2: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append7030698103840186580nt_int @ Xs1 @ Us2 ) )
=> ( ( append7030698103840186580nt_int @ Xs @ Ys )
= ( append7030698103840186580nt_int @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_91_rev__swap,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( ( rev_Pr2923690841345412895nt_int @ Xs )
= Ys )
= ( Xs
= ( rev_Pr2923690841345412895nt_int @ Ys ) ) ) ).
% rev_swap
thf(fact_92_list__nonempty__induct,axiom,
! [Xs: list_P5707943133018811711nt_int,P: list_P5707943133018811711nt_int > $o] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ! [X5: product_prod_int_int] : ( P @ ( cons_P3334398858971670639nt_int @ X5 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [X5: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( ( Xs2 != nil_Pr2300489316682597567nt_int )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_P3334398858971670639nt_int @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_93_list__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X5: int] : ( P @ ( cons_int @ X5 @ nil_int ) )
=> ( ! [X5: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_int @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_94_list__induct2_H,axiom,
! [P: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( P @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int )
=> ( ! [X5: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] : ( P @ ( cons_P3334398858971670639nt_int @ X5 @ Xs2 ) @ nil_Pr2300489316682597567nt_int )
=> ( ! [Y2: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] : ( P @ nil_Pr2300489316682597567nt_int @ ( cons_P3334398858971670639nt_int @ Y2 @ Ys2 ) )
=> ( ! [X5: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Y2: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_P3334398858971670639nt_int @ X5 @ Xs2 ) @ ( cons_P3334398858971670639nt_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_95_list__induct2_H,axiom,
! [P: list_P5707943133018811711nt_int > list_int > $o,Xs: list_P5707943133018811711nt_int,Ys: list_int] :
( ( P @ nil_Pr2300489316682597567nt_int @ nil_int )
=> ( ! [X5: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] : ( P @ ( cons_P3334398858971670639nt_int @ X5 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P @ nil_Pr2300489316682597567nt_int @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X5: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Y2: int,Ys2: list_int] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_P3334398858971670639nt_int @ X5 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_96_list__induct2_H,axiom,
! [P: list_int > list_P5707943133018811711nt_int > $o,Xs: list_int,Ys: list_P5707943133018811711nt_int] :
( ( P @ nil_int @ nil_Pr2300489316682597567nt_int )
=> ( ! [X5: int,Xs2: list_int] : ( P @ ( cons_int @ X5 @ Xs2 ) @ nil_Pr2300489316682597567nt_int )
=> ( ! [Y2: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] : ( P @ nil_int @ ( cons_P3334398858971670639nt_int @ Y2 @ Ys2 ) )
=> ( ! [X5: int,Xs2: list_int,Y2: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_int @ X5 @ Xs2 ) @ ( cons_P3334398858971670639nt_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_97_list__induct2_H,axiom,
! [P: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P @ nil_int @ nil_int )
=> ( ! [X5: int,Xs2: list_int] : ( P @ ( cons_int @ X5 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P @ nil_int @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X5: int,Xs2: list_int,Y2: int,Ys2: list_int] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_int @ X5 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_98_neq__Nil__conv,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
= ( ? [Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
( Xs
= ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_99_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y3: int,Ys3: list_int] :
( Xs
= ( cons_int @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_100_remdups__adj_Ocases,axiom,
! [X: list_P5707943133018811711nt_int] :
( ( X != nil_Pr2300489316682597567nt_int )
=> ( ! [X5: product_prod_int_int] :
( X
!= ( cons_P3334398858971670639nt_int @ X5 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [X5: product_prod_int_int,Y2: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( X
!= ( cons_P3334398858971670639nt_int @ X5 @ ( cons_P3334398858971670639nt_int @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_101_remdups__adj_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ( ! [X5: int] :
( X
!= ( cons_int @ X5 @ nil_int ) )
=> ~ ! [X5: int,Y2: int,Xs2: list_int] :
( X
!= ( cons_int @ X5 @ ( cons_int @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_102_transpose_Ocases,axiom,
! [X: list_l1670014477004246597nt_int] :
( ( X != nil_li8670148097206105925nt_int )
=> ( ! [Xss: list_l1670014477004246597nt_int] :
( X
!= ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ Xss ) )
=> ~ ! [X5: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Xss: list_l1670014477004246597nt_int] :
( X
!= ( cons_l7309679040211256053nt_int @ ( cons_P3334398858971670639nt_int @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_103_transpose_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X5: int,Xs2: list_int,Xss: list_list_int] :
( X
!= ( cons_list_int @ ( cons_int @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_104_min__list_Ocases,axiom,
! [X: list_int] :
( ! [X5: int,Xs2: list_int] :
( X
!= ( cons_int @ X5 @ Xs2 ) )
=> ( X = nil_int ) ) ).
% min_list.cases
thf(fact_105_list_Oexhaust,axiom,
! [Y: list_P5707943133018811711nt_int] :
( ( Y != nil_Pr2300489316682597567nt_int )
=> ~ ! [X212: product_prod_int_int,X222: list_P5707943133018811711nt_int] :
( Y
!= ( cons_P3334398858971670639nt_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_106_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X222: list_int] :
( Y
!= ( cons_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_107_list_OdiscI,axiom,
! [List: list_P5707943133018811711nt_int,X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
( ( List
= ( cons_P3334398858971670639nt_int @ X21 @ X22 ) )
=> ( List != nil_Pr2300489316682597567nt_int ) ) ).
% list.discI
thf(fact_108_list_OdiscI,axiom,
! [List: list_int,X21: int,X22: list_int] :
( ( List
= ( cons_int @ X21 @ X22 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_109_list_Odistinct_I1_J,axiom,
! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
( nil_Pr2300489316682597567nt_int
!= ( cons_P3334398858971670639nt_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_110_list_Odistinct_I1_J,axiom,
! [X21: int,X22: list_int] :
( nil_int
!= ( cons_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_111_Cons__eq__appendI,axiom,
! [X: product_prod_int_int,Xs1: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int] :
( ( ( cons_P3334398858971670639nt_int @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append7030698103840186580nt_int @ Xs1 @ Zs ) )
=> ( ( cons_P3334398858971670639nt_int @ X @ Xs )
= ( append7030698103840186580nt_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_112_Cons__eq__appendI,axiom,
! [X: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_int @ Xs1 @ Zs ) )
=> ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_113_append__Cons,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ Ys )
= ( cons_P3334398858971670639nt_int @ X @ ( append7030698103840186580nt_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_114_append__Cons,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( append_int @ ( cons_int @ X @ Xs ) @ Ys )
= ( cons_int @ X @ ( append_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_115_eq__Nil__appendI,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( Xs = Ys )
=> ( Xs
= ( append7030698103840186580nt_int @ nil_Pr2300489316682597567nt_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_116_append_Oleft__neutral,axiom,
! [A: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ nil_Pr2300489316682597567nt_int @ A )
= A ) ).
% append.left_neutral
thf(fact_117_append__Nil,axiom,
! [Ys: list_P5707943133018811711nt_int] :
( ( append7030698103840186580nt_int @ nil_Pr2300489316682597567nt_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_118_rev_Osimps_I1_J,axiom,
( ( rev_Pr2923690841345412895nt_int @ nil_Pr2300489316682597567nt_int )
= nil_Pr2300489316682597567nt_int ) ).
% rev.simps(1)
thf(fact_119_list_Osel_I1_J,axiom,
! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
( ( hd_Pro282112905867057956nt_int @ ( cons_P3334398858971670639nt_int @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_120_list_Osel_I1_J,axiom,
! [X21: int,X22: list_int] :
( ( hd_int @ ( cons_int @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_121_rev__nonempty__induct,axiom,
! [Xs: list_P5707943133018811711nt_int,P: list_P5707943133018811711nt_int > $o] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ! [X5: product_prod_int_int] : ( P @ ( cons_P3334398858971670639nt_int @ X5 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [X5: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( ( Xs2 != nil_Pr2300489316682597567nt_int )
=> ( ( P @ Xs2 )
=> ( P @ ( append7030698103840186580nt_int @ Xs2 @ ( cons_P3334398858971670639nt_int @ X5 @ nil_Pr2300489316682597567nt_int ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_122_rev__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X5: int] : ( P @ ( cons_int @ X5 @ nil_int ) )
=> ( ! [X5: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P @ Xs2 )
=> ( P @ ( append_int @ Xs2 @ ( cons_int @ X5 @ nil_int ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_123_append__eq__Cons__conv,axiom,
! [Ys: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int,X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( ( append7030698103840186580nt_int @ Ys @ Zs )
= ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( ( ( Ys = nil_Pr2300489316682597567nt_int )
& ( Zs
= ( cons_P3334398858971670639nt_int @ X @ Xs ) ) )
| ? [Ys4: list_P5707943133018811711nt_int] :
( ( Ys
= ( cons_P3334398858971670639nt_int @ X @ Ys4 ) )
& ( ( append7030698103840186580nt_int @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_124_append__eq__Cons__conv,axiom,
! [Ys: list_int,Zs: list_int,X: int,Xs: list_int] :
( ( ( append_int @ Ys @ Zs )
= ( cons_int @ X @ Xs ) )
= ( ( ( Ys = nil_int )
& ( Zs
= ( cons_int @ X @ Xs ) ) )
| ? [Ys4: list_int] :
( ( Ys
= ( cons_int @ X @ Ys4 ) )
& ( ( append_int @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_125_Cons__eq__append__conv,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int] :
( ( ( cons_P3334398858971670639nt_int @ X @ Xs )
= ( append7030698103840186580nt_int @ Ys @ Zs ) )
= ( ( ( Ys = nil_Pr2300489316682597567nt_int )
& ( ( cons_P3334398858971670639nt_int @ X @ Xs )
= Zs ) )
| ? [Ys4: list_P5707943133018811711nt_int] :
( ( ( cons_P3334398858971670639nt_int @ X @ Ys4 )
= Ys )
& ( Xs
= ( append7030698103840186580nt_int @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_126_Cons__eq__append__conv,axiom,
! [X: int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ( ( Ys = nil_int )
& ( ( cons_int @ X @ Xs )
= Zs ) )
| ? [Ys4: list_int] :
( ( ( cons_int @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_int @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_127_rev__exhaust,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ~ ! [Ys2: list_P5707943133018811711nt_int,Y2: product_prod_int_int] :
( Xs
!= ( append7030698103840186580nt_int @ Ys2 @ ( cons_P3334398858971670639nt_int @ Y2 @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% rev_exhaust
thf(fact_128_rev__exhaust,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ~ ! [Ys2: list_int,Y2: int] :
( Xs
!= ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) ) ) ).
% rev_exhaust
thf(fact_129_rev__induct,axiom,
! [P: list_P5707943133018811711nt_int > $o,Xs: list_P5707943133018811711nt_int] :
( ( P @ nil_Pr2300489316682597567nt_int )
=> ( ! [X5: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( ( P @ Xs2 )
=> ( P @ ( append7030698103840186580nt_int @ Xs2 @ ( cons_P3334398858971670639nt_int @ X5 @ nil_Pr2300489316682597567nt_int ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_130_rev__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ( P @ nil_int )
=> ( ! [X5: int,Xs2: list_int] :
( ( P @ Xs2 )
=> ( P @ ( append_int @ Xs2 @ ( cons_int @ X5 @ nil_int ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_131_last_Osimps,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( ( Xs = nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) ) ) ).
% last.simps
thf(fact_132_last_Osimps,axiom,
! [Xs: list_int,X: int] :
( ( ( Xs = nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= ( last_int @ Xs ) ) ) ) ).
% last.simps
thf(fact_133_last__ConsL,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( Xs = nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_134_last__ConsL,axiom,
! [Xs: list_int,X: int] :
( ( Xs = nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_135_last__ConsR,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) ) ).
% last_ConsR
thf(fact_136_last__ConsR,axiom,
! [Xs: list_int,X: int] :
( ( Xs != nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= ( last_int @ Xs ) ) ) ).
% last_ConsR
thf(fact_137_hd__append,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( ( Xs = nil_Pr2300489316682597567nt_int )
=> ( ( hd_Pro282112905867057956nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
= ( hd_Pro282112905867057956nt_int @ Ys ) ) )
& ( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( ( hd_Pro282112905867057956nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
= ( hd_Pro282112905867057956nt_int @ Xs ) ) ) ) ).
% hd_append
thf(fact_138_longest__common__prefix,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
? [Ps4: list_P5707943133018811711nt_int,Xs3: list_P5707943133018811711nt_int,Ys5: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Ps4 @ Xs3 ) )
& ( Ys
= ( append7030698103840186580nt_int @ Ps4 @ Ys5 ) )
& ( ( Xs3 = nil_Pr2300489316682597567nt_int )
| ( Ys5 = nil_Pr2300489316682597567nt_int )
| ( ( hd_Pro282112905867057956nt_int @ Xs3 )
!= ( hd_Pro282112905867057956nt_int @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_139_last__append,axiom,
! [Ys: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
( ( ( Ys = nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) )
& ( ( Ys != nil_Pr2300489316682597567nt_int )
=> ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
= ( last_P3305686521732843992nt_int @ Ys ) ) ) ) ).
% last_append
thf(fact_140_longest__common__suffix,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
? [Ss: list_P5707943133018811711nt_int,Xs3: list_P5707943133018811711nt_int,Ys5: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Xs3 @ Ss ) )
& ( Ys
= ( append7030698103840186580nt_int @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_Pr2300489316682597567nt_int )
| ( Ys5 = nil_Pr2300489316682597567nt_int )
| ( ( last_P3305686521732843992nt_int @ Xs3 )
!= ( last_P3305686521732843992nt_int @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_141_hd__Nil__eq__last,axiom,
( ( hd_Pro282112905867057956nt_int @ nil_Pr2300489316682597567nt_int )
= ( last_P3305686521732843992nt_int @ nil_Pr2300489316682597567nt_int ) ) ).
% hd_Nil_eq_last
thf(fact_142_hd__rev,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( hd_Pro282112905867057956nt_int @ ( rev_Pr2923690841345412895nt_int @ Xs ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) ).
% hd_rev
thf(fact_143_last__rev,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( last_P3305686521732843992nt_int @ ( rev_Pr2923690841345412895nt_int @ Xs ) )
= ( hd_Pro282112905867057956nt_int @ Xs ) ) ).
% last_rev
thf(fact_144_rev_Osimps_I2_J,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( rev_Pr2923690841345412895nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( append7030698103840186580nt_int @ ( rev_Pr2923690841345412895nt_int @ Xs ) @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) ) ).
% rev.simps(2)
thf(fact_145_rev_Osimps_I2_J,axiom,
! [X: int,Xs: list_int] :
( ( rev_int @ ( cons_int @ X @ Xs ) )
= ( append_int @ ( rev_int @ Xs ) @ ( cons_int @ X @ nil_int ) ) ) ).
% rev.simps(2)
thf(fact_146__092_060open_062hd_A_Idrop_Ak_Aps_J_A_061_A_I1_M_A1_J_092_060close_062,axiom,
( ( hd_Pro282112905867057956nt_int @ ( drop_P5690361596310759935nt_int @ k @ ps ) )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% \<open>hd (drop k ps) = (1, 1)\<close>
thf(fact_147__092_060open_062_I1_M_A3_J_A_092_060in_062_Aset_Aps_092_060_094sub_062r_092_060close_062,axiom,
member5262025264175285858nt_int @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) @ ( set_Pr2470121279949933262nt_int @ ps_r ) ).
% \<open>(1, 3) \<in> set ps\<^sub>r\<close>
thf(fact_148__092_060open_062_I1_M_A2_J_A_092_060in_062_Aset_Aps_092_060_094sub_062r_092_060close_062,axiom,
member5262025264175285858nt_int @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( set_Pr2470121279949933262nt_int @ ps_r ) ).
% \<open>(1, 2) \<in> set ps\<^sub>r\<close>
thf(fact_149_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_150__092_060open_062_I1_M_A1_J_A_092_060in_062_Aset_Aps_092_060close_062,axiom,
member5262025264175285858nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( set_Pr2470121279949933262nt_int @ ps ) ).
% \<open>(1, 1) \<in> set ps\<close>
thf(fact_151__092_060open_062_I1_M_A1_J_A_092_060in_062_Aset_Aps_092_060_094sub_062r_092_060close_062,axiom,
member5262025264175285858nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( set_Pr2470121279949933262nt_int @ ps_r ) ).
% \<open>(1, 1) \<in> set ps\<^sub>r\<close>
thf(fact_152_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_153__092_060open_062s_092_060_094sub_062k_A_092_060in_062_Aset_Aps_092_060_094sub_062r_092_060close_062,axiom,
member5262025264175285858nt_int @ s_k @ ( set_Pr2470121279949933262nt_int @ ps_r ) ).
% \<open>s\<^sub>k \<in> set ps\<^sub>r\<close>
thf(fact_154__092_060open_062s_092_060_094sub_062j_A_092_060in_062_Aset_Aps_092_060_094sub_062r_092_060close_062,axiom,
member5262025264175285858nt_int @ s_j @ ( set_Pr2470121279949933262nt_int @ ps_r ) ).
% \<open>s\<^sub>j \<in> set ps\<^sub>r\<close>
thf(fact_155_assms_I2_J,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( ord_min_nat @ n @ m ) ).
% assms(2)
thf(fact_156_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_157_verit__eq__simplify_I8_J,axiom,
! [X23: num,Y23: num] :
( ( ( bit0 @ X23 )
= ( bit0 @ Y23 ) )
= ( X23 = Y23 ) ) ).
% verit_eq_simplify(8)
thf(fact_158_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_159_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_160_set__rev,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( set_Pr2470121279949933262nt_int @ ( rev_Pr2923690841345412895nt_int @ Xs ) )
= ( set_Pr2470121279949933262nt_int @ Xs ) ) ).
% set_rev
thf(fact_161_min__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ U ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_min_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ V ) ) ) ) ).
% min_number_of(1)
thf(fact_162_min__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_min_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ U ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_min_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ V ) ) ) ) ).
% min_number_of(1)
thf(fact_163_min__0__1_I5_J,axiom,
! [X: num] :
( ( ord_min_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= one_one_int ) ).
% min_0_1(5)
thf(fact_164_min__0__1_I5_J,axiom,
! [X: num] :
( ( ord_min_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= one_one_nat ) ).
% min_0_1(5)
thf(fact_165_min__0__1_I6_J,axiom,
! [X: num] :
( ( ord_min_int @ ( numeral_numeral_int @ X ) @ one_one_int )
= one_one_int ) ).
% min_0_1(6)
thf(fact_166_min__0__1_I6_J,axiom,
! [X: num] :
( ( ord_min_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
= one_one_nat ) ).
% min_0_1(6)
thf(fact_167_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_168_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_169_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_170_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_171_in__set__dropD,axiom,
! [X: product_prod_int_int,N: nat,Xs: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) ) )
=> ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) ) ) ).
% in_set_dropD
thf(fact_172_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_173_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_174_verit__comp__simplify1_I2_J,axiom,
! [A: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_175_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_176_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_177_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_178_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_179_drop__Nil,axiom,
! [N: nat] :
( ( drop_P5690361596310759935nt_int @ N @ nil_Pr2300489316682597567nt_int )
= nil_Pr2300489316682597567nt_int ) ).
% drop_Nil
thf(fact_180_list_Oset__intros_I2_J,axiom,
! [Y: product_prod_int_int,X22: list_P5707943133018811711nt_int,X21: product_prod_int_int] :
( ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ X22 ) )
=> ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_181_list_Oset__intros_I2_J,axiom,
! [Y: int,X22: list_int,X21: int] :
( ( member_int @ Y @ ( set_int2 @ X22 ) )
=> ( member_int @ Y @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_182_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] : ( member5262025264175285858nt_int @ X21 @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_183_list_Oset__intros_I1_J,axiom,
! [X21: int,X22: list_int] : ( member_int @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_184_list_Oset__cases,axiom,
! [E: product_prod_int_int,A: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ E @ ( set_Pr2470121279949933262nt_int @ A ) )
=> ( ! [Z2: list_P5707943133018811711nt_int] :
( A
!= ( cons_P3334398858971670639nt_int @ E @ Z2 ) )
=> ~ ! [Z1: product_prod_int_int,Z2: list_P5707943133018811711nt_int] :
( ( A
= ( cons_P3334398858971670639nt_int @ Z1 @ Z2 ) )
=> ~ ( member5262025264175285858nt_int @ E @ ( set_Pr2470121279949933262nt_int @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_185_list_Oset__cases,axiom,
! [E: int,A: list_int] :
( ( member_int @ E @ ( set_int2 @ A ) )
=> ( ! [Z2: list_int] :
( A
!= ( cons_int @ E @ Z2 ) )
=> ~ ! [Z1: int,Z2: list_int] :
( ( A
= ( cons_int @ Z1 @ Z2 ) )
=> ~ ( member_int @ E @ ( set_int2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_186_set__ConsD,axiom,
! [Y: product_prod_int_int,X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_187_set__ConsD,axiom,
! [Y: int,X: int,Xs: list_int] :
( ( member_int @ Y @ ( set_int2 @ ( cons_int @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_int @ Y @ ( set_int2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_188_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_189_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_190_knights__path__set__eq,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int] :
( ( knights_path @ B @ Ps2 )
=> ( ( set_Pr2470121279949933262nt_int @ Ps2 )
= B ) ) ).
% knights_path_set_eq
thf(fact_191_knights__path__board__m__n__geq__1,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int] :
( ( knights_path @ ( board @ N @ M ) @ Ps2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( ord_min_nat @ N @ M ) ) ) ).
% knights_path_board_m_n_geq_1
thf(fact_192_split__list,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
=> ? [Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( Xs
= ( append7030698103840186580nt_int @ Ys2 @ ( cons_P3334398858971670639nt_int @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_193_split__list,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [Ys2: list_int,Zs2: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_194_split__list__last,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
=> ? [Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Ys2 @ ( cons_P3334398858971670639nt_int @ X @ Zs2 ) ) )
& ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_195_split__list__last,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [Ys2: list_int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X @ Zs2 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_196_split__list__prop,axiom,
! [Xs: list_P5707943133018811711nt_int,P: product_prod_int_int > $o] :
( ? [X6: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X6 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys2: list_P5707943133018811711nt_int,X5: product_prod_int_int] :
( ? [Zs2: list_P5707943133018811711nt_int] :
( Xs
= ( append7030698103840186580nt_int @ Ys2 @ ( cons_P3334398858971670639nt_int @ X5 @ Zs2 ) ) )
& ( P @ X5 ) ) ) ).
% split_list_prop
thf(fact_197_split__list__prop,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys2: list_int,X5: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X5 @ Zs2 ) ) )
& ( P @ X5 ) ) ) ).
% split_list_prop
thf(fact_198_split__list__first,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
=> ? [Ys2: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Ys2 @ ( cons_P3334398858971670639nt_int @ X @ Zs2 ) ) )
& ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_199_split__list__first,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
=> ? [Ys2: list_int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X @ Zs2 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_200_split__list__propE,axiom,
! [Xs: list_P5707943133018811711nt_int,P: product_prod_int_int > $o] :
( ? [X6: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X6 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys2: list_P5707943133018811711nt_int,X5: product_prod_int_int] :
( ? [Zs2: list_P5707943133018811711nt_int] :
( Xs
= ( append7030698103840186580nt_int @ Ys2 @ ( cons_P3334398858971670639nt_int @ X5 @ Zs2 ) ) )
=> ~ ( P @ X5 ) ) ) ).
% split_list_propE
thf(fact_201_split__list__propE,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys2: list_int,X5: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X5 @ Zs2 ) ) )
=> ~ ( P @ X5 ) ) ) ).
% split_list_propE
thf(fact_202_append__Cons__eq__iff,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Xs4: list_P5707943133018811711nt_int,Ys6: list_P5707943133018811711nt_int] :
( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
=> ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Ys ) )
=> ( ( ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ Ys ) )
= ( append7030698103840186580nt_int @ Xs4 @ ( cons_P3334398858971670639nt_int @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_203_append__Cons__eq__iff,axiom,
! [X: int,Xs: list_int,Ys: list_int,Xs4: list_int,Ys6: list_int] :
( ~ ( member_int @ X @ ( set_int2 @ Xs ) )
=> ( ~ ( member_int @ X @ ( set_int2 @ Ys ) )
=> ( ( ( append_int @ Xs @ ( cons_int @ X @ Ys ) )
= ( append_int @ Xs4 @ ( cons_int @ X @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_204_in__set__conv__decomp,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
= ( ? [Ys3: list_P5707943133018811711nt_int,Zs3: list_P5707943133018811711nt_int] :
( Xs
= ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_205_in__set__conv__decomp,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [Ys3: list_int,Zs3: list_int] :
( Xs
= ( append_int @ Ys3 @ ( cons_int @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_206_split__list__last__prop,axiom,
! [Xs: list_P5707943133018811711nt_int,P: product_prod_int_int > $o] :
( ? [X6: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X6 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys2: list_P5707943133018811711nt_int,X5: product_prod_int_int,Zs2: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Ys2 @ ( cons_P3334398858971670639nt_int @ X5 @ Zs2 ) ) )
& ( P @ X5 )
& ! [Xa: product_prod_int_int] :
( ( member5262025264175285858nt_int @ Xa @ ( set_Pr2470121279949933262nt_int @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_207_split__list__last__prop,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys2: list_int,X5: int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X5 @ Zs2 ) ) )
& ( P @ X5 )
& ! [Xa: int] :
( ( member_int @ Xa @ ( set_int2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_208_split__list__first__prop,axiom,
! [Xs: list_P5707943133018811711nt_int,P: product_prod_int_int > $o] :
( ? [X6: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X6 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys2: list_P5707943133018811711nt_int,X5: product_prod_int_int] :
( ? [Zs2: list_P5707943133018811711nt_int] :
( Xs
= ( append7030698103840186580nt_int @ Ys2 @ ( cons_P3334398858971670639nt_int @ X5 @ Zs2 ) ) )
& ( P @ X5 )
& ! [Xa: product_prod_int_int] :
( ( member5262025264175285858nt_int @ Xa @ ( set_Pr2470121279949933262nt_int @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_209_split__list__first__prop,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P @ X6 ) )
=> ? [Ys2: list_int,X5: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X5 @ Zs2 ) ) )
& ( P @ X5 )
& ! [Xa: int] :
( ( member_int @ Xa @ ( set_int2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_210_split__list__last__propE,axiom,
! [Xs: list_P5707943133018811711nt_int,P: product_prod_int_int > $o] :
( ? [X6: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X6 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys2: list_P5707943133018811711nt_int,X5: product_prod_int_int,Zs2: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Ys2 @ ( cons_P3334398858971670639nt_int @ X5 @ Zs2 ) ) )
=> ( ( P @ X5 )
=> ~ ! [Xa: product_prod_int_int] :
( ( member5262025264175285858nt_int @ Xa @ ( set_Pr2470121279949933262nt_int @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_211_split__list__last__propE,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys2: list_int,X5: int,Zs2: list_int] :
( ( Xs
= ( append_int @ Ys2 @ ( cons_int @ X5 @ Zs2 ) ) )
=> ( ( P @ X5 )
=> ~ ! [Xa: int] :
( ( member_int @ Xa @ ( set_int2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_212_split__list__first__propE,axiom,
! [Xs: list_P5707943133018811711nt_int,P: product_prod_int_int > $o] :
( ? [X6: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X6 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys2: list_P5707943133018811711nt_int,X5: product_prod_int_int] :
( ? [Zs2: list_P5707943133018811711nt_int] :
( Xs
= ( append7030698103840186580nt_int @ Ys2 @ ( cons_P3334398858971670639nt_int @ X5 @ Zs2 ) ) )
=> ( ( P @ X5 )
=> ~ ! [Xa: product_prod_int_int] :
( ( member5262025264175285858nt_int @ Xa @ ( set_Pr2470121279949933262nt_int @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_213_split__list__first__propE,axiom,
! [Xs: list_int,P: int > $o] :
( ? [X6: int] :
( ( member_int @ X6 @ ( set_int2 @ Xs ) )
& ( P @ X6 ) )
=> ~ ! [Ys2: list_int,X5: int] :
( ? [Zs2: list_int] :
( Xs
= ( append_int @ Ys2 @ ( cons_int @ X5 @ Zs2 ) ) )
=> ( ( P @ X5 )
=> ~ ! [Xa: int] :
( ( member_int @ Xa @ ( set_int2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_214_in__set__conv__decomp__last,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
= ( ? [Ys3: list_P5707943133018811711nt_int,Zs3: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X @ Zs3 ) ) )
& ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_215_in__set__conv__decomp__last,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [Ys3: list_int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys3 @ ( cons_int @ X @ Zs3 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_216_in__set__conv__decomp__first,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
= ( ? [Ys3: list_P5707943133018811711nt_int,Zs3: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X @ Zs3 ) ) )
& ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_217_in__set__conv__decomp__first,axiom,
! [X: int,Xs: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs ) )
= ( ? [Ys3: list_int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys3 @ ( cons_int @ X @ Zs3 ) ) )
& ~ ( member_int @ X @ ( set_int2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_218_split__list__last__prop__iff,axiom,
! [Xs: list_P5707943133018811711nt_int,P: product_prod_int_int > $o] :
( ( ? [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_P5707943133018811711nt_int,X4: product_prod_int_int,Zs3: list_P5707943133018811711nt_int] :
( ( Xs
= ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: product_prod_int_int] :
( ( member5262025264175285858nt_int @ Y3 @ ( set_Pr2470121279949933262nt_int @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_219_split__list__last__prop__iff,axiom,
! [Xs: list_int,P: int > $o] :
( ( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_int,X4: int,Zs3: list_int] :
( ( Xs
= ( append_int @ Ys3 @ ( cons_int @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: int] :
( ( member_int @ Y3 @ ( set_int2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_220_split__list__first__prop__iff,axiom,
! [Xs: list_P5707943133018811711nt_int,P: product_prod_int_int > $o] :
( ( ? [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_P5707943133018811711nt_int,X4: product_prod_int_int] :
( ? [Zs3: list_P5707943133018811711nt_int] :
( Xs
= ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: product_prod_int_int] :
( ( member5262025264175285858nt_int @ Y3 @ ( set_Pr2470121279949933262nt_int @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_221_split__list__first__prop__iff,axiom,
! [Xs: list_int,P: int > $o] :
( ( ? [X4: int] :
( ( member_int @ X4 @ ( set_int2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_int,X4: int] :
( ? [Zs3: list_int] :
( Xs
= ( append_int @ Ys3 @ ( cons_int @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: int] :
( ( member_int @ Y3 @ ( set_int2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_222_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_223_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_224_hd__in__set,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( Xs != nil_Pr2300489316682597567nt_int )
=> ( member5262025264175285858nt_int @ ( hd_Pro282112905867057956nt_int @ Xs ) @ ( set_Pr2470121279949933262nt_int @ Xs ) ) ) ).
% hd_in_set
thf(fact_225_list_Oset__sel_I1_J,axiom,
! [A: list_P5707943133018811711nt_int] :
( ( A != nil_Pr2300489316682597567nt_int )
=> ( member5262025264175285858nt_int @ ( hd_Pro282112905867057956nt_int @ A ) @ ( set_Pr2470121279949933262nt_int @ A ) ) ) ).
% list.set_sel(1)
thf(fact_226_last__in__set,axiom,
! [As: list_P5707943133018811711nt_int] :
( ( As != nil_Pr2300489316682597567nt_int )
=> ( member5262025264175285858nt_int @ ( last_P3305686521732843992nt_int @ As ) @ ( set_Pr2470121279949933262nt_int @ As ) ) ) ).
% last_in_set
thf(fact_227_verit__eq__simplify_I10_J,axiom,
! [X23: num] :
( one
!= ( bit0 @ X23 ) ) ).
% verit_eq_simplify(10)
thf(fact_228_verit__eq__simplify_I14_J,axiom,
! [X23: num,X32: num] :
( ( bit0 @ X23 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_229_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_230_min_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_231_min_Oabsorb1,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_min_num @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_232_min_Oabsorb1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_min_int @ A @ B )
= A ) ) ).
% min.absorb1
thf(fact_233_min_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_234_min_Oabsorb2,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_min_num @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_235_min_Oabsorb2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_min_int @ A @ B )
= B ) ) ).
% min.absorb2
thf(fact_236_min_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_237_min_Obounded__iff,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( ord_min_num @ B @ C ) )
= ( ( ord_less_eq_num @ A @ B )
& ( ord_less_eq_num @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_238_min_Obounded__iff,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ A @ C ) ) ) ).
% min.bounded_iff
thf(fact_239_min_Oidem,axiom,
! [A: nat] :
( ( ord_min_nat @ A @ A )
= A ) ).
% min.idem
thf(fact_240_min_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B ) )
= ( ord_min_nat @ A @ B ) ) ).
% min.left_idem
thf(fact_241_min_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ B )
= ( ord_min_nat @ A @ B ) ) ).
% min.right_idem
thf(fact_242_old_Oprod_Oinject,axiom,
! [A: int,B: int,A3: int,B3: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_243_old_Oprod_Oinject,axiom,
! [A: product_prod_int_int,B: product_prod_int_int,A3: product_prod_int_int,B3: product_prod_int_int] :
( ( ( produc3646306378393792727nt_int @ A @ B )
= ( produc3646306378393792727nt_int @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_244_old_Oprod_Oinject,axiom,
! [A: set_Pr958786334691620121nt_int,B: list_P5707943133018811711nt_int,A3: set_Pr958786334691620121nt_int,B3: list_P5707943133018811711nt_int] :
( ( ( produc2261658324281137661nt_int @ A @ B )
= ( produc2261658324281137661nt_int @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_245_old_Oprod_Oinject,axiom,
! [A: num,B: num,A3: num,B3: num] :
( ( ( product_Pair_num_num @ A @ B )
= ( product_Pair_num_num @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_246_prod_Oinject,axiom,
! [X1: int,X23: int,Y1: int,Y23: int] :
( ( ( product_Pair_int_int @ X1 @ X23 )
= ( product_Pair_int_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_247_prod_Oinject,axiom,
! [X1: product_prod_int_int,X23: product_prod_int_int,Y1: product_prod_int_int,Y23: product_prod_int_int] :
( ( ( produc3646306378393792727nt_int @ X1 @ X23 )
= ( produc3646306378393792727nt_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_248_prod_Oinject,axiom,
! [X1: set_Pr958786334691620121nt_int,X23: list_P5707943133018811711nt_int,Y1: set_Pr958786334691620121nt_int,Y23: list_P5707943133018811711nt_int] :
( ( ( produc2261658324281137661nt_int @ X1 @ X23 )
= ( produc2261658324281137661nt_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_249_prod_Oinject,axiom,
! [X1: num,X23: num,Y1: num,Y23: num] :
( ( ( product_Pair_num_num @ X1 @ X23 )
= ( product_Pair_num_num @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_250_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_251_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_252_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_253_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_254_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_255_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_256_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_257_subset__code_I1_J,axiom,
! [Xs: list_P5707943133018811711nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( set_Pr2470121279949933262nt_int @ Xs ) @ B4 )
= ( ! [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
=> ( member5262025264175285858nt_int @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_258_step__checker_Ocases,axiom,
! [X: produc1219242969750017639nt_int] :
~ ! [I: int,J: int,I2: int,J2: int] :
( X
!= ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ I @ J ) @ ( product_Pair_int_int @ I2 @ J2 ) ) ) ).
% step_checker.cases
thf(fact_259_successively_Ocases,axiom,
! [X: produc1050408459402128056nt_int] :
( ! [P2: product_prod_int_int > product_prod_int_int > $o] :
( X
!= ( produc3328129369365053992nt_int @ P2 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [P2: product_prod_int_int > product_prod_int_int > $o,X5: product_prod_int_int] :
( X
!= ( produc3328129369365053992nt_int @ P2 @ ( cons_P3334398858971670639nt_int @ X5 @ nil_Pr2300489316682597567nt_int ) ) )
=> ~ ! [P2: product_prod_int_int > product_prod_int_int > $o,X5: product_prod_int_int,Y2: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
( X
!= ( produc3328129369365053992nt_int @ P2 @ ( cons_P3334398858971670639nt_int @ X5 @ ( cons_P3334398858971670639nt_int @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_260_successively_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P2: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P2 @ nil_int ) )
=> ( ! [P2: int > int > $o,X5: int] :
( X
!= ( produc8618682346314911123st_int @ P2 @ ( cons_int @ X5 @ nil_int ) ) )
=> ~ ! [P2: int > int > $o,X5: int,Y2: int,Xs2: list_int] :
( X
!= ( produc8618682346314911123st_int @ P2 @ ( cons_int @ X5 @ ( cons_int @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_261_sorted__wrt_Ocases,axiom,
! [X: produc1050408459402128056nt_int] :
( ! [P2: product_prod_int_int > product_prod_int_int > $o] :
( X
!= ( produc3328129369365053992nt_int @ P2 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [P2: product_prod_int_int > product_prod_int_int > $o,X5: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc3328129369365053992nt_int @ P2 @ ( cons_P3334398858971670639nt_int @ X5 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_262_sorted__wrt_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P2: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P2 @ nil_int ) )
=> ~ ! [P2: int > int > $o,X5: int,Ys2: list_int] :
( X
!= ( produc8618682346314911123st_int @ P2 @ ( cons_int @ X5 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_263_shuffles_Ocases,axiom,
! [X: produc1089560213143673063nt_int] :
( ! [Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys2 ) )
=> ( ! [Xs2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ Xs2 @ nil_Pr2300489316682597567nt_int ) )
=> ~ ! [X5: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Y2: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X5 @ Xs2 ) @ ( cons_P3334398858971670639nt_int @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_264_shuffles_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys2: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys2 ) )
=> ( ! [Xs2: list_int] :
( X
!= ( produc364263696895485585st_int @ Xs2 @ nil_int ) )
=> ~ ! [X5: int,Xs2: list_int,Y2: int,Ys2: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X5 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_265_splice_Ocases,axiom,
! [X: produc1089560213143673063nt_int] :
( ! [Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys2 ) )
=> ~ ! [X5: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( X
!= ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X5 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_266_splice_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys2: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys2 ) )
=> ~ ! [X5: int,Xs2: list_int,Ys2: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X5 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_267_set__subset__Cons,axiom,
! [Xs: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_268_set__subset__Cons,axiom,
! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( set_Pr2470121279949933262nt_int @ Xs ) @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_269_set__drop__subset,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] : ( ord_le2843351958646193337nt_int @ ( set_Pr2470121279949933262nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) ) @ ( set_Pr2470121279949933262nt_int @ Xs ) ) ).
% set_drop_subset
thf(fact_270_board__leq__subset,axiom,
! [N_1: nat,N_2: nat,M_1: nat,M_2: nat] :
( ( ( ord_less_eq_nat @ N_1 @ N_2 )
& ( ord_less_eq_nat @ M_1 @ M_2 ) )
=> ( ord_le2843351958646193337nt_int @ ( board @ N_1 @ M_1 ) @ ( board @ N_2 @ M_2 ) ) ) ).
% board_leq_subset
thf(fact_271_path__checker_Ocases,axiom,
! [X: produc2007852851243229709nt_int] :
( ! [B5: set_Pr958786334691620121nt_int] :
( X
!= ( produc2261658324281137661nt_int @ B5 @ nil_Pr2300489316682597567nt_int ) )
=> ( ! [B5: set_Pr958786334691620121nt_int,S_i2: product_prod_int_int] :
( X
!= ( produc2261658324281137661nt_int @ B5 @ ( cons_P3334398858971670639nt_int @ S_i2 @ nil_Pr2300489316682597567nt_int ) ) )
=> ~ ! [B5: set_Pr958786334691620121nt_int,S_i2: product_prod_int_int,S_j: product_prod_int_int,Ps4: list_P5707943133018811711nt_int] :
( X
!= ( produc2261658324281137661nt_int @ B5 @ ( cons_P3334398858971670639nt_int @ S_i2 @ ( cons_P3334398858971670639nt_int @ S_j @ Ps4 ) ) ) ) ) ) ).
% path_checker.cases
thf(fact_272_knights__path__subset,axiom,
! [B_1: set_Pr958786334691620121nt_int,Ps_1: list_P5707943133018811711nt_int,B_2: set_Pr958786334691620121nt_int,Ps_2: list_P5707943133018811711nt_int] :
( ( knights_path @ B_1 @ Ps_1 )
=> ( ( knights_path @ B_2 @ Ps_2 )
=> ( ( ord_le2843351958646193337nt_int @ ( set_Pr2470121279949933262nt_int @ Ps_1 ) @ ( set_Pr2470121279949933262nt_int @ Ps_2 ) )
= ( ord_le2843351958646193337nt_int @ B_1 @ B_2 ) ) ) ) ).
% knights_path_subset
thf(fact_273_xor__num_Ocases,axiom,
! [X: product_prod_num_num] :
( ( X
!= ( product_Pair_num_num @ one @ one ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
=> ( ! [M2: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) )
=> ( ! [M2: num,N2: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) ) )
=> ( ! [M2: num,N2: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) ) )
=> ( ! [M2: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) )
=> ( ! [M2: num,N2: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) ) )
=> ~ ! [M2: num,N2: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_274_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A4: int,B5: int] :
( Y
!= ( product_Pair_int_int @ A4 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_275_old_Oprod_Oexhaust,axiom,
! [Y: produc1219242969750017639nt_int] :
~ ! [A4: product_prod_int_int,B5: product_prod_int_int] :
( Y
!= ( produc3646306378393792727nt_int @ A4 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_276_old_Oprod_Oexhaust,axiom,
! [Y: produc2007852851243229709nt_int] :
~ ! [A4: set_Pr958786334691620121nt_int,B5: list_P5707943133018811711nt_int] :
( Y
!= ( produc2261658324281137661nt_int @ A4 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_277_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_num_num] :
~ ! [A4: num,B5: num] :
( Y
!= ( product_Pair_num_num @ A4 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_278_surj__pair,axiom,
! [P3: product_prod_int_int] :
? [X5: int,Y2: int] :
( P3
= ( product_Pair_int_int @ X5 @ Y2 ) ) ).
% surj_pair
thf(fact_279_surj__pair,axiom,
! [P3: produc1219242969750017639nt_int] :
? [X5: product_prod_int_int,Y2: product_prod_int_int] :
( P3
= ( produc3646306378393792727nt_int @ X5 @ Y2 ) ) ).
% surj_pair
thf(fact_280_surj__pair,axiom,
! [P3: produc2007852851243229709nt_int] :
? [X5: set_Pr958786334691620121nt_int,Y2: list_P5707943133018811711nt_int] :
( P3
= ( produc2261658324281137661nt_int @ X5 @ Y2 ) ) ).
% surj_pair
thf(fact_281_surj__pair,axiom,
! [P3: product_prod_num_num] :
? [X5: num,Y2: num] :
( P3
= ( product_Pair_num_num @ X5 @ Y2 ) ) ).
% surj_pair
thf(fact_282_prod__cases,axiom,
! [P: product_prod_int_int > $o,P3: product_prod_int_int] :
( ! [A4: int,B5: int] : ( P @ ( product_Pair_int_int @ A4 @ B5 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_283_prod__cases,axiom,
! [P: produc1219242969750017639nt_int > $o,P3: produc1219242969750017639nt_int] :
( ! [A4: product_prod_int_int,B5: product_prod_int_int] : ( P @ ( produc3646306378393792727nt_int @ A4 @ B5 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_284_prod__cases,axiom,
! [P: produc2007852851243229709nt_int > $o,P3: produc2007852851243229709nt_int] :
( ! [A4: set_Pr958786334691620121nt_int,B5: list_P5707943133018811711nt_int] : ( P @ ( produc2261658324281137661nt_int @ A4 @ B5 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_285_prod__cases,axiom,
! [P: product_prod_num_num > $o,P3: product_prod_num_num] :
( ! [A4: num,B5: num] : ( P @ ( product_Pair_num_num @ A4 @ B5 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_286_Pair__inject,axiom,
! [A: int,B: int,A3: int,B3: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_287_Pair__inject,axiom,
! [A: product_prod_int_int,B: product_prod_int_int,A3: product_prod_int_int,B3: product_prod_int_int] :
( ( ( produc3646306378393792727nt_int @ A @ B )
= ( produc3646306378393792727nt_int @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_288_Pair__inject,axiom,
! [A: set_Pr958786334691620121nt_int,B: list_P5707943133018811711nt_int,A3: set_Pr958786334691620121nt_int,B3: list_P5707943133018811711nt_int] :
( ( ( produc2261658324281137661nt_int @ A @ B )
= ( produc2261658324281137661nt_int @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_289_Pair__inject,axiom,
! [A: num,B: num,A3: num,B3: num] :
( ( ( product_Pair_num_num @ A @ B )
= ( product_Pair_num_num @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_290_prod__cases3,axiom,
! [Y: produc1219242969750017639nt_int] :
~ ! [A4: product_prod_int_int,B5: int,C2: int] :
( Y
!= ( produc3646306378393792727nt_int @ A4 @ ( product_Pair_int_int @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_291_prod__induct3,axiom,
! [P: produc1219242969750017639nt_int > $o,X: produc1219242969750017639nt_int] :
( ! [A4: product_prod_int_int,B5: int,C2: int] : ( P @ ( produc3646306378393792727nt_int @ A4 @ ( product_Pair_int_int @ B5 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_292_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_293_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_294_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_le2843351958646193337nt_int @ ( set_Pr2470121279949933262nt_int @ ( drop_P5690361596310759935nt_int @ M @ Xs ) ) @ ( set_Pr2470121279949933262nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_295_min_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_min_nat @ B @ ( ord_min_nat @ A @ C ) )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.left_commute
thf(fact_296_min_Ocommute,axiom,
( ord_min_nat
= ( ^ [A5: nat,B2: nat] : ( ord_min_nat @ B2 @ A5 ) ) ) ).
% min.commute
thf(fact_297_min_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ C )
= ( ord_min_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ).
% min.assoc
thf(fact_298_min__le__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
| ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_299_min__le__iff__disj,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ ( ord_min_num @ X @ Y ) @ Z )
= ( ( ord_less_eq_num @ X @ Z )
| ( ord_less_eq_num @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_300_min__le__iff__disj,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ ( ord_min_int @ X @ Y ) @ Z )
= ( ( ord_less_eq_int @ X @ Z )
| ( ord_less_eq_int @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_301_min_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_302_min_OcoboundedI2,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_303_min_OcoboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C ) ) ).
% min.coboundedI2
thf(fact_304_min_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_305_min_OcoboundedI1,axiom,
! [A: num,C: num,B: num] :
( ( ord_less_eq_num @ A @ C )
=> ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_306_min_OcoboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C ) ) ).
% min.coboundedI1
thf(fact_307_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A5: nat] :
( ( ord_min_nat @ A5 @ B2 )
= B2 ) ) ) ).
% min.absorb_iff2
thf(fact_308_min_Oabsorb__iff2,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A5: num] :
( ( ord_min_num @ A5 @ B2 )
= B2 ) ) ) ).
% min.absorb_iff2
thf(fact_309_min_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A5: int] :
( ( ord_min_int @ A5 @ B2 )
= B2 ) ) ) ).
% min.absorb_iff2
thf(fact_310_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B2: nat] :
( ( ord_min_nat @ A5 @ B2 )
= A5 ) ) ) ).
% min.absorb_iff1
thf(fact_311_min_Oabsorb__iff1,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B2: num] :
( ( ord_min_num @ A5 @ B2 )
= A5 ) ) ) ).
% min.absorb_iff1
thf(fact_312_min_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B2: int] :
( ( ord_min_int @ A5 @ B2 )
= A5 ) ) ) ).
% min.absorb_iff1
thf(fact_313_min_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_314_min_Ocobounded2,axiom,
! [A: num,B: num] : ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_315_min_Ocobounded2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ B ) ).
% min.cobounded2
thf(fact_316_min_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_317_min_Ocobounded1,axiom,
! [A: num,B: num] : ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_318_min_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ A ) ).
% min.cobounded1
thf(fact_319_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B2: nat] :
( A5
= ( ord_min_nat @ A5 @ B2 ) ) ) ) ).
% min.order_iff
thf(fact_320_min_Oorder__iff,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B2: num] :
( A5
= ( ord_min_num @ A5 @ B2 ) ) ) ) ).
% min.order_iff
thf(fact_321_min_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B2: int] :
( A5
= ( ord_min_int @ A5 @ B2 ) ) ) ) ).
% min.order_iff
thf(fact_322_min_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_323_min_OboundedI,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ A @ C )
=> ( ord_less_eq_num @ A @ ( ord_min_num @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_324_min_OboundedI,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) ) ) ) ).
% min.boundedI
thf(fact_325_min_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% min.boundedE
thf(fact_326_min_OboundedE,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( ord_min_num @ B @ C ) )
=> ~ ( ( ord_less_eq_num @ A @ B )
=> ~ ( ord_less_eq_num @ A @ C ) ) ) ).
% min.boundedE
thf(fact_327_min_OboundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) )
=> ~ ( ( ord_less_eq_int @ A @ B )
=> ~ ( ord_less_eq_int @ A @ C ) ) ) ).
% min.boundedE
thf(fact_328_min_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_min_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% min.orderI
thf(fact_329_min_OorderI,axiom,
! [A: num,B: num] :
( ( A
= ( ord_min_num @ A @ B ) )
=> ( ord_less_eq_num @ A @ B ) ) ).
% min.orderI
thf(fact_330_min_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( ord_min_int @ A @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% min.orderI
thf(fact_331_min_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( ord_min_nat @ A @ B ) ) ) ).
% min.orderE
thf(fact_332_min_OorderE,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( A
= ( ord_min_num @ A @ B ) ) ) ).
% min.orderE
thf(fact_333_min_OorderE,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( A
= ( ord_min_int @ A @ B ) ) ) ).
% min.orderE
thf(fact_334_min_Omono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C @ D ) ) ) ) ).
% min.mono
thf(fact_335_min_Omono,axiom,
! [A: num,C: num,B: num,D: num] :
( ( ord_less_eq_num @ A @ C )
=> ( ( ord_less_eq_num @ B @ D )
=> ( ord_less_eq_num @ ( ord_min_num @ A @ B ) @ ( ord_min_num @ C @ D ) ) ) ) ).
% min.mono
thf(fact_336_min_Omono,axiom,
! [A: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ B @ D )
=> ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ ( ord_min_int @ C @ D ) ) ) ) ).
% min.mono
thf(fact_337_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_338_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_339_dual__order_Orefl,axiom,
! [A: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A @ A ) ).
% dual_order.refl
thf(fact_340_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_341_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_342_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_343_order__refl,axiom,
! [X: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ X @ X ) ).
% order_refl
thf(fact_344_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_345__092_060open_062knights__path_A_Iboard_An_Am_A_N_A_123_I1_M_A1_J_125_J_A_Is_092_060_094sub_062k_A_D_Arev_Aps_H_A_064_A_091s_092_060_094sub_062j_093_J_092_060close_062,axiom,
knights_path @ ( minus_1052850069191792384nt_int @ ( board @ n @ m ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ bot_bo1796632182523588997nt_int ) ) @ ( cons_P3334398858971670639nt_int @ s_k @ ( append7030698103840186580nt_int @ ( rev_Pr2923690841345412895nt_int @ ps2 ) @ ( cons_P3334398858971670639nt_int @ s_j @ nil_Pr2300489316682597567nt_int ) ) ) ).
% \<open>knights_path (board n m - {(1, 1)}) (s\<^sub>k # rev ps' @ [s\<^sub>j])\<close>
thf(fact_346__092_060open_062knights__path_A_Iboard_An_Am_A_N_A_123_I1_M_A1_J_125_J_A_Irev_A_Is_092_060_094sub_062j_A_D_Aps_H_A_064_A_091s_092_060_094sub_062k_093_J_J_092_060close_062,axiom,
knights_path @ ( minus_1052850069191792384nt_int @ ( board @ n @ m ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ bot_bo1796632182523588997nt_int ) ) @ ( rev_Pr2923690841345412895nt_int @ ( cons_P3334398858971670639nt_int @ s_j @ ( append7030698103840186580nt_int @ ps2 @ ( cons_P3334398858971670639nt_int @ s_k @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% \<open>knights_path (board n m - {(1, 1)}) (rev (s\<^sub>j # ps' @ [s\<^sub>k]))\<close>
thf(fact_347_the__elem__set,axiom,
! [X: product_prod_int_int] :
( ( the_el8326832613380209454nt_int @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) )
= X ) ).
% the_elem_set
thf(fact_348_the__elem__set,axiom,
! [X: int] :
( ( the_elem_int @ ( set_int2 @ ( cons_int @ X @ nil_int ) ) )
= X ) ).
% the_elem_set
thf(fact_349_kp_H,axiom,
knights_path @ ( minus_1052850069191792384nt_int @ ( board @ n @ m ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ bot_bo1796632182523588997nt_int ) ) @ ( cons_P3334398858971670639nt_int @ s_j @ ( append7030698103840186580nt_int @ ps2 @ ( cons_P3334398858971670639nt_int @ s_k @ nil_Pr2300489316682597567nt_int ) ) ) ).
% kp'
thf(fact_350_min__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_min_nat @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_351_min__absorb2,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_min_num @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_352_min__absorb2,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( ( ord_mi3891036259616950784nt_int @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_353_min__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_min_int @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_354_min__bot,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( ord_mi3891036259616950784nt_int @ bot_bo1796632182523588997nt_int @ X )
= bot_bo1796632182523588997nt_int ) ).
% min_bot
thf(fact_355_min__bot,axiom,
! [X: nat] :
( ( ord_min_nat @ bot_bot_nat @ X )
= bot_bot_nat ) ).
% min_bot
thf(fact_356_min__bot2,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( ord_mi3891036259616950784nt_int @ X @ bot_bo1796632182523588997nt_int )
= bot_bo1796632182523588997nt_int ) ).
% min_bot2
thf(fact_357_min__bot2,axiom,
! [X: nat] :
( ( ord_min_nat @ X @ bot_bot_nat )
= bot_bot_nat ) ).
% min_bot2
thf(fact_358__092_060open_062_I1_M_A1_J_A_092_060notin_062_Aboard_An_Am_A_N_A_123_I1_M_A1_J_125_092_060close_062,axiom,
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( minus_1052850069191792384nt_int @ ( board @ n @ m ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% \<open>(1, 1) \<notin> board n m - {(1, 1)}\<close>
thf(fact_359_set__empty2,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( bot_bo1796632182523588997nt_int
= ( set_Pr2470121279949933262nt_int @ Xs ) )
= ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% set_empty2
thf(fact_360_set__empty,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( ( set_Pr2470121279949933262nt_int @ Xs )
= bot_bo1796632182523588997nt_int )
= ( Xs = nil_Pr2300489316682597567nt_int ) ) ).
% set_empty
thf(fact_361_list_Osimps_I15_J,axiom,
! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
( ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X21 @ X22 ) )
= ( insert5033312907999012233nt_int @ X21 @ ( set_Pr2470121279949933262nt_int @ X22 ) ) ) ).
% list.simps(15)
thf(fact_362_list_Osimps_I15_J,axiom,
! [X21: int,X22: list_int] :
( ( set_int2 @ ( cons_int @ X21 @ X22 ) )
= ( insert_int @ X21 @ ( set_int2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_363_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_364_bot_Oextremum,axiom,
! [A: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ bot_bo1796632182523588997nt_int @ A ) ).
% bot.extremum
thf(fact_365_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_366_bot_Oextremum__unique,axiom,
! [A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ bot_bo1796632182523588997nt_int )
= ( A = bot_bo1796632182523588997nt_int ) ) ).
% bot.extremum_unique
thf(fact_367_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_368_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ bot_bo1796632182523588997nt_int )
=> ( A = bot_bo1796632182523588997nt_int ) ) ).
% bot.extremum_uniqueI
thf(fact_369_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_370_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_371_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_372_circuit__checker_Ocases,axiom,
! [X: produc2007852851243229709nt_int] :
~ ! [B5: set_Pr958786334691620121nt_int,Ps4: list_P5707943133018811711nt_int] :
( X
!= ( produc2261658324281137661nt_int @ B5 @ Ps4 ) ) ).
% circuit_checker.cases
thf(fact_373_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_374_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_375_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_376_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_377_knights__path__intro__rev,axiom,
! [S_i: product_prod_int_int,B: set_Pr958786334691620121nt_int,S_j2: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ S_i @ B )
=> ( ( valid_step @ S_i @ S_j2 )
=> ( ( knights_path @ ( minus_1052850069191792384nt_int @ B @ ( insert5033312907999012233nt_int @ S_i @ bot_bo1796632182523588997nt_int ) ) @ ( cons_P3334398858971670639nt_int @ S_j2 @ Ps2 ) )
=> ( knights_path @ B @ ( cons_P3334398858971670639nt_int @ S_i @ ( cons_P3334398858971670639nt_int @ S_j2 @ Ps2 ) ) ) ) ) ) ).
% knights_path_intro_rev
thf(fact_378_min__diff__distrib__left,axiom,
! [X: int,Y: int,Z: int] :
( ( minus_minus_int @ ( ord_min_int @ X @ Y ) @ Z )
= ( ord_min_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).
% min_diff_distrib_left
thf(fact_379_knights__path__board__non__empty,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int] :
( ( knights_path @ B @ Ps2 )
=> ( B != bot_bo1796632182523588997nt_int ) ) ).
% knights_path_board_non_empty
thf(fact_380_knights__path_Ointros_I1_J,axiom,
! [S_i: product_prod_int_int] : ( knights_path @ ( insert5033312907999012233nt_int @ S_i @ bot_bo1796632182523588997nt_int ) @ ( cons_P3334398858971670639nt_int @ S_i @ nil_Pr2300489316682597567nt_int ) ) ).
% knights_path.intros(1)
thf(fact_381_empty__set,axiom,
( bot_bo1796632182523588997nt_int
= ( set_Pr2470121279949933262nt_int @ nil_Pr2300489316682597567nt_int ) ) ).
% empty_set
thf(fact_382_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_383_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_384_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_385_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_386_le__cases3,axiom,
! [X: num,Y: num,Z: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z ) )
=> ( ( ( ord_less_eq_num @ X @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y ) )
=> ( ( ( ord_less_eq_num @ Z @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z )
=> ~ ( ord_less_eq_num @ Z @ X ) )
=> ~ ( ( ord_less_eq_num @ Z @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_387_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_388_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_389_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z3: num] : ( Y4 = Z3 ) )
= ( ^ [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
& ( ord_less_eq_num @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_390_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr958786334691620121nt_int,Z3: set_Pr958786334691620121nt_int] : ( Y4 = Z3 ) )
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X4 @ Y3 )
& ( ord_le2843351958646193337nt_int @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_391_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_392_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_393_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_394_ord__eq__le__trans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( A = B )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ord_le2843351958646193337nt_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_395_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_396_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_397_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_398_ord__le__eq__trans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( B = C )
=> ( ord_le2843351958646193337nt_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_399_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_400_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_401_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_402_order__antisym,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_403_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_404_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_405_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_406_order_Otrans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ord_le2843351958646193337nt_int @ A @ C ) ) ) ).
% order.trans
thf(fact_407_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_408_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_409_order__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_eq_num @ X @ Z ) ) ) ).
% order_trans
thf(fact_410_order__trans,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ Y @ Z )
=> ( ord_le2843351958646193337nt_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_411_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_412_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_413_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B5: num] :
( ( ord_less_eq_num @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: num,B5: num] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_414_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: int,B5: int] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_415_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A5: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A5 )
& ( ord_less_eq_nat @ A5 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_416_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: num,Z3: num] : ( Y4 = Z3 ) )
= ( ^ [A5: num,B2: num] :
( ( ord_less_eq_num @ B2 @ A5 )
& ( ord_less_eq_num @ A5 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_417_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr958786334691620121nt_int,Z3: set_Pr958786334691620121nt_int] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B2 @ A5 )
& ( ord_le2843351958646193337nt_int @ A5 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_418_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A5: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A5 )
& ( ord_less_eq_int @ A5 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_419_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_420_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_421_dual__order_Oantisym,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_422_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_423_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_424_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_425_dual__order_Otrans,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( ord_le2843351958646193337nt_int @ C @ B )
=> ( ord_le2843351958646193337nt_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_426_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_427_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_428_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_429_antisym,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_430_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_431_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A5: nat,B2: nat] :
( ( ord_less_eq_nat @ A5 @ B2 )
& ( ord_less_eq_nat @ B2 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_432_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z3: num] : ( Y4 = Z3 ) )
= ( ^ [A5: num,B2: num] :
( ( ord_less_eq_num @ A5 @ B2 )
& ( ord_less_eq_num @ B2 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_433_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr958786334691620121nt_int,Z3: set_Pr958786334691620121nt_int] : ( Y4 = Z3 ) )
= ( ^ [A5: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A5 @ B2 )
& ( ord_le2843351958646193337nt_int @ B2 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_434_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A5: int,B2: int] :
( ( ord_less_eq_int @ A5 @ B2 )
& ( ord_less_eq_int @ B2 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_435_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_436_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_437_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_438_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_439_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_440_order__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_441_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_442_order__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_443_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_444_order__subst1,axiom,
! [A: nat,F: set_Pr958786334691620121nt_int > nat,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ! [X5: set_Pr958786334691620121nt_int,Y2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_445_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_446_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_447_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_448_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_449_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_450_order__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_451_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_452_order__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_453_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_454_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_le2843351958646193337nt_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_le2843351958646193337nt_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_455_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_456_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_457_order__eq__refl,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( X = Y )
=> ( ord_le2843351958646193337nt_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_458_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_459_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_460_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_461_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_462_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_463_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_464_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_465_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_466_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_467_ord__eq__le__subst,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_468_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_469_ord__eq__le__subst,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_470_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_471_ord__eq__le__subst,axiom,
! [A: set_Pr958786334691620121nt_int,F: nat > set_Pr958786334691620121nt_int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_le2843351958646193337nt_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_le2843351958646193337nt_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_472_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_473_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_474_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_475_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_476_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_477_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_478_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_479_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_480_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_481_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_le2843351958646193337nt_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_le2843351958646193337nt_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_482_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_483_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_484_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_485_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_486_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_487_order__antisym__conv,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( ( ord_le2843351958646193337nt_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_488_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_489_min__def,axiom,
( ord_min_nat
= ( ^ [A5: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B2 ) @ A5 @ B2 ) ) ) ).
% min_def
thf(fact_490_min__def,axiom,
( ord_min_num
= ( ^ [A5: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B2 ) @ A5 @ B2 ) ) ) ).
% min_def
thf(fact_491_min__def,axiom,
( ord_mi3891036259616950784nt_int
= ( ^ [A5: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : ( if_set4441254072200386271nt_int @ ( ord_le2843351958646193337nt_int @ A5 @ B2 ) @ A5 @ B2 ) ) ) ).
% min_def
thf(fact_492_min__def,axiom,
( ord_min_int
= ( ^ [A5: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B2 ) @ A5 @ B2 ) ) ) ).
% min_def
thf(fact_493_min__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_min_nat @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_494_min__absorb1,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_min_num @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_495_min__absorb1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_mi3891036259616950784nt_int @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_496_min__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_min_int @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_497__092_060open_062knights__path_A_Iboard_An_Am_A_N_A_123_I1_M_A1_J_125_A_092_060union_062_A_123_I1_M_A1_J_125_J_A_I_I1_M_A1_J_A_D_As_092_060_094sub_062k_A_D_Arev_Aps_H_A_064_A_091s_092_060_094sub_062j_093_J_092_060close_062,axiom,
knights_path @ ( sup_su6024340866399070445nt_int @ ( minus_1052850069191792384nt_int @ ( board @ n @ m ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ bot_bo1796632182523588997nt_int ) ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ bot_bo1796632182523588997nt_int ) ) @ ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( cons_P3334398858971670639nt_int @ s_k @ ( append7030698103840186580nt_int @ ( rev_Pr2923690841345412895nt_int @ ps2 ) @ ( cons_P3334398858971670639nt_int @ s_j @ nil_Pr2300489316682597567nt_int ) ) ) ) ).
% \<open>knights_path (board n m - {(1, 1)} \<union> {(1, 1)}) ((1, 1) # s\<^sub>k # rev ps' @ [s\<^sub>j])\<close>
thf(fact_498_the__elem__eq,axiom,
! [X: product_prod_int_int] :
( ( the_el8326832613380209454nt_int @ ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) )
= X ) ).
% the_elem_eq
thf(fact_499_insert__Diff__single,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ( insert5033312907999012233nt_int @ A @ ( minus_1052850069191792384nt_int @ A2 @ ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) )
= ( insert5033312907999012233nt_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_500_Diff__eq__empty__iff,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ( minus_1052850069191792384nt_int @ A2 @ B4 )
= bot_bo1796632182523588997nt_int )
= ( ord_le2843351958646193337nt_int @ A2 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_501_singleton__insert__inj__eq,axiom,
! [B: product_prod_int_int,A: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ( ( insert5033312907999012233nt_int @ B @ bot_bo1796632182523588997nt_int )
= ( insert5033312907999012233nt_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_le2843351958646193337nt_int @ A2 @ ( insert5033312907999012233nt_int @ B @ bot_bo1796632182523588997nt_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_502_singleton__insert__inj__eq_H,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B: product_prod_int_int] :
( ( ( insert5033312907999012233nt_int @ A @ A2 )
= ( insert5033312907999012233nt_int @ B @ bot_bo1796632182523588997nt_int ) )
= ( ( A = B )
& ( ord_le2843351958646193337nt_int @ A2 @ ( insert5033312907999012233nt_int @ B @ bot_bo1796632182523588997nt_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_503_Diff__insert0,axiom,
! [X: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ~ ( member5262025264175285858nt_int @ X @ A2 )
=> ( ( minus_1052850069191792384nt_int @ A2 @ ( insert5033312907999012233nt_int @ X @ B4 ) )
= ( minus_1052850069191792384nt_int @ A2 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_504_empty__Collect__eq,axiom,
! [P: product_prod_int_int > $o] :
( ( bot_bo1796632182523588997nt_int
= ( collec213857154873943460nt_int @ P ) )
= ( ! [X4: product_prod_int_int] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_505_Collect__empty__eq,axiom,
! [P: product_prod_int_int > $o] :
( ( ( collec213857154873943460nt_int @ P )
= bot_bo1796632182523588997nt_int )
= ( ! [X4: product_prod_int_int] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_506_all__not__in__conv,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( ! [X4: product_prod_int_int] :
~ ( member5262025264175285858nt_int @ X4 @ A2 ) )
= ( A2 = bot_bo1796632182523588997nt_int ) ) ).
% all_not_in_conv
thf(fact_507_empty__iff,axiom,
! [C: product_prod_int_int] :
~ ( member5262025264175285858nt_int @ C @ bot_bo1796632182523588997nt_int ) ).
% empty_iff
thf(fact_508_subset__antisym,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B4 )
=> ( ( ord_le2843351958646193337nt_int @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% subset_antisym
thf(fact_509_subsetI,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ! [X5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X5 @ A2 )
=> ( member5262025264175285858nt_int @ X5 @ B4 ) )
=> ( ord_le2843351958646193337nt_int @ A2 @ B4 ) ) ).
% subsetI
thf(fact_510_insert__absorb2,axiom,
! [X: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ( insert5033312907999012233nt_int @ X @ ( insert5033312907999012233nt_int @ X @ A2 ) )
= ( insert5033312907999012233nt_int @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_511_insert__iff,axiom,
! [A: product_prod_int_int,B: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ A @ ( insert5033312907999012233nt_int @ B @ A2 ) )
= ( ( A = B )
| ( member5262025264175285858nt_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_512_insertCI,axiom,
! [A: product_prod_int_int,B4: set_Pr958786334691620121nt_int,B: product_prod_int_int] :
( ( ~ ( member5262025264175285858nt_int @ A @ B4 )
=> ( A = B ) )
=> ( member5262025264175285858nt_int @ A @ ( insert5033312907999012233nt_int @ B @ B4 ) ) ) ).
% insertCI
thf(fact_513_sup_Oidem,axiom,
! [A: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A @ A )
= A ) ).
% sup.idem
thf(fact_514_sup__idem,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ X )
= X ) ).
% sup_idem
thf(fact_515_sup_Oleft__idem,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A @ ( sup_su6024340866399070445nt_int @ A @ B ) )
= ( sup_su6024340866399070445nt_int @ A @ B ) ) ).
% sup.left_idem
thf(fact_516_sup__left__idem,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ ( sup_su6024340866399070445nt_int @ X @ Y ) )
= ( sup_su6024340866399070445nt_int @ X @ Y ) ) ).
% sup_left_idem
thf(fact_517_sup_Oright__idem,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( sup_su6024340866399070445nt_int @ A @ B ) @ B )
= ( sup_su6024340866399070445nt_int @ A @ B ) ) ).
% sup.right_idem
thf(fact_518_Diff__idemp,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) @ B4 )
= ( minus_1052850069191792384nt_int @ A2 @ B4 ) ) ).
% Diff_idemp
thf(fact_519_Diff__iff,axiom,
! [C: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) )
= ( ( member5262025264175285858nt_int @ C @ A2 )
& ~ ( member5262025264175285858nt_int @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_520_DiffI,axiom,
! [C: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ A2 )
=> ( ~ ( member5262025264175285858nt_int @ C @ B4 )
=> ( member5262025264175285858nt_int @ C @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_521_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_522_sup_Obounded__iff,axiom,
! [B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ B @ C ) @ A )
= ( ( ord_le2843351958646193337nt_int @ B @ A )
& ( ord_le2843351958646193337nt_int @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_523_sup_Obounded__iff,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_524_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_525_le__sup__iff,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ X @ Y ) @ Z )
= ( ( ord_le2843351958646193337nt_int @ X @ Z )
& ( ord_le2843351958646193337nt_int @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_526_le__sup__iff,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z )
= ( ( ord_less_eq_int @ X @ Z )
& ( ord_less_eq_int @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_527_empty__subsetI,axiom,
! [A2: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ bot_bo1796632182523588997nt_int @ A2 ) ).
% empty_subsetI
thf(fact_528_subset__empty,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ bot_bo1796632182523588997nt_int )
= ( A2 = bot_bo1796632182523588997nt_int ) ) ).
% subset_empty
thf(fact_529_sup__bot__left,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ bot_bo1796632182523588997nt_int @ X )
= X ) ).
% sup_bot_left
thf(fact_530_sup__bot__right,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ bot_bo1796632182523588997nt_int )
= X ) ).
% sup_bot_right
thf(fact_531_bot__eq__sup__iff,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( bot_bo1796632182523588997nt_int
= ( sup_su6024340866399070445nt_int @ X @ Y ) )
= ( ( X = bot_bo1796632182523588997nt_int )
& ( Y = bot_bo1796632182523588997nt_int ) ) ) ).
% bot_eq_sup_iff
thf(fact_532_sup__eq__bot__iff,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ( sup_su6024340866399070445nt_int @ X @ Y )
= bot_bo1796632182523588997nt_int )
= ( ( X = bot_bo1796632182523588997nt_int )
& ( Y = bot_bo1796632182523588997nt_int ) ) ) ).
% sup_eq_bot_iff
thf(fact_533_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ( sup_su6024340866399070445nt_int @ A @ B )
= bot_bo1796632182523588997nt_int )
= ( ( A = bot_bo1796632182523588997nt_int )
& ( B = bot_bo1796632182523588997nt_int ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_534_sup__bot_Oleft__neutral,axiom,
! [A: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ bot_bo1796632182523588997nt_int @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_535_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( bot_bo1796632182523588997nt_int
= ( sup_su6024340866399070445nt_int @ A @ B ) )
= ( ( A = bot_bo1796632182523588997nt_int )
& ( B = bot_bo1796632182523588997nt_int ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_536_sup__bot_Oright__neutral,axiom,
! [A: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A @ bot_bo1796632182523588997nt_int )
= A ) ).
% sup_bot.right_neutral
thf(fact_537_singletonI,axiom,
! [A: product_prod_int_int] : ( member5262025264175285858nt_int @ A @ ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) ).
% singletonI
thf(fact_538_insert__subset,axiom,
! [X: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( insert5033312907999012233nt_int @ X @ A2 ) @ B4 )
= ( ( member5262025264175285858nt_int @ X @ B4 )
& ( ord_le2843351958646193337nt_int @ A2 @ B4 ) ) ) ).
% insert_subset
thf(fact_539_Un__empty,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ( sup_su6024340866399070445nt_int @ A2 @ B4 )
= bot_bo1796632182523588997nt_int )
= ( ( A2 = bot_bo1796632182523588997nt_int )
& ( B4 = bot_bo1796632182523588997nt_int ) ) ) ).
% Un_empty
thf(fact_540_Diff__cancel,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ A2 @ A2 )
= bot_bo1796632182523588997nt_int ) ).
% Diff_cancel
thf(fact_541_empty__Diff,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ bot_bo1796632182523588997nt_int @ A2 )
= bot_bo1796632182523588997nt_int ) ).
% empty_Diff
thf(fact_542_Diff__empty,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ A2 @ bot_bo1796632182523588997nt_int )
= A2 ) ).
% Diff_empty
thf(fact_543_Un__subset__iff,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ A2 @ B4 ) @ C3 )
= ( ( ord_le2843351958646193337nt_int @ A2 @ C3 )
& ( ord_le2843351958646193337nt_int @ B4 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_544_Un__insert__left,axiom,
! [A: product_prod_int_int,B4: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( insert5033312907999012233nt_int @ A @ B4 ) @ C3 )
= ( insert5033312907999012233nt_int @ A @ ( sup_su6024340866399070445nt_int @ B4 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_545_Un__insert__right,axiom,
! [A2: set_Pr958786334691620121nt_int,A: product_prod_int_int,B4: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A2 @ ( insert5033312907999012233nt_int @ A @ B4 ) )
= ( insert5033312907999012233nt_int @ A @ ( sup_su6024340866399070445nt_int @ A2 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_546_insert__Diff1,axiom,
! [X: product_prod_int_int,B4: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ X @ B4 )
=> ( ( minus_1052850069191792384nt_int @ ( insert5033312907999012233nt_int @ X @ A2 ) @ B4 )
= ( minus_1052850069191792384nt_int @ A2 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_547_Un__Diff__cancel,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A2 @ ( minus_1052850069191792384nt_int @ B4 @ A2 ) )
= ( sup_su6024340866399070445nt_int @ A2 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_548_Un__Diff__cancel2,axiom,
! [B4: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( minus_1052850069191792384nt_int @ B4 @ A2 ) @ A2 )
= ( sup_su6024340866399070445nt_int @ B4 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_549_set__append,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( set_Pr2470121279949933262nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
= ( sup_su6024340866399070445nt_int @ ( set_Pr2470121279949933262nt_int @ Xs ) @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).
% set_append
thf(fact_550_drop__Cons__numeral,axiom,
! [V: num,X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( drop_P5690361596310759935nt_int @ ( numeral_numeral_nat @ V ) @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
= ( drop_P5690361596310759935nt_int @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_551_drop__Cons__numeral,axiom,
! [V: num,X: int,Xs: list_int] :
( ( drop_int @ ( numeral_numeral_nat @ V ) @ ( cons_int @ X @ Xs ) )
= ( drop_int @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_552_Un__mono,axiom,
! [A2: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,D2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ C3 )
=> ( ( ord_le2843351958646193337nt_int @ B4 @ D2 )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ A2 @ B4 ) @ ( sup_su6024340866399070445nt_int @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_553_Un__least,axiom,
! [A2: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ C3 )
=> ( ( ord_le2843351958646193337nt_int @ B4 @ C3 )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ A2 @ B4 ) @ C3 ) ) ) ).
% Un_least
thf(fact_554_Un__upper1,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A2 @ ( sup_su6024340866399070445nt_int @ A2 @ B4 ) ) ).
% Un_upper1
thf(fact_555_Un__upper2,axiom,
! [B4: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ B4 @ ( sup_su6024340866399070445nt_int @ A2 @ B4 ) ) ).
% Un_upper2
thf(fact_556_Un__absorb1,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B4 )
=> ( ( sup_su6024340866399070445nt_int @ A2 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_557_Un__absorb2,axiom,
! [B4: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B4 @ A2 )
=> ( ( sup_su6024340866399070445nt_int @ A2 @ B4 )
= A2 ) ) ).
% Un_absorb2
thf(fact_558_subset__UnE,axiom,
! [C3: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ C3 @ ( sup_su6024340866399070445nt_int @ A2 @ B4 ) )
=> ~ ! [A6: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A6 @ A2 )
=> ! [B6: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B6 @ B4 )
=> ( C3
!= ( sup_su6024340866399070445nt_int @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_559_subset__Un__eq,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A7: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A7 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_560_Un__Diff,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ ( sup_su6024340866399070445nt_int @ A2 @ B4 ) @ C3 )
= ( sup_su6024340866399070445nt_int @ ( minus_1052850069191792384nt_int @ A2 @ C3 ) @ ( minus_1052850069191792384nt_int @ B4 @ C3 ) ) ) ).
% Un_Diff
thf(fact_561_bot__set__def,axiom,
( bot_bo1796632182523588997nt_int
= ( collec213857154873943460nt_int @ bot_bo8147686125503663512_int_o ) ) ).
% bot_set_def
thf(fact_562_Un__empty__left,axiom,
! [B4: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ bot_bo1796632182523588997nt_int @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_563_Un__empty__right,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A2 @ bot_bo1796632182523588997nt_int )
= A2 ) ).
% Un_empty_right
thf(fact_564_inf__sup__aci_I8_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ ( sup_su6024340866399070445nt_int @ X @ Y ) )
= ( sup_su6024340866399070445nt_int @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_565_inf__sup__aci_I7_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ ( sup_su6024340866399070445nt_int @ Y @ Z ) )
= ( sup_su6024340866399070445nt_int @ Y @ ( sup_su6024340866399070445nt_int @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_566_inf__sup__aci_I6_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( sup_su6024340866399070445nt_int @ X @ Y ) @ Z )
= ( sup_su6024340866399070445nt_int @ X @ ( sup_su6024340866399070445nt_int @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_567_inf__sup__aci_I5_J,axiom,
( sup_su6024340866399070445nt_int
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] : ( sup_su6024340866399070445nt_int @ Y3 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_568_sup_Oassoc,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( sup_su6024340866399070445nt_int @ A @ B ) @ C )
= ( sup_su6024340866399070445nt_int @ A @ ( sup_su6024340866399070445nt_int @ B @ C ) ) ) ).
% sup.assoc
thf(fact_569_sup__assoc,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( sup_su6024340866399070445nt_int @ X @ Y ) @ Z )
= ( sup_su6024340866399070445nt_int @ X @ ( sup_su6024340866399070445nt_int @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_570_sup_Ocommute,axiom,
( sup_su6024340866399070445nt_int
= ( ^ [A5: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : ( sup_su6024340866399070445nt_int @ B2 @ A5 ) ) ) ).
% sup.commute
thf(fact_571_sup__commute,axiom,
( sup_su6024340866399070445nt_int
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] : ( sup_su6024340866399070445nt_int @ Y3 @ X4 ) ) ) ).
% sup_commute
thf(fact_572_sup_Oleft__commute,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ B @ ( sup_su6024340866399070445nt_int @ A @ C ) )
= ( sup_su6024340866399070445nt_int @ A @ ( sup_su6024340866399070445nt_int @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_573_sup__left__commute,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ ( sup_su6024340866399070445nt_int @ Y @ Z ) )
= ( sup_su6024340866399070445nt_int @ Y @ ( sup_su6024340866399070445nt_int @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_574_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_575_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ Y @ ( sup_su6024340866399070445nt_int @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_576_inf__sup__ord_I4_J,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_577_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_578_inf__sup__ord_I3_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ X @ ( sup_su6024340866399070445nt_int @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_579_inf__sup__ord_I3_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_580_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_581_le__supE,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ A @ B ) @ X )
=> ~ ( ( ord_le2843351958646193337nt_int @ A @ X )
=> ~ ( ord_le2843351958646193337nt_int @ B @ X ) ) ) ).
% le_supE
thf(fact_582_le__supE,axiom,
! [A: int,B: int,X: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_int @ A @ X )
=> ~ ( ord_less_eq_int @ B @ X ) ) ) ).
% le_supE
thf(fact_583_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_584_le__supI,axiom,
! [A: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ X )
=> ( ( ord_le2843351958646193337nt_int @ B @ X )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_585_le__supI,axiom,
! [A: int,X: int,B: int] :
( ( ord_less_eq_int @ A @ X )
=> ( ( ord_less_eq_int @ B @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_586_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_587_sup__ge1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ X @ ( sup_su6024340866399070445nt_int @ X @ Y ) ) ).
% sup_ge1
thf(fact_588_sup__ge1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge1
thf(fact_589_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_590_sup__ge2,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ Y @ ( sup_su6024340866399070445nt_int @ X @ Y ) ) ).
% sup_ge2
thf(fact_591_sup__ge2,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge2
thf(fact_592_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_593_le__supI1,axiom,
! [X: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ A )
=> ( ord_le2843351958646193337nt_int @ X @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% le_supI1
thf(fact_594_le__supI1,axiom,
! [X: int,A: int,B: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% le_supI1
thf(fact_595_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_596_le__supI2,axiom,
! [X: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ B )
=> ( ord_le2843351958646193337nt_int @ X @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% le_supI2
thf(fact_597_le__supI2,axiom,
! [X: int,B: int,A: int] :
( ( ord_less_eq_int @ X @ B )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).
% le_supI2
thf(fact_598_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_599_sup_Omono,axiom,
! [C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,D: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ C @ A )
=> ( ( ord_le2843351958646193337nt_int @ D @ B )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ C @ D ) @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_600_sup_Omono,axiom,
! [C: int,A: int,D: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ( ord_less_eq_int @ D @ B )
=> ( ord_less_eq_int @ ( sup_sup_int @ C @ D ) @ ( sup_sup_int @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_601_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_602_sup__mono,axiom,
! [A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,D: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ C )
=> ( ( ord_le2843351958646193337nt_int @ B @ D )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ A @ B ) @ ( sup_su6024340866399070445nt_int @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_603_sup__mono,axiom,
! [A: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ B @ D )
=> ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ ( sup_sup_int @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_604_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_605_sup__least,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( ( ord_le2843351958646193337nt_int @ Z @ X )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_606_sup__least,axiom,
! [Y: int,X: int,Z: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ Z @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_607_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y3: nat] :
( ( sup_sup_nat @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_608_le__iff__sup,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_609_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y3: int] :
( ( sup_sup_int @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_610_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_611_sup_OorderE,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( A
= ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% sup.orderE
thf(fact_612_sup_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( sup_sup_int @ A @ B ) ) ) ).
% sup.orderE
thf(fact_613_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_614_sup_OorderI,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( A
= ( sup_su6024340866399070445nt_int @ A @ B ) )
=> ( ord_le2843351958646193337nt_int @ B @ A ) ) ).
% sup.orderI
thf(fact_615_sup_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( sup_sup_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% sup.orderI
thf(fact_616_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X5: nat,Y2: nat] : ( ord_less_eq_nat @ X5 @ ( F @ X5 @ Y2 ) )
=> ( ! [X5: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X5 @ Y2 ) )
=> ( ! [X5: nat,Y2: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y2 @ X5 )
=> ( ( ord_less_eq_nat @ Z4 @ X5 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z4 ) @ X5 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_617_sup__unique,axiom,
! [F: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ! [X5: set_Pr958786334691620121nt_int,Y2: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ X5 @ ( F @ X5 @ Y2 ) )
=> ( ! [X5: set_Pr958786334691620121nt_int,Y2: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ Y2 @ ( F @ X5 @ Y2 ) )
=> ( ! [X5: set_Pr958786334691620121nt_int,Y2: set_Pr958786334691620121nt_int,Z4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y2 @ X5 )
=> ( ( ord_le2843351958646193337nt_int @ Z4 @ X5 )
=> ( ord_le2843351958646193337nt_int @ ( F @ Y2 @ Z4 ) @ X5 ) ) )
=> ( ( sup_su6024340866399070445nt_int @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_618_sup__unique,axiom,
! [F: int > int > int,X: int,Y: int] :
( ! [X5: int,Y2: int] : ( ord_less_eq_int @ X5 @ ( F @ X5 @ Y2 ) )
=> ( ! [X5: int,Y2: int] : ( ord_less_eq_int @ Y2 @ ( F @ X5 @ Y2 ) )
=> ( ! [X5: int,Y2: int,Z4: int] :
( ( ord_less_eq_int @ Y2 @ X5 )
=> ( ( ord_less_eq_int @ Z4 @ X5 )
=> ( ord_less_eq_int @ ( F @ Y2 @ Z4 ) @ X5 ) ) )
=> ( ( sup_sup_int @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_619_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_620_sup_Oabsorb1,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( sup_su6024340866399070445nt_int @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_621_sup_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( sup_sup_int @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_622_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_623_sup_Oabsorb2,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( sup_su6024340866399070445nt_int @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_624_sup_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( sup_sup_int @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_625_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_626_sup__absorb1,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( ( sup_su6024340866399070445nt_int @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_627_sup__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( sup_sup_int @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_628_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_629_sup__absorb2,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( sup_su6024340866399070445nt_int @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_630_sup__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( sup_sup_int @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_631_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_632_sup_OboundedE,axiom,
! [B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ B @ C ) @ A )
=> ~ ( ( ord_le2843351958646193337nt_int @ B @ A )
=> ~ ( ord_le2843351958646193337nt_int @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_633_sup_OboundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_int @ B @ A )
=> ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_634_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_635_sup_OboundedI,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( ord_le2843351958646193337nt_int @ C @ A )
=> ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_636_sup_OboundedI,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_637_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_638_sup_Oorder__iff,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [B2: set_Pr958786334691620121nt_int,A5: set_Pr958786334691620121nt_int] :
( A5
= ( sup_su6024340866399070445nt_int @ A5 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_639_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A5: int] :
( A5
= ( sup_sup_int @ A5 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_640_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_641_sup_Ocobounded1,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ).
% sup.cobounded1
thf(fact_642_sup_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B ) ) ).
% sup.cobounded1
thf(fact_643_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_644_sup_Ocobounded2,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ B @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ).
% sup.cobounded2
thf(fact_645_sup_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A @ B ) ) ).
% sup.cobounded2
thf(fact_646_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B2 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_647_sup_Oabsorb__iff1,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [B2: set_Pr958786334691620121nt_int,A5: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A5 @ B2 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_648_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A5: int] :
( ( sup_sup_int @ A5 @ B2 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_649_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B2: nat] :
( ( sup_sup_nat @ A5 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_650_sup_Oabsorb__iff2,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A5: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ A5 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_651_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B2: int] :
( ( sup_sup_int @ A5 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_652_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_653_sup_OcoboundedI1,axiom,
! [C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ C @ A )
=> ( ord_le2843351958646193337nt_int @ C @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_654_sup_OcoboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_655_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_656_sup_OcoboundedI2,axiom,
! [C: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ C @ B )
=> ( ord_le2843351958646193337nt_int @ C @ ( sup_su6024340866399070445nt_int @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_657_sup_OcoboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_658_singleton__Un__iff,axiom,
! [X: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int )
= ( sup_su6024340866399070445nt_int @ A2 @ B4 ) )
= ( ( ( A2 = bot_bo1796632182523588997nt_int )
& ( B4
= ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) ) )
| ( ( A2
= ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) )
& ( B4 = bot_bo1796632182523588997nt_int ) )
| ( ( A2
= ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) )
& ( B4
= ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_659_Un__singleton__iff,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,X: product_prod_int_int] :
( ( ( sup_su6024340866399070445nt_int @ A2 @ B4 )
= ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) )
= ( ( ( A2 = bot_bo1796632182523588997nt_int )
& ( B4
= ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) ) )
| ( ( A2
= ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) )
& ( B4 = bot_bo1796632182523588997nt_int ) )
| ( ( A2
= ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) )
& ( B4
= ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_660_insert__is__Un,axiom,
( insert5033312907999012233nt_int
= ( ^ [A5: product_prod_int_int] : ( sup_su6024340866399070445nt_int @ ( insert5033312907999012233nt_int @ A5 @ bot_bo1796632182523588997nt_int ) ) ) ) ).
% insert_is_Un
thf(fact_661_Diff__subset__conv,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) @ C3 )
= ( ord_le2843351958646193337nt_int @ A2 @ ( sup_su6024340866399070445nt_int @ B4 @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_662_Diff__partition,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B4 )
=> ( ( sup_su6024340866399070445nt_int @ A2 @ ( minus_1052850069191792384nt_int @ B4 @ A2 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_663_ex__in__conv,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( ? [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ A2 ) )
= ( A2 != bot_bo1796632182523588997nt_int ) ) ).
% ex_in_conv
thf(fact_664_equals0I,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ! [Y2: product_prod_int_int] :
~ ( member5262025264175285858nt_int @ Y2 @ A2 )
=> ( A2 = bot_bo1796632182523588997nt_int ) ) ).
% equals0I
thf(fact_665_equals0D,axiom,
! [A2: set_Pr958786334691620121nt_int,A: product_prod_int_int] :
( ( A2 = bot_bo1796632182523588997nt_int )
=> ~ ( member5262025264175285858nt_int @ A @ A2 ) ) ).
% equals0D
thf(fact_666_emptyE,axiom,
! [A: product_prod_int_int] :
~ ( member5262025264175285858nt_int @ A @ bot_bo1796632182523588997nt_int ) ).
% emptyE
thf(fact_667_Collect__mono__iff,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) )
= ( ! [X4: product_prod_int_int] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_668_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr958786334691620121nt_int,Z3: set_Pr958786334691620121nt_int] : ( Y4 = Z3 ) )
= ( ^ [A7: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A7 @ B7 )
& ( ord_le2843351958646193337nt_int @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_669_subset__trans,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B4 )
=> ( ( ord_le2843351958646193337nt_int @ B4 @ C3 )
=> ( ord_le2843351958646193337nt_int @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_670_Collect__mono,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ! [X5: product_prod_int_int] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).
% Collect_mono
thf(fact_671_subset__refl,axiom,
! [A2: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_672_subset__iff,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A7: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int] :
! [T: product_prod_int_int] :
( ( member5262025264175285858nt_int @ T @ A7 )
=> ( member5262025264175285858nt_int @ T @ B7 ) ) ) ) ).
% subset_iff
thf(fact_673_equalityD2,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( A2 = B4 )
=> ( ord_le2843351958646193337nt_int @ B4 @ A2 ) ) ).
% equalityD2
thf(fact_674_equalityD1,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( A2 = B4 )
=> ( ord_le2843351958646193337nt_int @ A2 @ B4 ) ) ).
% equalityD1
thf(fact_675_subset__eq,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A7: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int] :
! [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ A7 )
=> ( member5262025264175285858nt_int @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_676_equalityE,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( A2 = B4 )
=> ~ ( ( ord_le2843351958646193337nt_int @ A2 @ B4 )
=> ~ ( ord_le2843351958646193337nt_int @ B4 @ A2 ) ) ) ).
% equalityE
thf(fact_677_subsetD,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,C: product_prod_int_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B4 )
=> ( ( member5262025264175285858nt_int @ C @ A2 )
=> ( member5262025264175285858nt_int @ C @ B4 ) ) ) ).
% subsetD
thf(fact_678_in__mono,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,X: product_prod_int_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B4 )
=> ( ( member5262025264175285858nt_int @ X @ A2 )
=> ( member5262025264175285858nt_int @ X @ B4 ) ) ) ).
% in_mono
thf(fact_679_mk__disjoint__insert,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ A @ A2 )
=> ? [B8: set_Pr958786334691620121nt_int] :
( ( A2
= ( insert5033312907999012233nt_int @ A @ B8 ) )
& ~ ( member5262025264175285858nt_int @ A @ B8 ) ) ) ).
% mk_disjoint_insert
thf(fact_680_insert__commute,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ( insert5033312907999012233nt_int @ X @ ( insert5033312907999012233nt_int @ Y @ A2 ) )
= ( insert5033312907999012233nt_int @ Y @ ( insert5033312907999012233nt_int @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_681_insert__eq__iff,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B: product_prod_int_int,B4: set_Pr958786334691620121nt_int] :
( ~ ( member5262025264175285858nt_int @ A @ A2 )
=> ( ~ ( member5262025264175285858nt_int @ B @ B4 )
=> ( ( ( insert5033312907999012233nt_int @ A @ A2 )
= ( insert5033312907999012233nt_int @ B @ B4 ) )
= ( ( ( A = B )
=> ( A2 = B4 ) )
& ( ( A != B )
=> ? [C4: set_Pr958786334691620121nt_int] :
( ( A2
= ( insert5033312907999012233nt_int @ B @ C4 ) )
& ~ ( member5262025264175285858nt_int @ B @ C4 )
& ( B4
= ( insert5033312907999012233nt_int @ A @ C4 ) )
& ~ ( member5262025264175285858nt_int @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_682_insert__absorb,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ A @ A2 )
=> ( ( insert5033312907999012233nt_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_683_insert__ident,axiom,
! [X: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ~ ( member5262025264175285858nt_int @ X @ A2 )
=> ( ~ ( member5262025264175285858nt_int @ X @ B4 )
=> ( ( ( insert5033312907999012233nt_int @ X @ A2 )
= ( insert5033312907999012233nt_int @ X @ B4 ) )
= ( A2 = B4 ) ) ) ) ).
% insert_ident
thf(fact_684_Set_Oset__insert,axiom,
! [X: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ X @ A2 )
=> ~ ! [B8: set_Pr958786334691620121nt_int] :
( ( A2
= ( insert5033312907999012233nt_int @ X @ B8 ) )
=> ( member5262025264175285858nt_int @ X @ B8 ) ) ) ).
% Set.set_insert
thf(fact_685_insertI2,axiom,
! [A: product_prod_int_int,B4: set_Pr958786334691620121nt_int,B: product_prod_int_int] :
( ( member5262025264175285858nt_int @ A @ B4 )
=> ( member5262025264175285858nt_int @ A @ ( insert5033312907999012233nt_int @ B @ B4 ) ) ) ).
% insertI2
thf(fact_686_insertI1,axiom,
! [A: product_prod_int_int,B4: set_Pr958786334691620121nt_int] : ( member5262025264175285858nt_int @ A @ ( insert5033312907999012233nt_int @ A @ B4 ) ) ).
% insertI1
thf(fact_687_insertE,axiom,
! [A: product_prod_int_int,B: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ A @ ( insert5033312907999012233nt_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member5262025264175285858nt_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_688_DiffD2,axiom,
! [C: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) )
=> ~ ( member5262025264175285858nt_int @ C @ B4 ) ) ).
% DiffD2
thf(fact_689_DiffD1,axiom,
! [C: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) )
=> ( member5262025264175285858nt_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_690_DiffE,axiom,
! [C: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ C @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) )
=> ~ ( ( member5262025264175285858nt_int @ C @ A2 )
=> ( member5262025264175285858nt_int @ C @ B4 ) ) ) ).
% DiffE
thf(fact_691_knights__path_Ointros_I2_J,axiom,
! [S_i: product_prod_int_int,B: set_Pr958786334691620121nt_int,S_j2: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
( ~ ( member5262025264175285858nt_int @ S_i @ B )
=> ( ( valid_step @ S_i @ S_j2 )
=> ( ( knights_path @ B @ ( cons_P3334398858971670639nt_int @ S_j2 @ Ps2 ) )
=> ( knights_path @ ( sup_su6024340866399070445nt_int @ B @ ( insert5033312907999012233nt_int @ S_i @ bot_bo1796632182523588997nt_int ) ) @ ( cons_P3334398858971670639nt_int @ S_i @ ( cons_P3334398858971670639nt_int @ S_j2 @ Ps2 ) ) ) ) ) ) ).
% knights_path.intros(2)
thf(fact_692_singleton__inject,axiom,
! [A: product_prod_int_int,B: product_prod_int_int] :
( ( ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int )
= ( insert5033312907999012233nt_int @ B @ bot_bo1796632182523588997nt_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_693_insert__not__empty,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ( insert5033312907999012233nt_int @ A @ A2 )
!= bot_bo1796632182523588997nt_int ) ).
% insert_not_empty
thf(fact_694_doubleton__eq__iff,axiom,
! [A: product_prod_int_int,B: product_prod_int_int,C: product_prod_int_int,D: product_prod_int_int] :
( ( ( insert5033312907999012233nt_int @ A @ ( insert5033312907999012233nt_int @ B @ bot_bo1796632182523588997nt_int ) )
= ( insert5033312907999012233nt_int @ C @ ( insert5033312907999012233nt_int @ D @ bot_bo1796632182523588997nt_int ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_695_singleton__iff,axiom,
! [B: product_prod_int_int,A: product_prod_int_int] :
( ( member5262025264175285858nt_int @ B @ ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_696_singletonD,axiom,
! [B: product_prod_int_int,A: product_prod_int_int] :
( ( member5262025264175285858nt_int @ B @ ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_697_subset__insertI2,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,B: product_prod_int_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B4 )
=> ( ord_le2843351958646193337nt_int @ A2 @ ( insert5033312907999012233nt_int @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_698_subset__insertI,axiom,
! [B4: set_Pr958786334691620121nt_int,A: product_prod_int_int] : ( ord_le2843351958646193337nt_int @ B4 @ ( insert5033312907999012233nt_int @ A @ B4 ) ) ).
% subset_insertI
thf(fact_699_subset__insert,axiom,
! [X: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ~ ( member5262025264175285858nt_int @ X @ A2 )
=> ( ( ord_le2843351958646193337nt_int @ A2 @ ( insert5033312907999012233nt_int @ X @ B4 ) )
= ( ord_le2843351958646193337nt_int @ A2 @ B4 ) ) ) ).
% subset_insert
thf(fact_700_insert__mono,axiom,
! [C3: set_Pr958786334691620121nt_int,D2: set_Pr958786334691620121nt_int,A: product_prod_int_int] :
( ( ord_le2843351958646193337nt_int @ C3 @ D2 )
=> ( ord_le2843351958646193337nt_int @ ( insert5033312907999012233nt_int @ A @ C3 ) @ ( insert5033312907999012233nt_int @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_701_double__diff,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B4 )
=> ( ( ord_le2843351958646193337nt_int @ B4 @ C3 )
=> ( ( minus_1052850069191792384nt_int @ B4 @ ( minus_1052850069191792384nt_int @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_702_Diff__subset,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) @ A2 ) ).
% Diff_subset
thf(fact_703_Diff__mono,axiom,
! [A2: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int,D2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ C3 )
=> ( ( ord_le2843351958646193337nt_int @ D2 @ B4 )
=> ( ord_le2843351958646193337nt_int @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) @ ( minus_1052850069191792384nt_int @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_704_insert__Diff__if,axiom,
! [X: product_prod_int_int,B4: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int] :
( ( ( member5262025264175285858nt_int @ X @ B4 )
=> ( ( minus_1052850069191792384nt_int @ ( insert5033312907999012233nt_int @ X @ A2 ) @ B4 )
= ( minus_1052850069191792384nt_int @ A2 @ B4 ) ) )
& ( ~ ( member5262025264175285858nt_int @ X @ B4 )
=> ( ( minus_1052850069191792384nt_int @ ( insert5033312907999012233nt_int @ X @ A2 ) @ B4 )
= ( insert5033312907999012233nt_int @ X @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_705_knights__path_Ocases,axiom,
! [A1: set_Pr958786334691620121nt_int,A22: list_P5707943133018811711nt_int] :
( ( knights_path @ A1 @ A22 )
=> ( ! [S_i2: product_prod_int_int] :
( ( A1
= ( insert5033312907999012233nt_int @ S_i2 @ bot_bo1796632182523588997nt_int ) )
=> ( A22
!= ( cons_P3334398858971670639nt_int @ S_i2 @ nil_Pr2300489316682597567nt_int ) ) )
=> ~ ! [S_i2: product_prod_int_int,B5: set_Pr958786334691620121nt_int] :
( ( A1
= ( sup_su6024340866399070445nt_int @ B5 @ ( insert5033312907999012233nt_int @ S_i2 @ bot_bo1796632182523588997nt_int ) ) )
=> ! [S_j: product_prod_int_int,Ps4: list_P5707943133018811711nt_int] :
( ( A22
= ( cons_P3334398858971670639nt_int @ S_i2 @ ( cons_P3334398858971670639nt_int @ S_j @ Ps4 ) ) )
=> ( ~ ( member5262025264175285858nt_int @ S_i2 @ B5 )
=> ( ( valid_step @ S_i2 @ S_j )
=> ~ ( knights_path @ B5 @ ( cons_P3334398858971670639nt_int @ S_j @ Ps4 ) ) ) ) ) ) ) ) ).
% knights_path.cases
thf(fact_706_knights__path_Osimps,axiom,
( knights_path
= ( ^ [A12: set_Pr958786334691620121nt_int,A23: list_P5707943133018811711nt_int] :
( ? [S_i3: product_prod_int_int] :
( ( A12
= ( insert5033312907999012233nt_int @ S_i3 @ bot_bo1796632182523588997nt_int ) )
& ( A23
= ( cons_P3334398858971670639nt_int @ S_i3 @ nil_Pr2300489316682597567nt_int ) ) )
| ? [S_i3: product_prod_int_int,B2: set_Pr958786334691620121nt_int,S_j3: product_prod_int_int,Ps3: list_P5707943133018811711nt_int] :
( ( A12
= ( sup_su6024340866399070445nt_int @ B2 @ ( insert5033312907999012233nt_int @ S_i3 @ bot_bo1796632182523588997nt_int ) ) )
& ( A23
= ( cons_P3334398858971670639nt_int @ S_i3 @ ( cons_P3334398858971670639nt_int @ S_j3 @ Ps3 ) ) )
& ~ ( member5262025264175285858nt_int @ S_i3 @ B2 )
& ( valid_step @ S_i3 @ S_j3 )
& ( knights_path @ B2 @ ( cons_P3334398858971670639nt_int @ S_j3 @ Ps3 ) ) ) ) ) ) ).
% knights_path.simps
thf(fact_707_subset__singleton__iff,axiom,
! [X7: set_Pr958786334691620121nt_int,A: product_prod_int_int] :
( ( ord_le2843351958646193337nt_int @ X7 @ ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) )
= ( ( X7 = bot_bo1796632182523588997nt_int )
| ( X7
= ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_708_subset__singletonD,axiom,
! [A2: set_Pr958786334691620121nt_int,X: product_prod_int_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) )
=> ( ( A2 = bot_bo1796632182523588997nt_int )
| ( A2
= ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) ) ) ) ).
% subset_singletonD
thf(fact_709_Diff__insert__absorb,axiom,
! [X: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ~ ( member5262025264175285858nt_int @ X @ A2 )
=> ( ( minus_1052850069191792384nt_int @ ( insert5033312907999012233nt_int @ X @ A2 ) @ ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_710_Diff__insert2,axiom,
! [A2: set_Pr958786334691620121nt_int,A: product_prod_int_int,B4: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ A2 @ ( insert5033312907999012233nt_int @ A @ B4 ) )
= ( minus_1052850069191792384nt_int @ ( minus_1052850069191792384nt_int @ A2 @ ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_711_insert__Diff,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ A @ A2 )
=> ( ( insert5033312907999012233nt_int @ A @ ( minus_1052850069191792384nt_int @ A2 @ ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_712_Diff__insert,axiom,
! [A2: set_Pr958786334691620121nt_int,A: product_prod_int_int,B4: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ A2 @ ( insert5033312907999012233nt_int @ A @ B4 ) )
= ( minus_1052850069191792384nt_int @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) @ ( insert5033312907999012233nt_int @ A @ bot_bo1796632182523588997nt_int ) ) ) ).
% Diff_insert
thf(fact_713_subset__Diff__insert,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,X: product_prod_int_int,C3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ ( minus_1052850069191792384nt_int @ B4 @ ( insert5033312907999012233nt_int @ X @ C3 ) ) )
= ( ( ord_le2843351958646193337nt_int @ A2 @ ( minus_1052850069191792384nt_int @ B4 @ C3 ) )
& ~ ( member5262025264175285858nt_int @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_714_Diff__single__insert,axiom,
! [A2: set_Pr958786334691620121nt_int,X: product_prod_int_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( minus_1052850069191792384nt_int @ A2 @ ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) ) @ B4 )
=> ( ord_le2843351958646193337nt_int @ A2 @ ( insert5033312907999012233nt_int @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_715_subset__insert__iff,axiom,
! [A2: set_Pr958786334691620121nt_int,X: product_prod_int_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ ( insert5033312907999012233nt_int @ X @ B4 ) )
= ( ( ( member5262025264175285858nt_int @ X @ A2 )
=> ( ord_le2843351958646193337nt_int @ ( minus_1052850069191792384nt_int @ A2 @ ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) ) @ B4 ) )
& ( ~ ( member5262025264175285858nt_int @ X @ A2 )
=> ( ord_le2843351958646193337nt_int @ A2 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_716_set__union,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( set_Pr2470121279949933262nt_int @ ( union_56799373549498035nt_int @ Xs @ Ys ) )
= ( sup_su6024340866399070445nt_int @ ( set_Pr2470121279949933262nt_int @ Xs ) @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).
% set_union
thf(fact_717_diff__shunt__var,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ( minus_1052850069191792384nt_int @ X @ Y )
= bot_bo1796632182523588997nt_int )
= ( ord_le2843351958646193337nt_int @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_718_knights__path__append,axiom,
! [B_1: set_Pr958786334691620121nt_int,Ps_1: list_P5707943133018811711nt_int,B_2: set_Pr958786334691620121nt_int,Ps_2: list_P5707943133018811711nt_int] :
( ( knights_path @ B_1 @ Ps_1 )
=> ( ( knights_path @ B_2 @ Ps_2 )
=> ( ( ( inf_in2269163501485487111nt_int @ B_1 @ B_2 )
= bot_bo1796632182523588997nt_int )
=> ( ( valid_step @ ( last_P3305686521732843992nt_int @ Ps_1 ) @ ( hd_Pro282112905867057956nt_int @ Ps_2 ) )
=> ( knights_path @ ( sup_su6024340866399070445nt_int @ B_1 @ B_2 ) @ ( append7030698103840186580nt_int @ Ps_1 @ Ps_2 ) ) ) ) ) ) ).
% knights_path_append
thf(fact_719_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_720_is__singleton__the__elem,axiom,
( is_sin8895854488172861613nt_int
= ( ^ [A7: set_Pr958786334691620121nt_int] :
( A7
= ( insert5033312907999012233nt_int @ ( the_el8326832613380209454nt_int @ A7 ) @ bot_bo1796632182523588997nt_int ) ) ) ) ).
% is_singleton_the_elem
thf(fact_721_less__by__empty,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( A2 = bot_bo1796632182523588997nt_int )
=> ( ord_le2843351958646193337nt_int @ A2 @ B4 ) ) ).
% less_by_empty
thf(fact_722_inf__right__idem,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ ( inf_in2269163501485487111nt_int @ X @ Y ) @ Y )
= ( inf_in2269163501485487111nt_int @ X @ Y ) ) ).
% inf_right_idem
thf(fact_723_inf__right__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_724_inf_Oright__idem,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ B )
= ( inf_in2269163501485487111nt_int @ A @ B ) ) ).
% inf.right_idem
thf(fact_725_inf_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_726_inf__left__idem,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X @ ( inf_in2269163501485487111nt_int @ X @ Y ) )
= ( inf_in2269163501485487111nt_int @ X @ Y ) ) ).
% inf_left_idem
thf(fact_727_inf__left__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_728_inf_Oleft__idem,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ A @ ( inf_in2269163501485487111nt_int @ A @ B ) )
= ( inf_in2269163501485487111nt_int @ A @ B ) ) ).
% inf.left_idem
thf(fact_729_inf_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_730_inf__idem,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X @ X )
= X ) ).
% inf_idem
thf(fact_731_inf__idem,axiom,
! [X: nat] :
( ( inf_inf_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_732_inf_Oidem,axiom,
! [A: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ A @ A )
= A ) ).
% inf.idem
thf(fact_733_inf_Oidem,axiom,
! [A: nat] :
( ( inf_inf_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_734_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_735_le__inf__iff,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ ( inf_in2269163501485487111nt_int @ Y @ Z ) )
= ( ( ord_le2843351958646193337nt_int @ X @ Y )
& ( ord_le2843351958646193337nt_int @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_736_le__inf__iff,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) )
= ( ( ord_less_eq_int @ X @ Y )
& ( ord_less_eq_int @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_737_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_738_inf_Obounded__iff,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ ( inf_in2269163501485487111nt_int @ B @ C ) )
= ( ( ord_le2843351958646193337nt_int @ A @ B )
& ( ord_le2843351958646193337nt_int @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_739_inf_Obounded__iff,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_740_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ bot_bo1796632182523588997nt_int @ X )
= bot_bo1796632182523588997nt_int ) ).
% boolean_algebra.conj_zero_left
thf(fact_741_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X @ bot_bo1796632182523588997nt_int )
= bot_bo1796632182523588997nt_int ) ).
% boolean_algebra.conj_zero_right
thf(fact_742_inf__bot__right,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X @ bot_bo1796632182523588997nt_int )
= bot_bo1796632182523588997nt_int ) ).
% inf_bot_right
thf(fact_743_inf__bot__left,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ bot_bo1796632182523588997nt_int @ X )
= bot_bo1796632182523588997nt_int ) ).
% inf_bot_left
thf(fact_744_sup__inf__absorb,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_745_sup__inf__absorb,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ ( inf_in2269163501485487111nt_int @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_746_inf__sup__absorb,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_747_inf__sup__absorb,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X @ ( sup_su6024340866399070445nt_int @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_748_Int__subset__iff,axiom,
! [C3: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ C3 @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) )
= ( ( ord_le2843351958646193337nt_int @ C3 @ A2 )
& ( ord_le2843351958646193337nt_int @ C3 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_749_Int__insert__right__if1,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ A @ A2 )
=> ( ( inf_in2269163501485487111nt_int @ A2 @ ( insert5033312907999012233nt_int @ A @ B4 ) )
= ( insert5033312907999012233nt_int @ A @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_750_Int__insert__right__if0,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ~ ( member5262025264175285858nt_int @ A @ A2 )
=> ( ( inf_in2269163501485487111nt_int @ A2 @ ( insert5033312907999012233nt_int @ A @ B4 ) )
= ( inf_in2269163501485487111nt_int @ A2 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_751_insert__inter__insert,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ ( insert5033312907999012233nt_int @ A @ A2 ) @ ( insert5033312907999012233nt_int @ A @ B4 ) )
= ( insert5033312907999012233nt_int @ A @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_752_Int__insert__left__if1,axiom,
! [A: product_prod_int_int,C3: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ A @ C3 )
=> ( ( inf_in2269163501485487111nt_int @ ( insert5033312907999012233nt_int @ A @ B4 ) @ C3 )
= ( insert5033312907999012233nt_int @ A @ ( inf_in2269163501485487111nt_int @ B4 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_753_Int__insert__left__if0,axiom,
! [A: product_prod_int_int,C3: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ~ ( member5262025264175285858nt_int @ A @ C3 )
=> ( ( inf_in2269163501485487111nt_int @ ( insert5033312907999012233nt_int @ A @ B4 ) @ C3 )
= ( inf_in2269163501485487111nt_int @ B4 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_754_disjoint__insert_I2_J,axiom,
! [A2: set_Pr958786334691620121nt_int,B: product_prod_int_int,B4: set_Pr958786334691620121nt_int] :
( ( bot_bo1796632182523588997nt_int
= ( inf_in2269163501485487111nt_int @ A2 @ ( insert5033312907999012233nt_int @ B @ B4 ) ) )
= ( ~ ( member5262025264175285858nt_int @ B @ A2 )
& ( bot_bo1796632182523588997nt_int
= ( inf_in2269163501485487111nt_int @ A2 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_755_disjoint__insert_I1_J,axiom,
! [B4: set_Pr958786334691620121nt_int,A: product_prod_int_int,A2: set_Pr958786334691620121nt_int] :
( ( ( inf_in2269163501485487111nt_int @ B4 @ ( insert5033312907999012233nt_int @ A @ A2 ) )
= bot_bo1796632182523588997nt_int )
= ( ~ ( member5262025264175285858nt_int @ A @ B4 )
& ( ( inf_in2269163501485487111nt_int @ B4 @ A2 )
= bot_bo1796632182523588997nt_int ) ) ) ).
% disjoint_insert(1)
thf(fact_756_insert__disjoint_I2_J,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( bot_bo1796632182523588997nt_int
= ( inf_in2269163501485487111nt_int @ ( insert5033312907999012233nt_int @ A @ A2 ) @ B4 ) )
= ( ~ ( member5262025264175285858nt_int @ A @ B4 )
& ( bot_bo1796632182523588997nt_int
= ( inf_in2269163501485487111nt_int @ A2 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_757_insert__disjoint_I1_J,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ( inf_in2269163501485487111nt_int @ ( insert5033312907999012233nt_int @ A @ A2 ) @ B4 )
= bot_bo1796632182523588997nt_int )
= ( ~ ( member5262025264175285858nt_int @ A @ B4 )
& ( ( inf_in2269163501485487111nt_int @ A2 @ B4 )
= bot_bo1796632182523588997nt_int ) ) ) ).
% insert_disjoint(1)
thf(fact_758_Diff__disjoint,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ A2 @ ( minus_1052850069191792384nt_int @ B4 @ A2 ) )
= bot_bo1796632182523588997nt_int ) ).
% Diff_disjoint
thf(fact_759_is__singletonI,axiom,
! [X: product_prod_int_int] : ( is_sin8895854488172861613nt_int @ ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) ) ).
% is_singletonI
thf(fact_760_diff__commute,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J3 ) ) ).
% diff_commute
thf(fact_761_inf__left__commute,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X @ ( inf_in2269163501485487111nt_int @ Y @ Z ) )
= ( inf_in2269163501485487111nt_int @ Y @ ( inf_in2269163501485487111nt_int @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_762_inf__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_763_inf_Oleft__commute,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ B @ ( inf_in2269163501485487111nt_int @ A @ C ) )
= ( inf_in2269163501485487111nt_int @ A @ ( inf_in2269163501485487111nt_int @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_764_inf_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_765_inf__commute,axiom,
( inf_in2269163501485487111nt_int
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] : ( inf_in2269163501485487111nt_int @ Y3 @ X4 ) ) ) ).
% inf_commute
thf(fact_766_inf__commute,axiom,
( inf_inf_nat
= ( ^ [X4: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X4 ) ) ) ).
% inf_commute
thf(fact_767_inf_Ocommute,axiom,
( inf_in2269163501485487111nt_int
= ( ^ [A5: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] : ( inf_in2269163501485487111nt_int @ B2 @ A5 ) ) ) ).
% inf.commute
thf(fact_768_inf_Ocommute,axiom,
( inf_inf_nat
= ( ^ [A5: nat,B2: nat] : ( inf_inf_nat @ B2 @ A5 ) ) ) ).
% inf.commute
thf(fact_769_inf__assoc,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ ( inf_in2269163501485487111nt_int @ X @ Y ) @ Z )
= ( inf_in2269163501485487111nt_int @ X @ ( inf_in2269163501485487111nt_int @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_770_inf__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_771_inf_Oassoc,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ C )
= ( inf_in2269163501485487111nt_int @ A @ ( inf_in2269163501485487111nt_int @ B @ C ) ) ) ).
% inf.assoc
thf(fact_772_inf_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_773_inf__sup__aci_I1_J,axiom,
( inf_in2269163501485487111nt_int
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] : ( inf_in2269163501485487111nt_int @ Y3 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_774_inf__sup__aci_I1_J,axiom,
( inf_inf_nat
= ( ^ [X4: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_775_inf__sup__aci_I2_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ ( inf_in2269163501485487111nt_int @ X @ Y ) @ Z )
= ( inf_in2269163501485487111nt_int @ X @ ( inf_in2269163501485487111nt_int @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_776_inf__sup__aci_I2_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_777_inf__sup__aci_I3_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X @ ( inf_in2269163501485487111nt_int @ Y @ Z ) )
= ( inf_in2269163501485487111nt_int @ Y @ ( inf_in2269163501485487111nt_int @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_778_inf__sup__aci_I3_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_779_inf__sup__aci_I4_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X @ ( inf_in2269163501485487111nt_int @ X @ Y ) )
= ( inf_in2269163501485487111nt_int @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_780_inf__sup__aci_I4_J,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_781_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_782_inf_OcoboundedI2,axiom,
! [B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_783_inf_OcoboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_784_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_785_inf_OcoboundedI1,axiom,
! [A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ C )
=> ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_786_inf_OcoboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_787_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_788_inf_Oabsorb__iff2,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [B2: set_Pr958786334691620121nt_int,A5: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ A5 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_789_inf_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A5: int] :
( ( inf_inf_int @ A5 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_790_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B2: nat] :
( ( inf_inf_nat @ A5 @ B2 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_791_inf_Oabsorb__iff1,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A5: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ A5 @ B2 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_792_inf_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B2: int] :
( ( inf_inf_int @ A5 @ B2 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_793_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_794_inf_Ocobounded2,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_795_inf_Ocobounded2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_796_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_797_inf_Ocobounded1,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_798_inf_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_799_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B2: nat] :
( A5
= ( inf_inf_nat @ A5 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_800_inf_Oorder__iff,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A5: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( A5
= ( inf_in2269163501485487111nt_int @ A5 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_801_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B2: int] :
( A5
= ( inf_inf_int @ A5 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_802_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_803_inf__greatest,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ X @ Z )
=> ( ord_le2843351958646193337nt_int @ X @ ( inf_in2269163501485487111nt_int @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_804_inf__greatest,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Z )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_805_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_806_inf_OboundedI,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ A @ C )
=> ( ord_le2843351958646193337nt_int @ A @ ( inf_in2269163501485487111nt_int @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_807_inf_OboundedI,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_808_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_809_inf_OboundedE,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ ( inf_in2269163501485487111nt_int @ B @ C ) )
=> ~ ( ( ord_le2843351958646193337nt_int @ A @ B )
=> ~ ( ord_le2843351958646193337nt_int @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_810_inf_OboundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
=> ~ ( ( ord_less_eq_int @ A @ B )
=> ~ ( ord_less_eq_int @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_811_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_812_inf__absorb2,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ( ( inf_in2269163501485487111nt_int @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_813_inf__absorb2,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( inf_inf_int @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_814_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_815_inf__absorb1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( inf_in2269163501485487111nt_int @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_816_inf__absorb1,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( inf_inf_int @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_817_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_818_inf_Oabsorb2,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( inf_in2269163501485487111nt_int @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_819_inf_Oabsorb2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( inf_inf_int @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_820_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_821_inf_Oabsorb1,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( inf_in2269163501485487111nt_int @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_822_inf_Oabsorb1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( inf_inf_int @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_823_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y3: nat] :
( ( inf_inf_nat @ X4 @ Y3 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_824_le__iff__inf,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X4 @ Y3 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_825_le__iff__inf,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y3: int] :
( ( inf_inf_int @ X4 @ Y3 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_826_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X5: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y2 ) @ X5 )
=> ( ! [X5: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y2 ) @ Y2 )
=> ( ! [X5: nat,Y2: nat,Z4: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ( ord_less_eq_nat @ X5 @ Z4 )
=> ( ord_less_eq_nat @ X5 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_827_inf__unique,axiom,
! [F: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ! [X5: set_Pr958786334691620121nt_int,Y2: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( F @ X5 @ Y2 ) @ X5 )
=> ( ! [X5: set_Pr958786334691620121nt_int,Y2: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( F @ X5 @ Y2 ) @ Y2 )
=> ( ! [X5: set_Pr958786334691620121nt_int,Y2: set_Pr958786334691620121nt_int,Z4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X5 @ Y2 )
=> ( ( ord_le2843351958646193337nt_int @ X5 @ Z4 )
=> ( ord_le2843351958646193337nt_int @ X5 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_in2269163501485487111nt_int @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_828_inf__unique,axiom,
! [F: int > int > int,X: int,Y: int] :
( ! [X5: int,Y2: int] : ( ord_less_eq_int @ ( F @ X5 @ Y2 ) @ X5 )
=> ( ! [X5: int,Y2: int] : ( ord_less_eq_int @ ( F @ X5 @ Y2 ) @ Y2 )
=> ( ! [X5: int,Y2: int,Z4: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ( ord_less_eq_int @ X5 @ Z4 )
=> ( ord_less_eq_int @ X5 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf_int @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_829_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_830_inf_OorderI,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( A
= ( inf_in2269163501485487111nt_int @ A @ B ) )
=> ( ord_le2843351958646193337nt_int @ A @ B ) ) ).
% inf.orderI
thf(fact_831_inf_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( inf_inf_int @ A @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% inf.orderI
thf(fact_832_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_833_inf_OorderE,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( A
= ( inf_in2269163501485487111nt_int @ A @ B ) ) ) ).
% inf.orderE
thf(fact_834_inf_OorderE,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( A
= ( inf_inf_int @ A @ B ) ) ) ).
% inf.orderE
thf(fact_835_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_836_le__infI2,axiom,
! [B: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ X )
=> ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_837_le__infI2,axiom,
! [B: int,X: int,A: int] :
( ( ord_less_eq_int @ B @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_838_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_839_le__infI1,axiom,
! [A: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ X )
=> ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_840_le__infI1,axiom,
! [A: int,X: int,B: int] :
( ( ord_less_eq_int @ A @ X )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_841_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_842_inf__mono,axiom,
! [A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,D: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ C )
=> ( ( ord_le2843351958646193337nt_int @ B @ D )
=> ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ ( inf_in2269163501485487111nt_int @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_843_inf__mono,axiom,
! [A: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ B @ D )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ ( inf_inf_int @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_844_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_845_le__infI,axiom,
! [X: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ A )
=> ( ( ord_le2843351958646193337nt_int @ X @ B )
=> ( ord_le2843351958646193337nt_int @ X @ ( inf_in2269163501485487111nt_int @ A @ B ) ) ) ) ).
% le_infI
thf(fact_846_le__infI,axiom,
! [X: int,A: int,B: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ X @ B )
=> ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B ) ) ) ) ).
% le_infI
thf(fact_847_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_848_le__infE,axiom,
! [X: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ ( inf_in2269163501485487111nt_int @ A @ B ) )
=> ~ ( ( ord_le2843351958646193337nt_int @ X @ A )
=> ~ ( ord_le2843351958646193337nt_int @ X @ B ) ) ) ).
% le_infE
thf(fact_849_le__infE,axiom,
! [X: int,A: int,B: int] :
( ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B ) )
=> ~ ( ( ord_less_eq_int @ X @ A )
=> ~ ( ord_less_eq_int @ X @ B ) ) ) ).
% le_infE
thf(fact_850_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_851_inf__le2,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_852_inf__le2,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_853_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_854_inf__le1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_855_inf__le1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_856_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_857_inf__sup__ord_I1_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_858_inf__sup__ord_I1_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_859_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_860_inf__sup__ord_I2_J,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_861_inf__sup__ord_I2_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_862_sup__inf__distrib2,axiom,
! [Y: nat,Z: nat,X: nat] :
( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z ) @ X )
= ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_863_sup__inf__distrib2,axiom,
! [Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( inf_in2269163501485487111nt_int @ Y @ Z ) @ X )
= ( inf_in2269163501485487111nt_int @ ( sup_su6024340866399070445nt_int @ Y @ X ) @ ( sup_su6024340866399070445nt_int @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_864_sup__inf__distrib1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_865_sup__inf__distrib1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ ( inf_in2269163501485487111nt_int @ Y @ Z ) )
= ( inf_in2269163501485487111nt_int @ ( sup_su6024340866399070445nt_int @ X @ Y ) @ ( sup_su6024340866399070445nt_int @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_866_inf__sup__distrib2,axiom,
! [Y: nat,Z: nat,X: nat] :
( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z ) @ X )
= ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_867_inf__sup__distrib2,axiom,
! [Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ ( sup_su6024340866399070445nt_int @ Y @ Z ) @ X )
= ( sup_su6024340866399070445nt_int @ ( inf_in2269163501485487111nt_int @ Y @ X ) @ ( inf_in2269163501485487111nt_int @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_868_inf__sup__distrib1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_869_inf__sup__distrib1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X @ ( sup_su6024340866399070445nt_int @ Y @ Z ) )
= ( sup_su6024340866399070445nt_int @ ( inf_in2269163501485487111nt_int @ X @ Y ) @ ( inf_in2269163501485487111nt_int @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_870_distrib__imp2,axiom,
! [X: nat,Y: nat,Z: nat] :
( ! [X5: nat,Y2: nat,Z4: nat] :
( ( sup_sup_nat @ X5 @ ( inf_inf_nat @ Y2 @ Z4 ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X5 @ Y2 ) @ ( sup_sup_nat @ X5 @ Z4 ) ) )
=> ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_871_distrib__imp2,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ! [X5: set_Pr958786334691620121nt_int,Y2: set_Pr958786334691620121nt_int,Z4: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X5 @ ( inf_in2269163501485487111nt_int @ Y2 @ Z4 ) )
= ( inf_in2269163501485487111nt_int @ ( sup_su6024340866399070445nt_int @ X5 @ Y2 ) @ ( sup_su6024340866399070445nt_int @ X5 @ Z4 ) ) )
=> ( ( inf_in2269163501485487111nt_int @ X @ ( sup_su6024340866399070445nt_int @ Y @ Z ) )
= ( sup_su6024340866399070445nt_int @ ( inf_in2269163501485487111nt_int @ X @ Y ) @ ( inf_in2269163501485487111nt_int @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_872_distrib__imp1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ! [X5: nat,Y2: nat,Z4: nat] :
( ( inf_inf_nat @ X5 @ ( sup_sup_nat @ Y2 @ Z4 ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X5 @ Y2 ) @ ( inf_inf_nat @ X5 @ Z4 ) ) )
=> ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_873_distrib__imp1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ! [X5: set_Pr958786334691620121nt_int,Y2: set_Pr958786334691620121nt_int,Z4: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X5 @ ( sup_su6024340866399070445nt_int @ Y2 @ Z4 ) )
= ( sup_su6024340866399070445nt_int @ ( inf_in2269163501485487111nt_int @ X5 @ Y2 ) @ ( inf_in2269163501485487111nt_int @ X5 @ Z4 ) ) )
=> ( ( sup_su6024340866399070445nt_int @ X @ ( inf_in2269163501485487111nt_int @ Y @ Z ) )
= ( inf_in2269163501485487111nt_int @ ( sup_su6024340866399070445nt_int @ X @ Y ) @ ( sup_su6024340866399070445nt_int @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_874_disjoint__iff__not__equal,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ( inf_in2269163501485487111nt_int @ A2 @ B4 )
= bot_bo1796632182523588997nt_int )
= ( ! [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ A2 )
=> ! [Y3: product_prod_int_int] :
( ( member5262025264175285858nt_int @ Y3 @ B4 )
=> ( X4 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_875_Int__empty__right,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ A2 @ bot_bo1796632182523588997nt_int )
= bot_bo1796632182523588997nt_int ) ).
% Int_empty_right
thf(fact_876_Int__empty__left,axiom,
! [B4: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ bot_bo1796632182523588997nt_int @ B4 )
= bot_bo1796632182523588997nt_int ) ).
% Int_empty_left
thf(fact_877_disjoint__iff,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ( inf_in2269163501485487111nt_int @ A2 @ B4 )
= bot_bo1796632182523588997nt_int )
= ( ! [X4: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X4 @ A2 )
=> ~ ( member5262025264175285858nt_int @ X4 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_878_Int__emptyI,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ! [X5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X5 @ A2 )
=> ~ ( member5262025264175285858nt_int @ X5 @ B4 ) )
=> ( ( inf_in2269163501485487111nt_int @ A2 @ B4 )
= bot_bo1796632182523588997nt_int ) ) ).
% Int_emptyI
thf(fact_879_Int__Collect__mono,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ A2 @ B4 )
=> ( ! [X5: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X5 @ A2 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ A2 @ ( collec213857154873943460nt_int @ P ) ) @ ( inf_in2269163501485487111nt_int @ B4 @ ( collec213857154873943460nt_int @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_880_Int__greatest,axiom,
! [C3: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ C3 @ A2 )
=> ( ( ord_le2843351958646193337nt_int @ C3 @ B4 )
=> ( ord_le2843351958646193337nt_int @ C3 @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_881_Int__absorb2,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B4 )
=> ( ( inf_in2269163501485487111nt_int @ A2 @ B4 )
= A2 ) ) ).
% Int_absorb2
thf(fact_882_Int__absorb1,axiom,
! [B4: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B4 @ A2 )
=> ( ( inf_in2269163501485487111nt_int @ A2 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_883_Int__lower2,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_884_Int__lower1,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) @ A2 ) ).
% Int_lower1
thf(fact_885_Int__mono,axiom,
! [A2: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,D2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ C3 )
=> ( ( ord_le2843351958646193337nt_int @ B4 @ D2 )
=> ( ord_le2843351958646193337nt_int @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) @ ( inf_in2269163501485487111nt_int @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_886_Int__insert__right,axiom,
! [A: product_prod_int_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ( member5262025264175285858nt_int @ A @ A2 )
=> ( ( inf_in2269163501485487111nt_int @ A2 @ ( insert5033312907999012233nt_int @ A @ B4 ) )
= ( insert5033312907999012233nt_int @ A @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) ) ) )
& ( ~ ( member5262025264175285858nt_int @ A @ A2 )
=> ( ( inf_in2269163501485487111nt_int @ A2 @ ( insert5033312907999012233nt_int @ A @ B4 ) )
= ( inf_in2269163501485487111nt_int @ A2 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_887_Int__insert__left,axiom,
! [A: product_prod_int_int,C3: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ( member5262025264175285858nt_int @ A @ C3 )
=> ( ( inf_in2269163501485487111nt_int @ ( insert5033312907999012233nt_int @ A @ B4 ) @ C3 )
= ( insert5033312907999012233nt_int @ A @ ( inf_in2269163501485487111nt_int @ B4 @ C3 ) ) ) )
& ( ~ ( member5262025264175285858nt_int @ A @ C3 )
=> ( ( inf_in2269163501485487111nt_int @ ( insert5033312907999012233nt_int @ A @ B4 ) @ C3 )
= ( inf_in2269163501485487111nt_int @ B4 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_888_inf__min,axiom,
inf_inf_nat = ord_min_nat ).
% inf_min
thf(fact_889_Diff__Int__distrib2,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) @ C3 )
= ( minus_1052850069191792384nt_int @ ( inf_in2269163501485487111nt_int @ A2 @ C3 ) @ ( inf_in2269163501485487111nt_int @ B4 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_890_Diff__Int__distrib,axiom,
! [C3: set_Pr958786334691620121nt_int,A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ C3 @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) )
= ( minus_1052850069191792384nt_int @ ( inf_in2269163501485487111nt_int @ C3 @ A2 ) @ ( inf_in2269163501485487111nt_int @ C3 @ B4 ) ) ) ).
% Diff_Int_distrib
thf(fact_891_Diff__Diff__Int,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ A2 @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) )
= ( inf_in2269163501485487111nt_int @ A2 @ B4 ) ) ).
% Diff_Diff_Int
thf(fact_892_Diff__Int2,axiom,
! [A2: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ ( inf_in2269163501485487111nt_int @ A2 @ C3 ) @ ( inf_in2269163501485487111nt_int @ B4 @ C3 ) )
= ( minus_1052850069191792384nt_int @ ( inf_in2269163501485487111nt_int @ A2 @ C3 ) @ B4 ) ) ).
% Diff_Int2
thf(fact_893_Int__Diff,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) @ C3 )
= ( inf_in2269163501485487111nt_int @ A2 @ ( minus_1052850069191792384nt_int @ B4 @ C3 ) ) ) ).
% Int_Diff
thf(fact_894_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_895_distrib__sup__le,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ X @ ( inf_in2269163501485487111nt_int @ Y @ Z ) ) @ ( inf_in2269163501485487111nt_int @ ( sup_su6024340866399070445nt_int @ X @ Y ) @ ( sup_su6024340866399070445nt_int @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_896_distrib__sup__le,axiom,
! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z ) ) @ ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_897_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_898_distrib__inf__le,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] : ( ord_le2843351958646193337nt_int @ ( sup_su6024340866399070445nt_int @ ( inf_in2269163501485487111nt_int @ X @ Y ) @ ( inf_in2269163501485487111nt_int @ X @ Z ) ) @ ( inf_in2269163501485487111nt_int @ X @ ( sup_su6024340866399070445nt_int @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_899_distrib__inf__le,axiom,
! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z ) ) @ ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_900_Diff__triv,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ( inf_in2269163501485487111nt_int @ A2 @ B4 )
= bot_bo1796632182523588997nt_int )
=> ( ( minus_1052850069191792384nt_int @ A2 @ B4 )
= A2 ) ) ).
% Diff_triv
thf(fact_901_Int__Diff__disjoint,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) )
= bot_bo1796632182523588997nt_int ) ).
% Int_Diff_disjoint
thf(fact_902_Un__Int__assoc__eq,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( ( sup_su6024340866399070445nt_int @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) @ C3 )
= ( inf_in2269163501485487111nt_int @ A2 @ ( sup_su6024340866399070445nt_int @ B4 @ C3 ) ) )
= ( ord_le2843351958646193337nt_int @ C3 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_903_Un__Diff__Int,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_904_Int__Diff__Un,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ ( inf_in2269163501485487111nt_int @ A2 @ B4 ) @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_905_Diff__Int,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ A2 @ ( inf_in2269163501485487111nt_int @ B4 @ C3 ) )
= ( sup_su6024340866399070445nt_int @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) @ ( minus_1052850069191792384nt_int @ A2 @ C3 ) ) ) ).
% Diff_Int
thf(fact_906_Diff__Un,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int,C3: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ A2 @ ( sup_su6024340866399070445nt_int @ B4 @ C3 ) )
= ( inf_in2269163501485487111nt_int @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) @ ( minus_1052850069191792384nt_int @ A2 @ C3 ) ) ) ).
% Diff_Un
thf(fact_907_is__singletonI_H,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( A2 != bot_bo1796632182523588997nt_int )
=> ( ! [X5: product_prod_int_int,Y2: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X5 @ A2 )
=> ( ( member5262025264175285858nt_int @ Y2 @ A2 )
=> ( X5 = Y2 ) ) )
=> ( is_sin8895854488172861613nt_int @ A2 ) ) ) ).
% is_singletonI'
thf(fact_908_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_909_le__trans,axiom,
! [I3: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_910_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_911_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_912_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_913_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_914_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_915_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_916_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_917_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_918_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_919_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_920_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_921_min__diff,axiom,
! [M: nat,I3: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I3 ) @ ( minus_minus_nat @ N @ I3 ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I3 ) ) ).
% min_diff
thf(fact_922_is__singletonE,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( is_sin8895854488172861613nt_int @ A2 )
=> ~ ! [X5: product_prod_int_int] :
( A2
!= ( insert5033312907999012233nt_int @ X5 @ bot_bo1796632182523588997nt_int ) ) ) ).
% is_singletonE
thf(fact_923_is__singleton__def,axiom,
( is_sin8895854488172861613nt_int
= ( ^ [A7: set_Pr958786334691620121nt_int] :
? [X4: product_prod_int_int] :
( A7
= ( insert5033312907999012233nt_int @ X4 @ bot_bo1796632182523588997nt_int ) ) ) ) ).
% is_singleton_def
thf(fact_924_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( sup_su6024340866399070445nt_int @ X @ bot_bo1796632182523588997nt_int )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_925_Collect__empty__eq__bot,axiom,
! [P: product_prod_int_int > $o] :
( ( ( collec213857154873943460nt_int @ P )
= bot_bo1796632182523588997nt_int )
= ( P = bot_bo8147686125503663512_int_o ) ) ).
% Collect_empty_eq_bot
thf(fact_926_bot__empty__eq,axiom,
( bot_bo8147686125503663512_int_o
= ( ^ [X4: product_prod_int_int] : ( member5262025264175285858nt_int @ X4 @ bot_bo1796632182523588997nt_int ) ) ) ).
% bot_empty_eq
thf(fact_927_subrelI,axiom,
! [R: set_Pr2560585780119916871nt_int,S: set_Pr2560585780119916871nt_int] :
( ! [X5: product_prod_int_int,Y2: product_prod_int_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X5 @ Y2 ) @ R )
=> ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X5 @ Y2 ) @ S ) )
=> ( ord_le6090609446090860775nt_int @ R @ S ) ) ).
% subrelI
thf(fact_928_subrelI,axiom,
! [R: set_Pr2166573435379693421nt_int,S: set_Pr2166573435379693421nt_int] :
( ! [X5: set_Pr958786334691620121nt_int,Y2: list_P5707943133018811711nt_int] :
( ( member6077610525077772982nt_int @ ( produc2261658324281137661nt_int @ X5 @ Y2 ) @ R )
=> ( member6077610525077772982nt_int @ ( produc2261658324281137661nt_int @ X5 @ Y2 ) @ S ) )
=> ( ord_le7621015491206151949nt_int @ R @ S ) ) ).
% subrelI
thf(fact_929_subrelI,axiom,
! [R: set_Pr8218934625190621173um_num,S: set_Pr8218934625190621173um_num] :
( ! [X5: num,Y2: num] :
( ( member7279096912039735102um_num @ ( product_Pair_num_num @ X5 @ Y2 ) @ R )
=> ( member7279096912039735102um_num @ ( product_Pair_num_num @ X5 @ Y2 ) @ S ) )
=> ( ord_le880128212290418581um_num @ R @ S ) ) ).
% subrelI
thf(fact_930_subrelI,axiom,
! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ! [X5: int,Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y2 ) @ R )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y2 ) @ S ) )
=> ( ord_le2843351958646193337nt_int @ R @ S ) ) ).
% subrelI
thf(fact_931_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_932_set__removeAll,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( set_Pr2470121279949933262nt_int @ ( remove494893200766290885nt_int @ X @ Xs ) )
= ( minus_1052850069191792384nt_int @ ( set_Pr2470121279949933262nt_int @ Xs ) @ ( insert5033312907999012233nt_int @ X @ bot_bo1796632182523588997nt_int ) ) ) ).
% set_removeAll
thf(fact_933_insert__subsetI,axiom,
! [X: product_prod_int_int,A2: set_Pr958786334691620121nt_int,X7: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ X @ A2 )
=> ( ( ord_le2843351958646193337nt_int @ X7 @ A2 )
=> ( ord_le2843351958646193337nt_int @ ( insert5033312907999012233nt_int @ X @ X7 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_934_subset__emptyI,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ! [X5: product_prod_int_int] :
~ ( member5262025264175285858nt_int @ X5 @ A2 )
=> ( ord_le2843351958646193337nt_int @ A2 @ bot_bo1796632182523588997nt_int ) ) ).
% subset_emptyI
thf(fact_935_removeAll__id,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
=> ( ( remove494893200766290885nt_int @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_936_removeAll__append,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( remove494893200766290885nt_int @ X @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
= ( append7030698103840186580nt_int @ ( remove494893200766290885nt_int @ X @ Xs ) @ ( remove494893200766290885nt_int @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_937_removeAll_Osimps_I2_J,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( ( X = Y )
=> ( ( remove494893200766290885nt_int @ X @ ( cons_P3334398858971670639nt_int @ Y @ Xs ) )
= ( remove494893200766290885nt_int @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( remove494893200766290885nt_int @ X @ ( cons_P3334398858971670639nt_int @ Y @ Xs ) )
= ( cons_P3334398858971670639nt_int @ Y @ ( remove494893200766290885nt_int @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_938_removeAll_Osimps_I2_J,axiom,
! [X: int,Y: int,Xs: list_int] :
( ( ( X = Y )
=> ( ( removeAll_int @ X @ ( cons_int @ Y @ Xs ) )
= ( removeAll_int @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( removeAll_int @ X @ ( cons_int @ Y @ Xs ) )
= ( cons_int @ Y @ ( removeAll_int @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_939_removeAll_Osimps_I1_J,axiom,
! [X: product_prod_int_int] :
( ( remove494893200766290885nt_int @ X @ nil_Pr2300489316682597567nt_int )
= nil_Pr2300489316682597567nt_int ) ).
% removeAll.simps(1)
thf(fact_940_ssubst__Pair__rhs,axiom,
! [R: int,S: int,R2: set_Pr958786334691620121nt_int,S2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_941_ssubst__Pair__rhs,axiom,
! [R: product_prod_int_int,S: product_prod_int_int,R2: set_Pr2560585780119916871nt_int,S2: product_prod_int_int] :
( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_942_ssubst__Pair__rhs,axiom,
! [R: set_Pr958786334691620121nt_int,S: list_P5707943133018811711nt_int,R2: set_Pr2166573435379693421nt_int,S2: list_P5707943133018811711nt_int] :
( ( member6077610525077772982nt_int @ ( produc2261658324281137661nt_int @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member6077610525077772982nt_int @ ( produc2261658324281137661nt_int @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_943_ssubst__Pair__rhs,axiom,
! [R: num,S: num,R2: set_Pr8218934625190621173um_num,S2: num] :
( ( member7279096912039735102um_num @ ( product_Pair_num_num @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member7279096912039735102um_num @ ( product_Pair_num_num @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_944__092_060open_062k_A_060_Alength_Aps_092_060close_062,axiom,
ord_less_nat @ k @ ( size_s5157815400016825771nt_int @ ps ) ).
% \<open>k < length ps\<close>
thf(fact_945_path__checker_Osimps_I2_J,axiom,
! [B: set_Pr958786334691620121nt_int,S_i: product_prod_int_int] :
( ( path_checker @ B @ ( cons_P3334398858971670639nt_int @ S_i @ nil_Pr2300489316682597567nt_int ) )
= ( ( insert5033312907999012233nt_int @ S_i @ bot_bo1796632182523588997nt_int )
= B ) ) ).
% path_checker.simps(2)
thf(fact_946_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_947_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_948_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_949_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_950_Compl__subset__Compl__iff,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ A2 ) @ ( uminus6221592323253981072nt_int @ B4 ) )
= ( ord_le2843351958646193337nt_int @ B4 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_951_Compl__anti__mono,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ B4 )
=> ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ B4 ) @ ( uminus6221592323253981072nt_int @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_952__092_060open_0623_A_060_Alength_Aps_092_060_094sub_062r_092_060close_062,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( size_s5157815400016825771nt_int @ ps_r ) ).
% \<open>3 < length ps\<^sub>r\<close>
thf(fact_953_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_954_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_955_compl__le__compl__iff,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ X ) @ ( uminus6221592323253981072nt_int @ Y ) )
= ( ord_le2843351958646193337nt_int @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_956_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_957_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_958_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_959_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_960_Compl__disjoint2,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ ( uminus6221592323253981072nt_int @ A2 ) @ A2 )
= bot_bo1796632182523588997nt_int ) ).
% Compl_disjoint2
thf(fact_961_Compl__disjoint,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ A2 @ ( uminus6221592323253981072nt_int @ A2 ) )
= bot_bo1796632182523588997nt_int ) ).
% Compl_disjoint
thf(fact_962_min_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_min_nat @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_963_min_Oabsorb3,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_min_num @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_964_min_Oabsorb3,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_min_int @ A @ B )
= A ) ) ).
% min.absorb3
thf(fact_965_min_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_min_nat @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_966_min_Oabsorb4,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_min_num @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_967_min_Oabsorb4,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_min_int @ A @ B )
= B ) ) ).
% min.absorb4
thf(fact_968_min__less__iff__conj,axiom,
! [Z: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z @ X )
& ( ord_less_nat @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_969_min__less__iff__conj,axiom,
! [Z: num,X: num,Y: num] :
( ( ord_less_num @ Z @ ( ord_min_num @ X @ Y ) )
= ( ( ord_less_num @ Z @ X )
& ( ord_less_num @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_970_min__less__iff__conj,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ Z @ ( ord_min_int @ X @ Y ) )
= ( ( ord_less_int @ Z @ X )
& ( ord_less_int @ Z @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_971_append__eq__append__conv,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Us2: list_P5707943133018811711nt_int,Vs: list_P5707943133018811711nt_int] :
( ( ( ( size_s5157815400016825771nt_int @ Xs )
= ( size_s5157815400016825771nt_int @ Ys ) )
| ( ( size_s5157815400016825771nt_int @ Us2 )
= ( size_s5157815400016825771nt_int @ Vs ) ) )
=> ( ( ( append7030698103840186580nt_int @ Xs @ Us2 )
= ( append7030698103840186580nt_int @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_972_Diff__Compl,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( minus_1052850069191792384nt_int @ A2 @ ( uminus6221592323253981072nt_int @ B4 ) )
= ( inf_in2269163501485487111nt_int @ A2 @ B4 ) ) ).
% Diff_Compl
thf(fact_973_Compl__Diff__eq,axiom,
! [A2: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
( ( uminus6221592323253981072nt_int @ ( minus_1052850069191792384nt_int @ A2 @ B4 ) )
= ( sup_su6024340866399070445nt_int @ ( uminus6221592323253981072nt_int @ A2 ) @ B4 ) ) ).
% Compl_Diff_eq
thf(fact_974_length__rev,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( size_s5157815400016825771nt_int @ ( rev_Pr2923690841345412895nt_int @ Xs ) )
= ( size_s5157815400016825771nt_int @ Xs ) ) ).
% length_rev
thf(fact_975__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062k_O_A_092_060lbrakk_062hd_A_Idrop_Ak_Aps_J_A_061_A_I1_M_A1_J_059_Ak_A_060_Alength_Aps_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [K2: nat] :
( ( ( hd_Pro282112905867057956nt_int @ ( drop_P5690361596310759935nt_int @ K2 @ ps ) )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) )
=> ~ ( ord_less_nat @ K2 @ ( size_s5157815400016825771nt_int @ ps ) ) ) ).
% \<open>\<And>thesis. (\<And>k. \<lbrakk>hd (drop k ps) = (1, 1); k < length ps\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_976_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_977_boolean__algebra_Oconj__cancel__right,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X @ ( uminus6221592323253981072nt_int @ X ) )
= bot_bo1796632182523588997nt_int ) ).
% boolean_algebra.conj_cancel_right
thf(fact_978_boolean__algebra_Oconj__cancel__left,axiom,
! [X: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ ( uminus6221592323253981072nt_int @ X ) @ X )
= bot_bo1796632182523588997nt_int ) ).
% boolean_algebra.conj_cancel_left
thf(fact_979_inf__compl__bot__right,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X @ ( inf_in2269163501485487111nt_int @ Y @ ( uminus6221592323253981072nt_int @ X ) ) )
= bot_bo1796632182523588997nt_int ) ).
% inf_compl_bot_right
thf(fact_980_inf__compl__bot__left2,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ X @ ( inf_in2269163501485487111nt_int @ ( uminus6221592323253981072nt_int @ X ) @ Y ) )
= bot_bo1796632182523588997nt_int ) ).
% inf_compl_bot_left2
thf(fact_981_inf__compl__bot__left1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( inf_in2269163501485487111nt_int @ ( uminus6221592323253981072nt_int @ X ) @ ( inf_in2269163501485487111nt_int @ X @ Y ) )
= bot_bo1796632182523588997nt_int ) ).
% inf_compl_bot_left1
thf(fact_982_subset__Compl__singleton,axiom,
! [A2: set_Pr958786334691620121nt_int,B: product_prod_int_int] :
( ( ord_le2843351958646193337nt_int @ A2 @ ( uminus6221592323253981072nt_int @ ( insert5033312907999012233nt_int @ B @ bot_bo1796632182523588997nt_int ) ) )
= ( ~ ( member5262025264175285858nt_int @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_983_length__drop,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( size_s5157815400016825771nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) )
= ( minus_minus_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N ) ) ).
% length_drop
thf(fact_984_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_985_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_986_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_987_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_988_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_989_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_990_min__number__of_I2_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_min_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( numeral_numeral_int @ U ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_min_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ) ) ).
% min_number_of(2)
thf(fact_991_min__number__of_I3_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_min_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_min_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ V ) ) ) ) ).
% min_number_of(3)
thf(fact_992_min__number__of_I4_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_min_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_min_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ) ) ).
% min_number_of(4)
thf(fact_993_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( nil_Pr2300489316682597567nt_int
= ( drop_P5690361596310759935nt_int @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_994_drop__eq__Nil,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ( drop_P5690361596310759935nt_int @ N @ Xs )
= nil_Pr2300489316682597567nt_int )
= ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_995_drop__all,axiom,
! [Xs: list_P5707943133018811711nt_int,N: nat] :
( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N )
=> ( ( drop_P5690361596310759935nt_int @ N @ Xs )
= nil_Pr2300489316682597567nt_int ) ) ).
% drop_all
thf(fact_996_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_997_drop__append,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( drop_P5690361596310759935nt_int @ N @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
= ( append7030698103840186580nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) @ ( drop_P5690361596310759935nt_int @ ( minus_minus_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_998_last__drop,axiom,
! [N: nat,Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ( ( last_P3305686521732843992nt_int @ ( drop_P5690361596310759935nt_int @ N @ Xs ) )
= ( last_P3305686521732843992nt_int @ Xs ) ) ) ).
% last_drop
thf(fact_999_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_1000_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_1001_diff__numeral__special_I11_J,axiom,
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_1002_diff__numeral__special_I10_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_1003_inf_Ostrict__coboundedI2,axiom,
! [B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B @ C )
=> ( ord_le7563427860532173253nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1004_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1005_inf_Ostrict__coboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ B @ C )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1006_inf_Ostrict__coboundedI1,axiom,
! [A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A @ C )
=> ( ord_le7563427860532173253nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1007_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1008_inf_Ostrict__coboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ C )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1009_inf_Ostrict__order__iff,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [A5: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( A5
= ( inf_in2269163501485487111nt_int @ A5 @ B2 ) )
& ( A5 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1010_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A5: nat,B2: nat] :
( ( A5
= ( inf_inf_nat @ A5 @ B2 ) )
& ( A5 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1011_inf_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A5: int,B2: int] :
( ( A5
= ( inf_inf_int @ A5 @ B2 ) )
& ( A5 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1012_inf_Ostrict__boundedE,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A @ ( inf_in2269163501485487111nt_int @ B @ C ) )
=> ~ ( ( ord_le7563427860532173253nt_int @ A @ B )
=> ~ ( ord_le7563427860532173253nt_int @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1013_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1014_inf_Ostrict__boundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ ( inf_inf_int @ B @ C ) )
=> ~ ( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1015_inf_Oabsorb4,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B @ A )
=> ( ( inf_in2269163501485487111nt_int @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_1016_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_1017_inf_Oabsorb4,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( inf_inf_int @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_1018_inf_Oabsorb3,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A @ B )
=> ( ( inf_in2269163501485487111nt_int @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_1019_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_1020_inf_Oabsorb3,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( inf_inf_int @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_1021_less__infI2,axiom,
! [B: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B @ X )
=> ( ord_le7563427860532173253nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_1022_less__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_nat @ B @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_1023_less__infI2,axiom,
! [B: int,X: int,A: int] :
( ( ord_less_int @ B @ X )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_1024_less__infI1,axiom,
! [A: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A @ X )
=> ( ord_le7563427860532173253nt_int @ ( inf_in2269163501485487111nt_int @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_1025_less__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_1026_less__infI1,axiom,
! [A: int,X: int,B: int] :
( ( ord_less_int @ A @ X )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_1027_length__removeAll__less,axiom,
! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
=> ( ord_less_nat @ ( size_s5157815400016825771nt_int @ ( remove494893200766290885nt_int @ X @ Xs ) ) @ ( size_s5157815400016825771nt_int @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_1028_compl__mono,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ Y ) @ ( uminus6221592323253981072nt_int @ X ) ) ) ).
% compl_mono
thf(fact_1029_compl__le__swap1,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ ( uminus6221592323253981072nt_int @ X ) )
=> ( ord_le2843351958646193337nt_int @ X @ ( uminus6221592323253981072nt_int @ Y ) ) ) ).
% compl_le_swap1
thf(fact_1030_compl__le__swap2,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ Y ) @ X )
=> ( ord_le2843351958646193337nt_int @ ( uminus6221592323253981072nt_int @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_1031_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1032_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J3: nat] :
( ! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1033_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1034_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1035_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1036_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1037_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1038_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1039_less__imp__diff__less,axiom,
! [J3: nat,K: nat,N: nat] :
( ( ord_less_nat @ J3 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1040_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1041_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1042_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_1043_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_less_numeral
thf(fact_1044_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_1045_list__len__g__1__split,axiom,
! [Xs: list_int] :
( ( ord_less_nat @ one_one_nat @ ( size_size_list_int @ Xs ) )
=> ? [X_1: int,X_2: int,Xs3: list_int] :
( Xs
= ( cons_int @ X_1 @ ( cons_int @ X_2 @ Xs3 ) ) ) ) ).
% list_len_g_1_split
thf(fact_1046_list__len__g__1__split,axiom,
! [Xs: list_P5707943133018811711nt_int] :
( ( ord_less_nat @ one_one_nat @ ( size_s5157815400016825771nt_int @ Xs ) )
=> ? [X_1: product_prod_int_int,X_2: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
( Xs
= ( cons_P3334398858971670639nt_int @ X_1 @ ( cons_P3334398858971670639nt_int @ X_2 @ Xs3 ) ) ) ) ).
% list_len_g_1_split
thf(fact_1047_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_1048_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_1049_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_1050_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_1051_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_1052_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_1053_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_1054_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1055_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_1056_leD,axiom,
! [Y: set_Pr958786334691620121nt_int,X: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ Y @ X )
=> ~ ( ord_le7563427860532173253nt_int @ X @ Y ) ) ).
% leD
thf(fact_1057_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_1058_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1059_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_1060_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_1061_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1062_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1063_nless__le,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ~ ( ord_le7563427860532173253nt_int @ A @ B ) )
= ( ~ ( ord_le2843351958646193337nt_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1064_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1065_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1066_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1067_antisym__conv1,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ~ ( ord_le7563427860532173253nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1068_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1069_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1070_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1071_antisym__conv2,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ~ ( ord_le7563427860532173253nt_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1072_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1073_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_1074_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
& ~ ( ord_less_eq_num @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_1075_less__le__not__le,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X4 @ Y3 )
& ~ ( ord_le2843351958646193337nt_int @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_1076_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_1077_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1078_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1079_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1080_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B2: nat] :
( ( ord_less_nat @ A5 @ B2 )
| ( A5 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1081_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A5: num,B2: num] :
( ( ord_less_num @ A5 @ B2 )
| ( A5 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1082_order_Oorder__iff__strict,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [A5: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A5 @ B2 )
| ( A5 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1083_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B2: int] :
( ( ord_less_int @ A5 @ B2 )
| ( A5 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1084_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B2: nat] :
( ( ord_less_eq_nat @ A5 @ B2 )
& ( A5 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1085_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A5: num,B2: num] :
( ( ord_less_eq_num @ A5 @ B2 )
& ( A5 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1086_order_Ostrict__iff__order,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [A5: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A5 @ B2 )
& ( A5 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1087_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B2: int] :
( ( ord_less_eq_int @ A5 @ B2 )
& ( A5 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1088_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1089_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1090_order_Ostrict__trans1,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( ord_le7563427860532173253nt_int @ B @ C )
=> ( ord_le7563427860532173253nt_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1091_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1092_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1093_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1094_order_Ostrict__trans2,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ B @ C )
=> ( ord_le7563427860532173253nt_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1095_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1096_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B2: nat] :
( ( ord_less_eq_nat @ A5 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1097_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A5: num,B2: num] :
( ( ord_less_eq_num @ A5 @ B2 )
& ~ ( ord_less_eq_num @ B2 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1098_order_Ostrict__iff__not,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [A5: set_Pr958786334691620121nt_int,B2: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A5 @ B2 )
& ~ ( ord_le2843351958646193337nt_int @ B2 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1099_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B2: int] :
( ( ord_less_eq_int @ A5 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1100_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A5: nat] :
( ( ord_less_nat @ B2 @ A5 )
| ( A5 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1101_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A5: num] :
( ( ord_less_num @ B2 @ A5 )
| ( A5 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1102_dual__order_Oorder__iff__strict,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [B2: set_Pr958786334691620121nt_int,A5: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B2 @ A5 )
| ( A5 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1103_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A5: int] :
( ( ord_less_int @ B2 @ A5 )
| ( A5 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1104_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A5: nat] :
( ( ord_less_eq_nat @ B2 @ A5 )
& ( A5 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1105_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B2: num,A5: num] :
( ( ord_less_eq_num @ B2 @ A5 )
& ( A5 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1106_dual__order_Ostrict__iff__order,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [B2: set_Pr958786334691620121nt_int,A5: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B2 @ A5 )
& ( A5 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1107_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A5: int] :
( ( ord_less_eq_int @ B2 @ A5 )
& ( A5 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1108_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1109_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1110_dual__order_Ostrict__trans1,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B @ A )
=> ( ( ord_le7563427860532173253nt_int @ C @ B )
=> ( ord_le7563427860532173253nt_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1111_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1112_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1113_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1114_dual__order_Ostrict__trans2,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B @ A )
=> ( ( ord_le2843351958646193337nt_int @ C @ B )
=> ( ord_le7563427860532173253nt_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1115_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1116_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A5: nat] :
( ( ord_less_eq_nat @ B2 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1117_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B2: num,A5: num] :
( ( ord_less_eq_num @ B2 @ A5 )
& ~ ( ord_less_eq_num @ A5 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1118_dual__order_Ostrict__iff__not,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [B2: set_Pr958786334691620121nt_int,A5: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ B2 @ A5 )
& ~ ( ord_le2843351958646193337nt_int @ A5 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1119_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A5: int] :
( ( ord_less_eq_int @ B2 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1120_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1121_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1122_order_Ostrict__implies__order,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ A @ B )
=> ( ord_le2843351958646193337nt_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1123_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1124_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1125_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1126_dual__order_Ostrict__implies__order,axiom,
! [B: set_Pr958786334691620121nt_int,A: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ B @ A )
=> ( ord_le2843351958646193337nt_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1127_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1128_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1129_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X4: num,Y3: num] :
( ( ord_less_num @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1130_order__le__less,axiom,
( ord_le2843351958646193337nt_int
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1131_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y3: int] :
( ( ord_less_int @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_1132_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1133_order__less__le,axiom,
( ord_less_num
= ( ^ [X4: num,Y3: num] :
( ( ord_less_eq_num @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1134_order__less__le,axiom,
( ord_le7563427860532173253nt_int
= ( ^ [X4: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1135_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y3: int] :
( ( ord_less_eq_int @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_1136_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1137_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1138_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1139_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1140_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1141_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1142_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1143_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1144_order__less__imp__le,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ X @ Y )
=> ( ord_le2843351958646193337nt_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1145_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1146_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1147_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1148_order__le__neq__trans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ( A != B )
=> ( ord_le7563427860532173253nt_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1149_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1150_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1151_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1152_order__neq__le__trans,axiom,
! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] :
( ( A != B )
=> ( ( ord_le2843351958646193337nt_int @ A @ B )
=> ( ord_le7563427860532173253nt_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1153_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1154_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1155_order__le__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1156_order__le__less__trans,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le2843351958646193337nt_int @ X @ Y )
=> ( ( ord_le7563427860532173253nt_int @ Y @ Z )
=> ( ord_le7563427860532173253nt_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1157_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1158_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1159_order__less__le__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1160_order__less__le__trans,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int,Z: set_Pr958786334691620121nt_int] :
( ( ord_le7563427860532173253nt_int @ X @ Y )
=> ( ( ord_le2843351958646193337nt_int @ Y @ Z )
=> ( ord_le7563427860532173253nt_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1161_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1162_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1163_order__le__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_num @ X5 @ Y2 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1164_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1165_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ord_less_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1166_order__le__less__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_num @ X5 @ Y2 )
=> ( ord_less_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1167_order__le__less__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1168_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1169_order__le__less__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_num @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1170_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1171_order__le__less__subst1,axiom,
! [A: set_Pr958786334691620121nt_int,F: nat > set_Pr958786334691620121nt_int,B: nat,C: nat] :
( ( ord_le2843351958646193337nt_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ord_le7563427860532173253nt_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_le7563427860532173253nt_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1172_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1173_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1174_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1175_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1176_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1177_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1178_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1179_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1180_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1181_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le7563427860532173253nt_int @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_le2843351958646193337nt_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_le7563427860532173253nt_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1182_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1183_order__less__le__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1184_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1185_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1186_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1187_order__less__le__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_eq_num @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1188_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1189_order__less__le__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1190_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_eq_int @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1191_order__less__le__subst1,axiom,
! [A: set_Pr958786334691620121nt_int,F: nat > set_Pr958786334691620121nt_int,B: nat,C: nat] :
( ( ord_le7563427860532173253nt_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_eq_nat @ X5 @ Y2 )
=> ( ord_le2843351958646193337nt_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_le7563427860532173253nt_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1192_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ord_less_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1193_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_num @ X5 @ Y2 )
=> ( ord_less_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1194_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > num,C: num] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_num @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1195_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ord_le7563427860532173253nt_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_le7563427860532173253nt_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1196_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_num @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ ( F @ B ) @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_num @ X5 @ Y2 )
=> ( ord_le7563427860532173253nt_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_le7563427860532173253nt_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1197_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > set_Pr958786334691620121nt_int,C: set_Pr958786334691620121nt_int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le2843351958646193337nt_int @ ( F @ B ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_le7563427860532173253nt_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_le7563427860532173253nt_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1198_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X5: nat,Y2: nat] :
( ( ord_less_nat @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1199_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X5: num,Y2: num] :
( ( ord_less_num @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1200_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1201_path__checker__correct,axiom,
path_checker = knights_path ).
% path_checker_correct
thf(fact_1202_path__checker_Osimps_I1_J,axiom,
! [B: set_Pr958786334691620121nt_int] :
~ ( path_checker @ B @ nil_Pr2300489316682597567nt_int ) ).
% path_checker.simps(1)
thf(fact_1203_knights__circuit__rotate__to,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int,K: nat,S_i: product_prod_int_int] :
( ( knights_circuit @ B @ Ps2 )
=> ( ( ( hd_Pro282112905867057956nt_int @ ( drop_P5690361596310759935nt_int @ K @ Ps2 ) )
= S_i )
=> ( ( ord_less_nat @ K @ ( size_s5157815400016825771nt_int @ Ps2 ) )
=> ? [Ps: list_P5707943133018811711nt_int] :
( ( knights_circuit @ B @ Ps )
& ( ( hd_Pro282112905867057956nt_int @ Ps )
= S_i ) ) ) ) ) ).
% knights_circuit_rotate_to
thf(fact_1204_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1205_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1206__092_060open_0623_A_060_Acard_A_Iboard_An_Am_J_092_060close_062,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( finite6756421564338198497nt_int @ ( board @ n @ m ) ) ).
% \<open>3 < card (board n m)\<close>
thf(fact_1207_path__checker_Oelims_I1_J,axiom,
! [X: set_Pr958786334691620121nt_int,Xa2: list_P5707943133018811711nt_int,Y: $o] :
( ( ( path_checker @ X @ Xa2 )
= Y )
=> ( ( ( Xa2 = nil_Pr2300489316682597567nt_int )
=> Y )
=> ( ! [S_i2: product_prod_int_int] :
( ( Xa2
= ( cons_P3334398858971670639nt_int @ S_i2 @ nil_Pr2300489316682597567nt_int ) )
=> ( Y
= ( ( insert5033312907999012233nt_int @ S_i2 @ bot_bo1796632182523588997nt_int )
!= X ) ) )
=> ~ ! [S_i2: product_prod_int_int,S_j: product_prod_int_int,Ps4: list_P5707943133018811711nt_int] :
( ( Xa2
= ( cons_P3334398858971670639nt_int @ S_i2 @ ( cons_P3334398858971670639nt_int @ S_j @ Ps4 ) ) )
=> ( Y
= ( ~ ( ( member5262025264175285858nt_int @ S_i2 @ X )
& ( step_checker @ S_i2 @ S_j )
& ( path_checker @ ( minus_1052850069191792384nt_int @ X @ ( insert5033312907999012233nt_int @ S_i2 @ bot_bo1796632182523588997nt_int ) ) @ ( cons_P3334398858971670639nt_int @ S_j @ Ps4 ) ) ) ) ) ) ) ) ) ).
% path_checker.elims(1)
thf(fact_1208_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_1209_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_1210_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_1211_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_1212_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_1213_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1214_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_1215_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_1216_step__checker__rev,axiom,
! [I3: int,J3: int,I5: int,J4: int] :
( ( step_checker @ ( product_Pair_int_int @ I3 @ J3 ) @ ( product_Pair_int_int @ I5 @ J4 ) )
=> ( step_checker @ ( product_Pair_int_int @ I5 @ J4 ) @ ( product_Pair_int_int @ I3 @ J3 ) ) ) ).
% step_checker_rev
thf(fact_1217_step__checker__correct,axiom,
step_checker = valid_step ).
% step_checker_correct
thf(fact_1218_knights__path__length,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int] :
( ( knights_path @ B @ Ps2 )
=> ( ( size_s5157815400016825771nt_int @ Ps2 )
= ( finite6756421564338198497nt_int @ B ) ) ) ).
% knights_path_length
thf(fact_1219_path__checker_Osimps_I3_J,axiom,
! [B: set_Pr958786334691620121nt_int,S_i: product_prod_int_int,S_j2: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
( ( path_checker @ B @ ( cons_P3334398858971670639nt_int @ S_i @ ( cons_P3334398858971670639nt_int @ S_j2 @ Ps2 ) ) )
= ( ( member5262025264175285858nt_int @ S_i @ B )
& ( step_checker @ S_i @ S_j2 )
& ( path_checker @ ( minus_1052850069191792384nt_int @ B @ ( insert5033312907999012233nt_int @ S_i @ bot_bo1796632182523588997nt_int ) ) @ ( cons_P3334398858971670639nt_int @ S_j2 @ Ps2 ) ) ) ) ).
% path_checker.simps(3)
thf(fact_1220_path__checker_Oelims_I3_J,axiom,
! [X: set_Pr958786334691620121nt_int,Xa2: list_P5707943133018811711nt_int] :
( ~ ( path_checker @ X @ Xa2 )
=> ( ( Xa2 != nil_Pr2300489316682597567nt_int )
=> ( ! [S_i2: product_prod_int_int] :
( ( Xa2
= ( cons_P3334398858971670639nt_int @ S_i2 @ nil_Pr2300489316682597567nt_int ) )
=> ( ( insert5033312907999012233nt_int @ S_i2 @ bot_bo1796632182523588997nt_int )
= X ) )
=> ~ ! [S_i2: product_prod_int_int,S_j: product_prod_int_int,Ps4: list_P5707943133018811711nt_int] :
( ( Xa2
= ( cons_P3334398858971670639nt_int @ S_i2 @ ( cons_P3334398858971670639nt_int @ S_j @ Ps4 ) ) )
=> ( ( member5262025264175285858nt_int @ S_i2 @ X )
& ( step_checker @ S_i2 @ S_j )
& ( path_checker @ ( minus_1052850069191792384nt_int @ X @ ( insert5033312907999012233nt_int @ S_i2 @ bot_bo1796632182523588997nt_int ) ) @ ( cons_P3334398858971670639nt_int @ S_j @ Ps4 ) ) ) ) ) ) ) ).
% path_checker.elims(3)
thf(fact_1221_path__checker_Oelims_I2_J,axiom,
! [X: set_Pr958786334691620121nt_int,Xa2: list_P5707943133018811711nt_int] :
( ( path_checker @ X @ Xa2 )
=> ( ! [S_i2: product_prod_int_int] :
( ( Xa2
= ( cons_P3334398858971670639nt_int @ S_i2 @ nil_Pr2300489316682597567nt_int ) )
=> ( ( insert5033312907999012233nt_int @ S_i2 @ bot_bo1796632182523588997nt_int )
!= X ) )
=> ~ ! [S_i2: product_prod_int_int,S_j: product_prod_int_int,Ps4: list_P5707943133018811711nt_int] :
( ( Xa2
= ( cons_P3334398858971670639nt_int @ S_i2 @ ( cons_P3334398858971670639nt_int @ S_j @ Ps4 ) ) )
=> ~ ( ( member5262025264175285858nt_int @ S_i2 @ X )
& ( step_checker @ S_i2 @ S_j )
& ( path_checker @ ( minus_1052850069191792384nt_int @ X @ ( insert5033312907999012233nt_int @ S_i2 @ bot_bo1796632182523588997nt_int ) ) @ ( cons_P3334398858971670639nt_int @ S_j @ Ps4 ) ) ) ) ) ) ).
% path_checker.elims(2)
thf(fact_1222_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1223_circuit__checker_Oelims_I3_J,axiom,
! [X: set_Pr958786334691620121nt_int,Xa2: list_P5707943133018811711nt_int] :
( ~ ( circuit_checker @ X @ Xa2 )
=> ~ ( ( path_checker @ X @ Xa2 )
& ( step_checker @ ( last_P3305686521732843992nt_int @ Xa2 ) @ ( hd_Pro282112905867057956nt_int @ Xa2 ) ) ) ) ).
% circuit_checker.elims(3)
thf(fact_1224_circuit__checker_Oelims_I2_J,axiom,
! [X: set_Pr958786334691620121nt_int,Xa2: list_P5707943133018811711nt_int] :
( ( circuit_checker @ X @ Xa2 )
=> ( ( path_checker @ X @ Xa2 )
& ( step_checker @ ( last_P3305686521732843992nt_int @ Xa2 ) @ ( hd_Pro282112905867057956nt_int @ Xa2 ) ) ) ) ).
% circuit_checker.elims(2)
thf(fact_1225_circuit__checker__correct,axiom,
circuit_checker = knights_circuit ).
% circuit_checker_correct
thf(fact_1226_circuit__checker_Osimps,axiom,
( circuit_checker
= ( ^ [B2: set_Pr958786334691620121nt_int,Ps3: list_P5707943133018811711nt_int] :
( ( path_checker @ B2 @ Ps3 )
& ( step_checker @ ( last_P3305686521732843992nt_int @ Ps3 ) @ ( hd_Pro282112905867057956nt_int @ Ps3 ) ) ) ) ) ).
% circuit_checker.simps
thf(fact_1227_circuit__checker_Oelims_I1_J,axiom,
! [X: set_Pr958786334691620121nt_int,Xa2: list_P5707943133018811711nt_int,Y: $o] :
( ( ( circuit_checker @ X @ Xa2 )
= Y )
=> ( Y
= ( ( path_checker @ X @ Xa2 )
& ( step_checker @ ( last_P3305686521732843992nt_int @ Xa2 ) @ ( hd_Pro282112905867057956nt_int @ Xa2 ) ) ) ) ) ).
% circuit_checker.elims(1)
thf(fact_1228_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1229_knights__circuit__exec__simp,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int] :
( ( knights_circuit @ ( board @ N @ M ) @ Ps2 )
= ( circuit_checker @ ( board_exec @ N @ M ) @ Ps2 ) ) ).
% knights_circuit_exec_simp
thf(fact_1230_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1231_board__exec__correct,axiom,
board = board_exec ).
% board_exec_correct
thf(fact_1232_int__less__induct,axiom,
! [I3: int,K: int,P: int > $o] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_less_induct
thf(fact_1233_int__le__induct,axiom,
! [I3: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_le_induct
thf(fact_1234_knights__path__exec__simp,axiom,
! [N: nat,M: nat,Ps2: list_P5707943133018811711nt_int] :
( ( knights_path @ ( board @ N @ M ) @ Ps2 )
= ( path_checker @ ( board_exec @ N @ M ) @ Ps2 ) ) ).
% knights_path_exec_simp
thf(fact_1235_valid__step__1__1,axiom,
! [I3: int,J3: int] :
( ( valid_step @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ I3 @ J3 ) )
=> ( ( ord_less_int @ zero_zero_int @ I3 )
=> ( ( ord_less_int @ zero_zero_int @ J3 )
=> ( ( ( product_Pair_int_int @ I3 @ J3 )
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
| ( ( product_Pair_int_int @ I3 @ J3 )
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% valid_step_1_1
thf(fact_1236_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I6: int,J5: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J5 @ I6 ) @ Js @ ( upto_aux @ I6 @ ( minus_minus_int @ J5 @ one_one_int ) @ ( cons_int @ J5 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1237_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1238_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1239_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1240_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1241_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1242_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1243_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1244_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1245_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_1246_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% min_0L
thf(fact_1247_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1248_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1249_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1250_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1251_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1252_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1253_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1254_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1255_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1256_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1257_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1258_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1259_knights__path__drop,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int,K: nat] :
( ( knights_path @ B @ Ps2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ K @ ( size_s5157815400016825771nt_int @ Ps2 ) )
=> ( knights_path @ ( set_Pr2470121279949933262nt_int @ ( drop_P5690361596310759935nt_int @ K @ Ps2 ) ) @ ( drop_P5690361596310759935nt_int @ K @ Ps2 ) ) ) ) ) ).
% knights_path_drop
thf(fact_1260_knights__path__split,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int,K: nat] :
( ( knights_path @ B @ Ps2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ K @ ( size_s5157815400016825771nt_int @ Ps2 ) )
=> ? [B_12: set_Pr958786334691620121nt_int,B_22: set_Pr958786334691620121nt_int] :
( ( knights_path @ B_12 @ ( take_P8218740963776755879nt_int @ K @ Ps2 ) )
& ( knights_path @ B_22 @ ( drop_P5690361596310759935nt_int @ K @ Ps2 ) )
& ( ( sup_su6024340866399070445nt_int @ B_12 @ B_22 )
= B )
& ( ( inf_in2269163501485487111nt_int @ B_12 @ B_22 )
= bot_bo1796632182523588997nt_int ) ) ) ) ) ).
% knights_path_split
thf(fact_1261_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1262_knights__path__take,axiom,
! [B: set_Pr958786334691620121nt_int,Ps2: list_P5707943133018811711nt_int,K: nat] :
( ( knights_path @ B @ Ps2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ K @ ( size_s5157815400016825771nt_int @ Ps2 ) )
=> ( knights_path @ ( set_Pr2470121279949933262nt_int @ ( take_P8218740963776755879nt_int @ K @ Ps2 ) ) @ ( take_P8218740963776755879nt_int @ K @ Ps2 ) ) ) ) ) ).
% knights_path_take
thf(fact_1263_path__checker_Opelims_I2_J,axiom,
! [X: set_Pr958786334691620121nt_int,Xa2: list_P5707943133018811711nt_int] :
( ( path_checker @ X @ Xa2 )
=> ( ( accp_P5348991460554856342nt_int @ path_checker_rel @ ( produc2261658324281137661nt_int @ X @ Xa2 ) )
=> ( ! [S_i2: product_prod_int_int] :
( ( Xa2
= ( cons_P3334398858971670639nt_int @ S_i2 @ nil_Pr2300489316682597567nt_int ) )
=> ( ( accp_P5348991460554856342nt_int @ path_checker_rel @ ( produc2261658324281137661nt_int @ X @ ( cons_P3334398858971670639nt_int @ S_i2 @ nil_Pr2300489316682597567nt_int ) ) )
=> ( ( insert5033312907999012233nt_int @ S_i2 @ bot_bo1796632182523588997nt_int )
!= X ) ) )
=> ~ ! [S_i2: product_prod_int_int,S_j: product_prod_int_int,Ps4: list_P5707943133018811711nt_int] :
( ( Xa2
= ( cons_P3334398858971670639nt_int @ S_i2 @ ( cons_P3334398858971670639nt_int @ S_j @ Ps4 ) ) )
=> ( ( accp_P5348991460554856342nt_int @ path_checker_rel @ ( produc2261658324281137661nt_int @ X @ ( cons_P3334398858971670639nt_int @ S_i2 @ ( cons_P3334398858971670639nt_int @ S_j @ Ps4 ) ) ) )
=> ~ ( ( member5262025264175285858nt_int @ S_i2 @ X )
& ( step_checker @ S_i2 @ S_j )
& ( path_checker @ ( minus_1052850069191792384nt_int @ X @ ( insert5033312907999012233nt_int @ S_i2 @ bot_bo1796632182523588997nt_int ) ) @ ( cons_P3334398858971670639nt_int @ S_j @ Ps4 ) ) ) ) ) ) ) ) ).
% path_checker.pelims(2)
thf(fact_1264_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1265_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1266_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1267_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_1268_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1269_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1270_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_1271_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1272_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
% Helper facts (11)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( if_set4441254072200386271nt_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
! [X: set_Pr958786334691620121nt_int,Y: set_Pr958786334691620121nt_int] :
( ( if_set4441254072200386271nt_int @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [Ps5: list_P5707943133018811711nt_int] :
( ( knights_circuit @ ( board @ n @ m ) @ Ps5 )
& ( ( hd_Pro282112905867057956nt_int @ Ps5 )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) )
& ( ( last_P3305686521732843992nt_int @ Ps5 )
= ( product_Pair_int_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
%------------------------------------------------------------------------------