TPTP Problem File: ITP166^1.p
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%------------------------------------------------------------------------------
% File : ITP166^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer RelationalIncorrectness problem prob_274__5902014_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : RelationalIncorrectness/prob_274__5902014_1 [Des21]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 330 ( 125 unt; 108 typ; 0 def)
% Number of atoms : 601 ( 284 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 2463 ( 74 ~; 17 |; 97 &;2079 @)
% ( 0 <=>; 196 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 742 ( 742 >; 0 *; 0 +; 0 <<)
% Number of symbols : 82 ( 81 usr; 7 con; 0-6 aty)
% Number of variables : 897 ( 98 ^; 747 !; 52 ?; 897 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:33:22.688
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_J_J,type,
set_Pr1333062727ar_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_J,type,
produc1346207591ar_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_J_J,type,
set_li1762364675ar_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Typerep__Otyperep_J_Mt__List__Olist_It__Typerep__Otyperep_J_J_J,type,
set_Pr446800905yperep: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_J,type,
list_P1741976099ar_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_J,type,
set_Pr1148378109ar_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr1560408065st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Typerep__Otyperep_J_Mt__List__Olist_It__Typerep__Otyperep_J_J,type,
produc490010537yperep: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J,type,
produc1260470173ar_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc1699244961st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Typerep__Otyperep_Mt__Typerep__Otyperep_J_J,type,
set_Pr1097818217yperep: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1986765409at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Typerep__Otyperep_Mt__Typerep__Otyperep_J,type,
produc1788831497yperep: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Typerep__Otyperep_J_J,type,
set_list_typerep: $tType ).
thf(ty_n_t__List__Olist_It__Typerep__Otyperep_J,type,
list_typerep: $tType ).
thf(ty_n_t__Set__Oset_It__Typerep__Otyperep_J,type,
set_typerep: $tType ).
thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Typerep__Otyperep,type,
typerep: $tType ).
thf(ty_n_t__String__Oliteral,type,
literal: $tType ).
thf(ty_n_t__BExp__Obexp,type,
bexp: $tType ).
thf(ty_n_t__AExp__Oaexp,type,
aexp: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_t__Com__Ocom,type,
com: $tType ).
% Explicit typings (81)
thf(sy_c_BExp_Obval,type,
bval: bexp > ( list_char > int ) > $o ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Typerep__Otyperep,type,
bNF_Gr457912949yperep: set_list_typerep > typerep > set_list_typerep ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Typerep__Otyperep,type,
bNF_Gr1303086585yperep: set_list_typerep > list_typerep > set_typerep ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_001t__Com__Ocom_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J,type,
bNF_Gr202451329ar_int: set_Pr1148378109ar_int > ( produc1260470173ar_int > com ) > ( produc1260470173ar_int > list_char > int ) > set_Pr1148378109ar_int ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_Gr749752805at_nat: set_Pr1148378109ar_int > ( produc1260470173ar_int > nat ) > ( produc1260470173ar_int > nat ) > set_Pr1986765409at_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_Gr1247133574at_nat: set_Pr1986765409at_nat > ( nat > nat ) > set_Pr1986765409at_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_Gr808751971at_nat: set_nat > set_Pr1986765409at_nat > ( nat > nat ) > set_Pr1986765409at_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_001t__Nat__Onat,type,
bNF_Gr1143394412nt_nat: set_Pr1148378109ar_int > set_Pr1986765409at_nat > ( produc1260470173ar_int > nat ) > set_Pr1333062727ar_int ).
thf(sy_c_Big__Step_Obig__step,type,
big_big_step: produc1260470173ar_int > ( list_char > int ) > $o ).
thf(sy_c_Com_Ocom_OAssign,type,
assign: list_char > aexp > com ).
thf(sy_c_Com_Ocom_OIf,type,
if: bexp > com > com > com ).
thf(sy_c_Com_Ocom_OSKIP,type,
skip: com ).
thf(sy_c_Com_Ocom_OSeq,type,
seq: com > com > com ).
thf(sy_c_Com_Ocom_OWhile,type,
while: bexp > com > com ).
thf(sy_c_Com_Ocom_Osize__com,type,
size_com: com > nat ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
unique1592618082ux_nat: product_prod_nat_nat > $o ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Typerep__Otyperep,type,
append_typerep: list_typerep > list_typerep > list_typerep ).
thf(sy_c_List_Ogen__length_001t__Typerep__Otyperep,type,
gen_length_typerep: nat > list_typerep > nat ).
thf(sy_c_List_Olex_001t__Nat__Onat,type,
lex_nat: set_Pr1986765409at_nat > set_Pr1560408065st_nat ).
thf(sy_c_List_Olex_001t__Typerep__Otyperep,type,
lex_typerep: set_Pr1097818217yperep > set_Pr446800905yperep ).
thf(sy_c_List_Olexord_001t__Nat__Onat,type,
lexord_nat: set_Pr1986765409at_nat > set_Pr1560408065st_nat ).
thf(sy_c_List_Olexord_001t__Typerep__Otyperep,type,
lexord_typerep: set_Pr1097818217yperep > set_Pr446800905yperep ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J,type,
cons_P1254790739ar_int: produc1260470173ar_int > list_P1741976099ar_int > list_P1741976099ar_int ).
thf(sy_c_List_Olist_OCons_001t__Typerep__Otyperep,type,
cons_typerep: typerep > list_typerep > list_typerep ).
thf(sy_c_List_Olist_Osize__list_001t__Typerep__Otyperep,type,
size_list_typerep: ( typerep > nat ) > list_typerep > nat ).
thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
listrel1_nat: set_Pr1986765409at_nat > set_Pr1560408065st_nat ).
thf(sy_c_List_Olistrel1_001t__Typerep__Otyperep,type,
listrel1_typerep: set_Pr1097818217yperep > set_Pr446800905yperep ).
thf(sy_c_List_Oset__Cons_001t__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J,type,
set_Co1954412371ar_int: set_Pr1148378109ar_int > set_li1762364675ar_int > set_li1762364675ar_int ).
thf(sy_c_List_Oset__Cons_001t__Typerep__Otyperep,type,
set_Cons_typerep: set_typerep > set_list_typerep > set_list_typerep ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Com__Ocom,type,
size_size_com: com > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Typerep__Otyperep_J,type,
size_s416147576yperep: list_typerep > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Typerep__Otyperep,type,
size_size_typerep: typerep > nat ).
thf(sy_c_Product__Type_OPair_001t__Com__Ocom_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J,type,
produc1193801173ar_int: com > ( list_char > int ) > produc1260470173ar_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc1625736153st_nat: list_nat > list_nat > produc1699244961st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Typerep__Otyperep_J_001t__List__Olist_It__Typerep__Otyperep_J,type,
produc1695567329yperep: list_typerep > list_typerep > produc490010537yperep ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_001t__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J,type,
produc317451735ar_int: produc1260470173ar_int > produc1260470173ar_int > produc1346207591ar_int ).
thf(sy_c_Product__Type_OPair_001t__Typerep__Otyperep_001t__Typerep__Otyperep,type,
produc2071514689yperep: typerep > typerep > produc1788831497yperep ).
thf(sy_c_RelationalIncorrectness__Mirabelle__kefqvmgyto_Oflip,type,
relati186416454e_flip: ( ( list_char > int ) > ( list_char > int ) > $o ) > ( list_char > int ) > ( list_char > int ) > $o ).
thf(sy_c_RelationalIncorrectness__Mirabelle__kefqvmgyto_Oget__back_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J,type,
relati23543761ar_int: ( ( list_char > int ) > ( list_char > int ) > $o ) > bexp > com > nat > ( list_char > int ) > ( list_char > int ) > $o ).
thf(sy_c_RelationalIncorrectness__Mirabelle__kefqvmgyto_Oir__hoare,type,
relati2102071307_hoare: ( ( list_char > int ) > ( list_char > int ) > $o ) > com > com > ( ( list_char > int ) > ( list_char > int ) > $o ) > $o ).
thf(sy_c_RelationalIncorrectness__Mirabelle__kefqvmgyto_Oir__valid,type,
relati775585786_valid: ( ( list_char > int ) > ( list_char > int ) > $o ) > com > com > ( ( list_char > int ) > ( list_char > int ) > $o ) > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_J,type,
collec672902542ar_int: ( list_P1741976099ar_int > $o ) > set_li1762364675ar_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Typerep__Otyperep_J,type,
collect_list_typerep: ( list_typerep > $o ) > set_list_typerep ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J,type,
collec770796680ar_int: ( produc1260470173ar_int > $o ) > set_Pr1148378109ar_int ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_J,type,
collec1834387154ar_int: ( produc1346207591ar_int > $o ) > set_Pr1333062727ar_int ).
thf(sy_c_Typerep_Otyperep_OTyperep,type,
typerep2: literal > list_typerep > typerep ).
thf(sy_c_Typerep_Otyperep_Osize__typerep,type,
size_typerep: typerep > nat ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J_J,type,
member2052091212ar_int: list_P1741976099ar_int > set_li1762364675ar_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Typerep__Otyperep_J,type,
member_list_typerep: list_typerep > set_list_typerep > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J,type,
member92250950ar_int: produc1260470173ar_int > set_Pr1148378109ar_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member1926390090st_nat: produc1699244961st_nat > set_Pr1560408065st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Typerep__Otyperep_J_Mt__List__Olist_It__Typerep__Otyperep_J_J,type,
member1812711762yperep: produc490010537yperep > set_Pr446800905yperep > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Typerep__Otyperep_Mt__Typerep__Otyperep_J,type,
member100911538yperep: produc1788831497yperep > set_Pr1097818217yperep > $o ).
thf(sy_c_member_001t__Typerep__Otyperep,type,
member_typerep: typerep > set_typerep > $o ).
thf(sy_v_P,type,
p: ( list_char > int ) > ( list_char > int ) > $o ).
thf(sy_v_Q,type,
q: ( list_char > int ) > ( list_char > int ) > $o ).
thf(sy_v_b,type,
b: bexp ).
thf(sy_v_c,type,
c: com ).
thf(sy_v_c_H,type,
c2: com ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_ka____,type,
ka: nat ).
thf(sy_v_s_H____,type,
s: list_char > int ).
thf(sy_v_s_Ha____,type,
s_a: list_char > int ).
thf(sy_v_s____,type,
s2: list_char > int ).
thf(sy_v_sa____,type,
sa: list_char > int ).
thf(sy_v_t_H____,type,
t: list_char > int ).
thf(sy_v_t_Ha____,type,
t_a: list_char > int ).
thf(sy_v_t____,type,
t2: list_char > int ).
thf(sy_v_ta____,type,
ta: list_char > int ).
thf(sy_v_u____,type,
u: list_char > int ).
thf(sy_v_ua____,type,
ua: list_char > int ).
% Relevant facts (221)
thf(fact_0_Suc_Oprems_I6_J,axiom,
bval @ b @ sa ).
% Suc.prems(6)
thf(fact_1__092_060open_062bval_Ab_As_____092_060close_062,axiom,
bval @ b @ s2 ).
% \<open>bval b s__\<close>
thf(fact_2_Suc_Oprems_I3_J,axiom,
big_big_step @ ( produc1193801173ar_int @ c @ sa ) @ ta ).
% Suc.prems(3)
thf(fact_3_Suc_Oprems_I4_J,axiom,
big_big_step @ ( produc1193801173ar_int @ c2 @ s_a ) @ t_a ).
% Suc.prems(4)
thf(fact_4_Suc_Oprems_I7_J,axiom,
relati23543761ar_int @ p @ b @ c @ ( suc @ ka ) @ sa @ s_a ).
% Suc.prems(7)
thf(fact_5__092_060open_062_Ic_H_M_As_H_____J_A_092_060Rightarrow_062_At_H_____092_060close_062,axiom,
big_big_step @ ( produc1193801173ar_int @ c2 @ s ) @ t ).
% \<open>(c', s'__) \<Rightarrow> t'__\<close>
thf(fact_6__092_060open_062_Ic_M_As_____J_A_092_060Rightarrow_062_At_____092_060close_062,axiom,
big_big_step @ ( produc1193801173ar_int @ c @ s2 ) @ t2 ).
% \<open>(c, s__) \<Rightarrow> t__\<close>
thf(fact_7_Suc_Oprems_I5_J,axiom,
q @ ua @ t_a ).
% Suc.prems(5)
thf(fact_8__092_060open_062get__back_AP_Ab_Ac_Ak_As_____As_H_____092_060close_062,axiom,
relati23543761ar_int @ p @ b @ c @ k @ s2 @ s ).
% \<open>get_back P b c k s__ s'__\<close>
thf(fact_9_Suc_Oprems_I2_J,axiom,
big_big_step @ ( produc1193801173ar_int @ ( while @ b @ c ) @ ta ) @ ua ).
% Suc.prems(2)
thf(fact_10__092_060open_062_IWHILE_Ab_ADO_Ac_M_At_____J_A_092_060Rightarrow_062_Au_____092_060close_062,axiom,
big_big_step @ ( produc1193801173ar_int @ ( while @ b @ c ) @ t2 ) @ u ).
% \<open>(WHILE b DO c, t__) \<Rightarrow> u__\<close>
thf(fact_11_Suc_Oprems_I1_J,axiom,
relati775585786_valid @ ( relati23543761ar_int @ p @ b @ c @ ( suc @ ka ) ) @ ( while @ b @ c ) @ c2 @ q ).
% Suc.prems(1)
thf(fact_12_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_13_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_14_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_15_prod_Oinject,axiom,
! [X1: com,X2: list_char > int,Y1: com,Y2: list_char > int] :
( ( ( produc1193801173ar_int @ X1 @ X2 )
= ( produc1193801173ar_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
! [A: com,B: list_char > int,A2: com,B2: list_char > int] :
( ( ( produc1193801173ar_int @ A @ B )
= ( produc1193801173ar_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_18_Suc_OIH,axiom,
! [T: list_char > int,U: list_char > int,S: list_char > int,S2: list_char > int,T2: list_char > int] :
( ( relati775585786_valid @ ( relati23543761ar_int @ p @ b @ c @ ka ) @ ( while @ b @ c ) @ c2 @ q )
=> ( ( big_big_step @ ( produc1193801173ar_int @ ( while @ b @ c ) @ T ) @ U )
=> ( ( big_big_step @ ( produc1193801173ar_int @ c @ S ) @ T )
=> ( ( big_big_step @ ( produc1193801173ar_int @ c2 @ S2 ) @ T2 )
=> ( ( q @ U @ T2 )
=> ( ( bval @ b @ S )
=> ( ( relati23543761ar_int @ p @ b @ c @ ka @ S @ S2 )
=> ? [S3: list_char > int,S4: list_char > int] :
( ( relati23543761ar_int @ p @ b @ c @ ka @ S3 @ S4 )
& ( bval @ b @ S3 )
& ( big_big_step @ ( produc1193801173ar_int @ c @ S3 ) @ T )
& ( big_big_step @ ( produc1193801173ar_int @ c2 @ S4 ) @ T2 ) ) ) ) ) ) ) ) ) ).
% Suc.IH
thf(fact_19_sim__sym,axiom,
! [C: com,C2: com] :
( ( ! [S5: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C @ S5 ) @ T3 )
= ( big_big_step @ ( produc1193801173ar_int @ C2 @ S5 ) @ T3 ) ) )
= ( ! [S5: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C2 @ S5 ) @ T3 )
= ( big_big_step @ ( produc1193801173ar_int @ C @ S5 ) @ T3 ) ) ) ) ).
% sim_sym
thf(fact_20_sim__refl,axiom,
! [C: com,S6: list_char > int,T4: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C @ S6 ) @ T4 )
= ( big_big_step @ ( produc1193801173ar_int @ C @ S6 ) @ T4 ) ) ).
% sim_refl
thf(fact_21__092_060open_062Q_Au_____At_H_____092_060close_062,axiom,
q @ u @ t ).
% \<open>Q u__ t'__\<close>
thf(fact_22_disj__ir__valid,axiom,
! [P_1: ( list_char > int ) > ( list_char > int ) > $o,C: com,C2: com,Q_1: ( list_char > int ) > ( list_char > int ) > $o,P_2: ( list_char > int ) > ( list_char > int ) > $o,Q_2: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati775585786_valid @ P_1 @ C @ C2 @ Q_1 )
=> ( ( relati775585786_valid @ P_2 @ C @ C2 @ Q_2 )
=> ( relati775585786_valid
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( P_1 @ S5 @ S7 )
| ( P_2 @ S5 @ S7 ) )
@ C
@ C2
@ ^ [T3: list_char > int,T5: list_char > int] :
( ( Q_1 @ T3 @ T5 )
| ( Q_2 @ T3 @ T5 ) ) ) ) ) ).
% disj_ir_valid
thf(fact_23__092_060open_062ir__valid_A_Iget__back_AP_Ab_Ac_Ak_J_A_IWHILE_Ab_ADO_Ac_J_Ac_H_AQ_092_060close_062,axiom,
relati775585786_valid @ ( relati23543761ar_int @ p @ b @ c @ k ) @ ( while @ b @ c ) @ c2 @ q ).
% \<open>ir_valid (get_back P b c k) (WHILE b DO c) c' Q\<close>
thf(fact_24_conseq__ir__valid,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,P2: ( list_char > int ) > ( list_char > int ) > $o,Q2: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati775585786_valid @ P @ C @ C2 @ Q )
=> ( ! [S3: list_char > int,S4: list_char > int] :
( ( P @ S3 @ S4 )
=> ( P2 @ S3 @ S4 ) )
=> ( ! [S3: list_char > int,S4: list_char > int] :
( ( Q2 @ S3 @ S4 )
=> ( Q @ S3 @ S4 ) )
=> ( relati775585786_valid @ P2 @ C @ C2 @ Q2 ) ) ) ) ).
% conseq_ir_valid
thf(fact_25_ir__valid__get__back,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C: com,K: nat,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati775585786_valid @ ( relati23543761ar_int @ P @ B @ C @ ( suc @ K ) ) @ ( while @ B @ C ) @ C2 @ Q )
=> ( relati775585786_valid @ ( relati23543761ar_int @ P @ B @ C @ K ) @ ( while @ B @ C ) @ C2 @ Q ) ) ).
% ir_valid_get_back
thf(fact_26_sim__while__cong__aux,axiom,
! [B: bexp,C: com,S: list_char > int,T: list_char > int,C2: com] :
( ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C ) @ S ) @ T )
=> ( ! [S3: list_char > int,T6: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C @ S3 ) @ T6 )
= ( big_big_step @ ( produc1193801173ar_int @ C2 @ S3 ) @ T6 ) )
=> ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C2 ) @ S ) @ T ) ) ) ).
% sim_while_cong_aux
thf(fact_27_sim__while__cong,axiom,
! [C: com,C2: com,B: bexp] :
( ! [S3: list_char > int,T6: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C @ S3 ) @ T6 )
= ( big_big_step @ ( produc1193801173ar_int @ C2 @ S3 ) @ T6 ) )
=> ! [S6: list_char > int,T4: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C ) @ S6 ) @ T4 )
= ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C2 ) @ S6 ) @ T4 ) ) ) ).
% sim_while_cong
thf(fact_28_ir__valid__def,axiom,
( relati775585786_valid
= ( ^ [P3: ( list_char > int ) > ( list_char > int ) > $o,C3: com,C4: com,Q3: ( list_char > int ) > ( list_char > int ) > $o] :
! [T3: list_char > int,T5: list_char > int] :
( ( Q3 @ T3 @ T5 )
=> ? [S5: list_char > int,S7: list_char > int] :
( ( P3 @ S5 @ S7 )
& ( big_big_step @ ( produc1193801173ar_int @ C3 @ S5 ) @ T3 )
& ( big_big_step @ ( produc1193801173ar_int @ C4 @ S7 ) @ T5 ) ) ) ) ) ).
% ir_valid_def
thf(fact_29_WhileFalse,axiom,
! [B: bexp,S: list_char > int,C: com] :
( ~ ( bval @ B @ S )
=> ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C ) @ S ) @ S ) ) ).
% WhileFalse
thf(fact_30_WhileTrue,axiom,
! [B: bexp,S_1: list_char > int,C: com,S_2: list_char > int,S_3: list_char > int] :
( ( bval @ B @ S_1 )
=> ( ( big_big_step @ ( produc1193801173ar_int @ C @ S_1 ) @ S_2 )
=> ( ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C ) @ S_2 ) @ S_3 )
=> ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C ) @ S_1 ) @ S_3 ) ) ) ) ).
% WhileTrue
thf(fact_31_WhileE,axiom,
! [B: bexp,C: com,S: list_char > int,T: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C ) @ S ) @ T )
=> ( ( ( T = S )
=> ( bval @ B @ S ) )
=> ~ ( ( bval @ B @ S )
=> ! [S_22: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C @ S ) @ S_22 )
=> ~ ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C ) @ S_22 ) @ T ) ) ) ) ) ).
% WhileE
thf(fact_32_old_Oprod_Oinducts,axiom,
! [P: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_33_old_Oprod_Oinducts,axiom,
! [P: produc1260470173ar_int > $o,Prod: produc1260470173ar_int] :
( ! [A3: com,B3: list_char > int] : ( P @ ( produc1193801173ar_int @ A3 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_34_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A3: nat,B3: nat] :
( Y
!= ( product_Pair_nat_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_35_old_Oprod_Oexhaust,axiom,
! [Y: produc1260470173ar_int] :
~ ! [A3: com,B3: list_char > int] :
( Y
!= ( produc1193801173ar_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_36_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_37_Pair__inject,axiom,
! [A: com,B: list_char > int,A2: com,B2: list_char > int] :
( ( ( produc1193801173ar_int @ A @ B )
= ( produc1193801173ar_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_38_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_39_prod__cases,axiom,
! [P: produc1260470173ar_int > $o,P4: produc1260470173ar_int] :
( ! [A3: com,B3: list_char > int] : ( P @ ( produc1193801173ar_int @ A3 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_40_mem__Collect__eq,axiom,
! [A: produc1260470173ar_int,P: produc1260470173ar_int > $o] :
( ( member92250950ar_int @ A @ ( collec770796680ar_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_41_Collect__mem__eq,axiom,
! [A4: set_Pr1148378109ar_int] :
( ( collec770796680ar_int
@ ^ [X: produc1260470173ar_int] : ( member92250950ar_int @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_42_surj__pair,axiom,
! [P4: product_prod_nat_nat] :
? [X3: nat,Y3: nat] :
( P4
= ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_43_surj__pair,axiom,
! [P4: produc1260470173ar_int] :
? [X3: com,Y3: list_char > int] :
( P4
= ( produc1193801173ar_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_44_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_45_Suc__inject,axiom,
! [X4: nat,Y: nat] :
( ( ( suc @ X4 )
= ( suc @ Y ) )
=> ( X4 = Y ) ) ).
% Suc_inject
thf(fact_46_big__step__determ,axiom,
! [C: com,S: list_char > int,T: list_char > int,U: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C @ S ) @ T )
=> ( ( big_big_step @ ( produc1193801173ar_int @ C @ S ) @ U )
=> ( U = T ) ) ) ).
% big_step_determ
thf(fact_47_sim__trans,axiom,
! [C: com,C2: com,C5: com] :
( ! [S3: list_char > int,T6: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C @ S3 ) @ T6 )
= ( big_big_step @ ( produc1193801173ar_int @ C2 @ S3 ) @ T6 ) )
=> ( ! [S3: list_char > int,T6: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C2 @ S3 ) @ T6 )
= ( big_big_step @ ( produc1193801173ar_int @ C5 @ S3 ) @ T6 ) )
=> ! [S6: list_char > int,T4: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C @ S6 ) @ T4 )
= ( big_big_step @ ( produc1193801173ar_int @ C5 @ S6 ) @ T4 ) ) ) ) ).
% sim_trans
thf(fact_48_com_Oinject_I4_J,axiom,
! [X51: bexp,X52: com,Y51: bexp,Y52: com] :
( ( ( while @ X51 @ X52 )
= ( while @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% com.inject(4)
thf(fact_49_While__False__ir__valid,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,C: com] :
( ( relati775585786_valid
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ~ ( bval @ B @ S5 ) )
@ skip
@ C2
@ Q )
=> ( relati775585786_valid @ P @ ( while @ B @ C ) @ C2 @ Q ) ) ).
% While_False_ir_valid
thf(fact_50_While__True__ir__valid,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati775585786_valid
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ( bval @ B @ S5 ) )
@ ( seq @ C @ ( while @ B @ C ) )
@ C2
@ Q )
=> ( relati775585786_valid @ P @ ( while @ B @ C ) @ C2 @ Q ) ) ).
% While_True_ir_valid
thf(fact_51_get__back_Osimps_I2_J,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C: com,N: nat] :
( ( relati23543761ar_int @ P @ B @ C @ ( suc @ N ) )
= ( ^ [T3: list_char > int,T5: list_char > int] :
? [S5: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C @ S5 ) @ T3 )
& ( bval @ B @ S5 )
& ( relati23543761ar_int @ P @ B @ C @ N @ S5 @ T5 ) ) ) ) ).
% get_back.simps(2)
thf(fact_52_Skip__ir__valid,axiom,
! [Q: ( list_char > int ) > ( list_char > int ) > $o,P: ( list_char > int ) > ( list_char > int ) > $o,C2: com] :
( ! [T6: list_char > int,T7: list_char > int] :
( ( Q @ T6 @ T7 )
=> ? [S8: list_char > int] :
( ( P @ T6 @ S8 )
& ( big_big_step @ ( produc1193801173ar_int @ C2 @ S8 ) @ T7 ) ) )
=> ( relati775585786_valid @ P @ skip @ C2 @ Q ) ) ).
% Skip_ir_valid
thf(fact_53_While__backwards__frontier__ir__valid_H,axiom,
! [P: nat > ( list_char > int ) > ( list_char > int ) > $o,K: nat,B: bexp,C: com,Q: ( list_char > int ) > ( list_char > int ) > $o,C2: com,T: list_char > int,T2: list_char > int] :
( ! [N2: nat,T6: list_char > int,T7: list_char > int] :
( ( P @ ( plus_plus_nat @ K @ ( suc @ N2 ) ) @ T6 @ T7 )
=> ? [S6: list_char > int] :
( ( P @ ( plus_plus_nat @ K @ N2 ) @ S6 @ T7 )
& ( bval @ B @ S6 )
& ( big_big_step @ ( produc1193801173ar_int @ C @ S6 ) @ T6 ) ) )
=> ( ! [T6: list_char > int,T7: list_char > int] :
( ( Q @ T6 @ T7 )
=> ? [S6: list_char > int,S8: list_char > int] :
( ? [N3: nat] : ( P @ ( plus_plus_nat @ K @ N3 ) @ S6 @ S8 )
& ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C ) @ S6 ) @ T6 )
& ( big_big_step @ ( produc1193801173ar_int @ C2 @ S8 ) @ T7 ) ) )
=> ( ( Q @ T @ T2 )
=> ? [S3: list_char > int,S4: list_char > int] :
( ( P @ K @ S3 @ S4 )
& ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C ) @ S3 ) @ T )
& ( big_big_step @ ( produc1193801173ar_int @ C2 @ S4 ) @ T2 ) ) ) ) ) ).
% While_backwards_frontier_ir_valid'
thf(fact_54_sym__ir__valid,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C2: com,C: com,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati775585786_valid @ ( relati186416454e_flip @ P ) @ C2 @ C @ ( relati186416454e_flip @ Q ) )
=> ( relati775585786_valid @ P @ C @ C2 @ Q ) ) ).
% sym_ir_valid
thf(fact_55_pred__equals__eq2,axiom,
! [R: set_Pr1986765409at_nat,S9: set_Pr1986765409at_nat] :
( ( ( ^ [X: nat,Y4: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ R ) )
= ( ^ [X: nat,Y4: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ S9 ) ) )
= ( R = S9 ) ) ).
% pred_equals_eq2
thf(fact_56_pred__equals__eq2,axiom,
! [R: set_Pr1148378109ar_int,S9: set_Pr1148378109ar_int] :
( ( ( ^ [X: com,Y4: list_char > int] : ( member92250950ar_int @ ( produc1193801173ar_int @ X @ Y4 ) @ R ) )
= ( ^ [X: com,Y4: list_char > int] : ( member92250950ar_int @ ( produc1193801173ar_int @ X @ Y4 ) @ S9 ) ) )
= ( R = S9 ) ) ).
% pred_equals_eq2
thf(fact_57_com_Oinject_I2_J,axiom,
! [X31: com,X32: com,Y31: com,Y32: com] :
( ( ( seq @ X31 @ X32 )
= ( seq @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% com.inject(2)
thf(fact_58_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_59_Seq__ir__valid,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,D: com,D2: com,R: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati775585786_valid @ P @ C @ C2 @ Q )
=> ( ( relati775585786_valid @ Q @ D @ D2 @ R )
=> ( relati775585786_valid @ P @ ( seq @ C @ D ) @ ( seq @ C2 @ D2 ) @ R ) ) ) ).
% Seq_ir_valid
thf(fact_60_meh__simp,axiom,
! [S2: list_char > int,T2: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ skip @ S2 ) @ T2 )
= ( S2 = T2 ) ) ).
% meh_simp
thf(fact_61_Seq2__ir__valid,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C: com,Q: ( list_char > int ) > ( list_char > int ) > $o,D: com,C2: com,R: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati775585786_valid @ P @ C @ skip @ Q )
=> ( ( relati775585786_valid @ Q @ D @ C2 @ R )
=> ( relati775585786_valid @ P @ ( seq @ C @ D ) @ C2 @ R ) ) ) ).
% Seq2_ir_valid
thf(fact_62_Seq1__ir__valid,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,D: com,R: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati775585786_valid @ P @ C @ C2 @ Q )
=> ( ( relati775585786_valid @ Q @ D @ skip @ R )
=> ( relati775585786_valid @ P @ ( seq @ C @ D ) @ C2 @ R ) ) ) ).
% Seq1_ir_valid
thf(fact_63_com_Odistinct_I3_J,axiom,
! [X31: com,X32: com] :
( skip
!= ( seq @ X31 @ X32 ) ) ).
% com.distinct(3)
thf(fact_64_com_Odistinct_I17_J,axiom,
! [X31: com,X32: com,X51: bexp,X52: com] :
( ( seq @ X31 @ X32 )
!= ( while @ X51 @ X52 ) ) ).
% com.distinct(17)
thf(fact_65_com_Odistinct_I7_J,axiom,
! [X51: bexp,X52: com] :
( skip
!= ( while @ X51 @ X52 ) ) ).
% com.distinct(7)
thf(fact_66_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_67_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_68_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_69_ir__valid__Seq1_H,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C1: com,C22: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati775585786_valid @ P @ ( seq @ C1 @ C22 ) @ C2 @ Q )
=> ( relati775585786_valid
@ ^ [T3: list_char > int,T5: list_char > int] :
? [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ( big_big_step @ ( produc1193801173ar_int @ C1 @ S5 ) @ T3 )
& ( big_big_step @ ( produc1193801173ar_int @ C2 @ S7 ) @ T5 )
& ? [U2: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C22 @ T3 ) @ U2 )
& ( Q @ U2 @ T5 ) ) )
@ C22
@ skip
@ Q ) ) ).
% ir_valid_Seq1'
thf(fact_70_flip__def,axiom,
( relati186416454e_flip
= ( ^ [P3: ( list_char > int ) > ( list_char > int ) > $o,S5: list_char > int,S7: list_char > int] : ( P3 @ S7 @ S5 ) ) ) ).
% flip_def
thf(fact_71_ir__valid__Seq1,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C1: com,C22: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati775585786_valid @ P @ ( seq @ C1 @ C22 ) @ C2 @ Q )
=> ( relati775585786_valid @ P @ C1 @ C2
@ ^ [T3: list_char > int,T5: list_char > int] :
? [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ( big_big_step @ ( produc1193801173ar_int @ C1 @ S5 ) @ T3 )
& ( big_big_step @ ( produc1193801173ar_int @ C2 @ S7 ) @ T5 )
& ? [U2: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C22 @ T3 ) @ U2 )
& ( Q @ U2 @ T5 ) ) ) ) ) ).
% ir_valid_Seq1
thf(fact_72_SeqE,axiom,
! [C1: com,C22: com,S1: list_char > int,S32: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ ( seq @ C1 @ C22 ) @ S1 ) @ S32 )
=> ~ ! [S_22: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C1 @ S1 ) @ S_22 )
=> ~ ( big_big_step @ ( produc1193801173ar_int @ C22 @ S_22 ) @ S32 ) ) ) ).
% SeqE
thf(fact_73_Seq__assoc,axiom,
! [C1: com,C22: com,C32: com,S: list_char > int,S2: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ ( seq @ ( seq @ C1 @ C22 ) @ C32 ) @ S ) @ S2 )
= ( big_big_step @ ( produc1193801173ar_int @ ( seq @ C1 @ ( seq @ C22 @ C32 ) ) @ S ) @ S2 ) ) ).
% Seq_assoc
thf(fact_74_Seq,axiom,
! [C_1: com,S_1: list_char > int,S_2: list_char > int,C_2: com,S_3: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ C_1 @ S_1 ) @ S_2 )
=> ( ( big_big_step @ ( produc1193801173ar_int @ C_2 @ S_2 ) @ S_3 )
=> ( big_big_step @ ( produc1193801173ar_int @ ( seq @ C_1 @ C_2 ) @ S_1 ) @ S_3 ) ) ) ).
% Seq
thf(fact_75_SkipE,axiom,
! [S: list_char > int,T: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ skip @ S ) @ T )
=> ( T = S ) ) ).
% SkipE
thf(fact_76_Skip,axiom,
! [S: list_char > int] : ( big_big_step @ ( produc1193801173ar_int @ skip @ S ) @ S ) ).
% Skip
thf(fact_77_ir__valid__track__history,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,S: list_char > int,S2: list_char > int] :
( ( relati775585786_valid @ P @ C @ C2 @ Q )
=> ( relati775585786_valid @ P @ C @ C2
@ ^ [T3: list_char > int,T5: list_char > int] :
( ( Q @ S @ S2 )
& ? [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ( big_big_step @ ( produc1193801173ar_int @ C @ S5 ) @ T3 )
& ( big_big_step @ ( produc1193801173ar_int @ C2 @ S7 ) @ T5 ) ) ) ) ) ).
% ir_valid_track_history
thf(fact_78_ir__valid__While1,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C: com,K: nat,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati775585786_valid @ ( relati23543761ar_int @ P @ B @ C @ K ) @ ( while @ B @ C ) @ C2 @ Q )
=> ( relati775585786_valid
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( relati23543761ar_int @ P @ B @ C @ K @ S5 @ S7 )
& ( bval @ B @ S5 ) )
@ C
@ skip
@ ^ [T3: list_char > int,T5: list_char > int] :
( ( relati23543761ar_int @ P @ B @ C @ ( suc @ K ) @ T3 @ T5 )
& ? [U2: list_char > int,U3: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C ) @ T3 ) @ U2 )
& ( big_big_step @ ( produc1193801173ar_int @ C2 @ T5 ) @ U3 )
& ( Q @ U2 @ U3 ) ) ) ) ) ).
% ir_valid_While1
thf(fact_79_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_80_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_81_ir__valid__If,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C1: com,C22: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati775585786_valid @ P @ ( if @ B @ C1 @ C22 ) @ C2 @ Q )
=> ( ( relati775585786_valid
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ( bval @ B @ S5 ) )
@ C1
@ C2
@ ^ [T3: list_char > int,T5: list_char > int] :
( ( Q @ T3 @ T5 )
& ? [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ( big_big_step @ ( produc1193801173ar_int @ C1 @ S5 ) @ T3 )
& ( big_big_step @ ( produc1193801173ar_int @ C2 @ S7 ) @ T5 )
& ( bval @ B @ S5 ) ) ) )
& ( relati775585786_valid
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ~ ( bval @ B @ S5 ) )
@ C22
@ C2
@ ^ [T3: list_char > int,T5: list_char > int] :
( ( Q @ T3 @ T5 )
& ? [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ( big_big_step @ ( produc1193801173ar_int @ C22 @ S5 ) @ T3 )
& ( big_big_step @ ( produc1193801173ar_int @ C2 @ S7 ) @ T5 )
& ~ ( bval @ B @ S5 ) ) ) ) ) ) ).
% ir_valid_If
thf(fact_82_while__unfold,axiom,
! [B: bexp,C: com,S6: list_char > int,T4: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ ( while @ B @ C ) @ S6 ) @ T4 )
= ( big_big_step @ ( produc1193801173ar_int @ ( if @ B @ ( seq @ C @ ( while @ B @ C ) ) @ skip ) @ S6 ) @ T4 ) ) ).
% while_unfold
thf(fact_83_If__True__ir__valid,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C_1: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,C_2: com] :
( ( relati775585786_valid
@ ^ [A5: list_char > int,C3: list_char > int] :
( ( P @ A5 @ C3 )
& ( bval @ B @ A5 ) )
@ C_1
@ C2
@ Q )
=> ( relati775585786_valid @ P @ ( if @ B @ C_1 @ C_2 ) @ C2 @ Q ) ) ).
% If_True_ir_valid
thf(fact_84_If__False__ir__valid,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C_2: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,C_1: com] :
( ( relati775585786_valid
@ ^ [A5: list_char > int,C3: list_char > int] :
( ( P @ A5 @ C3 )
& ~ ( bval @ B @ A5 ) )
@ C_2
@ C2
@ Q )
=> ( relati775585786_valid @ P @ ( if @ B @ C_1 @ C_2 ) @ C2 @ Q ) ) ).
% If_False_ir_valid
thf(fact_85_ir__While__True,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati2102071307_hoare
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ( bval @ B @ S5 ) )
@ ( seq @ C @ ( while @ B @ C ) )
@ C2
@ Q )
=> ( relati2102071307_hoare @ P @ ( while @ B @ C ) @ C2 @ Q ) ) ).
% ir_While_True
thf(fact_86_com_Oinject_I3_J,axiom,
! [X41: bexp,X42: com,X43: com,Y41: bexp,Y42: com,Y43: com] :
( ( ( if @ X41 @ X42 @ X43 )
= ( if @ Y41 @ Y42 @ Y43 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 )
& ( X43 = Y43 ) ) ) ).
% com.inject(3)
thf(fact_87_ir__Skip__Skip,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o] : ( relati2102071307_hoare @ P @ skip @ skip @ P ) ).
% ir_Skip_Skip
thf(fact_88_ir__hoare__Skip__Skip,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati2102071307_hoare @ P @ skip @ skip @ Q )
= ( ! [S5: list_char > int,S7: list_char > int] :
( ( Q @ S5 @ S7 )
=> ( P @ S5 @ S7 ) ) ) ) ).
% ir_hoare_Skip_Skip
thf(fact_89_ir__disj,axiom,
! [P_1: ( list_char > int ) > ( list_char > int ) > $o,C: com,C2: com,Q_1: ( list_char > int ) > ( list_char > int ) > $o,P_2: ( list_char > int ) > ( list_char > int ) > $o,Q_2: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati2102071307_hoare @ P_1 @ C @ C2 @ Q_1 )
=> ( ( relati2102071307_hoare @ P_2 @ C @ C2 @ Q_2 )
=> ( relati2102071307_hoare
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( P_1 @ S5 @ S7 )
| ( P_2 @ S5 @ S7 ) )
@ C
@ C2
@ ^ [T3: list_char > int,T5: list_char > int] :
( ( Q_1 @ T3 @ T5 )
| ( Q_2 @ T3 @ T5 ) ) ) ) ) ).
% ir_disj
thf(fact_90_ir__pre,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,P2: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati2102071307_hoare @ P @ C @ C2 @ Q )
=> ( ! [S3: list_char > int,S4: list_char > int] :
( ( P @ S3 @ S4 )
=> ( P2 @ S3 @ S4 ) )
=> ( relati2102071307_hoare @ P2 @ C @ C2 @ Q ) ) ) ).
% ir_pre
thf(fact_91_ir__post,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,Q2: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati2102071307_hoare @ P @ C @ C2 @ Q )
=> ( ! [S3: list_char > int,S4: list_char > int] :
( ( Q2 @ S3 @ S4 )
=> ( Q @ S3 @ S4 ) )
=> ( relati2102071307_hoare @ P @ C @ C2 @ Q2 ) ) ) ).
% ir_post
thf(fact_92_ir__conseq,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,P2: ( list_char > int ) > ( list_char > int ) > $o,Q2: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati2102071307_hoare @ P @ C @ C2 @ Q )
=> ( ! [S3: list_char > int,S4: list_char > int] :
( ( P @ S3 @ S4 )
=> ( P2 @ S3 @ S4 ) )
=> ( ! [S3: list_char > int,S4: list_char > int] :
( ( Q2 @ S3 @ S4 )
=> ( Q @ S3 @ S4 ) )
=> ( relati2102071307_hoare @ P2 @ C @ C2 @ Q2 ) ) ) ) ).
% ir_conseq
thf(fact_93_ir__If__False,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C_2: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,C_1: com] :
( ( relati2102071307_hoare
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ~ ( bval @ B @ S5 ) )
@ C_2
@ C2
@ Q )
=> ( relati2102071307_hoare @ P @ ( if @ B @ C_1 @ C_2 ) @ C2 @ Q ) ) ).
% ir_If_False
thf(fact_94_ir__If__True,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C_1: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,C_2: com] :
( ( relati2102071307_hoare
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ( bval @ B @ S5 ) )
@ C_1
@ C2
@ Q )
=> ( relati2102071307_hoare @ P @ ( if @ B @ C_1 @ C_2 ) @ C2 @ Q ) ) ).
% ir_If_True
thf(fact_95_soundness,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati2102071307_hoare @ P @ C @ C2 @ Q )
=> ( relati775585786_valid @ P @ C @ C2 @ Q ) ) ).
% soundness
thf(fact_96_com_Odistinct_I19_J,axiom,
! [X41: bexp,X42: com,X43: com,X51: bexp,X52: com] :
( ( if @ X41 @ X42 @ X43 )
!= ( while @ X51 @ X52 ) ) ).
% com.distinct(19)
thf(fact_97_com_Odistinct_I5_J,axiom,
! [X41: bexp,X42: com,X43: com] :
( skip
!= ( if @ X41 @ X42 @ X43 ) ) ).
% com.distinct(5)
thf(fact_98_com_Odistinct_I15_J,axiom,
! [X31: com,X32: com,X41: bexp,X42: com,X43: com] :
( ( seq @ X31 @ X32 )
!= ( if @ X41 @ X42 @ X43 ) ) ).
% com.distinct(15)
thf(fact_99_ir__sym,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati2102071307_hoare @ ( relati186416454e_flip @ P ) @ C @ C2 @ ( relati186416454e_flip @ Q ) )
=> ( relati2102071307_hoare @ P @ C2 @ C @ Q ) ) ).
% ir_sym
thf(fact_100_ir__Seq1,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,D: com,R: ( list_char > int ) > ( list_char > int ) > $o] :
( ( relati2102071307_hoare @ P @ C @ C2 @ Q )
=> ( ( relati2102071307_hoare @ Q @ D @ skip @ R )
=> ( relati2102071307_hoare @ P @ ( seq @ C @ D ) @ C2 @ R ) ) ) ).
% ir_Seq1
thf(fact_101_commute__if,axiom,
! [B1: bexp,B22: bexp,C11: com,C12: com,C22: com,S6: list_char > int,T4: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ ( if @ B1 @ ( if @ B22 @ C11 @ C12 ) @ C22 ) @ S6 ) @ T4 )
= ( big_big_step @ ( produc1193801173ar_int @ ( if @ B22 @ ( if @ B1 @ C11 @ C22 ) @ ( if @ B1 @ C12 @ C22 ) ) @ S6 ) @ T4 ) ) ).
% commute_if
thf(fact_102_triv__if,axiom,
! [B: bexp,C: com,S6: list_char > int,T4: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ ( if @ B @ C @ C ) @ S6 ) @ T4 )
= ( big_big_step @ ( produc1193801173ar_int @ C @ S6 ) @ T4 ) ) ).
% triv_if
thf(fact_103_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_104_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_105_group__cancel_Oadd1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_106_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_107_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_108_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B5: nat] : ( plus_plus_nat @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_109_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_110_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_111_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_112_ir__Skip,axiom,
! [Q: ( list_char > int ) > ( list_char > int ) > $o,P: ( list_char > int ) > ( list_char > int ) > $o,C2: com] :
( ! [T6: list_char > int,T7: list_char > int] :
( ( Q @ T6 @ T7 )
=> ? [S8: list_char > int] :
( ( P @ T6 @ S8 )
& ( big_big_step @ ( produc1193801173ar_int @ C2 @ S8 ) @ T7 ) ) )
=> ( relati2102071307_hoare @ P @ skip @ C2 @ Q ) ) ).
% ir_Skip
thf(fact_113_IfFalse,axiom,
! [B: bexp,S: list_char > int,C_2: com,T: list_char > int,C_1: com] :
( ~ ( bval @ B @ S )
=> ( ( big_big_step @ ( produc1193801173ar_int @ C_2 @ S ) @ T )
=> ( big_big_step @ ( produc1193801173ar_int @ ( if @ B @ C_1 @ C_2 ) @ S ) @ T ) ) ) ).
% IfFalse
thf(fact_114_IfTrue,axiom,
! [B: bexp,S: list_char > int,C_1: com,T: list_char > int,C_2: com] :
( ( bval @ B @ S )
=> ( ( big_big_step @ ( produc1193801173ar_int @ C_1 @ S ) @ T )
=> ( big_big_step @ ( produc1193801173ar_int @ ( if @ B @ C_1 @ C_2 ) @ S ) @ T ) ) ) ).
% IfTrue
thf(fact_115_IfE,axiom,
! [B: bexp,C1: com,C22: com,S: list_char > int,T: list_char > int] :
( ( big_big_step @ ( produc1193801173ar_int @ ( if @ B @ C1 @ C22 ) @ S ) @ T )
=> ( ( ( bval @ B @ S )
=> ~ ( big_big_step @ ( produc1193801173ar_int @ C1 @ S ) @ T ) )
=> ~ ( ~ ( bval @ B @ S )
=> ~ ( big_big_step @ ( produc1193801173ar_int @ C22 @ S ) @ T ) ) ) ) ).
% IfE
thf(fact_116_ir__While__False,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o,C: com] :
( ( relati2102071307_hoare
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( P @ S5 @ S7 )
& ~ ( bval @ B @ S5 ) )
@ skip
@ C2
@ Q )
=> ( relati2102071307_hoare @ P @ ( while @ B @ C ) @ C2 @ Q ) ) ).
% ir_While_False
thf(fact_117_com_Oexhaust,axiom,
! [Y: com] :
( ( Y != skip )
=> ( ! [X21: list_char,X22: aexp] :
( Y
!= ( assign @ X21 @ X22 ) )
=> ( ! [X312: com,X322: com] :
( Y
!= ( seq @ X312 @ X322 ) )
=> ( ! [X412: bexp,X422: com,X432: com] :
( Y
!= ( if @ X412 @ X422 @ X432 ) )
=> ~ ! [X512: bexp,X522: com] :
( Y
!= ( while @ X512 @ X522 ) ) ) ) ) ) ).
% com.exhaust
thf(fact_118_com_Oinduct,axiom,
! [P: com > $o,Com: com] :
( ( P @ skip )
=> ( ! [X12: list_char,X23: aexp] : ( P @ ( assign @ X12 @ X23 ) )
=> ( ! [X12: com,X23: com] :
( ( P @ X12 )
=> ( ( P @ X23 )
=> ( P @ ( seq @ X12 @ X23 ) ) ) )
=> ( ! [X12: bexp,X23: com,X33: com] :
( ( P @ X23 )
=> ( ( P @ X33 )
=> ( P @ ( if @ X12 @ X23 @ X33 ) ) ) )
=> ( ! [X12: bexp,X23: com] :
( ( P @ X23 )
=> ( P @ ( while @ X12 @ X23 ) ) )
=> ( P @ Com ) ) ) ) ) ) ).
% com.induct
thf(fact_119_image2__def,axiom,
( bNF_Gr202451329ar_int
= ( ^ [A6: set_Pr1148378109ar_int,F: produc1260470173ar_int > com,G: produc1260470173ar_int > list_char > int] :
( collec770796680ar_int
@ ^ [Uu: produc1260470173ar_int] :
? [A5: produc1260470173ar_int] :
( ( Uu
= ( produc1193801173ar_int @ ( F @ A5 ) @ ( G @ A5 ) ) )
& ( member92250950ar_int @ A5 @ A6 ) ) ) ) ) ).
% image2_def
thf(fact_120_image2__def,axiom,
( bNF_Gr749752805at_nat
= ( ^ [A6: set_Pr1148378109ar_int,F: produc1260470173ar_int > nat,G: produc1260470173ar_int > nat] :
( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [A5: produc1260470173ar_int] :
( ( Uu
= ( product_Pair_nat_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
& ( member92250950ar_int @ A5 @ A6 ) ) ) ) ) ).
% image2_def
thf(fact_121_While__backwards__frontier__ir__valid,axiom,
! [P: nat > ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ! [N2: nat] :
( relati775585786_valid
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( P @ N2 @ S5 @ S7 )
& ( bval @ B @ S5 ) )
@ C
@ skip
@ ( P @ ( suc @ N2 ) ) )
=> ( ( relati775585786_valid
@ ^ [S5: list_char > int,S7: list_char > int] :
? [N4: nat] : ( P @ N4 @ S5 @ S7 )
@ ( while @ B @ C )
@ C2
@ Q )
=> ( relati775585786_valid @ ( P @ zero_zero_nat ) @ ( while @ B @ C ) @ C2 @ Q ) ) ) ).
% While_backwards_frontier_ir_valid
thf(fact_122_com_Oinject_I1_J,axiom,
! [X212: list_char,X222: aexp,Y21: list_char,Y22: aexp] :
( ( ( assign @ X212 @ X222 )
= ( assign @ Y21 @ Y22 ) )
= ( ( X212 = Y21 )
& ( X222 = Y22 ) ) ) ).
% com.inject(1)
thf(fact_123_add_Oleft__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add.left_neutral
thf(fact_124_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_125_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_126_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_127_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_128_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_129_add__eq__0__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_130_zero__eq__add__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X4 @ Y ) )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_131_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_132_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_133_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_134_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_135_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_136_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_137_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_138_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_139_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_140_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_141_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_142_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_143_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_144_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_145_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_146_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_147_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_148_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_149_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_150_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_151_image2__eqI,axiom,
! [B: com,F2: produc1260470173ar_int > com,X4: produc1260470173ar_int,C: list_char > int,G2: produc1260470173ar_int > list_char > int,A4: set_Pr1148378109ar_int] :
( ( B
= ( F2 @ X4 ) )
=> ( ( C
= ( G2 @ X4 ) )
=> ( ( member92250950ar_int @ X4 @ A4 )
=> ( member92250950ar_int @ ( produc1193801173ar_int @ B @ C ) @ ( bNF_Gr202451329ar_int @ A4 @ F2 @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_152_image2__eqI,axiom,
! [B: nat,F2: produc1260470173ar_int > nat,X4: produc1260470173ar_int,C: nat,G2: produc1260470173ar_int > nat,A4: set_Pr1148378109ar_int] :
( ( B
= ( F2 @ X4 ) )
=> ( ( C
= ( G2 @ X4 ) )
=> ( ( member92250950ar_int @ X4 @ A4 )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( bNF_Gr749752805at_nat @ A4 @ F2 @ G2 ) ) ) ) ) ).
% image2_eqI
thf(fact_153_com_Odistinct_I13_J,axiom,
! [X212: list_char,X222: aexp,X51: bexp,X52: com] :
( ( assign @ X212 @ X222 )
!= ( while @ X51 @ X52 ) ) ).
% com.distinct(13)
thf(fact_154_com_Odistinct_I1_J,axiom,
! [X212: list_char,X222: aexp] :
( skip
!= ( assign @ X212 @ X222 ) ) ).
% com.distinct(1)
thf(fact_155_com_Odistinct_I9_J,axiom,
! [X212: list_char,X222: aexp,X31: com,X32: com] :
( ( assign @ X212 @ X222 )
!= ( seq @ X31 @ X32 ) ) ).
% com.distinct(9)
thf(fact_156_com_Odistinct_I11_J,axiom,
! [X212: list_char,X222: aexp,X41: bexp,X42: com,X43: com] :
( ( assign @ X212 @ X222 )
!= ( if @ X41 @ X42 @ X43 ) ) ).
% com.distinct(11)
thf(fact_157_get__back_Osimps_I1_J,axiom,
! [P: ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C: com] :
( ( relati23543761ar_int @ P @ B @ C @ zero_zero_nat )
= P ) ).
% get_back.simps(1)
thf(fact_158_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_159_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_160_ir__While__backwards__frontier,axiom,
! [P: nat > ( list_char > int ) > ( list_char > int ) > $o,B: bexp,C: com,C2: com,Q: ( list_char > int ) > ( list_char > int ) > $o] :
( ! [N2: nat] :
( relati2102071307_hoare
@ ^ [S5: list_char > int,S7: list_char > int] :
( ( P @ N2 @ S5 @ S7 )
& ( bval @ B @ S5 ) )
@ C
@ skip
@ ( P @ ( suc @ N2 ) ) )
=> ( ( relati2102071307_hoare
@ ^ [S5: list_char > int,S7: list_char > int] :
? [N4: nat] : ( P @ N4 @ S5 @ S7 )
@ ( while @ B @ C )
@ C2
@ Q )
=> ( relati2102071307_hoare @ ( P @ zero_zero_nat ) @ ( while @ B @ C ) @ C2 @ Q ) ) ) ).
% ir_While_backwards_frontier
thf(fact_161_com_Osize__gen_I4_J,axiom,
! [X41: bexp,X42: com,X43: com] :
( ( size_com @ ( if @ X41 @ X42 @ X43 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_com @ X42 ) @ ( size_com @ X43 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size_gen(4)
thf(fact_162_com_Osize__gen_I3_J,axiom,
! [X31: com,X32: com] :
( ( size_com @ ( seq @ X31 @ X32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_com @ X31 ) @ ( size_com @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size_gen(3)
thf(fact_163_com_Osize__gen_I5_J,axiom,
! [X51: bexp,X52: com] :
( ( size_com @ ( while @ X51 @ X52 ) )
= ( plus_plus_nat @ ( size_com @ X52 ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size_gen(5)
thf(fact_164_com_Osize_I9_J,axiom,
! [X41: bexp,X42: com,X43: com] :
( ( size_size_com @ ( if @ X41 @ X42 @ X43 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_com @ X42 ) @ ( size_size_com @ X43 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size(9)
thf(fact_165_size__neq__size__imp__neq,axiom,
! [X4: com,Y: com] :
( ( ( size_size_com @ X4 )
!= ( size_size_com @ Y ) )
=> ( X4 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_166_size__neq__size__imp__neq,axiom,
! [X4: typerep,Y: typerep] :
( ( ( size_size_typerep @ X4 )
!= ( size_size_typerep @ Y ) )
=> ( X4 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_167_com_Osize_I6_J,axiom,
( ( size_size_com @ skip )
= zero_zero_nat ) ).
% com.size(6)
thf(fact_168_com_Osize_I7_J,axiom,
! [X212: list_char,X222: aexp] :
( ( size_size_com @ ( assign @ X212 @ X222 ) )
= zero_zero_nat ) ).
% com.size(7)
thf(fact_169_com_Osize__gen_I1_J,axiom,
( ( size_com @ skip )
= zero_zero_nat ) ).
% com.size_gen(1)
thf(fact_170_com_Osize__gen_I2_J,axiom,
! [X212: list_char,X222: aexp] :
( ( size_com @ ( assign @ X212 @ X222 ) )
= zero_zero_nat ) ).
% com.size_gen(2)
thf(fact_171_com_Osize_I10_J,axiom,
! [X51: bexp,X52: com] :
( ( size_size_com @ ( while @ X51 @ X52 ) )
= ( plus_plus_nat @ ( size_size_com @ X52 ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size(10)
thf(fact_172_com_Osize_I8_J,axiom,
! [X31: com,X32: com] :
( ( size_size_com @ ( seq @ X31 @ X32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_com @ X31 ) @ ( size_size_com @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size(8)
thf(fact_173_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
= ( P @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_174_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_175_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_176_relImage__def,axiom,
( bNF_Gr1247133574at_nat
= ( ^ [R2: set_Pr1986765409at_nat,F: nat > nat] :
( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [A1: nat,A22: nat] :
( ( Uu
= ( product_Pair_nat_nat @ ( F @ A1 ) @ ( F @ A22 ) ) )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R2 ) ) ) ) ) ).
% relImage_def
thf(fact_177_relInvImage__def,axiom,
( bNF_Gr1143394412nt_nat
= ( ^ [A6: set_Pr1148378109ar_int,R2: set_Pr1986765409at_nat,F: produc1260470173ar_int > nat] :
( collec1834387154ar_int
@ ^ [Uu: produc1346207591ar_int] :
? [A1: produc1260470173ar_int,A22: produc1260470173ar_int] :
( ( Uu
= ( produc317451735ar_int @ A1 @ A22 ) )
& ( member92250950ar_int @ A1 @ A6 )
& ( member92250950ar_int @ A22 @ A6 )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ ( F @ A1 ) @ ( F @ A22 ) ) @ R2 ) ) ) ) ) ).
% relInvImage_def
thf(fact_178_relInvImage__def,axiom,
( bNF_Gr808751971at_nat
= ( ^ [A6: set_nat,R2: set_Pr1986765409at_nat,F: nat > nat] :
( collec7649004at_nat
@ ^ [Uu: product_prod_nat_nat] :
? [A1: nat,A22: nat] :
( ( Uu
= ( product_Pair_nat_nat @ A1 @ A22 ) )
& ( member_nat @ A1 @ A6 )
& ( member_nat @ A22 @ A6 )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ ( F @ A1 ) @ ( F @ A22 ) ) @ R2 ) ) ) ) ) ).
% relInvImage_def
thf(fact_179_divides__aux__eq,axiom,
! [Q4: nat,R3: nat] :
( ( unique1592618082ux_nat @ ( product_Pair_nat_nat @ Q4 @ R3 ) )
= ( R3 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_180_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_181_typerep_Osize__neq,axiom,
! [X4: typerep] :
( ( size_size_typerep @ X4 )
!= zero_zero_nat ) ).
% typerep.size_neq
thf(fact_182_ssubst__Pair__rhs,axiom,
! [R3: nat,S: nat,R: set_Pr1986765409at_nat,S2: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ R3 @ S ) @ R )
=> ( ( S2 = S )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ R3 @ S2 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_183_ssubst__Pair__rhs,axiom,
! [R3: com,S: list_char > int,R: set_Pr1148378109ar_int,S2: list_char > int] :
( ( member92250950ar_int @ ( produc1193801173ar_int @ R3 @ S ) @ R )
=> ( ( S2 = S )
=> ( member92250950ar_int @ ( produc1193801173ar_int @ R3 @ S2 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_184_typerep_Osize_I2_J,axiom,
! [X1: literal,X2: list_typerep] :
( ( size_size_typerep @ ( typerep2 @ X1 @ X2 ) )
= ( plus_plus_nat @ ( size_list_typerep @ size_size_typerep @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% typerep.size(2)
thf(fact_185_typerep_Osize__gen,axiom,
! [X1: literal,X2: list_typerep] :
( ( size_typerep @ ( typerep2 @ X1 @ X2 ) )
= ( plus_plus_nat @ ( size_list_typerep @ size_typerep @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% typerep.size_gen
thf(fact_186_list_Osize__gen_I2_J,axiom,
! [X4: typerep > nat,X212: typerep,X222: list_typerep] :
( ( size_list_typerep @ X4 @ ( cons_typerep @ X212 @ X222 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( X4 @ X212 ) @ ( size_list_typerep @ X4 @ X222 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size_gen(2)
thf(fact_187_length__Suc__conv,axiom,
! [Xs: list_typerep,N: nat] :
( ( ( size_s416147576yperep @ Xs )
= ( suc @ N ) )
= ( ? [Y4: typerep,Ys: list_typerep] :
( ( Xs
= ( cons_typerep @ Y4 @ Ys ) )
& ( ( size_s416147576yperep @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_188_Suc__length__conv,axiom,
! [N: nat,Xs: list_typerep] :
( ( ( suc @ N )
= ( size_s416147576yperep @ Xs ) )
= ( ? [Y4: typerep,Ys: list_typerep] :
( ( Xs
= ( cons_typerep @ Y4 @ Ys ) )
& ( ( size_s416147576yperep @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_189_length__Cons,axiom,
! [X4: typerep,Xs: list_typerep] :
( ( size_s416147576yperep @ ( cons_typerep @ X4 @ Xs ) )
= ( suc @ ( size_s416147576yperep @ Xs ) ) ) ).
% length_Cons
thf(fact_190_list_Osize_I4_J,axiom,
! [X212: typerep,X222: list_typerep] :
( ( size_s416147576yperep @ ( cons_typerep @ X212 @ X222 ) )
= ( plus_plus_nat @ ( size_s416147576yperep @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_191_set__Cons__def,axiom,
( set_Co1954412371ar_int
= ( ^ [A6: set_Pr1148378109ar_int,XS: set_li1762364675ar_int] :
( collec672902542ar_int
@ ^ [Z: list_P1741976099ar_int] :
? [X: produc1260470173ar_int,Xs2: list_P1741976099ar_int] :
( ( Z
= ( cons_P1254790739ar_int @ X @ Xs2 ) )
& ( member92250950ar_int @ X @ A6 )
& ( member2052091212ar_int @ Xs2 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_192_set__Cons__def,axiom,
( set_Cons_typerep
= ( ^ [A6: set_typerep,XS: set_list_typerep] :
( collect_list_typerep
@ ^ [Z: list_typerep] :
? [X: typerep,Xs2: list_typerep] :
( ( Z
= ( cons_typerep @ X @ Xs2 ) )
& ( member_typerep @ X @ A6 )
& ( member_list_typerep @ Xs2 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_193_Cons__in__lex,axiom,
! [X4: typerep,Xs: list_typerep,Y: typerep,Ys2: list_typerep,R3: set_Pr1097818217yperep] :
( ( member1812711762yperep @ ( produc1695567329yperep @ ( cons_typerep @ X4 @ Xs ) @ ( cons_typerep @ Y @ Ys2 ) ) @ ( lex_typerep @ R3 ) )
= ( ( ( member100911538yperep @ ( produc2071514689yperep @ X4 @ Y ) @ R3 )
& ( ( size_s416147576yperep @ Xs )
= ( size_s416147576yperep @ Ys2 ) ) )
| ( ( X4 = Y )
& ( member1812711762yperep @ ( produc1695567329yperep @ Xs @ Ys2 ) @ ( lex_typerep @ R3 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_194_Cons__in__lex,axiom,
! [X4: nat,Xs: list_nat,Y: nat,Ys2: list_nat,R3: set_Pr1986765409at_nat] :
( ( member1926390090st_nat @ ( produc1625736153st_nat @ ( cons_nat @ X4 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( lex_nat @ R3 ) )
= ( ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y ) @ R3 )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) )
| ( ( X4 = Y )
& ( member1926390090st_nat @ ( produc1625736153st_nat @ Xs @ Ys2 ) @ ( lex_nat @ R3 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_195_gen__length__code_I2_J,axiom,
! [N: nat,X4: typerep,Xs: list_typerep] :
( ( gen_length_typerep @ N @ ( cons_typerep @ X4 @ Xs ) )
= ( gen_length_typerep @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_196_lexord__cons__cons,axiom,
! [A: typerep,X4: list_typerep,B: typerep,Y: list_typerep,R3: set_Pr1097818217yperep] :
( ( member1812711762yperep @ ( produc1695567329yperep @ ( cons_typerep @ A @ X4 ) @ ( cons_typerep @ B @ Y ) ) @ ( lexord_typerep @ R3 ) )
= ( ( member100911538yperep @ ( produc2071514689yperep @ A @ B ) @ R3 )
| ( ( A = B )
& ( member1812711762yperep @ ( produc1695567329yperep @ X4 @ Y ) @ ( lexord_typerep @ R3 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_197_lexord__cons__cons,axiom,
! [A: nat,X4: list_nat,B: nat,Y: list_nat,R3: set_Pr1986765409at_nat] :
( ( member1926390090st_nat @ ( produc1625736153st_nat @ ( cons_nat @ A @ X4 ) @ ( cons_nat @ B @ Y ) ) @ ( lexord_nat @ R3 ) )
= ( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
| ( ( A = B )
& ( member1926390090st_nat @ ( produc1625736153st_nat @ X4 @ Y ) @ ( lexord_nat @ R3 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_198_lexord__irreflexive,axiom,
! [R3: set_Pr1986765409at_nat,Xs: list_nat] :
( ! [X3: nat] :
~ ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R3 )
=> ~ ( member1926390090st_nat @ ( produc1625736153st_nat @ Xs @ Xs ) @ ( lexord_nat @ R3 ) ) ) ).
% lexord_irreflexive
thf(fact_199_lexord__linear,axiom,
! [R3: set_Pr1986765409at_nat,X4: list_nat,Y: list_nat] :
( ! [A3: nat,B3: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R3 )
| ( A3 = B3 )
| ( member701585322at_nat @ ( product_Pair_nat_nat @ B3 @ A3 ) @ R3 ) )
=> ( ( member1926390090st_nat @ ( produc1625736153st_nat @ X4 @ Y ) @ ( lexord_nat @ R3 ) )
| ( X4 = Y )
| ( member1926390090st_nat @ ( produc1625736153st_nat @ Y @ X4 ) @ ( lexord_nat @ R3 ) ) ) ) ).
% lexord_linear
thf(fact_200_Shift__def,axiom,
( bNF_Gr457912949yperep
= ( ^ [Kl: set_list_typerep,K2: typerep] :
( collect_list_typerep
@ ^ [Kl2: list_typerep] : ( member_list_typerep @ ( cons_typerep @ K2 @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_201_ShiftD,axiom,
! [Kl3: list_typerep,Kl4: set_list_typerep,K: typerep] :
( ( member_list_typerep @ Kl3 @ ( bNF_Gr457912949yperep @ Kl4 @ K ) )
=> ( member_list_typerep @ ( cons_typerep @ K @ Kl3 ) @ Kl4 ) ) ).
% ShiftD
thf(fact_202_Succ__Shift,axiom,
! [Kl4: set_list_typerep,K: typerep,Kl3: list_typerep] :
( ( bNF_Gr1303086585yperep @ ( bNF_Gr457912949yperep @ Kl4 @ K ) @ Kl3 )
= ( bNF_Gr1303086585yperep @ Kl4 @ ( cons_typerep @ K @ Kl3 ) ) ) ).
% Succ_Shift
thf(fact_203_Cons__listrel1__Cons,axiom,
! [X4: typerep,Xs: list_typerep,Y: typerep,Ys2: list_typerep,R3: set_Pr1097818217yperep] :
( ( member1812711762yperep @ ( produc1695567329yperep @ ( cons_typerep @ X4 @ Xs ) @ ( cons_typerep @ Y @ Ys2 ) ) @ ( listrel1_typerep @ R3 ) )
= ( ( ( member100911538yperep @ ( produc2071514689yperep @ X4 @ Y ) @ R3 )
& ( Xs = Ys2 ) )
| ( ( X4 = Y )
& ( member1812711762yperep @ ( produc1695567329yperep @ Xs @ Ys2 ) @ ( listrel1_typerep @ R3 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_204_Cons__listrel1__Cons,axiom,
! [X4: nat,Xs: list_nat,Y: nat,Ys2: list_nat,R3: set_Pr1986765409at_nat] :
( ( member1926390090st_nat @ ( produc1625736153st_nat @ ( cons_nat @ X4 @ Xs ) @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel1_nat @ R3 ) )
= ( ( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y ) @ R3 )
& ( Xs = Ys2 ) )
| ( ( X4 = Y )
& ( member1926390090st_nat @ ( produc1625736153st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R3 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_205_listrel1I1,axiom,
! [X4: typerep,Y: typerep,R3: set_Pr1097818217yperep,Xs: list_typerep] :
( ( member100911538yperep @ ( produc2071514689yperep @ X4 @ Y ) @ R3 )
=> ( member1812711762yperep @ ( produc1695567329yperep @ ( cons_typerep @ X4 @ Xs ) @ ( cons_typerep @ Y @ Xs ) ) @ ( listrel1_typerep @ R3 ) ) ) ).
% listrel1I1
thf(fact_206_listrel1I1,axiom,
! [X4: nat,Y: nat,R3: set_Pr1986765409at_nat,Xs: list_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y ) @ R3 )
=> ( member1926390090st_nat @ ( produc1625736153st_nat @ ( cons_nat @ X4 @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R3 ) ) ) ).
% listrel1I1
thf(fact_207_Cons__listrel1E1,axiom,
! [X4: typerep,Xs: list_typerep,Ys2: list_typerep,R3: set_Pr1097818217yperep] :
( ( member1812711762yperep @ ( produc1695567329yperep @ ( cons_typerep @ X4 @ Xs ) @ Ys2 ) @ ( listrel1_typerep @ R3 ) )
=> ( ! [Y3: typerep] :
( ( Ys2
= ( cons_typerep @ Y3 @ Xs ) )
=> ~ ( member100911538yperep @ ( produc2071514689yperep @ X4 @ Y3 ) @ R3 ) )
=> ~ ! [Zs: list_typerep] :
( ( Ys2
= ( cons_typerep @ X4 @ Zs ) )
=> ~ ( member1812711762yperep @ ( produc1695567329yperep @ Xs @ Zs ) @ ( listrel1_typerep @ R3 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_208_Cons__listrel1E1,axiom,
! [X4: nat,Xs: list_nat,Ys2: list_nat,R3: set_Pr1986765409at_nat] :
( ( member1926390090st_nat @ ( produc1625736153st_nat @ ( cons_nat @ X4 @ Xs ) @ Ys2 ) @ ( listrel1_nat @ R3 ) )
=> ( ! [Y3: nat] :
( ( Ys2
= ( cons_nat @ Y3 @ Xs ) )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R3 ) )
=> ~ ! [Zs: list_nat] :
( ( Ys2
= ( cons_nat @ X4 @ Zs ) )
=> ~ ( member1926390090st_nat @ ( produc1625736153st_nat @ Xs @ Zs ) @ ( listrel1_nat @ R3 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_209_Cons__listrel1E2,axiom,
! [Xs: list_typerep,Y: typerep,Ys2: list_typerep,R3: set_Pr1097818217yperep] :
( ( member1812711762yperep @ ( produc1695567329yperep @ Xs @ ( cons_typerep @ Y @ Ys2 ) ) @ ( listrel1_typerep @ R3 ) )
=> ( ! [X3: typerep] :
( ( Xs
= ( cons_typerep @ X3 @ Ys2 ) )
=> ~ ( member100911538yperep @ ( produc2071514689yperep @ X3 @ Y ) @ R3 ) )
=> ~ ! [Zs: list_typerep] :
( ( Xs
= ( cons_typerep @ Y @ Zs ) )
=> ~ ( member1812711762yperep @ ( produc1695567329yperep @ Zs @ Ys2 ) @ ( listrel1_typerep @ R3 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_210_Cons__listrel1E2,axiom,
! [Xs: list_nat,Y: nat,Ys2: list_nat,R3: set_Pr1986765409at_nat] :
( ( member1926390090st_nat @ ( produc1625736153st_nat @ Xs @ ( cons_nat @ Y @ Ys2 ) ) @ ( listrel1_nat @ R3 ) )
=> ( ! [X3: nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys2 ) )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R3 ) )
=> ~ ! [Zs: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Zs ) )
=> ~ ( member1926390090st_nat @ ( produc1625736153st_nat @ Zs @ Ys2 ) @ ( listrel1_nat @ R3 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_211_lexord__append__left__rightI,axiom,
! [A: typerep,B: typerep,R3: set_Pr1097818217yperep,U: list_typerep,X4: list_typerep,Y: list_typerep] :
( ( member100911538yperep @ ( produc2071514689yperep @ A @ B ) @ R3 )
=> ( member1812711762yperep @ ( produc1695567329yperep @ ( append_typerep @ U @ ( cons_typerep @ A @ X4 ) ) @ ( append_typerep @ U @ ( cons_typerep @ B @ Y ) ) ) @ ( lexord_typerep @ R3 ) ) ) ).
% lexord_append_left_rightI
thf(fact_212_lexord__append__left__rightI,axiom,
! [A: nat,B: nat,R3: set_Pr1986765409at_nat,U: list_nat,X4: list_nat,Y: list_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
=> ( member1926390090st_nat @ ( produc1625736153st_nat @ ( append_nat @ U @ ( cons_nat @ A @ X4 ) ) @ ( append_nat @ U @ ( cons_nat @ B @ Y ) ) ) @ ( lexord_nat @ R3 ) ) ) ).
% lexord_append_left_rightI
thf(fact_213_listrel1I,axiom,
! [X4: typerep,Y: typerep,R3: set_Pr1097818217yperep,Xs: list_typerep,Us: list_typerep,Vs: list_typerep,Ys2: list_typerep] :
( ( member100911538yperep @ ( produc2071514689yperep @ X4 @ Y ) @ R3 )
=> ( ( Xs
= ( append_typerep @ Us @ ( cons_typerep @ X4 @ Vs ) ) )
=> ( ( Ys2
= ( append_typerep @ Us @ ( cons_typerep @ Y @ Vs ) ) )
=> ( member1812711762yperep @ ( produc1695567329yperep @ Xs @ Ys2 ) @ ( listrel1_typerep @ R3 ) ) ) ) ) ).
% listrel1I
thf(fact_214_listrel1I,axiom,
! [X4: nat,Y: nat,R3: set_Pr1986765409at_nat,Xs: list_nat,Us: list_nat,Vs: list_nat,Ys2: list_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y ) @ R3 )
=> ( ( Xs
= ( append_nat @ Us @ ( cons_nat @ X4 @ Vs ) ) )
=> ( ( Ys2
= ( append_nat @ Us @ ( cons_nat @ Y @ Vs ) ) )
=> ( member1926390090st_nat @ ( produc1625736153st_nat @ Xs @ Ys2 ) @ ( listrel1_nat @ R3 ) ) ) ) ) ).
% listrel1I
thf(fact_215_append__eq__append__conv,axiom,
! [Xs: list_typerep,Ys2: list_typerep,Us: list_typerep,Vs: list_typerep] :
( ( ( ( size_s416147576yperep @ Xs )
= ( size_s416147576yperep @ Ys2 ) )
| ( ( size_s416147576yperep @ Us )
= ( size_s416147576yperep @ Vs ) ) )
=> ( ( ( append_typerep @ Xs @ Us )
= ( append_typerep @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_216_length__append,axiom,
! [Xs: list_typerep,Ys2: list_typerep] :
( ( size_s416147576yperep @ ( append_typerep @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_s416147576yperep @ Xs ) @ ( size_s416147576yperep @ Ys2 ) ) ) ).
% length_append
thf(fact_217_size__list__append,axiom,
! [F2: typerep > nat,Xs: list_typerep,Ys2: list_typerep] :
( ( size_list_typerep @ F2 @ ( append_typerep @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_list_typerep @ F2 @ Xs ) @ ( size_list_typerep @ F2 @ Ys2 ) ) ) ).
% size_list_append
thf(fact_218_lexord__sufE,axiom,
! [Xs: list_typerep,Zs2: list_typerep,Ys2: list_typerep,Qs: list_typerep,R3: set_Pr1097818217yperep] :
( ( member1812711762yperep @ ( produc1695567329yperep @ ( append_typerep @ Xs @ Zs2 ) @ ( append_typerep @ Ys2 @ Qs ) ) @ ( lexord_typerep @ R3 ) )
=> ( ( Xs != Ys2 )
=> ( ( ( size_s416147576yperep @ Xs )
= ( size_s416147576yperep @ Ys2 ) )
=> ( ( ( size_s416147576yperep @ Zs2 )
= ( size_s416147576yperep @ Qs ) )
=> ( member1812711762yperep @ ( produc1695567329yperep @ Xs @ Ys2 ) @ ( lexord_typerep @ R3 ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_219_lex__append__leftD,axiom,
! [R3: set_Pr1097818217yperep,Xs: list_typerep,Ys2: list_typerep,Zs2: list_typerep] :
( ! [X3: typerep] :
~ ( member100911538yperep @ ( produc2071514689yperep @ X3 @ X3 ) @ R3 )
=> ( ( member1812711762yperep @ ( produc1695567329yperep @ ( append_typerep @ Xs @ Ys2 ) @ ( append_typerep @ Xs @ Zs2 ) ) @ ( lex_typerep @ R3 ) )
=> ( member1812711762yperep @ ( produc1695567329yperep @ Ys2 @ Zs2 ) @ ( lex_typerep @ R3 ) ) ) ) ).
% lex_append_leftD
thf(fact_220_lex__append__leftD,axiom,
! [R3: set_Pr1986765409at_nat,Xs: list_nat,Ys2: list_nat,Zs2: list_nat] :
( ! [X3: nat] :
~ ( member701585322at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R3 )
=> ( ( member1926390090st_nat @ ( produc1625736153st_nat @ ( append_nat @ Xs @ Ys2 ) @ ( append_nat @ Xs @ Zs2 ) ) @ ( lex_nat @ R3 ) )
=> ( member1926390090st_nat @ ( produc1625736153st_nat @ Ys2 @ Zs2 ) @ ( lex_nat @ R3 ) ) ) ) ).
% lex_append_leftD
% Conjectures (1)
thf(conj_0,conjecture,
? [S6: list_char > int,S8: list_char > int] :
( ( relati23543761ar_int @ p @ b @ c @ ( suc @ ka ) @ S6 @ S8 )
& ( bval @ b @ S6 )
& ( big_big_step @ ( produc1193801173ar_int @ c @ S6 ) @ ta )
& ( big_big_step @ ( produc1193801173ar_int @ c2 @ S8 ) @ t_a ) ) ).
%------------------------------------------------------------------------------