TPTP Problem File: SLH0688^1.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 : VYDRA_MDL/0010_Temporal/prob_01892_088877__16908848_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 674 ( 274 unt; 228 typ; 0 def)
% Number of atoms : 1032 ( 678 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 3825 ( 197 ~; 20 |; 91 &;3071 @)
% ( 0 <=>; 446 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 55 ( 54 usr)
% Number of type conns : 407 ( 407 >; 0 *; 0 +; 0 <<)
% Number of symbols : 176 ( 174 usr; 53 con; 0-11 aty)
% Number of variables : 1248 ( 20 ^;1154 !; 74 ?;1248 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:54:27.603
%------------------------------------------------------------------------------
% Could-be-implicit typings (54)
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thf(ty_n_t__Trace__Otrace_Itf__a_Mtf__d_J,type,
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thf(ty_n_t__MDL__Oregex_Itf__a_Mtf__d_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Option__Ooption_Itf__d_J,type,
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thf(ty_n_t__IArray__Oiarray_I_Eo_J,type,
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thf(ty_n_t__List__Olist_Itf__d_J,type,
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thf(ty_n_t__Set__Oset_Itf__d_J,type,
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thf(ty_n_t__List__Olist_I_Eo_J,type,
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thf(ty_n_t__String__Ochar,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__t,type,
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thf(ty_n_tf__e,type,
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thf(ty_n_tf__d,type,
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% Explicit typings (174)
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_HOL_Oundefined_001t__IArray__Oiarray_I_Eo_J,type,
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thf(sy_c_IArray_Oiarray_OIArray_001_Eo,type,
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thf(sy_c_Interval_OmemR_001tf__d,type,
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thf(sy_c_Interval_Omem_001tf__d,type,
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thf(sy_c_Interval_Oright_001tf__d,type,
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thf(sy_c_List_Oappend_001t__IArray__Oiarray_I_Eo_J,type,
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thf(sy_c_List_Oappend_001t__MDL__Oformula_Itf__a_Mtf__d_J,type,
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thf(sy_c_List_Olist_OCons_001t__IArray__Oiarray_I_Eo_J,type,
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thf(sy_c_List_Olist_OCons_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__d_J_J,type,
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thf(sy_c_List_Olist_ONil_001t__IArray__Oiarray_I_Eo_J,type,
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nth_d: list_d > nat > d ).
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thf(sy_c_List_Otake_001tf__d,type,
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thf(sy_c_MDL_Oeps_001tf__a_001tf__d,type,
eps_a_d: regex_a_d > $o ).
thf(sy_c_MDL_Oformula_OBool_001tf__a_001tf__d,type,
bool_a_d: $o > formula_a_d ).
thf(sy_c_MDL_Oformula_OMatchF_001tf__d_001tf__a,type,
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thf(sy_c_MDL_Oformula_OMatchP_001tf__d_001tf__a,type,
matchP_d_a: i_d > regex_a_d > formula_a_d ).
thf(sy_c_MDL_Oformula_ONeg_001tf__a_001tf__d,type,
neg_a_d: formula_a_d > formula_a_d ).
thf(sy_c_MDL_Oformula_ONext_001tf__d_001tf__a,type,
next_d_a: i_d > formula_a_d > formula_a_d ).
thf(sy_c_MDL_Oformula_OSince_001tf__a_001tf__d,type,
since_a_d: formula_a_d > i_d > formula_a_d > formula_a_d ).
thf(sy_c_MDL_Oformula_OUntil_001tf__a_001tf__d,type,
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thf(sy_c_MDL_Orderive_001tf__a_001tf__d,type,
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thf(sy_c_MDL_Oregex_OStar_001tf__a_001tf__d,type,
star_a_d: regex_a_d > regex_a_d ).
thf(sy_c_MDL_Oregex_OSymbol_001tf__a_001tf__d,type,
symbol_a_d: formula_a_d > regex_a_d ).
thf(sy_c_MDL_Oregex_OTimes_001tf__a_001tf__d,type,
times_a_d: regex_a_d > regex_a_d > regex_a_d ).
thf(sy_c_MDL_Owf__regex_001tf__a_001tf__d,type,
wf_regex_a_d: regex_a_d > $o ).
thf(sy_c_Nat_OSuc,type,
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thf(sy_v_e____,type,
e3: list_P8365718696540089856_nat_d ).
thf(sy_v_i,type,
i2: nat ).
thf(sy_v_i_H_H____,type,
i3: nat ).
thf(sy_v_i_H____,type,
i4: nat ).
thf(sy_v_j_H_H____,type,
j: nat ).
thf(sy_v_j_H____,type,
j2: nat ).
thf(sy_v_j____,type,
j3: nat ).
thf(sy_v_l____,type,
l: nat ).
thf(sy_v_r,type,
r: regex_a_d ).
thf(sy_v_rho,type,
rho: list_P1335534284205260522rray_o ).
thf(sy_v_rho_H,type,
rho2: list_P1335534284205260522rray_o ).
thf(sy_v_rho_H_H____,type,
rho3: list_P1335534284205260522rray_o ).
thf(sy_v_si_H_H_H____,type,
si: e ).
thf(sy_v_si_H____,type,
si2: e ).
thf(sy_v_si____,type,
si3: e ).
thf(sy_v_sj_H_H_H,type,
sj: e ).
thf(sy_v_sj_H_H____,type,
sj2: e ).
thf(sy_v_sj_H____,type,
sj3: e ).
thf(sy_v_sj____,type,
sj4: e ).
thf(sy_v_sub,type,
sub: e ).
thf(sy_v_t,type,
t2: d ).
thf(sy_v_t0,type,
t0: t ).
thf(sy_v_te____,type,
te: d ).
thf(sy_v_ti_H_H_H____,type,
ti: t ).
thf(sy_v_ti_H____,type,
ti2: t ).
thf(sy_v_ti____,type,
ti3: t ).
thf(sy_v_tj_H_H_H,type,
tj: t ).
thf(sy_v_tj_H_H____,type,
tj2: t ).
thf(sy_v_tj_H____,type,
tj3: t ).
thf(sy_v_tj____,type,
tj4: t ).
thf(sy_v_tm,type,
tm: d ).
thf(sy_v_tm_H____,type,
tm2: t ).
thf(sy_v_w,type,
w: window145566784895814656t_unit ).
thf(sy_v_w_H_H____,type,
w2: window145566784895814656t_unit ).
thf(sy_v_w_H____,type,
w3: window145566784895814656t_unit ).
% Relevant facts (445)
thf(fact_0_calculation,axiom,
( tm
= ( ts_at_d_iarray_o @ ( append1741259189927458815rray_o @ rho3 @ ( cons_P2929697372952323866rray_o @ ( produc6263459992454928980rray_o @ tm @ undefined_iarray_o ) @ nil_Pr7455003300710157162rray_o ) ) @ ( size_s1844703894572694870rray_o @ rho3 ) ) ) ).
% calculation
thf(fact_1_contr,axiom,
~ ( ord_less_nat @ l @ ( size_s1844703894572694870rray_o @ rho3 ) ) ).
% contr
thf(fact_2_ex__key_Ocases,axiom,
! [X: produc2588096255342480170ng_t_o] :
( ! [Time: d > $o,Accept: t > $o,Ac: mapping_t_o] :
( X
!= ( produc6084929014563660708ng_t_o @ nil_Product_prod_t_d @ ( produc2954693272173378896ng_t_o @ Time @ ( produc5633596923390089885ng_t_o @ Accept @ Ac ) ) ) )
=> ~ ! [Q: t,T: d,Qts: list_P7979455665520128861od_t_d,Time: d > $o,Accept: t > $o,Ac: mapping_t_o] :
( X
!= ( produc6084929014563660708ng_t_o @ ( cons_P472143266409286669od_t_d @ ( product_Pair_t_d @ Q @ T ) @ Qts ) @ ( produc2954693272173378896ng_t_o @ Time @ ( produc5633596923390089885ng_t_o @ Accept @ Ac ) ) ) ) ) ).
% ex_key.cases
thf(fact_3_tbj_H__def_I2_J,axiom,
( te
= ( ts_at_d_iarray_o @ ( append1741259189927458815rray_o @ rho3 @ ( cons_P2929697372952323866rray_o @ ( produc6263459992454928980rray_o @ tm @ undefined_iarray_o ) @ nil_Pr7455003300710157162rray_o ) ) @ j2 ) ) ).
% tbj'_def(2)
thf(fact_4_append1__eq__conv,axiom,
! [Xs: list_iarray_o,X: iarray_o,Ys: list_iarray_o,Y: iarray_o] :
( ( ( append_iarray_o @ Xs @ ( cons_iarray_o @ X @ nil_iarray_o ) )
= ( append_iarray_o @ Ys @ ( cons_iarray_o @ Y @ nil_iarray_o ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_5_append1__eq__conv,axiom,
! [Xs: list_d,X: d,Ys: list_d,Y: d] :
( ( ( append_d @ Xs @ ( cons_d @ X @ nil_d ) )
= ( append_d @ Ys @ ( cons_d @ Y @ nil_d ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_6_append1__eq__conv,axiom,
! [Xs: list_P7979455665520128861od_t_d,X: product_prod_t_d,Ys: list_P7979455665520128861od_t_d,Y: product_prod_t_d] :
( ( ( append7713784995604329970od_t_d @ Xs @ ( cons_P472143266409286669od_t_d @ X @ nil_Product_prod_t_d ) )
= ( append7713784995604329970od_t_d @ Ys @ ( cons_P472143266409286669od_t_d @ Y @ nil_Product_prod_t_d ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_7_append1__eq__conv,axiom,
! [Xs: list_formula_a_d,X: formula_a_d,Ys: list_formula_a_d,Y: formula_a_d] :
( ( ( append_formula_a_d @ Xs @ ( cons_formula_a_d @ X @ nil_formula_a_d ) )
= ( append_formula_a_d @ Ys @ ( cons_formula_a_d @ Y @ nil_formula_a_d ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_8_append1__eq__conv,axiom,
! [Xs: list_P1335534284205260522rray_o,X: produc4999605722385084260rray_o,Ys: list_P1335534284205260522rray_o,Y: produc4999605722385084260rray_o] :
( ( ( append1741259189927458815rray_o @ Xs @ ( cons_P2929697372952323866rray_o @ X @ nil_Pr7455003300710157162rray_o ) )
= ( append1741259189927458815rray_o @ Ys @ ( cons_P2929697372952323866rray_o @ Y @ nil_Pr7455003300710157162rray_o ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_9_append__eq__append__conv,axiom,
! [Xs: list_formula_a_d,Ys: list_formula_a_d,Us: list_formula_a_d,Vs: list_formula_a_d] :
( ( ( ( size_s7710211460959691191la_a_d @ Xs )
= ( size_s7710211460959691191la_a_d @ Ys ) )
| ( ( size_s7710211460959691191la_a_d @ Us )
= ( size_s7710211460959691191la_a_d @ Vs ) ) )
=> ( ( ( append_formula_a_d @ Xs @ Us )
= ( append_formula_a_d @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_10_append__eq__append__conv,axiom,
! [Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o,Us: list_P1335534284205260522rray_o,Vs: list_P1335534284205260522rray_o] :
( ( ( ( size_s1844703894572694870rray_o @ Xs )
= ( size_s1844703894572694870rray_o @ Ys ) )
| ( ( size_s1844703894572694870rray_o @ Us )
= ( size_s1844703894572694870rray_o @ Vs ) ) )
=> ( ( ( append1741259189927458815rray_o @ Xs @ Us )
= ( append1741259189927458815rray_o @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_11_append__eq__append__conv,axiom,
! [Xs: list_d,Ys: list_d,Us: list_d,Vs: list_d] :
( ( ( ( size_size_list_d @ Xs )
= ( size_size_list_d @ Ys ) )
| ( ( size_size_list_d @ Us )
= ( size_size_list_d @ Vs ) ) )
=> ( ( ( append_d @ Xs @ Us )
= ( append_d @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_12_append__eq__append__conv,axiom,
! [Xs: list_iarray_o,Ys: list_iarray_o,Us: list_iarray_o,Vs: list_iarray_o] :
( ( ( ( size_s3243743627286111600rray_o @ Xs )
= ( size_s3243743627286111600rray_o @ Ys ) )
| ( ( size_s3243743627286111600rray_o @ Us )
= ( size_s3243743627286111600rray_o @ Vs ) ) )
=> ( ( ( append_iarray_o @ Xs @ Us )
= ( append_iarray_o @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_13_append_Oright__neutral,axiom,
! [A: list_iarray_o] :
( ( append_iarray_o @ A @ nil_iarray_o )
= A ) ).
% append.right_neutral
thf(fact_14_append_Oright__neutral,axiom,
! [A: list_d] :
( ( append_d @ A @ nil_d )
= A ) ).
% append.right_neutral
thf(fact_15_append_Oright__neutral,axiom,
! [A: list_P7979455665520128861od_t_d] :
( ( append7713784995604329970od_t_d @ A @ nil_Product_prod_t_d )
= A ) ).
% append.right_neutral
thf(fact_16_append_Oright__neutral,axiom,
! [A: list_P1335534284205260522rray_o] :
( ( append1741259189927458815rray_o @ A @ nil_Pr7455003300710157162rray_o )
= A ) ).
% append.right_neutral
thf(fact_17_append_Oright__neutral,axiom,
! [A: list_formula_a_d] :
( ( append_formula_a_d @ A @ nil_formula_a_d )
= A ) ).
% append.right_neutral
thf(fact_18_append__Nil2,axiom,
! [Xs: list_iarray_o] :
( ( append_iarray_o @ Xs @ nil_iarray_o )
= Xs ) ).
% append_Nil2
thf(fact_19_append__Nil2,axiom,
! [Xs: list_d] :
( ( append_d @ Xs @ nil_d )
= Xs ) ).
% append_Nil2
thf(fact_20_append__Nil2,axiom,
! [Xs: list_P7979455665520128861od_t_d] :
( ( append7713784995604329970od_t_d @ Xs @ nil_Product_prod_t_d )
= Xs ) ).
% append_Nil2
thf(fact_21_append__Nil2,axiom,
! [Xs: list_P1335534284205260522rray_o] :
( ( append1741259189927458815rray_o @ Xs @ nil_Pr7455003300710157162rray_o )
= Xs ) ).
% append_Nil2
thf(fact_22_append__Nil2,axiom,
! [Xs: list_formula_a_d] :
( ( append_formula_a_d @ Xs @ nil_formula_a_d )
= Xs ) ).
% append_Nil2
thf(fact_23_append__self__conv,axiom,
! [Xs: list_iarray_o,Ys: list_iarray_o] :
( ( ( append_iarray_o @ Xs @ Ys )
= Xs )
= ( Ys = nil_iarray_o ) ) ).
% append_self_conv
thf(fact_24_append__self__conv,axiom,
! [Xs: list_d,Ys: list_d] :
( ( ( append_d @ Xs @ Ys )
= Xs )
= ( Ys = nil_d ) ) ).
% append_self_conv
thf(fact_25_append__self__conv,axiom,
! [Xs: list_P7979455665520128861od_t_d,Ys: list_P7979455665520128861od_t_d] :
( ( ( append7713784995604329970od_t_d @ Xs @ Ys )
= Xs )
= ( Ys = nil_Product_prod_t_d ) ) ).
% append_self_conv
thf(fact_26_append__self__conv,axiom,
! [Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o] :
( ( ( append1741259189927458815rray_o @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr7455003300710157162rray_o ) ) ).
% append_self_conv
thf(fact_27_append__self__conv,axiom,
! [Xs: list_formula_a_d,Ys: list_formula_a_d] :
( ( ( append_formula_a_d @ Xs @ Ys )
= Xs )
= ( Ys = nil_formula_a_d ) ) ).
% append_self_conv
thf(fact_28_self__append__conv,axiom,
! [Y: list_iarray_o,Ys: list_iarray_o] :
( ( Y
= ( append_iarray_o @ Y @ Ys ) )
= ( Ys = nil_iarray_o ) ) ).
% self_append_conv
thf(fact_29_self__append__conv,axiom,
! [Y: list_d,Ys: list_d] :
( ( Y
= ( append_d @ Y @ Ys ) )
= ( Ys = nil_d ) ) ).
% self_append_conv
thf(fact_30_self__append__conv,axiom,
! [Y: list_P7979455665520128861od_t_d,Ys: list_P7979455665520128861od_t_d] :
( ( Y
= ( append7713784995604329970od_t_d @ Y @ Ys ) )
= ( Ys = nil_Product_prod_t_d ) ) ).
% self_append_conv
thf(fact_31_self__append__conv,axiom,
! [Y: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o] :
( ( Y
= ( append1741259189927458815rray_o @ Y @ Ys ) )
= ( Ys = nil_Pr7455003300710157162rray_o ) ) ).
% self_append_conv
thf(fact_32_self__append__conv,axiom,
! [Y: list_formula_a_d,Ys: list_formula_a_d] :
( ( Y
= ( append_formula_a_d @ Y @ Ys ) )
= ( Ys = nil_formula_a_d ) ) ).
% self_append_conv
thf(fact_33_append__self__conv2,axiom,
! [Xs: list_iarray_o,Ys: list_iarray_o] :
( ( ( append_iarray_o @ Xs @ Ys )
= Ys )
= ( Xs = nil_iarray_o ) ) ).
% append_self_conv2
thf(fact_34_append__self__conv2,axiom,
! [Xs: list_d,Ys: list_d] :
( ( ( append_d @ Xs @ Ys )
= Ys )
= ( Xs = nil_d ) ) ).
% append_self_conv2
thf(fact_35_append__self__conv2,axiom,
! [Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o] :
( ( ( append1741259189927458815rray_o @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr7455003300710157162rray_o ) ) ).
% append_self_conv2
thf(fact_36_append__self__conv2,axiom,
! [Xs: list_formula_a_d,Ys: list_formula_a_d] :
( ( ( append_formula_a_d @ Xs @ Ys )
= Ys )
= ( Xs = nil_formula_a_d ) ) ).
% append_self_conv2
thf(fact_37_append__self__conv2,axiom,
! [Xs: list_P7979455665520128861od_t_d,Ys: list_P7979455665520128861od_t_d] :
( ( ( append7713784995604329970od_t_d @ Xs @ Ys )
= Ys )
= ( Xs = nil_Product_prod_t_d ) ) ).
% append_self_conv2
thf(fact_38_self__append__conv2,axiom,
! [Y: list_iarray_o,Xs: list_iarray_o] :
( ( Y
= ( append_iarray_o @ Xs @ Y ) )
= ( Xs = nil_iarray_o ) ) ).
% self_append_conv2
thf(fact_39_self__append__conv2,axiom,
! [Y: list_d,Xs: list_d] :
( ( Y
= ( append_d @ Xs @ Y ) )
= ( Xs = nil_d ) ) ).
% self_append_conv2
thf(fact_40_self__append__conv2,axiom,
! [Y: list_P1335534284205260522rray_o,Xs: list_P1335534284205260522rray_o] :
( ( Y
= ( append1741259189927458815rray_o @ Xs @ Y ) )
= ( Xs = nil_Pr7455003300710157162rray_o ) ) ).
% self_append_conv2
thf(fact_41_self__append__conv2,axiom,
! [Y: list_formula_a_d,Xs: list_formula_a_d] :
( ( Y
= ( append_formula_a_d @ Xs @ Y ) )
= ( Xs = nil_formula_a_d ) ) ).
% self_append_conv2
thf(fact_42_self__append__conv2,axiom,
! [Y: list_P7979455665520128861od_t_d,Xs: list_P7979455665520128861od_t_d] :
( ( Y
= ( append7713784995604329970od_t_d @ Xs @ Y ) )
= ( Xs = nil_Product_prod_t_d ) ) ).
% self_append_conv2
thf(fact_43_Nil__is__append__conv,axiom,
! [Xs: list_iarray_o,Ys: list_iarray_o] :
( ( nil_iarray_o
= ( append_iarray_o @ Xs @ Ys ) )
= ( ( Xs = nil_iarray_o )
& ( Ys = nil_iarray_o ) ) ) ).
% Nil_is_append_conv
thf(fact_44_Nil__is__append__conv,axiom,
! [Xs: list_d,Ys: list_d] :
( ( nil_d
= ( append_d @ Xs @ Ys ) )
= ( ( Xs = nil_d )
& ( Ys = nil_d ) ) ) ).
% Nil_is_append_conv
thf(fact_45_Nil__is__append__conv,axiom,
! [Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o] :
( ( nil_Pr7455003300710157162rray_o
= ( append1741259189927458815rray_o @ Xs @ Ys ) )
= ( ( Xs = nil_Pr7455003300710157162rray_o )
& ( Ys = nil_Pr7455003300710157162rray_o ) ) ) ).
% Nil_is_append_conv
thf(fact_46_Nil__is__append__conv,axiom,
! [Xs: list_formula_a_d,Ys: list_formula_a_d] :
( ( nil_formula_a_d
= ( append_formula_a_d @ Xs @ Ys ) )
= ( ( Xs = nil_formula_a_d )
& ( Ys = nil_formula_a_d ) ) ) ).
% Nil_is_append_conv
thf(fact_47_Nil__is__append__conv,axiom,
! [Xs: list_P7979455665520128861od_t_d,Ys: list_P7979455665520128861od_t_d] :
( ( nil_Product_prod_t_d
= ( append7713784995604329970od_t_d @ Xs @ Ys ) )
= ( ( Xs = nil_Product_prod_t_d )
& ( Ys = nil_Product_prod_t_d ) ) ) ).
% Nil_is_append_conv
thf(fact_48_append__is__Nil__conv,axiom,
! [Xs: list_iarray_o,Ys: list_iarray_o] :
( ( ( append_iarray_o @ Xs @ Ys )
= nil_iarray_o )
= ( ( Xs = nil_iarray_o )
& ( Ys = nil_iarray_o ) ) ) ).
% append_is_Nil_conv
thf(fact_49_append__is__Nil__conv,axiom,
! [Xs: list_d,Ys: list_d] :
( ( ( append_d @ Xs @ Ys )
= nil_d )
= ( ( Xs = nil_d )
& ( Ys = nil_d ) ) ) ).
% append_is_Nil_conv
thf(fact_50_append__is__Nil__conv,axiom,
! [Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o] :
( ( ( append1741259189927458815rray_o @ Xs @ Ys )
= nil_Pr7455003300710157162rray_o )
= ( ( Xs = nil_Pr7455003300710157162rray_o )
& ( Ys = nil_Pr7455003300710157162rray_o ) ) ) ).
% append_is_Nil_conv
thf(fact_51_append__is__Nil__conv,axiom,
! [Xs: list_formula_a_d,Ys: list_formula_a_d] :
( ( ( append_formula_a_d @ Xs @ Ys )
= nil_formula_a_d )
= ( ( Xs = nil_formula_a_d )
& ( Ys = nil_formula_a_d ) ) ) ).
% append_is_Nil_conv
thf(fact_52_append__is__Nil__conv,axiom,
! [Xs: list_P7979455665520128861od_t_d,Ys: list_P7979455665520128861od_t_d] :
( ( ( append7713784995604329970od_t_d @ Xs @ Ys )
= nil_Product_prod_t_d )
= ( ( Xs = nil_Product_prod_t_d )
& ( Ys = nil_Product_prod_t_d ) ) ) ).
% append_is_Nil_conv
thf(fact_53_same__length__different,axiom,
! [Xs: list_P7979455665520128861od_t_d,Ys: list_P7979455665520128861od_t_d] :
( ( Xs != Ys )
=> ( ( ( size_s1244821776335097289od_t_d @ Xs )
= ( size_s1244821776335097289od_t_d @ Ys ) )
=> ? [Pre: list_P7979455665520128861od_t_d,X2: product_prod_t_d,Xs2: list_P7979455665520128861od_t_d,Y2: product_prod_t_d,Ys2: list_P7979455665520128861od_t_d] :
( ( X2 != Y2 )
& ( Xs
= ( append7713784995604329970od_t_d @ Pre @ ( append7713784995604329970od_t_d @ ( cons_P472143266409286669od_t_d @ X2 @ nil_Product_prod_t_d ) @ Xs2 ) ) )
& ( Ys
= ( append7713784995604329970od_t_d @ Pre @ ( append7713784995604329970od_t_d @ ( cons_P472143266409286669od_t_d @ Y2 @ nil_Product_prod_t_d ) @ Ys2 ) ) ) ) ) ) ).
% same_length_different
thf(fact_54_same__length__different,axiom,
! [Xs: list_formula_a_d,Ys: list_formula_a_d] :
( ( Xs != Ys )
=> ( ( ( size_s7710211460959691191la_a_d @ Xs )
= ( size_s7710211460959691191la_a_d @ Ys ) )
=> ? [Pre: list_formula_a_d,X2: formula_a_d,Xs2: list_formula_a_d,Y2: formula_a_d,Ys2: list_formula_a_d] :
( ( X2 != Y2 )
& ( Xs
= ( append_formula_a_d @ Pre @ ( append_formula_a_d @ ( cons_formula_a_d @ X2 @ nil_formula_a_d ) @ Xs2 ) ) )
& ( Ys
= ( append_formula_a_d @ Pre @ ( append_formula_a_d @ ( cons_formula_a_d @ Y2 @ nil_formula_a_d ) @ Ys2 ) ) ) ) ) ) ).
% same_length_different
thf(fact_55_same__length__different,axiom,
! [Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o] :
( ( Xs != Ys )
=> ( ( ( size_s1844703894572694870rray_o @ Xs )
= ( size_s1844703894572694870rray_o @ Ys ) )
=> ? [Pre: list_P1335534284205260522rray_o,X2: produc4999605722385084260rray_o,Xs2: list_P1335534284205260522rray_o,Y2: produc4999605722385084260rray_o,Ys2: list_P1335534284205260522rray_o] :
( ( X2 != Y2 )
& ( Xs
= ( append1741259189927458815rray_o @ Pre @ ( append1741259189927458815rray_o @ ( cons_P2929697372952323866rray_o @ X2 @ nil_Pr7455003300710157162rray_o ) @ Xs2 ) ) )
& ( Ys
= ( append1741259189927458815rray_o @ Pre @ ( append1741259189927458815rray_o @ ( cons_P2929697372952323866rray_o @ Y2 @ nil_Pr7455003300710157162rray_o ) @ Ys2 ) ) ) ) ) ) ).
% same_length_different
thf(fact_56_same__length__different,axiom,
! [Xs: list_d,Ys: list_d] :
( ( Xs != Ys )
=> ( ( ( size_size_list_d @ Xs )
= ( size_size_list_d @ Ys ) )
=> ? [Pre: list_d,X2: d,Xs2: list_d,Y2: d,Ys2: list_d] :
( ( X2 != Y2 )
& ( Xs
= ( append_d @ Pre @ ( append_d @ ( cons_d @ X2 @ nil_d ) @ Xs2 ) ) )
& ( Ys
= ( append_d @ Pre @ ( append_d @ ( cons_d @ Y2 @ nil_d ) @ Ys2 ) ) ) ) ) ) ).
% same_length_different
thf(fact_57_same__length__different,axiom,
! [Xs: list_iarray_o,Ys: list_iarray_o] :
( ( Xs != Ys )
=> ( ( ( size_s3243743627286111600rray_o @ Xs )
= ( size_s3243743627286111600rray_o @ Ys ) )
=> ? [Pre: list_iarray_o,X2: iarray_o,Xs2: list_iarray_o,Y2: iarray_o,Ys2: list_iarray_o] :
( ( X2 != Y2 )
& ( Xs
= ( append_iarray_o @ Pre @ ( append_iarray_o @ ( cons_iarray_o @ X2 @ nil_iarray_o ) @ Xs2 ) ) )
& ( Ys
= ( append_iarray_o @ Pre @ ( append_iarray_o @ ( cons_iarray_o @ Y2 @ nil_iarray_o ) @ Ys2 ) ) ) ) ) ) ).
% same_length_different
thf(fact_58_list_Oinject,axiom,
! [X21: produc4999605722385084260rray_o,X22: list_P1335534284205260522rray_o,Y21: produc4999605722385084260rray_o,Y22: list_P1335534284205260522rray_o] :
( ( ( cons_P2929697372952323866rray_o @ X21 @ X22 )
= ( cons_P2929697372952323866rray_o @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_59_list_Oinject,axiom,
! [X21: product_prod_t_d,X22: list_P7979455665520128861od_t_d,Y21: product_prod_t_d,Y22: list_P7979455665520128861od_t_d] :
( ( ( cons_P472143266409286669od_t_d @ X21 @ X22 )
= ( cons_P472143266409286669od_t_d @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_60_list_Oinject,axiom,
! [X21: formula_a_d,X22: list_formula_a_d,Y21: formula_a_d,Y22: list_formula_a_d] :
( ( ( cons_formula_a_d @ X21 @ X22 )
= ( cons_formula_a_d @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_61_same__append__eq,axiom,
! [Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o,Zs: list_P1335534284205260522rray_o] :
( ( ( append1741259189927458815rray_o @ Xs @ Ys )
= ( append1741259189927458815rray_o @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_62_same__append__eq,axiom,
! [Xs: list_formula_a_d,Ys: list_formula_a_d,Zs: list_formula_a_d] :
( ( ( append_formula_a_d @ Xs @ Ys )
= ( append_formula_a_d @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_63_same__append__eq,axiom,
! [Xs: list_iarray_o,Ys: list_iarray_o,Zs: list_iarray_o] :
( ( ( append_iarray_o @ Xs @ Ys )
= ( append_iarray_o @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_64_same__append__eq,axiom,
! [Xs: list_d,Ys: list_d,Zs: list_d] :
( ( ( append_d @ Xs @ Ys )
= ( append_d @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_65_append__same__eq,axiom,
! [Ys: list_P1335534284205260522rray_o,Xs: list_P1335534284205260522rray_o,Zs: list_P1335534284205260522rray_o] :
( ( ( append1741259189927458815rray_o @ Ys @ Xs )
= ( append1741259189927458815rray_o @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_66_append__same__eq,axiom,
! [Ys: list_formula_a_d,Xs: list_formula_a_d,Zs: list_formula_a_d] :
( ( ( append_formula_a_d @ Ys @ Xs )
= ( append_formula_a_d @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_67_append__same__eq,axiom,
! [Ys: list_iarray_o,Xs: list_iarray_o,Zs: list_iarray_o] :
( ( ( append_iarray_o @ Ys @ Xs )
= ( append_iarray_o @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_68_append__same__eq,axiom,
! [Ys: list_d,Xs: list_d,Zs: list_d] :
( ( ( append_d @ Ys @ Xs )
= ( append_d @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_69_append__assoc,axiom,
! [Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o,Zs: list_P1335534284205260522rray_o] :
( ( append1741259189927458815rray_o @ ( append1741259189927458815rray_o @ Xs @ Ys ) @ Zs )
= ( append1741259189927458815rray_o @ Xs @ ( append1741259189927458815rray_o @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_70_append__assoc,axiom,
! [Xs: list_formula_a_d,Ys: list_formula_a_d,Zs: list_formula_a_d] :
( ( append_formula_a_d @ ( append_formula_a_d @ Xs @ Ys ) @ Zs )
= ( append_formula_a_d @ Xs @ ( append_formula_a_d @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_71_append__assoc,axiom,
! [Xs: list_iarray_o,Ys: list_iarray_o,Zs: list_iarray_o] :
( ( append_iarray_o @ ( append_iarray_o @ Xs @ Ys ) @ Zs )
= ( append_iarray_o @ Xs @ ( append_iarray_o @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_72_append__assoc,axiom,
! [Xs: list_d,Ys: list_d,Zs: list_d] :
( ( append_d @ ( append_d @ Xs @ Ys ) @ Zs )
= ( append_d @ Xs @ ( append_d @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_73_append_Oassoc,axiom,
! [A: list_P1335534284205260522rray_o,B: list_P1335534284205260522rray_o,C: list_P1335534284205260522rray_o] :
( ( append1741259189927458815rray_o @ ( append1741259189927458815rray_o @ A @ B ) @ C )
= ( append1741259189927458815rray_o @ A @ ( append1741259189927458815rray_o @ B @ C ) ) ) ).
% append.assoc
thf(fact_74_append_Oassoc,axiom,
! [A: list_formula_a_d,B: list_formula_a_d,C: list_formula_a_d] :
( ( append_formula_a_d @ ( append_formula_a_d @ A @ B ) @ C )
= ( append_formula_a_d @ A @ ( append_formula_a_d @ B @ C ) ) ) ).
% append.assoc
thf(fact_75_append_Oassoc,axiom,
! [A: list_iarray_o,B: list_iarray_o,C: list_iarray_o] :
( ( append_iarray_o @ ( append_iarray_o @ A @ B ) @ C )
= ( append_iarray_o @ A @ ( append_iarray_o @ B @ C ) ) ) ).
% append.assoc
thf(fact_76_append_Oassoc,axiom,
! [A: list_d,B: list_d,C: list_d] :
( ( append_d @ ( append_d @ A @ B ) @ C )
= ( append_d @ A @ ( append_d @ B @ C ) ) ) ).
% append.assoc
thf(fact_77_adv__last__bounds_I4_J,axiom,
j = j2 ).
% adv_last_bounds(4)
thf(fact_78_i_H__lt__j_H,axiom,
ord_less_nat @ i4 @ j2 ).
% i'_lt_j'
thf(fact_79_length__induct,axiom,
! [P: list_P1335534284205260522rray_o > $o,Xs: list_P1335534284205260522rray_o] :
( ! [Xs3: list_P1335534284205260522rray_o] :
( ! [Ys3: list_P1335534284205260522rray_o] :
( ( ord_less_nat @ ( size_s1844703894572694870rray_o @ Ys3 ) @ ( size_s1844703894572694870rray_o @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_80_length__induct,axiom,
! [P: list_d > $o,Xs: list_d] :
( ! [Xs3: list_d] :
( ! [Ys3: list_d] :
( ( ord_less_nat @ ( size_size_list_d @ Ys3 ) @ ( size_size_list_d @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_81_length__induct,axiom,
! [P: list_iarray_o > $o,Xs: list_iarray_o] :
( ! [Xs3: list_iarray_o] :
( ! [Ys3: list_iarray_o] :
( ( ord_less_nat @ ( size_s3243743627286111600rray_o @ Ys3 ) @ ( size_s3243743627286111600rray_o @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_82_successively_Ocases,axiom,
! [X: produc6315692651791551267rray_o] :
( ! [P2: produc4999605722385084260rray_o > produc4999605722385084260rray_o > $o] :
( X
!= ( produc7148991607918580371rray_o @ P2 @ nil_Pr7455003300710157162rray_o ) )
=> ( ! [P2: produc4999605722385084260rray_o > produc4999605722385084260rray_o > $o,X2: produc4999605722385084260rray_o] :
( X
!= ( produc7148991607918580371rray_o @ P2 @ ( cons_P2929697372952323866rray_o @ X2 @ nil_Pr7455003300710157162rray_o ) ) )
=> ~ ! [P2: produc4999605722385084260rray_o > produc4999605722385084260rray_o > $o,X2: produc4999605722385084260rray_o,Y2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o] :
( X
!= ( produc7148991607918580371rray_o @ P2 @ ( cons_P2929697372952323866rray_o @ X2 @ ( cons_P2929697372952323866rray_o @ Y2 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_83_successively_Ocases,axiom,
! [X: produc698650026130441430od_t_d] :
( ! [P2: product_prod_t_d > product_prod_t_d > $o] :
( X
!= ( produc3153264129863119558od_t_d @ P2 @ nil_Product_prod_t_d ) )
=> ( ! [P2: product_prod_t_d > product_prod_t_d > $o,X2: product_prod_t_d] :
( X
!= ( produc3153264129863119558od_t_d @ P2 @ ( cons_P472143266409286669od_t_d @ X2 @ nil_Product_prod_t_d ) ) )
=> ~ ! [P2: product_prod_t_d > product_prod_t_d > $o,X2: product_prod_t_d,Y2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d] :
( X
!= ( produc3153264129863119558od_t_d @ P2 @ ( cons_P472143266409286669od_t_d @ X2 @ ( cons_P472143266409286669od_t_d @ Y2 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_84_successively_Ocases,axiom,
! [X: produc114835076914812850la_a_d] :
( ! [P2: formula_a_d > formula_a_d > $o] :
( X
!= ( produc2569379974314500652la_a_d @ P2 @ nil_formula_a_d ) )
=> ( ! [P2: formula_a_d > formula_a_d > $o,X2: formula_a_d] :
( X
!= ( produc2569379974314500652la_a_d @ P2 @ ( cons_formula_a_d @ X2 @ nil_formula_a_d ) ) )
=> ~ ! [P2: formula_a_d > formula_a_d > $o,X2: formula_a_d,Y2: formula_a_d,Xs3: list_formula_a_d] :
( X
!= ( produc2569379974314500652la_a_d @ P2 @ ( cons_formula_a_d @ X2 @ ( cons_formula_a_d @ Y2 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_85_sorted__wrt_Ocases,axiom,
! [X: produc6315692651791551267rray_o] :
( ! [P2: produc4999605722385084260rray_o > produc4999605722385084260rray_o > $o] :
( X
!= ( produc7148991607918580371rray_o @ P2 @ nil_Pr7455003300710157162rray_o ) )
=> ~ ! [P2: produc4999605722385084260rray_o > produc4999605722385084260rray_o > $o,X2: produc4999605722385084260rray_o,Ys4: list_P1335534284205260522rray_o] :
( X
!= ( produc7148991607918580371rray_o @ P2 @ ( cons_P2929697372952323866rray_o @ X2 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_86_sorted__wrt_Ocases,axiom,
! [X: produc698650026130441430od_t_d] :
( ! [P2: product_prod_t_d > product_prod_t_d > $o] :
( X
!= ( produc3153264129863119558od_t_d @ P2 @ nil_Product_prod_t_d ) )
=> ~ ! [P2: product_prod_t_d > product_prod_t_d > $o,X2: product_prod_t_d,Ys4: list_P7979455665520128861od_t_d] :
( X
!= ( produc3153264129863119558od_t_d @ P2 @ ( cons_P472143266409286669od_t_d @ X2 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_87_sorted__wrt_Ocases,axiom,
! [X: produc114835076914812850la_a_d] :
( ! [P2: formula_a_d > formula_a_d > $o] :
( X
!= ( produc2569379974314500652la_a_d @ P2 @ nil_formula_a_d ) )
=> ~ ! [P2: formula_a_d > formula_a_d > $o,X2: formula_a_d,Ys4: list_formula_a_d] :
( X
!= ( produc2569379974314500652la_a_d @ P2 @ ( cons_formula_a_d @ X2 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_88_shuffles_Ocases,axiom,
! [X: produc2309011733990780583rray_o] :
( ! [Ys4: list_P1335534284205260522rray_o] :
( X
!= ( produc486054063986750871rray_o @ nil_Pr7455003300710157162rray_o @ Ys4 ) )
=> ( ! [Xs3: list_P1335534284205260522rray_o] :
( X
!= ( produc486054063986750871rray_o @ Xs3 @ nil_Pr7455003300710157162rray_o ) )
=> ~ ! [X2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o,Y2: produc4999605722385084260rray_o,Ys4: list_P1335534284205260522rray_o] :
( X
!= ( produc486054063986750871rray_o @ ( cons_P2929697372952323866rray_o @ X2 @ Xs3 ) @ ( cons_P2929697372952323866rray_o @ Y2 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_89_shuffles_Ocases,axiom,
! [X: produc5301187560997494631od_t_d] :
( ! [Ys4: list_P7979455665520128861od_t_d] :
( X
!= ( produc1277340417561686487od_t_d @ nil_Product_prod_t_d @ Ys4 ) )
=> ( ! [Xs3: list_P7979455665520128861od_t_d] :
( X
!= ( produc1277340417561686487od_t_d @ Xs3 @ nil_Product_prod_t_d ) )
=> ~ ! [X2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d,Y2: product_prod_t_d,Ys4: list_P7979455665520128861od_t_d] :
( X
!= ( produc1277340417561686487od_t_d @ ( cons_P472143266409286669od_t_d @ X2 @ Xs3 ) @ ( cons_P472143266409286669od_t_d @ Y2 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_90_shuffles_Ocases,axiom,
! [X: produc3582812622803015463la_a_d] :
( ! [Ys4: list_formula_a_d] :
( X
!= ( produc2153817203491026655la_a_d @ nil_formula_a_d @ Ys4 ) )
=> ( ! [Xs3: list_formula_a_d] :
( X
!= ( produc2153817203491026655la_a_d @ Xs3 @ nil_formula_a_d ) )
=> ~ ! [X2: formula_a_d,Xs3: list_formula_a_d,Y2: formula_a_d,Ys4: list_formula_a_d] :
( X
!= ( produc2153817203491026655la_a_d @ ( cons_formula_a_d @ X2 @ Xs3 ) @ ( cons_formula_a_d @ Y2 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_91_splice_Ocases,axiom,
! [X: produc2309011733990780583rray_o] :
( ! [Ys4: list_P1335534284205260522rray_o] :
( X
!= ( produc486054063986750871rray_o @ nil_Pr7455003300710157162rray_o @ Ys4 ) )
=> ~ ! [X2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o,Ys4: list_P1335534284205260522rray_o] :
( X
!= ( produc486054063986750871rray_o @ ( cons_P2929697372952323866rray_o @ X2 @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_92_splice_Ocases,axiom,
! [X: produc5301187560997494631od_t_d] :
( ! [Ys4: list_P7979455665520128861od_t_d] :
( X
!= ( produc1277340417561686487od_t_d @ nil_Product_prod_t_d @ Ys4 ) )
=> ~ ! [X2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d,Ys4: list_P7979455665520128861od_t_d] :
( X
!= ( produc1277340417561686487od_t_d @ ( cons_P472143266409286669od_t_d @ X2 @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_93_splice_Ocases,axiom,
! [X: produc3582812622803015463la_a_d] :
( ! [Ys4: list_formula_a_d] :
( X
!= ( produc2153817203491026655la_a_d @ nil_formula_a_d @ Ys4 ) )
=> ~ ! [X2: formula_a_d,Xs3: list_formula_a_d,Ys4: list_formula_a_d] :
( X
!= ( produc2153817203491026655la_a_d @ ( cons_formula_a_d @ X2 @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_94_pos_Ocases,axiom,
! [X: produc1240105006719930925rray_o] :
( ! [A2: produc4999605722385084260rray_o] :
( X
!= ( produc4089732580901332125rray_o @ A2 @ nil_Pr7455003300710157162rray_o ) )
=> ~ ! [A2: produc4999605722385084260rray_o,X2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o] :
( X
!= ( produc4089732580901332125rray_o @ A2 @ ( cons_P2929697372952323866rray_o @ X2 @ Xs3 ) ) ) ) ).
% pos.cases
thf(fact_95_pos_Ocases,axiom,
! [X: produc5263298794716405357od_t_d] :
( ! [A2: product_prod_t_d] :
( X
!= ( produc3699784390446185565od_t_d @ A2 @ nil_Product_prod_t_d ) )
=> ~ ! [A2: product_prod_t_d,X2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d] :
( X
!= ( produc3699784390446185565od_t_d @ A2 @ ( cons_P472143266409286669od_t_d @ X2 @ Xs3 ) ) ) ) ).
% pos.cases
thf(fact_96_pos_Ocases,axiom,
! [X: produc7235502628711444247la_a_d] :
( ! [A2: formula_a_d] :
( X
!= ( produc2574096460984752719la_a_d @ A2 @ nil_formula_a_d ) )
=> ~ ! [A2: formula_a_d,X2: formula_a_d,Xs3: list_formula_a_d] :
( X
!= ( produc2574096460984752719la_a_d @ A2 @ ( cons_formula_a_d @ X2 @ Xs3 ) ) ) ) ).
% pos.cases
thf(fact_97_not__Cons__self2,axiom,
! [X: produc4999605722385084260rray_o,Xs: list_P1335534284205260522rray_o] :
( ( cons_P2929697372952323866rray_o @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_98_not__Cons__self2,axiom,
! [X: product_prod_t_d,Xs: list_P7979455665520128861od_t_d] :
( ( cons_P472143266409286669od_t_d @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_99_not__Cons__self2,axiom,
! [X: formula_a_d,Xs: list_formula_a_d] :
( ( cons_formula_a_d @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_100_neq__if__length__neq,axiom,
! [Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o] :
( ( ( size_s1844703894572694870rray_o @ Xs )
!= ( size_s1844703894572694870rray_o @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_101_neq__if__length__neq,axiom,
! [Xs: list_d,Ys: list_d] :
( ( ( size_size_list_d @ Xs )
!= ( size_size_list_d @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_102_neq__if__length__neq,axiom,
! [Xs: list_iarray_o,Ys: list_iarray_o] :
( ( ( size_s3243743627286111600rray_o @ Xs )
!= ( size_s3243743627286111600rray_o @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_103_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_P1335534284205260522rray_o] :
( ( size_s1844703894572694870rray_o @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_104_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_d] :
( ( size_size_list_d @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_105_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_iarray_o] :
( ( size_s3243743627286111600rray_o @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_106_append__eq__append__conv2,axiom,
! [Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o,Zs: list_P1335534284205260522rray_o,Ts: list_P1335534284205260522rray_o] :
( ( ( append1741259189927458815rray_o @ Xs @ Ys )
= ( append1741259189927458815rray_o @ Zs @ Ts ) )
= ( ? [Us2: list_P1335534284205260522rray_o] :
( ( ( Xs
= ( append1741259189927458815rray_o @ Zs @ Us2 ) )
& ( ( append1741259189927458815rray_o @ Us2 @ Ys )
= Ts ) )
| ( ( ( append1741259189927458815rray_o @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append1741259189927458815rray_o @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_107_append__eq__append__conv2,axiom,
! [Xs: list_formula_a_d,Ys: list_formula_a_d,Zs: list_formula_a_d,Ts: list_formula_a_d] :
( ( ( append_formula_a_d @ Xs @ Ys )
= ( append_formula_a_d @ Zs @ Ts ) )
= ( ? [Us2: list_formula_a_d] :
( ( ( Xs
= ( append_formula_a_d @ Zs @ Us2 ) )
& ( ( append_formula_a_d @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_formula_a_d @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_formula_a_d @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_108_append__eq__append__conv2,axiom,
! [Xs: list_iarray_o,Ys: list_iarray_o,Zs: list_iarray_o,Ts: list_iarray_o] :
( ( ( append_iarray_o @ Xs @ Ys )
= ( append_iarray_o @ Zs @ Ts ) )
= ( ? [Us2: list_iarray_o] :
( ( ( Xs
= ( append_iarray_o @ Zs @ Us2 ) )
& ( ( append_iarray_o @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_iarray_o @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_iarray_o @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_109_append__eq__append__conv2,axiom,
! [Xs: list_d,Ys: list_d,Zs: list_d,Ts: list_d] :
( ( ( append_d @ Xs @ Ys )
= ( append_d @ Zs @ Ts ) )
= ( ? [Us2: list_d] :
( ( ( Xs
= ( append_d @ Zs @ Us2 ) )
& ( ( append_d @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_d @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_d @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_110_append__eq__appendI,axiom,
! [Xs: list_P1335534284205260522rray_o,Xs1: list_P1335534284205260522rray_o,Zs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o,Us: list_P1335534284205260522rray_o] :
( ( ( append1741259189927458815rray_o @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append1741259189927458815rray_o @ Xs1 @ Us ) )
=> ( ( append1741259189927458815rray_o @ Xs @ Ys )
= ( append1741259189927458815rray_o @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_111_append__eq__appendI,axiom,
! [Xs: list_formula_a_d,Xs1: list_formula_a_d,Zs: list_formula_a_d,Ys: list_formula_a_d,Us: list_formula_a_d] :
( ( ( append_formula_a_d @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_formula_a_d @ Xs1 @ Us ) )
=> ( ( append_formula_a_d @ Xs @ Ys )
= ( append_formula_a_d @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_112_append__eq__appendI,axiom,
! [Xs: list_iarray_o,Xs1: list_iarray_o,Zs: list_iarray_o,Ys: list_iarray_o,Us: list_iarray_o] :
( ( ( append_iarray_o @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_iarray_o @ Xs1 @ Us ) )
=> ( ( append_iarray_o @ Xs @ Ys )
= ( append_iarray_o @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_113_append__eq__appendI,axiom,
! [Xs: list_d,Xs1: list_d,Zs: list_d,Ys: list_d,Us: list_d] :
( ( ( append_d @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_d @ Xs1 @ Us ) )
=> ( ( append_d @ Xs @ Ys )
= ( append_d @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_114_list__nonempty__induct,axiom,
! [Xs: list_P1335534284205260522rray_o,P: list_P1335534284205260522rray_o > $o] :
( ( Xs != nil_Pr7455003300710157162rray_o )
=> ( ! [X2: produc4999605722385084260rray_o] : ( P @ ( cons_P2929697372952323866rray_o @ X2 @ nil_Pr7455003300710157162rray_o ) )
=> ( ! [X2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o] :
( ( Xs3 != nil_Pr7455003300710157162rray_o )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_P2929697372952323866rray_o @ X2 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_115_list__nonempty__induct,axiom,
! [Xs: list_P7979455665520128861od_t_d,P: list_P7979455665520128861od_t_d > $o] :
( ( Xs != nil_Product_prod_t_d )
=> ( ! [X2: product_prod_t_d] : ( P @ ( cons_P472143266409286669od_t_d @ X2 @ nil_Product_prod_t_d ) )
=> ( ! [X2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d] :
( ( Xs3 != nil_Product_prod_t_d )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_P472143266409286669od_t_d @ X2 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_116_list__nonempty__induct,axiom,
! [Xs: list_formula_a_d,P: list_formula_a_d > $o] :
( ( Xs != nil_formula_a_d )
=> ( ! [X2: formula_a_d] : ( P @ ( cons_formula_a_d @ X2 @ nil_formula_a_d ) )
=> ( ! [X2: formula_a_d,Xs3: list_formula_a_d] :
( ( Xs3 != nil_formula_a_d )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_formula_a_d @ X2 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_117_list__induct2_H,axiom,
! [P: list_P1335534284205260522rray_o > list_P1335534284205260522rray_o > $o,Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o] :
( ( P @ nil_Pr7455003300710157162rray_o @ nil_Pr7455003300710157162rray_o )
=> ( ! [X2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o] : ( P @ ( cons_P2929697372952323866rray_o @ X2 @ Xs3 ) @ nil_Pr7455003300710157162rray_o )
=> ( ! [Y2: produc4999605722385084260rray_o,Ys4: list_P1335534284205260522rray_o] : ( P @ nil_Pr7455003300710157162rray_o @ ( cons_P2929697372952323866rray_o @ Y2 @ Ys4 ) )
=> ( ! [X2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o,Y2: produc4999605722385084260rray_o,Ys4: list_P1335534284205260522rray_o] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_P2929697372952323866rray_o @ X2 @ Xs3 ) @ ( cons_P2929697372952323866rray_o @ Y2 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_118_list__induct2_H,axiom,
! [P: list_P1335534284205260522rray_o > list_P7979455665520128861od_t_d > $o,Xs: list_P1335534284205260522rray_o,Ys: list_P7979455665520128861od_t_d] :
( ( P @ nil_Pr7455003300710157162rray_o @ nil_Product_prod_t_d )
=> ( ! [X2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o] : ( P @ ( cons_P2929697372952323866rray_o @ X2 @ Xs3 ) @ nil_Product_prod_t_d )
=> ( ! [Y2: product_prod_t_d,Ys4: list_P7979455665520128861od_t_d] : ( P @ nil_Pr7455003300710157162rray_o @ ( cons_P472143266409286669od_t_d @ Y2 @ Ys4 ) )
=> ( ! [X2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o,Y2: product_prod_t_d,Ys4: list_P7979455665520128861od_t_d] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_P2929697372952323866rray_o @ X2 @ Xs3 ) @ ( cons_P472143266409286669od_t_d @ Y2 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_119_list__induct2_H,axiom,
! [P: list_P1335534284205260522rray_o > list_formula_a_d > $o,Xs: list_P1335534284205260522rray_o,Ys: list_formula_a_d] :
( ( P @ nil_Pr7455003300710157162rray_o @ nil_formula_a_d )
=> ( ! [X2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o] : ( P @ ( cons_P2929697372952323866rray_o @ X2 @ Xs3 ) @ nil_formula_a_d )
=> ( ! [Y2: formula_a_d,Ys4: list_formula_a_d] : ( P @ nil_Pr7455003300710157162rray_o @ ( cons_formula_a_d @ Y2 @ Ys4 ) )
=> ( ! [X2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o,Y2: formula_a_d,Ys4: list_formula_a_d] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_P2929697372952323866rray_o @ X2 @ Xs3 ) @ ( cons_formula_a_d @ Y2 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_120_list__induct2_H,axiom,
! [P: list_P7979455665520128861od_t_d > list_P1335534284205260522rray_o > $o,Xs: list_P7979455665520128861od_t_d,Ys: list_P1335534284205260522rray_o] :
( ( P @ nil_Product_prod_t_d @ nil_Pr7455003300710157162rray_o )
=> ( ! [X2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d] : ( P @ ( cons_P472143266409286669od_t_d @ X2 @ Xs3 ) @ nil_Pr7455003300710157162rray_o )
=> ( ! [Y2: produc4999605722385084260rray_o,Ys4: list_P1335534284205260522rray_o] : ( P @ nil_Product_prod_t_d @ ( cons_P2929697372952323866rray_o @ Y2 @ Ys4 ) )
=> ( ! [X2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d,Y2: produc4999605722385084260rray_o,Ys4: list_P1335534284205260522rray_o] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_P472143266409286669od_t_d @ X2 @ Xs3 ) @ ( cons_P2929697372952323866rray_o @ Y2 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_121_list__induct2_H,axiom,
! [P: list_P7979455665520128861od_t_d > list_P7979455665520128861od_t_d > $o,Xs: list_P7979455665520128861od_t_d,Ys: list_P7979455665520128861od_t_d] :
( ( P @ nil_Product_prod_t_d @ nil_Product_prod_t_d )
=> ( ! [X2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d] : ( P @ ( cons_P472143266409286669od_t_d @ X2 @ Xs3 ) @ nil_Product_prod_t_d )
=> ( ! [Y2: product_prod_t_d,Ys4: list_P7979455665520128861od_t_d] : ( P @ nil_Product_prod_t_d @ ( cons_P472143266409286669od_t_d @ Y2 @ Ys4 ) )
=> ( ! [X2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d,Y2: product_prod_t_d,Ys4: list_P7979455665520128861od_t_d] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_P472143266409286669od_t_d @ X2 @ Xs3 ) @ ( cons_P472143266409286669od_t_d @ Y2 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_122_list__induct2_H,axiom,
! [P: list_P7979455665520128861od_t_d > list_formula_a_d > $o,Xs: list_P7979455665520128861od_t_d,Ys: list_formula_a_d] :
( ( P @ nil_Product_prod_t_d @ nil_formula_a_d )
=> ( ! [X2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d] : ( P @ ( cons_P472143266409286669od_t_d @ X2 @ Xs3 ) @ nil_formula_a_d )
=> ( ! [Y2: formula_a_d,Ys4: list_formula_a_d] : ( P @ nil_Product_prod_t_d @ ( cons_formula_a_d @ Y2 @ Ys4 ) )
=> ( ! [X2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d,Y2: formula_a_d,Ys4: list_formula_a_d] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_P472143266409286669od_t_d @ X2 @ Xs3 ) @ ( cons_formula_a_d @ Y2 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_123_list__induct2_H,axiom,
! [P: list_formula_a_d > list_P1335534284205260522rray_o > $o,Xs: list_formula_a_d,Ys: list_P1335534284205260522rray_o] :
( ( P @ nil_formula_a_d @ nil_Pr7455003300710157162rray_o )
=> ( ! [X2: formula_a_d,Xs3: list_formula_a_d] : ( P @ ( cons_formula_a_d @ X2 @ Xs3 ) @ nil_Pr7455003300710157162rray_o )
=> ( ! [Y2: produc4999605722385084260rray_o,Ys4: list_P1335534284205260522rray_o] : ( P @ nil_formula_a_d @ ( cons_P2929697372952323866rray_o @ Y2 @ Ys4 ) )
=> ( ! [X2: formula_a_d,Xs3: list_formula_a_d,Y2: produc4999605722385084260rray_o,Ys4: list_P1335534284205260522rray_o] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_formula_a_d @ X2 @ Xs3 ) @ ( cons_P2929697372952323866rray_o @ Y2 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_124_list__induct2_H,axiom,
! [P: list_formula_a_d > list_P7979455665520128861od_t_d > $o,Xs: list_formula_a_d,Ys: list_P7979455665520128861od_t_d] :
( ( P @ nil_formula_a_d @ nil_Product_prod_t_d )
=> ( ! [X2: formula_a_d,Xs3: list_formula_a_d] : ( P @ ( cons_formula_a_d @ X2 @ Xs3 ) @ nil_Product_prod_t_d )
=> ( ! [Y2: product_prod_t_d,Ys4: list_P7979455665520128861od_t_d] : ( P @ nil_formula_a_d @ ( cons_P472143266409286669od_t_d @ Y2 @ Ys4 ) )
=> ( ! [X2: formula_a_d,Xs3: list_formula_a_d,Y2: product_prod_t_d,Ys4: list_P7979455665520128861od_t_d] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_formula_a_d @ X2 @ Xs3 ) @ ( cons_P472143266409286669od_t_d @ Y2 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_125_list__induct2_H,axiom,
! [P: list_formula_a_d > list_formula_a_d > $o,Xs: list_formula_a_d,Ys: list_formula_a_d] :
( ( P @ nil_formula_a_d @ nil_formula_a_d )
=> ( ! [X2: formula_a_d,Xs3: list_formula_a_d] : ( P @ ( cons_formula_a_d @ X2 @ Xs3 ) @ nil_formula_a_d )
=> ( ! [Y2: formula_a_d,Ys4: list_formula_a_d] : ( P @ nil_formula_a_d @ ( cons_formula_a_d @ Y2 @ Ys4 ) )
=> ( ! [X2: formula_a_d,Xs3: list_formula_a_d,Y2: formula_a_d,Ys4: list_formula_a_d] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_formula_a_d @ X2 @ Xs3 ) @ ( cons_formula_a_d @ Y2 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_126_neq__Nil__conv,axiom,
! [Xs: list_P1335534284205260522rray_o] :
( ( Xs != nil_Pr7455003300710157162rray_o )
= ( ? [Y3: produc4999605722385084260rray_o,Ys5: list_P1335534284205260522rray_o] :
( Xs
= ( cons_P2929697372952323866rray_o @ Y3 @ Ys5 ) ) ) ) ).
% neq_Nil_conv
thf(fact_127_neq__Nil__conv,axiom,
! [Xs: list_P7979455665520128861od_t_d] :
( ( Xs != nil_Product_prod_t_d )
= ( ? [Y3: product_prod_t_d,Ys5: list_P7979455665520128861od_t_d] :
( Xs
= ( cons_P472143266409286669od_t_d @ Y3 @ Ys5 ) ) ) ) ).
% neq_Nil_conv
thf(fact_128_neq__Nil__conv,axiom,
! [Xs: list_formula_a_d] :
( ( Xs != nil_formula_a_d )
= ( ? [Y3: formula_a_d,Ys5: list_formula_a_d] :
( Xs
= ( cons_formula_a_d @ Y3 @ Ys5 ) ) ) ) ).
% neq_Nil_conv
thf(fact_129_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_130_mem__Collect__eq,axiom,
! [A: d,P: d > $o] :
( ( member_d @ A @ ( collect_d @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_131_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_132_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_133_Collect__mem__eq,axiom,
! [A3: set_d] :
( ( collect_d
@ ^ [X3: d] : ( member_d @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
! [A3: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_135_Collect__cong,axiom,
! [P: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
( ! [X2: product_prod_nat_nat] :
( ( P @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collec3392354462482085612at_nat @ P )
= ( collec3392354462482085612at_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_136_remdups__adj_Ocases,axiom,
! [X: list_P1335534284205260522rray_o] :
( ( X != nil_Pr7455003300710157162rray_o )
=> ( ! [X2: produc4999605722385084260rray_o] :
( X
!= ( cons_P2929697372952323866rray_o @ X2 @ nil_Pr7455003300710157162rray_o ) )
=> ~ ! [X2: produc4999605722385084260rray_o,Y2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o] :
( X
!= ( cons_P2929697372952323866rray_o @ X2 @ ( cons_P2929697372952323866rray_o @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_137_remdups__adj_Ocases,axiom,
! [X: list_P7979455665520128861od_t_d] :
( ( X != nil_Product_prod_t_d )
=> ( ! [X2: product_prod_t_d] :
( X
!= ( cons_P472143266409286669od_t_d @ X2 @ nil_Product_prod_t_d ) )
=> ~ ! [X2: product_prod_t_d,Y2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d] :
( X
!= ( cons_P472143266409286669od_t_d @ X2 @ ( cons_P472143266409286669od_t_d @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_138_remdups__adj_Ocases,axiom,
! [X: list_formula_a_d] :
( ( X != nil_formula_a_d )
=> ( ! [X2: formula_a_d] :
( X
!= ( cons_formula_a_d @ X2 @ nil_formula_a_d ) )
=> ~ ! [X2: formula_a_d,Y2: formula_a_d,Xs3: list_formula_a_d] :
( X
!= ( cons_formula_a_d @ X2 @ ( cons_formula_a_d @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_139_transpose_Ocases,axiom,
! [X: list_l8465279369410806256rray_o] :
( ( X != nil_li7972630826580172784rray_o )
=> ( ! [Xss: list_l8465279369410806256rray_o] :
( X
!= ( cons_l8419465691382457248rray_o @ nil_Pr7455003300710157162rray_o @ Xss ) )
=> ~ ! [X2: produc4999605722385084260rray_o,Xs3: list_P1335534284205260522rray_o,Xss: list_l8465279369410806256rray_o] :
( X
!= ( cons_l8419465691382457248rray_o @ ( cons_P2929697372952323866rray_o @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_140_transpose_Ocases,axiom,
! [X: list_l7981870404175188963od_t_d] :
( ( X != nil_li5923887051818988131od_t_d )
=> ( ! [Xss: list_l7981870404175188963od_t_d] :
( X
!= ( cons_l7418867369989776915od_t_d @ nil_Product_prod_t_d @ Xss ) )
=> ~ ! [X2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d,Xss: list_l7981870404175188963od_t_d] :
( X
!= ( cons_l7418867369989776915od_t_d @ ( cons_P472143266409286669od_t_d @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_141_transpose_Ocases,axiom,
! [X: list_l905698018746532147la_a_d] :
( ( X != nil_list_formula_a_d )
=> ( ! [Xss: list_l905698018746532147la_a_d] :
( X
!= ( cons_l2592367353233851117la_a_d @ nil_formula_a_d @ Xss ) )
=> ~ ! [X2: formula_a_d,Xs3: list_formula_a_d,Xss: list_l905698018746532147la_a_d] :
( X
!= ( cons_l2592367353233851117la_a_d @ ( cons_formula_a_d @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_142_list_Oexhaust,axiom,
! [Y: list_P1335534284205260522rray_o] :
( ( Y != nil_Pr7455003300710157162rray_o )
=> ~ ! [X212: produc4999605722385084260rray_o,X222: list_P1335534284205260522rray_o] :
( Y
!= ( cons_P2929697372952323866rray_o @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_143_list_Oexhaust,axiom,
! [Y: list_P7979455665520128861od_t_d] :
( ( Y != nil_Product_prod_t_d )
=> ~ ! [X212: product_prod_t_d,X222: list_P7979455665520128861od_t_d] :
( Y
!= ( cons_P472143266409286669od_t_d @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_144_list_Oexhaust,axiom,
! [Y: list_formula_a_d] :
( ( Y != nil_formula_a_d )
=> ~ ! [X212: formula_a_d,X222: list_formula_a_d] :
( Y
!= ( cons_formula_a_d @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_145_list_OdiscI,axiom,
! [List: list_P1335534284205260522rray_o,X21: produc4999605722385084260rray_o,X22: list_P1335534284205260522rray_o] :
( ( List
= ( cons_P2929697372952323866rray_o @ X21 @ X22 ) )
=> ( List != nil_Pr7455003300710157162rray_o ) ) ).
% list.discI
thf(fact_146_list_OdiscI,axiom,
! [List: list_P7979455665520128861od_t_d,X21: product_prod_t_d,X22: list_P7979455665520128861od_t_d] :
( ( List
= ( cons_P472143266409286669od_t_d @ X21 @ X22 ) )
=> ( List != nil_Product_prod_t_d ) ) ).
% list.discI
thf(fact_147_list_OdiscI,axiom,
! [List: list_formula_a_d,X21: formula_a_d,X22: list_formula_a_d] :
( ( List
= ( cons_formula_a_d @ X21 @ X22 ) )
=> ( List != nil_formula_a_d ) ) ).
% list.discI
thf(fact_148_list_Odistinct_I1_J,axiom,
! [X21: produc4999605722385084260rray_o,X22: list_P1335534284205260522rray_o] :
( nil_Pr7455003300710157162rray_o
!= ( cons_P2929697372952323866rray_o @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_149_list_Odistinct_I1_J,axiom,
! [X21: product_prod_t_d,X22: list_P7979455665520128861od_t_d] :
( nil_Product_prod_t_d
!= ( cons_P472143266409286669od_t_d @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_150_list_Odistinct_I1_J,axiom,
! [X21: formula_a_d,X22: list_formula_a_d] :
( nil_formula_a_d
!= ( cons_formula_a_d @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_151_Cons__eq__appendI,axiom,
! [X: iarray_o,Xs1: list_iarray_o,Ys: list_iarray_o,Xs: list_iarray_o,Zs: list_iarray_o] :
( ( ( cons_iarray_o @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_iarray_o @ Xs1 @ Zs ) )
=> ( ( cons_iarray_o @ X @ Xs )
= ( append_iarray_o @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_152_Cons__eq__appendI,axiom,
! [X: d,Xs1: list_d,Ys: list_d,Xs: list_d,Zs: list_d] :
( ( ( cons_d @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_d @ Xs1 @ Zs ) )
=> ( ( cons_d @ X @ Xs )
= ( append_d @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_153_Cons__eq__appendI,axiom,
! [X: produc4999605722385084260rray_o,Xs1: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o,Xs: list_P1335534284205260522rray_o,Zs: list_P1335534284205260522rray_o] :
( ( ( cons_P2929697372952323866rray_o @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append1741259189927458815rray_o @ Xs1 @ Zs ) )
=> ( ( cons_P2929697372952323866rray_o @ X @ Xs )
= ( append1741259189927458815rray_o @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_154_Cons__eq__appendI,axiom,
! [X: product_prod_t_d,Xs1: list_P7979455665520128861od_t_d,Ys: list_P7979455665520128861od_t_d,Xs: list_P7979455665520128861od_t_d,Zs: list_P7979455665520128861od_t_d] :
( ( ( cons_P472143266409286669od_t_d @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append7713784995604329970od_t_d @ Xs1 @ Zs ) )
=> ( ( cons_P472143266409286669od_t_d @ X @ Xs )
= ( append7713784995604329970od_t_d @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_155_Cons__eq__appendI,axiom,
! [X: formula_a_d,Xs1: list_formula_a_d,Ys: list_formula_a_d,Xs: list_formula_a_d,Zs: list_formula_a_d] :
( ( ( cons_formula_a_d @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_formula_a_d @ Xs1 @ Zs ) )
=> ( ( cons_formula_a_d @ X @ Xs )
= ( append_formula_a_d @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_156_append__Cons,axiom,
! [X: iarray_o,Xs: list_iarray_o,Ys: list_iarray_o] :
( ( append_iarray_o @ ( cons_iarray_o @ X @ Xs ) @ Ys )
= ( cons_iarray_o @ X @ ( append_iarray_o @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_157_append__Cons,axiom,
! [X: d,Xs: list_d,Ys: list_d] :
( ( append_d @ ( cons_d @ X @ Xs ) @ Ys )
= ( cons_d @ X @ ( append_d @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_158_append__Cons,axiom,
! [X: produc4999605722385084260rray_o,Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o] :
( ( append1741259189927458815rray_o @ ( cons_P2929697372952323866rray_o @ X @ Xs ) @ Ys )
= ( cons_P2929697372952323866rray_o @ X @ ( append1741259189927458815rray_o @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_159_append__Cons,axiom,
! [X: product_prod_t_d,Xs: list_P7979455665520128861od_t_d,Ys: list_P7979455665520128861od_t_d] :
( ( append7713784995604329970od_t_d @ ( cons_P472143266409286669od_t_d @ X @ Xs ) @ Ys )
= ( cons_P472143266409286669od_t_d @ X @ ( append7713784995604329970od_t_d @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_160_append__Cons,axiom,
! [X: formula_a_d,Xs: list_formula_a_d,Ys: list_formula_a_d] :
( ( append_formula_a_d @ ( cons_formula_a_d @ X @ Xs ) @ Ys )
= ( cons_formula_a_d @ X @ ( append_formula_a_d @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_161_eq__Nil__appendI,axiom,
! [Xs: list_iarray_o,Ys: list_iarray_o] :
( ( Xs = Ys )
=> ( Xs
= ( append_iarray_o @ nil_iarray_o @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_162_eq__Nil__appendI,axiom,
! [Xs: list_d,Ys: list_d] :
( ( Xs = Ys )
=> ( Xs
= ( append_d @ nil_d @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_163_eq__Nil__appendI,axiom,
! [Xs: list_P1335534284205260522rray_o,Ys: list_P1335534284205260522rray_o] :
( ( Xs = Ys )
=> ( Xs
= ( append1741259189927458815rray_o @ nil_Pr7455003300710157162rray_o @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_164_eq__Nil__appendI,axiom,
! [Xs: list_formula_a_d,Ys: list_formula_a_d] :
( ( Xs = Ys )
=> ( Xs
= ( append_formula_a_d @ nil_formula_a_d @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_165_eq__Nil__appendI,axiom,
! [Xs: list_P7979455665520128861od_t_d,Ys: list_P7979455665520128861od_t_d] :
( ( Xs = Ys )
=> ( Xs
= ( append7713784995604329970od_t_d @ nil_Product_prod_t_d @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_166_append_Oleft__neutral,axiom,
! [A: list_iarray_o] :
( ( append_iarray_o @ nil_iarray_o @ A )
= A ) ).
% append.left_neutral
thf(fact_167_append_Oleft__neutral,axiom,
! [A: list_d] :
( ( append_d @ nil_d @ A )
= A ) ).
% append.left_neutral
thf(fact_168_append_Oleft__neutral,axiom,
! [A: list_P1335534284205260522rray_o] :
( ( append1741259189927458815rray_o @ nil_Pr7455003300710157162rray_o @ A )
= A ) ).
% append.left_neutral
thf(fact_169_append_Oleft__neutral,axiom,
! [A: list_formula_a_d] :
( ( append_formula_a_d @ nil_formula_a_d @ A )
= A ) ).
% append.left_neutral
thf(fact_170_append_Oleft__neutral,axiom,
! [A: list_P7979455665520128861od_t_d] :
( ( append7713784995604329970od_t_d @ nil_Product_prod_t_d @ A )
= A ) ).
% append.left_neutral
thf(fact_171_append__Nil,axiom,
! [Ys: list_iarray_o] :
( ( append_iarray_o @ nil_iarray_o @ Ys )
= Ys ) ).
% append_Nil
thf(fact_172_append__Nil,axiom,
! [Ys: list_d] :
( ( append_d @ nil_d @ Ys )
= Ys ) ).
% append_Nil
thf(fact_173_append__Nil,axiom,
! [Ys: list_P1335534284205260522rray_o] :
( ( append1741259189927458815rray_o @ nil_Pr7455003300710157162rray_o @ Ys )
= Ys ) ).
% append_Nil
thf(fact_174_append__Nil,axiom,
! [Ys: list_formula_a_d] :
( ( append_formula_a_d @ nil_formula_a_d @ Ys )
= Ys ) ).
% append_Nil
thf(fact_175_append__Nil,axiom,
! [Ys: list_P7979455665520128861od_t_d] :
( ( append7713784995604329970od_t_d @ nil_Product_prod_t_d @ Ys )
= Ys ) ).
% append_Nil
thf(fact_176_list__induct4,axiom,
! [Xs: list_d,Ys: list_d,Zs: list_d,Ws: list_d,P: list_d > list_d > list_d > list_d > $o] :
( ( ( size_size_list_d @ Xs )
= ( size_size_list_d @ Ys ) )
=> ( ( ( size_size_list_d @ Ys )
= ( size_size_list_d @ Zs ) )
=> ( ( ( size_size_list_d @ Zs )
= ( size_size_list_d @ Ws ) )
=> ( ( P @ nil_d @ nil_d @ nil_d @ nil_d )
=> ( ! [X2: d,Xs3: list_d,Y2: d,Ys4: list_d,Z: d,Zs2: list_d,W: d,Ws2: list_d] :
( ( ( size_size_list_d @ Xs3 )
= ( size_size_list_d @ Ys4 ) )
=> ( ( ( size_size_list_d @ Ys4 )
= ( size_size_list_d @ Zs2 ) )
=> ( ( ( size_size_list_d @ Zs2 )
= ( size_size_list_d @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_d @ X2 @ Xs3 ) @ ( cons_d @ Y2 @ Ys4 ) @ ( cons_d @ Z @ Zs2 ) @ ( cons_d @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_177_list__induct4,axiom,
! [Xs: list_d,Ys: list_d,Zs: list_d,Ws: list_iarray_o,P: list_d > list_d > list_d > list_iarray_o > $o] :
( ( ( size_size_list_d @ Xs )
= ( size_size_list_d @ Ys ) )
=> ( ( ( size_size_list_d @ Ys )
= ( size_size_list_d @ Zs ) )
=> ( ( ( size_size_list_d @ Zs )
= ( size_s3243743627286111600rray_o @ Ws ) )
=> ( ( P @ nil_d @ nil_d @ nil_d @ nil_iarray_o )
=> ( ! [X2: d,Xs3: list_d,Y2: d,Ys4: list_d,Z: d,Zs2: list_d,W: iarray_o,Ws2: list_iarray_o] :
( ( ( size_size_list_d @ Xs3 )
= ( size_size_list_d @ Ys4 ) )
=> ( ( ( size_size_list_d @ Ys4 )
= ( size_size_list_d @ Zs2 ) )
=> ( ( ( size_size_list_d @ Zs2 )
= ( size_s3243743627286111600rray_o @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_d @ X2 @ Xs3 ) @ ( cons_d @ Y2 @ Ys4 ) @ ( cons_d @ Z @ Zs2 ) @ ( cons_iarray_o @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_178_list__induct4,axiom,
! [Xs: list_d,Ys: list_d,Zs: list_iarray_o,Ws: list_d,P: list_d > list_d > list_iarray_o > list_d > $o] :
( ( ( size_size_list_d @ Xs )
= ( size_size_list_d @ Ys ) )
=> ( ( ( size_size_list_d @ Ys )
= ( size_s3243743627286111600rray_o @ Zs ) )
=> ( ( ( size_s3243743627286111600rray_o @ Zs )
= ( size_size_list_d @ Ws ) )
=> ( ( P @ nil_d @ nil_d @ nil_iarray_o @ nil_d )
=> ( ! [X2: d,Xs3: list_d,Y2: d,Ys4: list_d,Z: iarray_o,Zs2: list_iarray_o,W: d,Ws2: list_d] :
( ( ( size_size_list_d @ Xs3 )
= ( size_size_list_d @ Ys4 ) )
=> ( ( ( size_size_list_d @ Ys4 )
= ( size_s3243743627286111600rray_o @ Zs2 ) )
=> ( ( ( size_s3243743627286111600rray_o @ Zs2 )
= ( size_size_list_d @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_d @ X2 @ Xs3 ) @ ( cons_d @ Y2 @ Ys4 ) @ ( cons_iarray_o @ Z @ Zs2 ) @ ( cons_d @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_179_list__induct4,axiom,
! [Xs: list_d,Ys: list_iarray_o,Zs: list_d,Ws: list_d,P: list_d > list_iarray_o > list_d > list_d > $o] :
( ( ( size_size_list_d @ Xs )
= ( size_s3243743627286111600rray_o @ Ys ) )
=> ( ( ( size_s3243743627286111600rray_o @ Ys )
= ( size_size_list_d @ Zs ) )
=> ( ( ( size_size_list_d @ Zs )
= ( size_size_list_d @ Ws ) )
=> ( ( P @ nil_d @ nil_iarray_o @ nil_d @ nil_d )
=> ( ! [X2: d,Xs3: list_d,Y2: iarray_o,Ys4: list_iarray_o,Z: d,Zs2: list_d,W: d,Ws2: list_d] :
( ( ( size_size_list_d @ Xs3 )
= ( size_s3243743627286111600rray_o @ Ys4 ) )
=> ( ( ( size_s3243743627286111600rray_o @ Ys4 )
= ( size_size_list_d @ Zs2 ) )
=> ( ( ( size_size_list_d @ Zs2 )
= ( size_size_list_d @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_d @ X2 @ Xs3 ) @ ( cons_iarray_o @ Y2 @ Ys4 ) @ ( cons_d @ Z @ Zs2 ) @ ( cons_d @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_180_list__induct4,axiom,
! [Xs: list_iarray_o,Ys: list_d,Zs: list_d,Ws: list_d,P: list_iarray_o > list_d > list_d > list_d > $o] :
( ( ( size_s3243743627286111600rray_o @ Xs )
= ( size_size_list_d @ Ys ) )
=> ( ( ( size_size_list_d @ Ys )
= ( size_size_list_d @ Zs ) )
=> ( ( ( size_size_list_d @ Zs )
= ( size_size_list_d @ Ws ) )
=> ( ( P @ nil_iarray_o @ nil_d @ nil_d @ nil_d )
=> ( ! [X2: iarray_o,Xs3: list_iarray_o,Y2: d,Ys4: list_d,Z: d,Zs2: list_d,W: d,Ws2: list_d] :
( ( ( size_s3243743627286111600rray_o @ Xs3 )
= ( size_size_list_d @ Ys4 ) )
=> ( ( ( size_size_list_d @ Ys4 )
= ( size_size_list_d @ Zs2 ) )
=> ( ( ( size_size_list_d @ Zs2 )
= ( size_size_list_d @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_iarray_o @ X2 @ Xs3 ) @ ( cons_d @ Y2 @ Ys4 ) @ ( cons_d @ Z @ Zs2 ) @ ( cons_d @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_181_list__induct4,axiom,
! [Xs: list_P7979455665520128861od_t_d,Ys: list_d,Zs: list_d,Ws: list_d,P: list_P7979455665520128861od_t_d > list_d > list_d > list_d > $o] :
( ( ( size_s1244821776335097289od_t_d @ Xs )
= ( size_size_list_d @ Ys ) )
=> ( ( ( size_size_list_d @ Ys )
= ( size_size_list_d @ Zs ) )
=> ( ( ( size_size_list_d @ Zs )
= ( size_size_list_d @ Ws ) )
=> ( ( P @ nil_Product_prod_t_d @ nil_d @ nil_d @ nil_d )
=> ( ! [X2: product_prod_t_d,Xs3: list_P7979455665520128861od_t_d,Y2: d,Ys4: list_d,Z: d,Zs2: list_d,W: d,Ws2: list_d] :
( ( ( size_s1244821776335097289od_t_d @ Xs3 )
= ( size_size_list_d @ Ys4 ) )
=> ( ( ( size_size_list_d @ Ys4 )
= ( size_size_list_d @ Zs2 ) )
=> ( ( ( size_size_list_d @ Zs2 )
= ( size_size_list_d @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_P472143266409286669od_t_d @ X2 @ Xs3 ) @ ( cons_d @ Y2 @ Ys4 ) @ ( cons_d @ Z @ Zs2 ) @ ( cons_d @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_182_list__induct4,axiom,
! [Xs: list_formula_a_d,Ys: list_d,Zs: list_d,Ws: list_d,P: list_formula_a_d > list_d > list_d > list_d > $o] :
( ( ( size_s7710211460959691191la_a_d @ Xs )
= ( size_size_list_d @ Ys ) )
=> ( ( ( size_size_list_d @ Ys )
= ( size_size_list_d @ Zs ) )
=> ( ( ( size_size_list_d @ Zs )
= ( size_size_list_d @ Ws ) )
=> ( ( P @ nil_formula_a_d @ nil_d @ nil_d @ nil_d )
=> ( ! [X2: formula_a_d,Xs3: list_formula_a_d,Y2: d,Ys4: list_d,Z: d,Zs2: list_d,W: d,Ws2: list_d] :
( ( ( size_s7710211460959691191la_a_d @ Xs3 )
= ( size_size_list_d @ Ys4 ) )
=> ( ( ( size_size_list_d @ Ys4 )
= ( size_size_list_d @ Zs2 ) )
=> ( ( ( size_size_list_d @ Zs2 )
= ( size_size_list_d @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_formula_a_d @ X2 @ Xs3 ) @ ( cons_d @ Y2 @ Ys4 ) @ ( cons_d @ Z @ Zs2 ) @ ( cons_d @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_183_list__induct4,axiom,
! [Xs: list_d,Ys: list_P7979455665520128861od_t_d,Zs: list_d,Ws: list_d,P: list_d > list_P7979455665520128861od_t_d > list_d > list_d > $o] :
( ( ( size_size_list_d @ Xs )
= ( size_s1244821776335097289od_t_d @ Ys ) )
=> ( ( ( size_s1244821776335097289od_t_d @ Ys )
= ( size_size_list_d @ Zs ) )
=> ( ( ( size_size_list_d @ Zs )
= ( size_size_list_d @ Ws ) )
=> ( ( P @ nil_d @ nil_Product_prod_t_d @ nil_d @ nil_d )
=> ( ! [X2: d,Xs3: list_d,Y2: product_prod_t_d,Ys4: list_P7979455665520128861od_t_d,Z: d,Zs2: list_d,W: d,Ws2: list_d] :
( ( ( size_size_list_d @ Xs3 )
= ( size_s1244821776335097289od_t_d @ Ys4 ) )
=> ( ( ( size_s1244821776335097289od_t_d @ Ys4 )
= ( size_size_list_d @ Zs2 ) )
=> ( ( ( size_size_list_d @ Zs2 )
= ( size_size_list_d @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_d @ X2 @ Xs3 ) @ ( cons_P472143266409286669od_t_d @ Y2 @ Ys4 ) @ ( cons_d @ Z @ Zs2 ) @ ( cons_d @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_184_list__induct4,axiom,
! [Xs: list_d,Ys: list_formula_a_d,Zs: list_d,Ws: list_d,P: list_d > list_formula_a_d > list_d > list_d > $o] :
( ( ( size_size_list_d @ Xs )
= ( size_s7710211460959691191la_a_d @ Ys ) )
=> ( ( ( size_s7710211460959691191la_a_d @ Ys )
= ( size_size_list_d @ Zs ) )
=> ( ( ( size_size_list_d @ Zs )
= ( size_size_list_d @ Ws ) )
=> ( ( P @ nil_d @ nil_formula_a_d @ nil_d @ nil_d )
=> ( ! [X2: d,Xs3: list_d,Y2: formula_a_d,Ys4: list_formula_a_d,Z: d,Zs2: list_d,W: d,Ws2: list_d] :
( ( ( size_size_list_d @ Xs3 )
= ( size_s7710211460959691191la_a_d @ Ys4 ) )
=> ( ( ( size_s7710211460959691191la_a_d @ Ys4 )
= ( size_size_list_d @ Zs2 ) )
=> ( ( ( size_size_list_d @ Zs2 )
= ( size_size_list_d @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_d @ X2 @ Xs3 ) @ ( cons_formula_a_d @ Y2 @ Ys4 ) @ ( cons_d @ Z @ Zs2 ) @ ( cons_d @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_185_list__induct4,axiom,
! [Xs: list_d,Ys: list_d,Zs: list_P7979455665520128861od_t_d,Ws: list_d,P: list_d > list_d > list_P7979455665520128861od_t_d > list_d > $o] :
( ( ( size_size_list_d @ Xs )
= ( size_size_list_d @ Ys ) )
=> ( ( ( size_size_list_d @ Ys )
= ( size_s1244821776335097289od_t_d @ Zs ) )
=> ( ( ( size_s1244821776335097289od_t_d @ Zs )
= ( size_size_list_d @ Ws ) )
=> ( ( P @ nil_d @ nil_d @ nil_Product_prod_t_d @ nil_d )
=> ( ! [X2: d,Xs3: list_d,Y2: d,Ys4: list_d,Z: product_prod_t_d,Zs2: list_P7979455665520128861od_t_d,W: d,Ws2: list_d] :
( ( ( size_size_list_d @ Xs3 )
= ( size_size_list_d @ Ys4 ) )
=> ( ( ( size_size_list_d @ Ys4 )
= ( size_s1244821776335097289od_t_d @ Zs2 ) )
=> ( ( ( size_s1244821776335097289od_t_d @ Zs2 )
= ( size_size_list_d @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_d @ X2 @ Xs3 ) @ ( cons_d @ Y2 @ Ys4 ) @ ( cons_P472143266409286669od_t_d @ Z @ Zs2 ) @ ( cons_d @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_186_rho_H_H__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s1844703894572694870rray_o @ rho3 ) )
=> ( ord_less_eq_d @ ( ts_at_d_iarray_o @ rho3 @ I ) @ ( ts_at_d_iarray_o @ rho3 @ J ) ) ) ) ).
% rho''_mono
thf(fact_187_j_H__le__rho_H_H,axiom,
ord_less_eq_nat @ j2 @ ( size_s1844703894572694870rray_o @ rho3 ) ).
% j'_le_rho''
thf(fact_188_valid__before_I2_J,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s1844703894572694870rray_o @ rho ) )
=> ( ord_less_eq_d @ ( ts_at_d_iarray_o @ rho @ I ) @ ( ts_at_d_iarray_o @ rho @ J ) ) ) ) ).
% valid_before(2)
thf(fact_189_l__def_I1_J,axiom,
ord_less_eq_nat @ i2 @ l ).
% l_def(1)
thf(fact_190_valid__after_I4_J,axiom,
i4 = i2 ).
% valid_after(4)
thf(fact_191_i__lt__j_H,axiom,
ord_less_nat @ i2 @ j2 ).
% i_lt_j'
thf(fact_192_i__le__j_H,axiom,
ord_less_eq_nat @ i2 @ j2 ).
% i_le_j'
thf(fact_193_i__lt__rho_H_H,axiom,
ord_less_nat @ i2 @ ( size_s1844703894572694870rray_o @ rho3 ) ).
% i_lt_rho''
thf(fact_194_i__le__rho,axiom,
ord_less_eq_nat @ i2 @ ( size_s1844703894572694870rray_o @ rho3 ) ).
% i_le_rho
thf(fact_195_tbi__def_I3_J,axiom,
( t2
= ( ts_at_d_iarray_o @ rho3 @ i2 ) ) ).
% tbi_def(3)
thf(fact_196_rho_H_H__def,axiom,
( rho3
= ( append1741259189927458815rray_o @ rho @ rho2 ) ) ).
% rho''_def
thf(fact_197_tbi__def_I4_J,axiom,
( b
= ( bs_at_d_iarray_o @ rho3 @ i2 ) ) ).
% tbi_def(4)
thf(fact_198_valid__before_I7_J,axiom,
( ( size_s1844703894572694870rray_o @ rho )
= j3 ) ).
% valid_before(7)
thf(fact_199_i__j,axiom,
ord_less_eq_nat @ i2 @ j3 ).
% i_j
thf(fact_200_t__def,axiom,
( t2
= ( tau_a_d @ sigma @ i2 ) ) ).
% t_def
thf(fact_201_i_H__set_I2_J,axiom,
~ ( memR_d @ ( ts_at_d_iarray_o @ rho3 @ i2 ) @ ( ts_at_d_iarray_o @ ( append1741259189927458815rray_o @ rho3 @ ( cons_P2929697372952323866rray_o @ ( produc6263459992454928980rray_o @ tm @ undefined_iarray_o ) @ nil_Pr7455003300710157162rray_o ) ) @ j2 ) @ i ) ).
% i'_set(2)
thf(fact_202_l__def_I2_J,axiom,
mem_d @ ( tau_a_d @ sigma @ i2 ) @ ( tau_a_d @ sigma @ l ) @ i ).
% l_def(2)
thf(fact_203_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_204_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_205_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_206_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_207_before__end_H,axiom,
~ ( memR_d @ t2 @ tm @ i ) ).
% before_end'
thf(fact_208_not__ets__te,axiom,
~ ( memR_d @ ( ts_at_d_iarray_o @ rho3 @ i2 ) @ te @ i ) ).
% not_ets_te
thf(fact_209_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_210_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_211_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_212_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_213_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less_nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_214_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_215_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_216_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_217_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 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_218_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_219_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_220_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_221_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_222_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_223_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_224_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_225_valid__after_I3_J,axiom,
! [L: nat] :
( ( member_nat @ L @ ( set_or4665077453230672383an_nat @ i2 @ j2 ) )
=> ( memR_d @ ( ts_at_d_iarray_o @ rho3 @ i2 ) @ ( ts_at_d_iarray_o @ rho3 @ L ) @ i ) ) ).
% valid_after(3)
thf(fact_226_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_227_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_228_match__le,axiom,
! [I: nat,J: nat,R: regex_a_d] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_d @ sigma @ R ) )
=> ( ord_less_eq_nat @ I @ J ) ) ).
% match_le
thf(fact_229_match__Star__unfold,axiom,
! [I: nat,J: nat,R: regex_a_d] :
( ( ord_less_nat @ I @ J )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_d @ sigma @ ( star_a_d @ R ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ I @ J ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ X2 ) @ ( match_a_d @ sigma @ ( star_a_d @ R ) ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ J ) @ ( match_a_d @ sigma @ R ) ) ) ) ) ).
% match_Star_unfold
thf(fact_230_match__refl__eps,axiom,
! [I: nat,R: regex_a_d] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ I ) @ ( match_a_d @ sigma @ R ) )
=> ( eps_a_d @ R ) ) ).
% match_refl_eps
thf(fact_231_l__def_I3_J,axiom,
member8440522571783428010at_nat @ ( product_Pair_nat_nat @ i2 @ ( suc @ l ) ) @ ( match_a_d @ sigma @ r ) ).
% l_def(3)
thf(fact_232__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062l_O_A_092_060lbrakk_062i_A_092_060le_062_Al_059_Amem_A_I_092_060tau_062_A_092_060sigma_062_Ai_J_A_I_092_060tau_062_A_092_060sigma_062_Al_J_AI_059_A_Ii_M_ASuc_Al_J_A_092_060in_062_Amatch_Ar_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [L2: nat] :
( ( ord_less_eq_nat @ i2 @ L2 )
=> ( ( mem_d @ ( tau_a_d @ sigma @ i2 ) @ ( tau_a_d @ sigma @ L2 ) @ i )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ i2 @ ( suc @ L2 ) ) @ ( match_a_d @ sigma @ r ) ) ) ) ).
% \<open>\<And>thesis. (\<And>l. \<lbrakk>i \<le> l; mem (\<tau> \<sigma> i) (\<tau> \<sigma> l) I; (i, Suc l) \<in> match r\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_233_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_234_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_235_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_236_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_237_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_238_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_239_local_Owf,axiom,
wf_regex_a_d @ r ).
% local.wf
thf(fact_240_adv__last__bounds_I1_J,axiom,
( i3
= ( suc @ i4 ) ) ).
% adv_last_bounds(1)
thf(fact_241_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_242_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_243_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_244_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_245_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_246_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_247_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_248_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_249_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_250_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_251_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_252_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_253_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_254_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_255_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_256_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_257_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_258_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_259_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_260_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X2: nat] : ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z: nat] :
( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ Z )
=> ( R2 @ X2 @ Z ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_261_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_262_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_263_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_264_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_265_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_266_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M5: nat] :
( M7
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_267_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_268_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_269_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_270_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_271_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_272_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_273_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_274_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_275_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_276_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_277_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_278_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_279_wf__regex__eps__match,axiom,
! [R: regex_a_d,I: nat] :
( ( wf_regex_a_d @ R )
=> ( ( eps_a_d @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ I ) @ ( match_a_d @ sigma @ R ) ) ) ) ).
% wf_regex_eps_match
thf(fact_280_match_Osimps_I5_J,axiom,
! [R: regex_a_d] :
( ( match_a_d @ sigma @ ( star_a_d @ R ) )
= ( transi2905341329935302413cl_nat @ ( match_a_d @ sigma @ R ) ) ) ).
% match.simps(5)
thf(fact_281_match__rderive,axiom,
! [R: regex_a_d,I: nat,J: nat] :
( ( wf_regex_a_d @ R )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) @ ( match_a_d @ sigma @ R ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_d @ sigma @ ( rderive_a_d @ R ) ) ) ) ) ) ).
% match_rderive
thf(fact_282__092_060open_062sat_A_IMatchF_AI_Ar_J_Ai_092_060close_062,axiom,
sat_a_d @ sigma @ ( matchF_d_a @ i @ r ) @ i2 ).
% \<open>sat (MatchF I r) i\<close>
thf(fact_283_sat_Osimps_I10_J,axiom,
! [I5: i_d,R: regex_a_d,I: nat] :
( ( sat_a_d @ sigma @ ( matchF_d_a @ I5 @ R ) @ I )
= ( ? [J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
& ( mem_d @ ( tau_a_d @ sigma @ I ) @ ( tau_a_d @ sigma @ J3 ) @ I5 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( suc @ J3 ) ) @ ( match_a_d @ sigma @ R ) ) ) ) ) ).
% sat.simps(10)
thf(fact_284_sat_Osimps_I9_J,axiom,
! [I5: i_d,R: regex_a_d,I: nat] :
( ( sat_a_d @ sigma @ ( matchP_d_a @ I5 @ R ) @ I )
= ( ? [J3: nat] :
( ( ord_less_eq_nat @ J3 @ I )
& ( mem_d @ ( tau_a_d @ sigma @ J3 ) @ ( tau_a_d @ sigma @ I ) @ I5 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ J3 @ ( suc @ I ) ) @ ( match_a_d @ sigma @ R ) ) ) ) ) ).
% sat.simps(9)
thf(fact_285_sat_Osimps_I6_J,axiom,
! [I5: i_d,Phi: formula_a_d,I: nat] :
( ( sat_a_d @ sigma @ ( next_d_a @ I5 @ Phi ) @ I )
= ( ( mem_d @ ( tau_a_d @ sigma @ I ) @ ( tau_a_d @ sigma @ ( suc @ I ) ) @ I5 )
& ( sat_a_d @ sigma @ Phi @ ( suc @ I ) ) ) ) ).
% sat.simps(6)
thf(fact_286_sat_Osimps_I3_J,axiom,
! [Phi: formula_a_d,I: nat] :
( ( sat_a_d @ sigma @ ( neg_a_d @ Phi ) @ I )
= ( ~ ( sat_a_d @ sigma @ Phi @ I ) ) ) ).
% sat.simps(3)
thf(fact_287_sat_Osimps_I1_J,axiom,
! [B: $o,I: nat] :
( ( sat_a_d @ sigma @ ( bool_a_d @ B ) @ I )
= B ) ).
% sat.simps(1)
thf(fact_288__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062te_O_A_092_060lbrakk_062w__read__t_Aargs_Atj_H_A_061_ASome_Ate_059_Ate_A_061_Ats__at_A_Irho_H_H_A_064_A_091_Itm_M_Aundefined_J_093_J_Aj_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Te: d] :
( ( ( w_read4082988946454912847t_unit @ args @ tj3 )
= ( some_d @ Te ) )
=> ( Te
!= ( ts_at_d_iarray_o @ ( append1741259189927458815rray_o @ rho3 @ ( cons_P2929697372952323866rray_o @ ( produc6263459992454928980rray_o @ tm @ undefined_iarray_o ) @ nil_Pr7455003300710157162rray_o ) ) @ j2 ) ) ) ).
% \<open>\<And>thesis. (\<And>te. \<lbrakk>w_read_t args tj' = Some te; te = ts_at (rho'' @ [(tm, undefined)]) j'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_289_run__t__read,axiom,
! [T2: t,T3: t,X: d] :
( ( ( w_run_6161204274118665960t_unit @ args @ T2 )
= ( some_P5970643471752873532od_t_d @ ( product_Pair_t_d @ T3 @ X ) ) )
=> ( ( w_read4082988946454912847t_unit @ args @ T2 )
= ( some_d @ X ) ) ) ).
% run_t_read
thf(fact_290_read__t__run,axiom,
! [T2: t,X: d] :
( ( ( w_read4082988946454912847t_unit @ args @ T2 )
= ( some_d @ X ) )
=> ? [T4: t] :
( ( w_run_6161204274118665960t_unit @ args @ T2 )
= ( some_P5970643471752873532od_t_d @ ( product_Pair_t_d @ T4 @ X ) ) ) ) ).
% read_t_run
thf(fact_291_tbj_H__def_I1_J,axiom,
( ( w_read4082988946454912847t_unit @ args @ tj3 )
= ( some_d @ te ) ) ).
% tbj'_def(1)
thf(fact_292_adv__last__bounds_I5_J,axiom,
tj2 = tj3 ).
% adv_last_bounds(5)
thf(fact_293_before__end_I4_J,axiom,
( ( w_read4082988946454912847t_unit @ args @ tj )
= ( some_d @ tm ) ) ).
% before_end(4)
thf(fact_294_tbi__def_I2_J,axiom,
( ( w_run_1349431270707325884t_unit @ args @ si3 )
= ( some_P1508204274909019528rray_o @ ( produc990979591218741651rray_o @ si @ b ) ) ) ).
% tbi_def(2)
thf(fact_295_tbi__def_I1_J,axiom,
( ( w_run_6161204274118665960t_unit @ args @ ti3 )
= ( some_P5970643471752873532od_t_d @ ( product_Pair_t_d @ ti @ t2 ) ) ) ).
% tbi_def(1)
thf(fact_296_read__ti,axiom,
( ( w_read4082988946454912847t_unit @ args @ ti3 )
= ( some_d @ t2 ) ) ).
% read_ti
thf(fact_297__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062ti_H_H_H_Asi_H_H_H_Ab_O_A_092_060lbrakk_062w__run__t_Aargs_Ati_A_061_ASome_A_Iti_H_H_H_M_At_J_059_Aw__run__sub_Aargs_Asi_A_061_ASome_A_Isi_H_H_H_M_Ab_J_059_At_A_061_Ats__at_Arho_H_H_Ai_059_Ab_A_061_Abs__at_Arho_H_H_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ( ? [Ti: t] :
( ( w_run_6161204274118665960t_unit @ args @ ti3 )
= ( some_P5970643471752873532od_t_d @ ( product_Pair_t_d @ Ti @ t2 ) ) )
=> ! [Si: e,B2: iarray_o] :
( ( ( w_run_1349431270707325884t_unit @ args @ si3 )
= ( some_P1508204274909019528rray_o @ ( produc990979591218741651rray_o @ Si @ B2 ) ) )
=> ( ( t2
= ( ts_at_d_iarray_o @ rho3 @ i2 ) )
=> ( B2
!= ( bs_at_d_iarray_o @ rho3 @ i2 ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>ti''' si''' b. \<lbrakk>w_run_t args ti = Some (ti''', t); w_run_sub args si = Some (si''', b); t = ts_at rho'' i; b = bs_at rho'' i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_298_valid__after_I5_J,axiom,
ti2 = ti3 ).
% valid_after(5)
thf(fact_299_valid__after_I6_J,axiom,
si2 = si3 ).
% valid_after(6)
thf(fact_300_before__end_I3_J,axiom,
( ( w_read4082988946454912847t_unit @ args @ ( w_ti_i9031400733183855247t_unit @ w ) )
= ( some_d @ t2 ) ) ).
% before_end(3)
thf(fact_301_valid__eval__matchP,axiom,
! [I5: i_d,Rho: list_P1335534284205260522rray_o,J: nat,W2: window145566784895814656t_unit,Tj: t,T2: d,Sj: e,B: iarray_o] :
( ( valid_4265144587182951976_d_t_e @ args @ I5 @ t0 @ sub @ Rho @ J @ W2 )
=> ( ( ( w_run_6161204274118665960t_unit @ args @ ( w_tj_i1823933927240583118t_unit @ W2 ) )
= ( some_P5970643471752873532od_t_d @ ( product_Pair_t_d @ Tj @ T2 ) ) )
=> ( ( ( w_run_1349431270707325884t_unit @ args @ ( w_sj_i5164955787286279501t_unit @ W2 ) )
= ( some_P1508204274909019528rray_o @ ( produc990979591218741651rray_o @ Sj @ B ) ) )
=> ( ( wf_regex_a_d @ r )
=> ? [W3: window145566784895814656t_unit] :
( ( ( eval_matchP_d_t_e @ args @ I5 @ W2 )
= ( some_P5929554187424712206t_unit @ ( produc8524775685493084185t_unit @ ( product_Pair_d_o @ ( tau_a_d @ sigma @ J ) @ ( sat_a_d @ sigma @ ( matchP_d_a @ I5 @ r ) @ J ) ) @ W3 ) ) )
& ( T2
= ( tau_a_d @ sigma @ J ) )
& ( valid_4265144587182951976_d_t_e @ args @ I5 @ t0 @ sub @ ( append1741259189927458815rray_o @ Rho @ ( cons_P2929697372952323866rray_o @ ( produc6263459992454928980rray_o @ T2 @ B ) @ nil_Pr7455003300710157162rray_o ) ) @ ( suc @ J ) @ W3 ) ) ) ) ) ) ).
% valid_eval_matchP
thf(fact_302_MDL__window__axioms,axiom,
mDL_window_a_d_t_e @ sigma @ r @ t0 @ sub @ args ).
% MDL_window_axioms
thf(fact_303_ti__def,axiom,
( ti3
= ( w_ti_i9031400733183855247t_unit @ w ) ) ).
% ti_def
thf(fact_304_valid__before_H,axiom,
valid_7832268287944518322_d_t_e @ args @ i @ t0 @ sub @ rho @ i2 @ w ).
% valid_before'
thf(fact_305_tj__def,axiom,
( tj4
= ( w_tj_i1823933927240583118t_unit @ w ) ) ).
% tj_def
thf(fact_306_sj__def,axiom,
( sj4
= ( w_sj_i5164955787286279501t_unit @ w ) ) ).
% sj_def
thf(fact_307_reach__window__i,axiom,
reach_8428144547757753332_d_t_e @ args @ t0 @ sub @ rho3 @ ( produc6011578857208472279od_t_e @ i2 @ ( produc2496349360114527592od_t_e @ ti3 @ ( produc4437257469937609000od_t_e @ si3 @ ( produc181612709947590831od_t_e @ ( size_s1844703894572694870rray_o @ rho3 ) @ ( product_Pair_t_e @ tj @ sj ) ) ) ) ) ).
% reach_window_i
thf(fact_308_adv__last__bounds_I2_J,axiom,
( ( w_ti_i9031400733183855247t_unit @ w2 )
= ti ) ).
% adv_last_bounds(2)
thf(fact_309_tj_H_H__def,axiom,
( tj2
= ( w_tj_i1823933927240583118t_unit @ w2 ) ) ).
% tj''_def
thf(fact_310_inv__before,axiom,
matchF8959259506797507443_d_t_e @ args @ i @ t0 @ sub @ rho3 @ i2 @ ti3 @ si3 @ tj @ sj @ w ).
% inv_before
thf(fact_311_sj_H_H__def,axiom,
( sj2
= ( w_sj_i5164955787286279501t_unit @ w2 ) ) ).
% sj''_def
thf(fact_312_reach__window__j,axiom,
reach_8428144547757753332_d_t_e @ args @ t0 @ sub @ rho3 @ ( produc6011578857208472279od_t_e @ j3 @ ( produc2496349360114527592od_t_e @ tj4 @ ( produc4437257469937609000od_t_e @ sj4 @ ( produc181612709947590831od_t_e @ ( size_s1844703894572694870rray_o @ rho3 ) @ ( product_Pair_t_e @ tj @ sj ) ) ) ) ) ).
% reach_window_j
thf(fact_313_valid__before_I1_J,axiom,
reach_8428144547757753332_d_t_e @ args @ t0 @ sub @ rho @ ( produc6011578857208472279od_t_e @ i2 @ ( produc2496349360114527592od_t_e @ ti3 @ ( produc4437257469937609000od_t_e @ si3 @ ( produc181612709947590831od_t_e @ j3 @ ( product_Pair_t_e @ tj4 @ sj4 ) ) ) ) ) ).
% valid_before(1)
thf(fact_314_valid__after_I2_J,axiom,
reach_8428144547757753332_d_t_e @ args @ t0 @ sub @ rho3 @ ( produc6011578857208472279od_t_e @ j2 @ ( produc2496349360114527592od_t_e @ tj3 @ ( produc4437257469937609000od_t_e @ sj3 @ ( produc181612709947590831od_t_e @ ( size_s1844703894572694870rray_o @ rho3 ) @ ( product_Pair_t_e @ tj @ sj ) ) ) ) ) ).
% valid_after(2)
thf(fact_315_adv__last__bounds_I3_J,axiom,
( ( w_si_i3149050556374775822t_unit @ w2 )
= si ) ).
% adv_last_bounds(3)
thf(fact_316_si__def,axiom,
( si3
= ( w_si_i3149050556374775822t_unit @ w ) ) ).
% si_def
thf(fact_317_j_H_H__def,axiom,
( j
= ( w_j_ia6176537919040997584t_unit @ w2 ) ) ).
% j''_def
thf(fact_318_i_H_H__def,axiom,
( i3
= ( w_i_ia4160632688129493905t_unit @ w2 ) ) ).
% i''_def
thf(fact_319_valid__before_I5_J,axiom,
( i2
= ( w_i_ia4160632688129493905t_unit @ w ) ) ).
% valid_before(5)
thf(fact_320_j__def,axiom,
( j3
= ( w_j_ia6176537919040997584t_unit @ w ) ) ).
% j_def
thf(fact_321_adv__last__bounds_I6_J,axiom,
sj2 = sj3 ).
% adv_last_bounds(6)
thf(fact_322_run__t__sound,axiom,
! [Ts: list_d,T2: t,T3: t,X: d] :
( ( reaches_on_t_d @ ( w_run_6161204274118665960t_unit @ args ) @ t0 @ Ts @ T2 )
=> ( ( ( w_run_6161204274118665960t_unit @ args @ T2 )
= ( some_P5970643471752873532od_t_d @ ( product_Pair_t_d @ T3 @ X ) ) )
=> ( X
= ( tau_a_d @ sigma @ ( size_size_list_d @ Ts ) ) ) ) ) ).
% run_t_sound
thf(fact_323_si_H__def,axiom,
( si2
= ( w_si_i3149050556374775822t_unit @ w3 ) ) ).
% si'_def
thf(fact_324_ti_H__def,axiom,
( ti2
= ( w_ti_i9031400733183855247t_unit @ w3 ) ) ).
% ti'_def
thf(fact_325_j_H__def,axiom,
( j2
= ( w_j_ia6176537919040997584t_unit @ w3 ) ) ).
% j'_def
thf(fact_326_tj_H__def,axiom,
( tj3
= ( w_tj_i1823933927240583118t_unit @ w3 ) ) ).
% tj'_def
thf(fact_327_run__ts,axiom,
! [T2: t,Ts: list_d,T3: t] :
( ( reaches_on_t_d @ ( w_run_6161204274118665960t_unit @ args ) @ T2 @ Ts @ T3 )
=> ( ( T2 = t0 )
=> ( chain_le_d @ Ts ) ) ) ).
% run_ts
thf(fact_328_sj_H__def,axiom,
( sj3
= ( w_sj_i5164955787286279501t_unit @ w3 ) ) ).
% sj'_def
thf(fact_329_i_H__def,axiom,
( i4
= ( w_i_ia4160632688129493905t_unit @ w3 ) ) ).
% i'_def
thf(fact_330_adv__start__last,axiom,
( w2
= ( adv_st2492704914681618616_d_t_e @ args @ w3 ) ) ).
% adv_start_last
thf(fact_331_run__t__sound_H,axiom,
! [Ts: list_d,T2: t,I: nat] :
( ( reaches_on_t_d @ ( w_run_6161204274118665960t_unit @ args ) @ t0 @ Ts @ T2 )
=> ( ( ord_less_nat @ I @ ( size_size_list_d @ Ts ) )
=> ( ( nth_d @ Ts @ I )
= ( tau_a_d @ sigma @ I ) ) ) ) ).
% run_t_sound'
thf(fact_332_valid__after_I1_J,axiom,
valid_298839866600460541_d_t_e @ args @ t0 @ sub @ ( take_P382030051730269010rray_o @ ( w_j_ia6176537919040997584t_unit @ w3 ) @ rho3 ) @ w3 ).
% valid_after(1)
thf(fact_333_w_H__def,axiom,
( ( while_5584694067979257635t_unit @ ( matchF1317222115672871843t_unit @ args @ i @ t2 ) @ ( adv_en5254224764142676017_d_t_e @ args ) @ w )
= ( some_w7886180785293430501t_unit @ w3 ) ) ).
% w'_def
thf(fact_334__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062w_H_O_Awhile__break_A_ImatchF__cond_AI_At_J_A_Iadv__end_Aargs_J_Aw_A_061_ASome_Aw_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [W3: window145566784895814656t_unit] :
( ( while_5584694067979257635t_unit @ ( matchF1317222115672871843t_unit @ args @ i @ t2 ) @ ( adv_en5254224764142676017_d_t_e @ args ) @ w )
!= ( some_w7886180785293430501t_unit @ W3 ) ) ).
% \<open>\<And>thesis. (\<And>w'. while_break (matchF_cond I t) (adv_end args) w = Some w' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_335_reach__tm,axiom,
reaches_on_t_d @ ( w_run_6161204274118665960t_unit @ args ) @ t0 @ ( map_Pr114866734410781409ay_o_d @ produc4433724027275887680rray_o @ ( append1741259189927458815rray_o @ rho3 @ ( cons_P2929697372952323866rray_o @ ( produc6263459992454928980rray_o @ tm @ undefined_iarray_o ) @ nil_Pr7455003300710157162rray_o ) ) ) @ tm2 ).
% reach_tm
thf(fact_336_before__end_I1_J,axiom,
reaches_on_t_d @ ( w_run_6161204274118665960t_unit @ args ) @ ( w_tj_i1823933927240583118t_unit @ w ) @ ( map_Pr114866734410781409ay_o_d @ produc4433724027275887680rray_o @ rho2 ) @ tj ).
% before_end(1)
thf(fact_337_before__end_I2_J,axiom,
reache5252139207361026560rray_o @ ( w_run_1349431270707325884t_unit @ args ) @ ( w_sj_i5164955787286279501t_unit @ w ) @ ( map_Pr7706330987264554334rray_o @ produc2103765176326027650rray_o @ rho2 ) @ sj ).
% before_end(2)
thf(fact_338_tj_H_H_H__eq,axiom,
! [Tj_cur: t] :
( ( reaches_on_t_d @ ( w_run_6161204274118665960t_unit @ args ) @ t0 @ ( map_Pr114866734410781409ay_o_d @ produc4433724027275887680rray_o @ rho3 ) @ Tj_cur )
=> ( Tj_cur = tj ) ) ).
% tj'''_eq
thf(fact_339_reach__tj_H_H_H,axiom,
reaches_on_t_d @ ( w_run_6161204274118665960t_unit @ args ) @ t0 @ ( map_Pr114866734410781409ay_o_d @ produc4433724027275887680rray_o @ rho3 ) @ tj ).
% reach_tj'''
thf(fact_340_reach__sj_H_H_H,axiom,
reache5252139207361026560rray_o @ ( w_run_1349431270707325884t_unit @ args ) @ sub @ ( map_Pr7706330987264554334rray_o @ produc2103765176326027650rray_o @ rho3 ) @ sj ).
% reach_sj'''
thf(fact_341_sj_H_H_H__eq,axiom,
! [Sj_cur: e] :
( ( reache5252139207361026560rray_o @ ( w_run_1349431270707325884t_unit @ args ) @ sub @ ( map_Pr7706330987264554334rray_o @ produc2103765176326027650rray_o @ rho3 ) @ Sj_cur )
=> ( Sj_cur = sj ) ) ).
% sj'''_eq
thf(fact_342_reach__ti,axiom,
reaches_on_t_d @ ( w_run_6161204274118665960t_unit @ args ) @ t0 @ ( take_d @ i2 @ ( map_Pr114866734410781409ay_o_d @ produc4433724027275887680rray_o @ rho3 ) ) @ ti3 ).
% reach_ti
thf(fact_343_reach__si,axiom,
reache5252139207361026560rray_o @ ( w_run_1349431270707325884t_unit @ args ) @ sub @ ( take_iarray_o @ i2 @ ( map_Pr7706330987264554334rray_o @ produc2103765176326027650rray_o @ rho3 ) ) @ si3 ).
% reach_si
thf(fact_344_reach__tj,axiom,
reaches_on_t_d @ ( w_run_6161204274118665960t_unit @ args ) @ t0 @ ( take_d @ j3 @ ( map_Pr114866734410781409ay_o_d @ produc4433724027275887680rray_o @ rho3 ) ) @ tj4 ).
% reach_tj
thf(fact_345_reach__sj,axiom,
reache5252139207361026560rray_o @ ( w_run_1349431270707325884t_unit @ args ) @ sub @ ( take_iarray_o @ j3 @ ( map_Pr7706330987264554334rray_o @ produc2103765176326027650rray_o @ rho3 ) ) @ sj4 ).
% reach_sj
thf(fact_346__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062tm_H_O_Areaches__on_A_Iw__run__t_Aargs_J_At0_A_Imap_Afst_A_Irho_H_H_A_064_A_091_Itm_M_Aundefined_J_093_J_J_Atm_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Tm: t] :
~ ( reaches_on_t_d @ ( w_run_6161204274118665960t_unit @ args ) @ t0 @ ( map_Pr114866734410781409ay_o_d @ produc4433724027275887680rray_o @ ( append1741259189927458815rray_o @ rho3 @ ( cons_P2929697372952323866rray_o @ ( produc6263459992454928980rray_o @ tm @ undefined_iarray_o ) @ nil_Pr7455003300710157162rray_o ) ) ) @ Tm ) ).
% \<open>\<And>thesis. (\<And>tm'. reaches_on (w_run_t args) t0 (map fst (rho'' @ [(tm, undefined)])) tm' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_347_valid__eval__matchF__Some,axiom,
! [I5: i_d,Rho: list_P1335534284205260522rray_o,I: nat,W2: window145566784895814656t_unit,T2: d,B: $o,W4: window145566784895814656t_unit] :
( ( valid_7832268287944518322_d_t_e @ args @ I5 @ t0 @ sub @ Rho @ I @ W2 )
=> ( ( ( eval_matchF_d_t_e @ args @ I5 @ W2 )
= ( some_P5929554187424712206t_unit @ ( produc8524775685493084185t_unit @ ( product_Pair_d_o @ T2 @ B ) @ W4 ) ) )
=> ( ( member_d @ ( right_d @ I5 ) @ tfin_tfin_d )
=> ? [Rho2: list_P1335534284205260522rray_o,Tm2: d] :
( ( reaches_on_t_d @ ( w_run_6161204274118665960t_unit @ args ) @ ( w_tj_i1823933927240583118t_unit @ W2 ) @ ( map_Pr114866734410781409ay_o_d @ produc4433724027275887680rray_o @ Rho2 ) @ ( w_tj_i1823933927240583118t_unit @ W4 ) )
& ( reache5252139207361026560rray_o @ ( w_run_1349431270707325884t_unit @ args ) @ ( w_sj_i5164955787286279501t_unit @ W2 ) @ ( map_Pr7706330987264554334rray_o @ produc2103765176326027650rray_o @ Rho2 ) @ ( w_sj_i5164955787286279501t_unit @ W4 ) )
& ( ( w_read4082988946454912847t_unit @ args @ ( w_ti_i9031400733183855247t_unit @ W2 ) )
= ( some_d @ T2 ) )
& ( ( w_read4082988946454912847t_unit @ args @ ( w_tj_i1823933927240583118t_unit @ W4 ) )
= ( some_d @ Tm2 ) )
& ~ ( memR_d @ T2 @ Tm2 @ I5 ) ) ) ) ) ).
% valid_eval_matchF_Some
thf(fact_348_t__tfin,axiom,
member_d @ t2 @ tfin_tfin_d ).
% t_tfin
thf(fact_349_loop,axiom,
( pred_o4930588667914135666t_unit
@ ^ [W5: window145566784895814656t_unit] :
( ( matchF8959259506797507443_d_t_e @ args @ i @ t0 @ sub @ rho3 @ i2 @ ti3 @ si3 @ tj @ sj @ W5 )
& ~ ( matchF1317222115672871843t_unit @ args @ i @ t2 @ W5 ) )
@ ( while_5584694067979257635t_unit @ ( matchF1317222115672871843t_unit @ args @ i @ t2 ) @ ( adv_en5254224764142676017_d_t_e @ args ) @ w ) ) ).
% loop
thf(fact_350_run__sub__sound,axiom,
! [Bs: list_iarray_o,S: e,S2: e,B: iarray_o] :
( ( reache5252139207361026560rray_o @ ( w_run_1349431270707325884t_unit @ args ) @ sub @ Bs @ S )
=> ( ( ( w_run_1349431270707325884t_unit @ args @ S )
= ( some_P1508204274909019528rray_o @ ( produc990979591218741651rray_o @ S2 @ B ) ) )
=> ( B
= ( iArray_o
@ ( map_formula_a_d_o
@ ^ [Phi2: formula_a_d] : ( sat_a_d @ sigma @ Phi2 @ ( size_s3243743627286111600rray_o @ Bs ) )
@ ( collect_subfmlas_a_d @ r @ nil_formula_a_d ) ) ) ) ) ) ).
% run_sub_sound
thf(fact_351_run__sub__sound_H,axiom,
! [Bs: list_iarray_o,S: e,I: nat] :
( ( reache5252139207361026560rray_o @ ( w_run_1349431270707325884t_unit @ args ) @ sub @ Bs @ S )
=> ( ( ord_less_nat @ I @ ( size_s3243743627286111600rray_o @ Bs ) )
=> ( ( nth_iarray_o @ Bs @ I )
= ( iArray_o
@ ( map_formula_a_d_o
@ ^ [Phi2: formula_a_d] : ( sat_a_d @ sigma @ Phi2 @ I )
@ ( collect_subfmlas_a_d @ r @ nil_formula_a_d ) ) ) ) ) ) ).
% run_sub_sound'
thf(fact_352_sat_Osimps_I8_J,axiom,
! [Phi: formula_a_d,I5: i_d,Psi: formula_a_d,I: nat] :
( ( sat_a_d @ sigma @ ( until_a_d @ Phi @ I5 @ Psi ) @ I )
= ( ? [J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
& ( mem_d @ ( tau_a_d @ sigma @ I ) @ ( tau_a_d @ sigma @ J3 ) @ I5 )
& ( sat_a_d @ sigma @ Psi @ J3 )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ I @ J3 ) )
=> ( sat_a_d @ sigma @ Phi @ X3 ) ) ) ) ) ).
% sat.simps(8)
thf(fact_353_valid__init__matchF,axiom,
! [I5: i_d] : ( valid_7832268287944518322_d_t_e @ args @ I5 @ t0 @ sub @ nil_Pr7455003300710157162rray_o @ zero_zero_nat @ ( init_w1645077234915333513_d_t_e @ args @ t0 @ sub ) ) ).
% valid_init_matchF
thf(fact_354_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_355_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_356_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_357_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_358_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_359_valid__init__matchP,axiom,
! [I5: i_d] : ( valid_4265144587182951976_d_t_e @ args @ I5 @ t0 @ sub @ nil_Pr7455003300710157162rray_o @ zero_zero_nat @ ( init_w1645077234915333513_d_t_e @ args @ t0 @ sub ) ) ).
% valid_init_matchP
thf(fact_360_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_361_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_362_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_363_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_364_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_365_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_366_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_367_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_368_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_369_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_370_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_371_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_372_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_373_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_374_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_375_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_376_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_377_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_378_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_379_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] :
( ( P @ X2 @ Y2 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_380_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_381_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_382_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_383_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_384_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_385_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_386_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_387_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_388_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_389_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_390_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_391_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ( P @ M ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_392_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M: nat] :
( ( ord_less_nat @ M @ N )
& ( P @ M ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_393_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_394_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_395_e_H__def,axiom,
( e2
= ( w_e_ia5320383801338254997t_unit @ w3 ) ) ).
% e'_def
thf(fact_396_e__def,axiom,
( e3
= ( w_e_ia5320383801338254997t_unit @ w ) ) ).
% e_def
thf(fact_397_trancl__Base,axiom,
! [P: nat > $o] :
( ( transi2905341329935302413cl_nat
@ ( collec3392354462482085612at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [I4: nat] :
( ( Uu
= ( product_Pair_nat_nat @ I4 @ ( suc @ I4 ) ) )
& ( P @ I4 ) ) ) )
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ I4 @ J3 ) )
=> ( P @ X3 ) ) ) ) ) ) ).
% trancl_Base
thf(fact_398_match_Osimps_I2_J,axiom,
! [Phi: formula_a_d] :
( ( match_a_d @ sigma @ ( symbol_a_d @ Phi ) )
= ( collec3392354462482085612at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [I4: nat] :
( ( Uu
= ( product_Pair_nat_nat @ I4 @ ( suc @ I4 ) ) )
& ( sat_a_d @ sigma @ Phi @ I4 ) ) ) ) ).
% match.simps(2)
thf(fact_399_until__rewrite,axiom,
! [Phi: formula_a_d,I5: i_d,Psi: formula_a_d,I: nat] :
( ( sat_a_d @ sigma @ ( until_a_d @ Phi @ I5 @ Psi ) @ I )
= ( sat_a_d @ sigma @ ( matchF_d_a @ I5 @ ( times_a_d @ ( star_a_d @ ( symbol_a_d @ Phi ) ) @ ( symbol_a_d @ Psi ) ) ) @ I ) ) ).
% until_rewrite
thf(fact_400_next__rewrite,axiom,
! [I5: i_d,Phi: formula_a_d,I: nat] :
( ( sat_a_d @ sigma @ ( next_d_a @ I5 @ Phi ) @ I )
= ( sat_a_d @ sigma @ ( matchF_d_a @ I5 @ ( times_a_d @ ( symbol_a_d @ ( bool_a_d @ $true ) ) @ ( symbol_a_d @ Phi ) ) ) @ I ) ) ).
% next_rewrite
thf(fact_401_match__Times,axiom,
! [I: nat,N: nat,R: regex_a_d,S: regex_a_d] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ N ) ) @ ( match_a_d @ sigma @ ( times_a_d @ R @ S ) ) )
= ( ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ K3 ) ) @ ( match_a_d @ sigma @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ I @ N ) ) @ ( match_a_d @ sigma @ S ) ) ) ) ) ).
% match_Times
thf(fact_402_since__rewrite,axiom,
! [Phi: formula_a_d,I5: i_d,Psi: formula_a_d,I: nat] :
( ( sat_a_d @ sigma @ ( since_a_d @ Phi @ I5 @ Psi ) @ I )
= ( sat_a_d @ sigma @ ( matchP_d_a @ I5 @ ( times_a_d @ ( symbol_a_d @ Psi ) @ ( star_a_d @ ( symbol_a_d @ Phi ) ) ) ) @ I ) ) ).
% since_rewrite
thf(fact_403_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_404_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_405_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_406_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_407_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_408_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_409_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q ) ) ) ) ).
% less_natE
thf(fact_410_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_411_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_412_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_413_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_414_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_415_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_416_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_417_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_418_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_419_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_420_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_421_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_422_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_423_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_424_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_425_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_426_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_427_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_428_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_429_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_430_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_431_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_432_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_433_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_434_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_435_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_436_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_437_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_438_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_439_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_440_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_441_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_442_match__Star,axiom,
! [I: nat,N: nat,R: regex_a_d] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ ( suc @ N ) ) ) @ ( match_a_d @ sigma @ ( star_a_d @ R ) ) )
= ( ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ ( plus_plus_nat @ I @ one_one_nat ) @ K3 ) ) @ ( match_a_d @ sigma @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( plus_plus_nat @ I @ one_one_nat ) @ K3 ) @ ( plus_plus_nat @ I @ ( suc @ N ) ) ) @ ( match_a_d @ sigma @ ( star_a_d @ R ) ) ) ) ) ) ).
% match_Star
thf(fact_443_sat_Osimps_I7_J,axiom,
! [Phi: formula_a_d,I5: i_d,Psi: formula_a_d,I: nat] :
( ( sat_a_d @ sigma @ ( since_a_d @ Phi @ I5 @ Psi ) @ I )
= ( ? [J3: nat] :
( ( ord_less_eq_nat @ J3 @ I )
& ( mem_d @ ( tau_a_d @ sigma @ J3 ) @ ( tau_a_d @ sigma @ I ) @ I5 )
& ( sat_a_d @ sigma @ Psi @ J3 )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or6659071591806873216st_nat @ J3 @ I ) )
=> ( sat_a_d @ sigma @ Phi @ X3 ) ) ) ) ) ).
% sat.simps(7)
thf(fact_444_Ball__atLeastLessThan__reindex,axiom,
! [J: nat,I: nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ J @ I ) )
=> ( P @ ( suc @ X3 ) ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or6659071591806873216st_nat @ J @ I ) )
=> ( P @ X3 ) ) ) ) ).
% Ball_atLeastLessThan_reindex
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_d @ ( ts_at_d_iarray_o @ ( append1741259189927458815rray_o @ rho3 @ ( cons_P2929697372952323866rray_o @ ( produc6263459992454928980rray_o @ tm @ undefined_iarray_o ) @ nil_Pr7455003300710157162rray_o ) ) @ ( size_s1844703894572694870rray_o @ rho3 ) ) @ ( tau_a_d @ sigma @ l ) ).
%------------------------------------------------------------------------------