TPTP Problem File: SWW519_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW519_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 272
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : hoare_272 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 188 ( 51 unt; 51 typ; 0 def)
% Number of atoms : 269 ( 106 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 172 ( 40 ~; 22 |; 13 &)
% ( 24 <=>; 73 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 37 ( 20 >; 17 *; 0 +; 0 <<)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-4 aty)
% Number of functors : 36 ( 36 usr; 14 con; 0-5 aty)
% Number of variables : 470 ( 429 !; 5 ?; 470 :)
% ( 36 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:22:31
%------------------------------------------------------------------------------
%----Should-be-implicit typings (9)
tff(ty_t_a,type,
a: $tType ).
tff(ty_tc_Com_Ocom,type,
com: $tType ).
tff(ty_tc_Com_Opname,type,
pname: $tType ).
tff(ty_tc_Com_Ostate,type,
state: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Hoare__Mirabelle__vtrypsmcwp_Otriple,type,
hoare_28830079triple: $tType > $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_Option_Ooption,type,
option: $tType > $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (42)
tff(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBI,type,
combi:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Com_Obody,type,
body1: fun(pname,option(com)) ).
tff(sy_c_Com_Ocom_OBODY,type,
body: fun(pname,com) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Ohoare__derivs,type,
hoare_992312373derivs:
!>[A: $tType] : ( ( fun(hoare_28830079triple(A),bool) * fun(hoare_28830079triple(A),bool) ) > $o ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple_Otriple,type,
hoare_1841697145triple:
!>[A: $tType] : fun(fun(A,fun(state,bool)),fun(com,fun(fun(A,fun(state,bool)),hoare_28830079triple(A)))) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple_Otriple__case,type,
hoare_376461865e_case:
!>[A: $tType,T1: $tType] : ( ( fun(fun(A,fun(state,bool)),fun(com,fun(fun(A,fun(state,bool)),T1))) * hoare_28830079triple(A) ) > T1 ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple_Otriple__rec,type,
hoare_678420151le_rec:
!>[A: $tType,T1: $tType] : ( ( fun(fun(A,fun(state,bool)),fun(com,fun(fun(A,fun(state,bool)),T1))) * hoare_28830079triple(A) ) > T1 ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple__valid,type,
hoare_1633586161_valid:
!>[A: $tType] : ( ( nat * hoare_28830079triple(A) ) > $o ) ).
tff(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri532925092at_aux:
!>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).
tff(sy_c_Natural_Oevaln,type,
evaln: ( com * state * nat * state ) > $o ).
tff(sy_c_Option_Othe,type,
the:
!>[A: $tType] : fun(option(A),A) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,bool) ) > fun(B,bool) ) ).
tff(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * fun(A,bool) ) > fun(A,bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fdisj,type,
fdisj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_fimplies,type,
fimplies: fun(bool,fun(bool,bool)) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : fun(A,fun(fun(A,bool),bool)) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_G,type,
g: fun(hoare_28830079triple(a),bool) ).
tff(sy_v_P,type,
p: fun(pname,fun(a,fun(state,bool))) ).
tff(sy_v_Procs,type,
procs: fun(pname,bool) ).
tff(sy_v_Q,type,
q: fun(pname,fun(a,fun(state,bool))) ).
tff(sy_v_na,type,
na: nat ).
tff(sy_v_x,type,
x: pname ).
%----Relevant facts (99)
tff(fact_0_triple_Oinject,axiom,
! [B: $tType,Fun22: fun(B,fun(state,bool)),Com2: com,Fun12: fun(B,fun(state,bool)),Fun21: fun(B,fun(state,bool)),Com1: com,Fun11: fun(B,fun(state,bool))] :
( ( aa(fun(B,fun(state,bool)),hoare_28830079triple(B),aa(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B)),aa(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B))),hoare_1841697145triple(B),Fun11),Com1),Fun21) = aa(fun(B,fun(state,bool)),hoare_28830079triple(B),aa(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B)),aa(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B))),hoare_1841697145triple(B),Fun12),Com2),Fun22) )
<=> ( ( Fun11 = Fun12 )
& ( Com1 = Com2 )
& ( Fun21 = Fun22 ) ) ) ).
tff(fact_1_Body__triple__valid__Suc,axiom,
! [B: $tType,Qa: fun(B,fun(state,bool)),Pn1: pname,Pa: fun(B,fun(state,bool)),N3: nat] :
( hoare_1633586161_valid(B,N3,aa(fun(B,fun(state,bool)),hoare_28830079triple(B),aa(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B)),aa(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B))),hoare_1841697145triple(B),Pa),aa(option(com),com,the(com),aa(pname,option(com),body1,Pn1))),Qa))
<=> hoare_1633586161_valid(B,suc(N3),aa(fun(B,fun(state,bool)),hoare_28830079triple(B),aa(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B)),aa(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B))),hoare_1841697145triple(B),Pa),aa(pname,com,body,Pn1)),Qa)) ) ).
tff(fact_2_triple__valid__Suc,axiom,
! [A: $tType,T: hoare_28830079triple(A),N1: nat] :
( hoare_1633586161_valid(A,suc(N1),T)
=> hoare_1633586161_valid(A,N1,T) ) ).
tff(fact_3_com_Osimps_I6_J,axiom,
! [Pname1: pname,Pname: pname] :
( ( aa(pname,com,body,Pname) = aa(pname,com,body,Pname1) )
<=> ( Pname = Pname1 ) ) ).
tff(fact_4_hoare__derivs_OBody,axiom,
! [B: $tType,Procsa: fun(pname,bool),Qa: fun(pname,fun(B,fun(state,bool))),Pa: fun(pname,fun(B,fun(state,bool))),Ga: fun(hoare_28830079triple(B),bool)] :
( hoare_992312373derivs(B,sup_sup(fun(hoare_28830079triple(B),bool),Ga,image(pname,hoare_28830079triple(B),combs(pname,fun(B,fun(state,bool)),hoare_28830079triple(B),combs(pname,com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B)),combb(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B))),pname,hoare_1841697145triple(B),Pa),body),Qa),Procsa)),image(pname,hoare_28830079triple(B),combs(pname,fun(B,fun(state,bool)),hoare_28830079triple(B),combs(pname,com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B)),combb(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B))),pname,hoare_1841697145triple(B),Pa),combb(option(com),com,pname,the(com),body1)),Qa),Procsa))
=> hoare_992312373derivs(B,Ga,image(pname,hoare_28830079triple(B),combs(pname,fun(B,fun(state,bool)),hoare_28830079triple(B),combs(pname,com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B)),combb(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B))),pname,hoare_1841697145triple(B),Pa),body),Qa),Procsa)) ) ).
tff(fact_5_Un__iff,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool),C1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),sup_sup(fun(B,bool),A1,B1)))
<=> ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),A1))
| pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),B1)) ) ) ).
tff(fact_6_UnE,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool),C1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),sup_sup(fun(B,bool),A1,B1)))
=> ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),B1)) ) ) ).
tff(fact_7_sup1E,axiom,
! [B: $tType,X3: B,B1: fun(B,bool),A1: fun(B,bool)] :
( pp(aa(B,bool,sup_sup(fun(B,bool),A1,B1),X3))
=> ( ~ pp(aa(B,bool,A1,X3))
=> pp(aa(B,bool,B1,X3)) ) ) ).
tff(fact_8_sup1CI,axiom,
! [B: $tType,A1: fun(B,bool),X3: B,B1: fun(B,bool)] :
( ( ~ pp(aa(B,bool,B1,X3))
=> pp(aa(B,bool,A1,X3)) )
=> pp(aa(B,bool,sup_sup(fun(B,bool),A1,B1),X3)) ) ).
tff(fact_9_UnCI,axiom,
! [B: $tType,A1: fun(B,bool),B1: fun(B,bool),C1: B] :
( ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),B1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),A1)) )
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),sup_sup(fun(B,bool),A1,B1))) ) ).
tff(fact_10_image__eqI,axiom,
! [B: $tType,C: $tType,A1: fun(C,bool),X3: C,F: fun(C,B),B2: B] :
( ( B2 = aa(C,B,F,X3) )
=> ( pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),X3),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B2),image(C,B,F,A1))) ) ) ).
tff(fact_11_nat_Oinject,axiom,
! [Nat1: nat,Nat: nat] :
( ( suc(Nat) = suc(Nat1) )
<=> ( Nat = Nat1 ) ) ).
tff(fact_12_sup_Oleft__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B3: A,A3: A] : sup_sup(A,A3,sup_sup(A,A3,B3)) = sup_sup(A,A3,B3) ) ).
tff(fact_13_sup__left__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X2: A] : sup_sup(A,X2,sup_sup(A,X2,Y)) = sup_sup(A,X2,Y) ) ).
tff(fact_14_cut,axiom,
! [B: $tType,Ga: fun(hoare_28830079triple(B),bool),Ts: fun(hoare_28830079triple(B),bool),G1: fun(hoare_28830079triple(B),bool)] :
( hoare_992312373derivs(B,G1,Ts)
=> ( hoare_992312373derivs(B,Ga,G1)
=> hoare_992312373derivs(B,Ga,Ts) ) ) ).
tff(fact_15_sup__assoc,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Z1: A,Y: A,X2: A] : sup_sup(A,sup_sup(A,X2,Y),Z1) = sup_sup(A,X2,sup_sup(A,Y,Z1)) ) ).
tff(fact_16_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Z1: A,Y: A,X2: A] : sup_sup(A,sup_sup(A,X2,Y),Z1) = sup_sup(A,X2,sup_sup(A,Y,Z1)) ) ).
tff(fact_17_sup_Oassoc,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C3: A,B3: A,A3: A] : sup_sup(A,sup_sup(A,A3,B3),C3) = sup_sup(A,A3,sup_sup(A,B3,C3)) ) ).
tff(fact_18_sup__left__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Z1: A,Y: A,X2: A] : sup_sup(A,X2,sup_sup(A,Y,Z1)) = sup_sup(A,Y,sup_sup(A,X2,Z1)) ) ).
tff(fact_19_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Z1: A,Y: A,X2: A] : sup_sup(A,X2,sup_sup(A,Y,Z1)) = sup_sup(A,Y,sup_sup(A,X2,Z1)) ) ).
tff(fact_20_sup_Oleft__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C3: A,A3: A,B3: A] : sup_sup(A,B3,sup_sup(A,A3,C3)) = sup_sup(A,A3,sup_sup(A,B3,C3)) ) ).
tff(fact_21_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Y: A,X2: A] : sup_sup(A,X2,sup_sup(A,X2,Y)) = sup_sup(A,X2,Y) ) ).
tff(fact_22_sup__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X2: A] : sup_sup(A,X2,Y) = sup_sup(A,Y,X2) ) ).
tff(fact_23_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Y: A,X2: A] : sup_sup(A,X2,Y) = sup_sup(A,Y,X2) ) ).
tff(fact_24_sup_Ocommute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B3: A,A3: A] : sup_sup(A,A3,B3) = sup_sup(A,B3,A3) ) ).
tff(fact_25_sup__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X2: A] : sup_sup(A,X2,X2) = X2 ) ).
tff(fact_26_sup_Oidem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A] : sup_sup(A,A3,A3) = A3 ) ).
tff(fact_27_Suc__inject,axiom,
! [Y: nat,X2: nat] :
( ( suc(X2) = suc(Y) )
=> ( X2 = Y ) ) ).
tff(fact_28_Suc__n__not__n,axiom,
! [N1: nat] : suc(N1) != N1 ).
tff(fact_29_n__not__Suc__n,axiom,
! [N1: nat] : N1 != suc(N1) ).
tff(fact_30_rev__image__eqI,axiom,
! [C: $tType,B: $tType,F: fun(B,C),B2: C,A1: fun(B,bool),X3: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),A1))
=> ( ( B2 = aa(B,C,F,X3) )
=> pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),B2),image(B,C,F,A1))) ) ) ).
tff(fact_31_imageI,axiom,
! [C: $tType,B: $tType,F: fun(B,C),A1: fun(B,bool),X3: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),A1))
=> pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),aa(B,C,F,X3)),image(B,C,F,A1))) ) ).
tff(fact_32_image__iff,axiom,
! [B: $tType,C: $tType,A1: fun(C,bool),F: fun(C,B),Z: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Z),image(C,B,F,A1)))
<=> ? [X4: C] :
( pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),X4),A1))
& ( Z = aa(C,B,F,X4) ) ) ) ).
tff(fact_33_UnI2,axiom,
! [B: $tType,A1: fun(B,bool),B1: fun(B,bool),C1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),B1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),sup_sup(fun(B,bool),A1,B1))) ) ).
tff(fact_34_UnI1,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool),C1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),sup_sup(fun(B,bool),A1,B1))) ) ).
tff(fact_35_sup1I2,axiom,
! [B: $tType,A1: fun(B,bool),X3: B,B1: fun(B,bool)] :
( pp(aa(B,bool,B1,X3))
=> pp(aa(B,bool,sup_sup(fun(B,bool),A1,B1),X3)) ) ).
tff(fact_36_sup1I1,axiom,
! [B: $tType,B1: fun(B,bool),X3: B,A1: fun(B,bool)] :
( pp(aa(B,bool,A1,X3))
=> pp(aa(B,bool,sup_sup(fun(B,bool),A1,B1),X3)) ) ).
tff(fact_37_ball__Un,axiom,
! [B: $tType,Pa: fun(B,bool),B1: fun(B,bool),A1: fun(B,bool)] :
( ! [X4: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),sup_sup(fun(B,bool),A1,B1)))
=> pp(aa(B,bool,Pa,X4)) )
<=> ( ! [X4: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),A1))
=> pp(aa(B,bool,Pa,X4)) )
& ! [X4: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),B1))
=> pp(aa(B,bool,Pa,X4)) ) ) ) ).
tff(fact_38_bex__Un,axiom,
! [B: $tType,Pa: fun(B,bool),B1: fun(B,bool),A1: fun(B,bool)] :
( ? [X4: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),sup_sup(fun(B,bool),A1,B1)))
& pp(aa(B,bool,Pa,X4)) )
<=> ( ? [X4: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),A1))
& pp(aa(B,bool,Pa,X4)) )
| ? [X4: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),B1))
& pp(aa(B,bool,Pa,X4)) ) ) ) ).
tff(fact_39_Un__assoc,axiom,
! [B: $tType,C2: fun(B,bool),B1: fun(B,bool),A1: fun(B,bool)] : sup_sup(fun(B,bool),sup_sup(fun(B,bool),A1,B1),C2) = sup_sup(fun(B,bool),A1,sup_sup(fun(B,bool),B1,C2)) ).
tff(fact_40_Un__left__commute,axiom,
! [B: $tType,C2: fun(B,bool),B1: fun(B,bool),A1: fun(B,bool)] : sup_sup(fun(B,bool),A1,sup_sup(fun(B,bool),B1,C2)) = sup_sup(fun(B,bool),B1,sup_sup(fun(B,bool),A1,C2)) ).
tff(fact_41_Un__left__absorb,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] : sup_sup(fun(B,bool),A1,sup_sup(fun(B,bool),A1,B1)) = sup_sup(fun(B,bool),A1,B1) ).
tff(fact_42_Un__commute,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] : sup_sup(fun(B,bool),A1,B1) = sup_sup(fun(B,bool),B1,A1) ).
tff(fact_43_Un__def,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] : sup_sup(fun(B,bool),A1,B1) = collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fdisj,combc(B,fun(B,bool),bool,member(B),A1)),combc(B,fun(B,bool),bool,member(B),B1))) ).
tff(fact_44_Un__absorb,axiom,
! [B: $tType,A1: fun(B,bool)] : sup_sup(fun(B,bool),A1,A1) = A1 ).
tff(fact_45_image__ident,axiom,
! [B: $tType,Y2: fun(B,bool)] : image(B,B,combi(B),Y2) = Y2 ).
tff(fact_46_image__image,axiom,
! [C: $tType,B: $tType,D1: $tType,A1: fun(D1,bool),G: fun(D1,C),F: fun(C,B)] : image(C,B,F,image(D1,C,G,A1)) = image(D1,B,combb(C,B,D1,F,G),A1) ).
tff(fact_47_sup__Un__eq,axiom,
! [B: $tType,S: fun(B,bool),R1: fun(B,bool),X: B] :
( pp(aa(B,bool,sup_sup(fun(B,bool),combc(B,fun(B,bool),bool,member(B),R1),combc(B,fun(B,bool),bool,member(B),S)),X))
<=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X),sup_sup(fun(B,bool),R1,S))) ) ).
tff(fact_48_Collect__disj__eq,axiom,
! [B: $tType,Qa: fun(B,bool),Pa: fun(B,bool)] : collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fdisj,Pa),Qa)) = sup_sup(fun(B,bool),collect(B,Pa),collect(B,Qa)) ).
tff(fact_49_sup__apply,axiom,
! [B: $tType,C: $tType] :
( lattice(B)
=> ! [X3: C,G: fun(C,B),F: fun(C,B)] : aa(C,B,sup_sup(fun(C,B),F,G),X3) = sup_sup(B,aa(C,B,F,X3),aa(C,B,G,X3)) ) ).
tff(fact_50_sup__fun__def,axiom,
! [C: $tType,B: $tType] :
( lattice(C)
=> ! [G: fun(B,C),F: fun(B,C),X: B] : aa(B,C,sup_sup(fun(B,C),F,G),X) = sup_sup(C,aa(B,C,F,X),aa(B,C,G,X)) ) ).
tff(fact_51_image__Un,axiom,
! [B: $tType,C: $tType,B1: fun(C,bool),A1: fun(C,bool),F: fun(C,B)] : image(C,B,F,sup_sup(fun(C,bool),A1,B1)) = sup_sup(fun(B,bool),image(C,B,F,A1),image(C,B,F,B1)) ).
tff(fact_52_imageE,axiom,
! [B: $tType,C: $tType,A1: fun(C,bool),F: fun(C,B),B2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B2),image(C,B,F,A1)))
=> ~ ! [X1: C] :
( ( B2 = aa(C,B,F,X1) )
=> ~ pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),X1),A1)) ) ) ).
tff(fact_53_triple_Osimps_I2_J,axiom,
! [B: $tType,C: $tType,Fun21: fun(C,fun(state,bool)),Com1: com,Fun11: fun(C,fun(state,bool)),F1: fun(fun(C,fun(state,bool)),fun(com,fun(fun(C,fun(state,bool)),B)))] : hoare_376461865e_case(C,B,F1,aa(fun(C,fun(state,bool)),hoare_28830079triple(C),aa(com,fun(fun(C,fun(state,bool)),hoare_28830079triple(C)),aa(fun(C,fun(state,bool)),fun(com,fun(fun(C,fun(state,bool)),hoare_28830079triple(C))),hoare_1841697145triple(C),Fun11),Com1),Fun21)) = aa(fun(C,fun(state,bool)),B,aa(com,fun(fun(C,fun(state,bool)),B),aa(fun(C,fun(state,bool)),fun(com,fun(fun(C,fun(state,bool)),B)),F1,Fun11),Com1),Fun21) ).
tff(fact_54_triple_Orecs,axiom,
! [B: $tType,C: $tType,Fun21: fun(C,fun(state,bool)),Com1: com,Fun11: fun(C,fun(state,bool)),F1: fun(fun(C,fun(state,bool)),fun(com,fun(fun(C,fun(state,bool)),B)))] : hoare_678420151le_rec(C,B,F1,aa(fun(C,fun(state,bool)),hoare_28830079triple(C),aa(com,fun(fun(C,fun(state,bool)),hoare_28830079triple(C)),aa(fun(C,fun(state,bool)),fun(com,fun(fun(C,fun(state,bool)),hoare_28830079triple(C))),hoare_1841697145triple(C),Fun11),Com1),Fun21)) = aa(fun(C,fun(state,bool)),B,aa(com,fun(fun(C,fun(state,bool)),B),aa(fun(C,fun(state,bool)),fun(com,fun(fun(C,fun(state,bool)),B)),F1,Fun11),Com1),Fun21) ).
tff(fact_55_of__nat__aux_Osimps_I2_J,axiom,
! [B: $tType] :
( semiring_1(B)
=> ! [I: B,N3: nat,Inc: fun(B,B)] : semiri532925092at_aux(B,Inc,suc(N3),I) = semiri532925092at_aux(B,Inc,N3,aa(B,B,Inc,I)) ) ).
tff(fact_56_triple_Oexhaust,axiom,
! [B: $tType,Y1: hoare_28830079triple(B)] :
~ ! [Fun1: fun(B,fun(state,bool)),Com: com,Fun2: fun(B,fun(state,bool))] : Y1 != aa(fun(B,fun(state,bool)),hoare_28830079triple(B),aa(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B)),aa(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B))),hoare_1841697145triple(B),Fun1),Com),Fun2) ).
tff(fact_57_Body1,axiom,
! [B: $tType,Pn1: pname,Procsa: fun(pname,bool),Qa: fun(pname,fun(B,fun(state,bool))),Pa: fun(pname,fun(B,fun(state,bool))),Ga: fun(hoare_28830079triple(B),bool)] :
( hoare_992312373derivs(B,sup_sup(fun(hoare_28830079triple(B),bool),Ga,image(pname,hoare_28830079triple(B),combs(pname,fun(B,fun(state,bool)),hoare_28830079triple(B),combs(pname,com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B)),combb(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B))),pname,hoare_1841697145triple(B),Pa),body),Qa),Procsa)),image(pname,hoare_28830079triple(B),combs(pname,fun(B,fun(state,bool)),hoare_28830079triple(B),combs(pname,com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B)),combb(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B))),pname,hoare_1841697145triple(B),Pa),combb(option(com),com,pname,the(com),body1)),Qa),Procsa))
=> ( pp(aa(fun(pname,bool),bool,aa(pname,fun(fun(pname,bool),bool),member(pname),Pn1),Procsa))
=> hoare_992312373derivs(B,Ga,insert(hoare_28830079triple(B),aa(fun(B,fun(state,bool)),hoare_28830079triple(B),aa(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B)),aa(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),hoare_28830079triple(B))),hoare_1841697145triple(B),aa(pname,fun(B,fun(state,bool)),Pa,Pn1)),aa(pname,com,body,Pn1)),aa(pname,fun(B,fun(state,bool)),Qa,Pn1)),bot_bot(fun(hoare_28830079triple(B),bool)))) ) ) ).
tff(fact_58_image__cong,axiom,
! [C: $tType,B: $tType,G: fun(B,C),F: fun(B,C),N2: fun(B,bool),M: fun(B,bool)] :
( ( M = N2 )
=> ( ! [X1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),N2))
=> ( aa(B,C,F,X1) = aa(B,C,G,X1) ) )
=> ( image(B,C,F,M) = image(B,C,G,N2) ) ) ) ).
tff(fact_59_evaln_OBody,axiom,
! [S1: state,N1: nat,S0: state,Pn: pname] :
( evaln(aa(option(com),com,the(com),aa(pname,option(com),body1,Pn)),S0,N1,S1)
=> evaln(aa(pname,com,body,Pn),S0,suc(N1),S1) ) ).
tff(fact_60_all__not__in__conv,axiom,
! [B: $tType,A1: fun(B,bool)] :
( ! [X4: B] : ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),A1))
<=> ( A1 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_61_empty__Collect__eq,axiom,
! [B: $tType,Pa: fun(B,bool)] :
( ( bot_bot(fun(B,bool)) = collect(B,Pa) )
<=> ! [X4: B] : ~ pp(aa(B,bool,Pa,X4)) ) ).
tff(fact_62_empty__iff,axiom,
! [B: $tType,C1: B] : ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),bot_bot(fun(B,bool)))) ).
tff(fact_63_Collect__empty__eq,axiom,
! [B: $tType,Pa: fun(B,bool)] :
( ( collect(B,Pa) = bot_bot(fun(B,bool)) )
<=> ! [X4: B] : ~ pp(aa(B,bool,Pa,X4)) ) ).
tff(fact_64_emptyE,axiom,
! [B: $tType,A2: B] : ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),bot_bot(fun(B,bool)))) ).
tff(fact_65_insert__absorb2,axiom,
! [B: $tType,A1: fun(B,bool),X3: B] : insert(B,X3,insert(B,X3,A1)) = insert(B,X3,A1) ).
tff(fact_66_insert__iff,axiom,
! [B: $tType,A1: fun(B,bool),B2: B,A2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),insert(B,B2,A1)))
<=> ( ( A2 = B2 )
| pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),A1)) ) ) ).
tff(fact_67_insertE,axiom,
! [B: $tType,A1: fun(B,bool),B2: B,A2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),insert(B,B2,A1)))
=> ( ( A2 != B2 )
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),A1)) ) ) ).
tff(fact_68_insertCI,axiom,
! [B: $tType,B2: B,B1: fun(B,bool),A2: B] :
( ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),B1))
=> ( A2 = B2 ) )
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),insert(B,B2,B1))) ) ).
tff(fact_69_sup__eq__bot__iff,axiom,
! [B: $tType] :
( bounded_lattice_bot(B)
=> ! [Y1: B,X3: B] :
( ( sup_sup(B,X3,Y1) = bot_bot(B) )
<=> ( ( X3 = bot_bot(B) )
& ( Y1 = bot_bot(B) ) ) ) ) ).
tff(fact_70_empty__is__image,axiom,
! [B: $tType,C: $tType,A1: fun(C,bool),F: fun(C,B)] :
( ( bot_bot(fun(B,bool)) = image(C,B,F,A1) )
<=> ( A1 = bot_bot(fun(C,bool)) ) ) ).
tff(fact_71_image__empty,axiom,
! [C: $tType,B: $tType,F: fun(C,B)] : image(C,B,F,bot_bot(fun(C,bool))) = bot_bot(fun(B,bool)) ).
tff(fact_72_image__is__empty,axiom,
! [B: $tType,C: $tType,A1: fun(C,bool),F: fun(C,B)] :
( ( image(C,B,F,A1) = bot_bot(fun(B,bool)) )
<=> ( A1 = bot_bot(fun(C,bool)) ) ) ).
tff(fact_73_Un__empty,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( ( sup_sup(fun(B,bool),A1,B1) = bot_bot(fun(B,bool)) )
<=> ( ( A1 = bot_bot(fun(B,bool)) )
& ( B1 = bot_bot(fun(B,bool)) ) ) ) ).
tff(fact_74_ext,axiom,
! [C: $tType,B: $tType,G: fun(B,C),F: fun(B,C)] :
( ! [X1: B] : aa(B,C,F,X1) = aa(B,C,G,X1)
=> ( F = G ) ) ).
tff(fact_75_mem__def,axiom,
! [B: $tType,A1: fun(B,bool),X3: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),A1))
<=> pp(aa(B,bool,A1,X3)) ) ).
tff(fact_76_Collect__def,axiom,
! [B: $tType,Pa: fun(B,bool)] : collect(B,Pa) = Pa ).
tff(fact_77_image__insert,axiom,
! [B: $tType,C: $tType,B1: fun(C,bool),A2: C,F: fun(C,B)] : image(C,B,F,insert(C,A2,B1)) = insert(B,aa(C,B,F,A2),image(C,B,F,B1)) ).
tff(fact_78_Un__insert__right,axiom,
! [B: $tType,B1: fun(B,bool),A2: B,A1: fun(B,bool)] : sup_sup(fun(B,bool),A1,insert(B,A2,B1)) = insert(B,A2,sup_sup(fun(B,bool),A1,B1)) ).
tff(fact_79_Un__insert__left,axiom,
! [B: $tType,C2: fun(B,bool),B1: fun(B,bool),A2: B] : sup_sup(fun(B,bool),insert(B,A2,B1),C2) = insert(B,A2,sup_sup(fun(B,bool),B1,C2)) ).
tff(fact_80_singleton__conv2,axiom,
! [B: $tType,A2: B] : collect(B,aa(B,fun(B,bool),fequal(B),A2)) = insert(B,A2,bot_bot(fun(B,bool))) ).
tff(fact_81_singleton__conv,axiom,
! [B: $tType,A2: B] : collect(B,combc(B,B,bool,fequal(B),A2)) = insert(B,A2,bot_bot(fun(B,bool))) ).
tff(fact_82_empty__not__insert,axiom,
! [B: $tType,A1: fun(B,bool),A2: B] : bot_bot(fun(B,bool)) != insert(B,A2,A1) ).
tff(fact_83_insert__not__empty,axiom,
! [B: $tType,A1: fun(B,bool),A2: B] : insert(B,A2,A1) != bot_bot(fun(B,bool)) ).
tff(fact_84_bot__empty__eq,axiom,
! [B: $tType,X: B] :
( pp(aa(B,bool,bot_bot(fun(B,bool)),X))
<=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X),bot_bot(fun(B,bool)))) ) ).
tff(fact_85_empty__def,axiom,
! [B: $tType] : bot_bot(fun(B,bool)) = collect(B,combk(bool,B,fFalse)) ).
tff(fact_86_insertI1,axiom,
! [B: $tType,B1: fun(B,bool),A2: B] : pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),insert(B,A2,B1))) ).
tff(fact_87_ex__in__conv,axiom,
! [B: $tType,A1: fun(B,bool)] :
( ? [X4: B] : pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),A1))
<=> ( A1 != bot_bot(fun(B,bool)) ) ) ).
tff(fact_88_Collect__conv__if2,axiom,
! [B: $tType,A2: B,Pa: fun(B,bool)] :
( ( pp(aa(B,bool,Pa,A2))
=> ( collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fconj,aa(B,fun(B,bool),fequal(B),A2)),Pa)) = insert(B,A2,bot_bot(fun(B,bool))) ) )
& ( ~ pp(aa(B,bool,Pa,A2))
=> ( collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fconj,aa(B,fun(B,bool),fequal(B),A2)),Pa)) = bot_bot(fun(B,bool)) ) ) ) ).
tff(fact_89_Collect__conv__if,axiom,
! [B: $tType,A2: B,Pa: fun(B,bool)] :
( ( pp(aa(B,bool,Pa,A2))
=> ( collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fconj,combc(B,B,bool,fequal(B),A2)),Pa)) = insert(B,A2,bot_bot(fun(B,bool))) ) )
& ( ~ pp(aa(B,bool,Pa,A2))
=> ( collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fconj,combc(B,B,bool,fequal(B),A2)),Pa)) = bot_bot(fun(B,bool)) ) ) ) ).
tff(fact_90_insert__compr,axiom,
! [B: $tType,B1: fun(B,bool),A2: B] : insert(B,A2,B1) = collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fdisj,combc(B,B,bool,fequal(B),A2)),combc(B,fun(B,bool),bool,member(B),B1))) ).
tff(fact_91_insert__is__Un,axiom,
! [B: $tType,A1: fun(B,bool),A2: B] : insert(B,A2,A1) = sup_sup(fun(B,bool),insert(B,A2,bot_bot(fun(B,bool))),A1) ).
tff(fact_92_insert__Collect,axiom,
! [B: $tType,Pa: fun(B,bool),A2: B] : insert(B,A2,collect(B,Pa)) = collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fimplies,combb(bool,bool,B,fNot,combc(B,B,bool,fequal(B),A2))),Pa)) ).
tff(fact_93_singleton__iff,axiom,
! [B: $tType,A2: B,B2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B2),insert(B,A2,bot_bot(fun(B,bool)))))
<=> ( B2 = A2 ) ) ).
tff(fact_94_insert__commute,axiom,
! [B: $tType,A1: fun(B,bool),Y1: B,X3: B] : insert(B,X3,insert(B,Y1,A1)) = insert(B,Y1,insert(B,X3,A1)) ).
tff(fact_95_doubleton__eq__iff,axiom,
! [B: $tType,D: B,C1: B,B2: B,A2: B] :
( ( insert(B,A2,insert(B,B2,bot_bot(fun(B,bool)))) = insert(B,C1,insert(B,D,bot_bot(fun(B,bool)))) )
<=> ( ( ( A2 = C1 )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C1 ) ) ) ) ).
tff(fact_96_insert__code,axiom,
! [B: $tType,X3: B,A1: fun(B,bool),Y1: B] :
( pp(aa(B,bool,insert(B,Y1,A1),X3))
<=> ( ( Y1 = X3 )
| pp(aa(B,bool,A1,X3)) ) ) ).
tff(fact_97_insert__compr__raw,axiom,
! [B: $tType,X: B,Xa: fun(B,bool)] : insert(B,X,Xa) = collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fdisj,combc(B,B,bool,fequal(B),X)),combc(B,fun(B,bool),bool,member(B),Xa))) ).
tff(fact_98_insert__ident,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool),X3: B] :
( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),A1))
=> ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),B1))
=> ( ( insert(B,X3,A1) = insert(B,X3,B1) )
<=> ( A1 = B1 ) ) ) ) ).
%----Arities (11)
tff(arity_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice(bool) ).
tff(arity_fun___Lattices_Obounded__lattice,axiom,
! [T_1: $tType,T_2: $tType] :
( bounded_lattice(T_2)
=> bounded_lattice(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Obounded__lattice__bot,axiom,
! [T_1: $tType,T_2: $tType] :
( bounded_lattice(T_2)
=> bounded_lattice_bot(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Osemilattice__sup,axiom,
! [T_1: $tType,T_2: $tType] :
( lattice(T_2)
=> semilattice_sup(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Olattice,axiom,
! [T_1: $tType,T_2: $tType] :
( lattice(T_2)
=> lattice(fun(T_1,T_2)) ) ).
tff(arity_Nat_Onat___Lattices_Osemilattice__sup,axiom,
semilattice_sup(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Lattices_Olattice,axiom,
lattice(nat) ).
tff(arity_HOL_Obool___Lattices_Obounded__lattice__bot,axiom,
bounded_lattice_bot(bool) ).
tff(arity_HOL_Obool___Lattices_Osemilattice__sup,axiom,
semilattice_sup(bool) ).
tff(arity_HOL_Obool___Lattices_Olattice,axiom,
lattice(bool) ).
%----Helper facts (22)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(aa(bool,bool,fNot,P))
| ~ pp(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(aa(bool,bool,fNot,P)) ) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,combb(B,C,A,P,Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBI_1_1_U,axiom,
! [A: $tType,P: A] : aa(A,A,combi(A),P) = P ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,combk(A,B,P),Q) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
tff(help_fdisj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q))
| pp(P)
| pp(Q) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,X2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X2),Y))
| ( X2 = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,Y: A,X2: A] :
( ( X2 != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X2),Y)) ) ).
tff(help_fimplies_1_1_U,axiom,
! [Q: bool,P: bool] :
( pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).
tff(help_fimplies_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).
tff(help_fimplies_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q))
| ~ pp(P)
| pp(Q) ) ).
%----Conjectures (5)
tff(conj_0,hypothesis,
! [N: nat] :
( ! [X1: hoare_28830079triple(a)] :
( pp(aa(fun(hoare_28830079triple(a),bool),bool,aa(hoare_28830079triple(a),fun(fun(hoare_28830079triple(a),bool),bool),member(hoare_28830079triple(a)),X1),sup_sup(fun(hoare_28830079triple(a),bool),g,image(pname,hoare_28830079triple(a),combs(pname,fun(a,fun(state,bool)),hoare_28830079triple(a),combs(pname,com,fun(fun(a,fun(state,bool)),hoare_28830079triple(a)),combb(fun(a,fun(state,bool)),fun(com,fun(fun(a,fun(state,bool)),hoare_28830079triple(a))),pname,hoare_1841697145triple(a),p),body),q),procs))))
=> hoare_1633586161_valid(a,N,X1) )
=> ! [X: pname] :
( pp(aa(fun(pname,bool),bool,aa(pname,fun(fun(pname,bool),bool),member(pname),X),procs))
=> hoare_1633586161_valid(a,N,aa(fun(a,fun(state,bool)),hoare_28830079triple(a),aa(com,fun(fun(a,fun(state,bool)),hoare_28830079triple(a)),aa(fun(a,fun(state,bool)),fun(com,fun(fun(a,fun(state,bool)),hoare_28830079triple(a))),hoare_1841697145triple(a),aa(pname,fun(a,fun(state,bool)),p,X)),aa(option(com),com,the(com),aa(pname,option(com),body1,X))),aa(pname,fun(a,fun(state,bool)),q,X))) ) ) ).
tff(conj_1,hypothesis,
( ! [X1: hoare_28830079triple(a)] :
( pp(aa(fun(hoare_28830079triple(a),bool),bool,aa(hoare_28830079triple(a),fun(fun(hoare_28830079triple(a),bool),bool),member(hoare_28830079triple(a)),X1),g))
=> hoare_1633586161_valid(a,na,X1) )
=> ! [X: pname] :
( pp(aa(fun(pname,bool),bool,aa(pname,fun(fun(pname,bool),bool),member(pname),X),procs))
=> hoare_1633586161_valid(a,na,aa(fun(a,fun(state,bool)),hoare_28830079triple(a),aa(com,fun(fun(a,fun(state,bool)),hoare_28830079triple(a)),aa(fun(a,fun(state,bool)),fun(com,fun(fun(a,fun(state,bool)),hoare_28830079triple(a))),hoare_1841697145triple(a),aa(pname,fun(a,fun(state,bool)),p,X)),aa(pname,com,body,X)),aa(pname,fun(a,fun(state,bool)),q,X))) ) ) ).
tff(conj_2,hypothesis,
! [X: hoare_28830079triple(a)] :
( pp(aa(fun(hoare_28830079triple(a),bool),bool,aa(hoare_28830079triple(a),fun(fun(hoare_28830079triple(a),bool),bool),member(hoare_28830079triple(a)),X),g))
=> hoare_1633586161_valid(a,suc(na),X) ) ).
tff(conj_3,hypothesis,
pp(aa(fun(pname,bool),bool,aa(pname,fun(fun(pname,bool),bool),member(pname),x),procs)) ).
tff(conj_4,conjecture,
hoare_1633586161_valid(a,suc(na),aa(fun(a,fun(state,bool)),hoare_28830079triple(a),aa(com,fun(fun(a,fun(state,bool)),hoare_28830079triple(a)),aa(fun(a,fun(state,bool)),fun(com,fun(fun(a,fun(state,bool)),hoare_28830079triple(a))),hoare_1841697145triple(a),aa(pname,fun(a,fun(state,bool)),p,x)),aa(pname,com,body,x)),aa(pname,fun(a,fun(state,bool)),q,x))) ).
%------------------------------------------------------------------------------