TPTP Problem File: SWW506_5.p
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%------------------------------------------------------------------------------
% File : SWW506_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 137
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : hoare_137 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.50 v7.3.0, 0.75 v7.1.0, 0.67 v6.4.0
% Syntax : Number of formulae : 164 ( 44 unt; 46 typ; 0 def)
% Number of atoms : 240 ( 73 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 168 ( 46 ~; 3 |; 4 &)
% ( 26 <=>; 89 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 38 ( 22 >; 16 *; 0 +; 0 <<)
% Number of predicates : 15 ( 14 usr; 0 prp; 1-4 aty)
% Number of functors : 25 ( 25 usr; 6 con; 0-4 aty)
% Number of variables : 264 ( 231 !; 7 ?; 264 :)
% ( 26 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:14:20
%------------------------------------------------------------------------------
%----Should-be-implicit typings (10)
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_Int_Oint,type,
int: $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 (36)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Osgn__if,type,
sgn_if:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_c_Com_Obody,type,
body1: pname > option(com) ).
tff(sy_c_Com_Ocom_OBODY,type,
body: pname > com ).
tff(sy_c_Com_Ocom_Ocom__size,type,
com_size: com > nat ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple_Otriple,type,
hoare_1841697145triple:
!>[A: $tType] : ( ( fun(A,fun(state,bool)) * com * fun(A,fun(state,bool)) ) > hoare_28830079triple(A) ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple_Otriple__case,type,
hoare_376461865e_case:
!>[A: $tType,T4: $tType] : ( ( fun(fun(A,fun(state,bool)),fun(com,fun(fun(A,fun(state,bool)),T4))) * hoare_28830079triple(A) ) > T4 ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple_Otriple__rec,type,
hoare_678420151le_rec:
!>[A: $tType,T4: $tType] : ( ( fun(fun(A,fun(state,bool)),fun(com,fun(fun(A,fun(state,bool)),T4))) * hoare_28830079triple(A) ) > T4 ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple_Otriple__size,type,
hoare_47506394e_size:
!>[A: $tType] : ( ( fun(A,nat) * hoare_28830079triple(A) ) > nat ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple__valid,type,
hoare_1633586161_valid:
!>[A: $tType] : ( ( nat * hoare_28830079triple(A) ) > $o ) ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri532925092at_aux:
!>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Natural_Oevalc,type,
evalc: ( com * state * state ) > $o ).
tff(sy_c_Natural_Oevaln,type,
evaln: ( com * state * nat * state ) > $o ).
tff(sy_c_Option_Othe,type,
the:
!>[A: $tType] : ( option(A) > A ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
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_fTrue,type,
fTrue: bool ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_P,type,
p: fun(a,fun(state,bool)) ).
tff(sy_v_Q,type,
q: fun(a,fun(state,bool)) ).
tff(sy_v_n,type,
n: nat ).
tff(sy_v_pn,type,
pn: pname ).
%----Relevant facts (97)
tff(fact_0_triple_Oinject,axiom,
! [B: $tType,Fun22: fun(B,fun(state,bool)),Com2: com,Fun12: fun(B,fun(state,bool)),Fun2: fun(B,fun(state,bool)),Com: com,Fun1: fun(B,fun(state,bool))] :
( ( hoare_1841697145triple(B,Fun1,Com,Fun2) = hoare_1841697145triple(B,Fun12,Com2,Fun22) )
<=> ( ( Fun1 = Fun12 )
& ( Com = Com2 )
& ( Fun2 = Fun22 ) ) ) ).
tff(fact_1_com_Osimps_I6_J,axiom,
! [Pname2: pname,Pname1: pname] :
( ( body(Pname1) = body(Pname2) )
<=> ( Pname1 = Pname2 ) ) ).
tff(fact_2_nat_Oinject,axiom,
! [Nat4: nat,Nat3: nat] :
( ( suc(Nat3) = suc(Nat4) )
<=> ( Nat3 = Nat4 ) ) ).
tff(fact_3_triple_Orecs,axiom,
! [B: $tType,C4: $tType,Fun2: fun(C4,fun(state,bool)),Com: com,Fun1: fun(C4,fun(state,bool)),F1: fun(fun(C4,fun(state,bool)),fun(com,fun(fun(C4,fun(state,bool)),B)))] : hoare_678420151le_rec(C4,B,F1,hoare_1841697145triple(C4,Fun1,Com,Fun2)) = aa(fun(C4,fun(state,bool)),B,aa(com,fun(fun(C4,fun(state,bool)),B),aa(fun(C4,fun(state,bool)),fun(com,fun(fun(C4,fun(state,bool)),B)),F1,Fun1),Com),Fun2) ).
tff(fact_4_triple_Osimps_I2_J,axiom,
! [B: $tType,C4: $tType,Fun2: fun(C4,fun(state,bool)),Com: com,Fun1: fun(C4,fun(state,bool)),F1: fun(fun(C4,fun(state,bool)),fun(com,fun(fun(C4,fun(state,bool)),B)))] : hoare_376461865e_case(C4,B,F1,hoare_1841697145triple(C4,Fun1,Com,Fun2)) = aa(fun(C4,fun(state,bool)),B,aa(com,fun(fun(C4,fun(state,bool)),B),aa(fun(C4,fun(state,bool)),fun(com,fun(fun(C4,fun(state,bool)),B)),F1,Fun1),Com),Fun2) ).
tff(fact_5_triple_Oexhaust,axiom,
! [B: $tType,Y1: hoare_28830079triple(B)] :
~ ! [Fun11: fun(B,fun(state,bool)),Com1: com,Fun21: fun(B,fun(state,bool))] : Y1 != hoare_1841697145triple(B,Fun11,Com1,Fun21) ).
tff(fact_6_evaln_OBody,axiom,
! [S1: state,N: nat,S0: state,Pn: pname] :
( evaln(the(com,body1(Pn)),S0,N,S1)
=> evaln(body(Pn),S0,suc(N),S1) ) ).
tff(fact_7_Body__triple__valid__0,axiom,
! [B: $tType,Qa: fun(B,fun(state,bool)),Pna: pname,Pa: fun(B,fun(state,bool))] : hoare_1633586161_valid(B,zero_zero(nat),hoare_1841697145triple(B,Pa,body(Pna),Qa)) ).
tff(fact_8_evalc__elim__cases_I6_J,axiom,
! [S1: state,S: state,P: pname] :
( evalc(body(P),S,S1)
=> evalc(the(com,body1(P)),S,S1) ) ).
tff(fact_9_evalc_OBody,axiom,
! [S1: state,S0: state,Pn: pname] :
( evalc(the(com,body1(Pn)),S0,S1)
=> evalc(body(Pn),S0,S1) ) ).
tff(fact_10_evaln__elim__cases_I6_J,axiom,
! [S1: state,N: nat,S: state,P: pname] :
( evaln(body(P),S,N,S1)
=> ~ ! [N1: nat] :
( ( N = suc(N1) )
=> ~ evaln(the(com,body1(P)),S,N1,S1) ) ) ).
tff(fact_11_n__not__Suc__n,axiom,
! [N: nat] : N != suc(N) ).
tff(fact_12_eval__eq,axiom,
! [T3: state,S6: state,C3: com] :
( evalc(C3,S6,T3)
<=> ? [N3: nat] : evaln(C3,S6,N3,T3) ) ).
tff(fact_13_com__det,axiom,
! [U: state,T: state,S: state,C: com] :
( evalc(C,S,T)
=> ( evalc(C,S,U)
=> ( U = T ) ) ) ).
tff(fact_14_evaln__evalc,axiom,
! [T: state,N: nat,S: state,C: com] :
( evaln(C,S,N,T)
=> evalc(C,S,T) ) ).
tff(fact_15_Suc__neq__Zero,axiom,
! [M: nat] : suc(M) != zero_zero(nat) ).
tff(fact_16_Zero__neq__Suc,axiom,
! [M: nat] : zero_zero(nat) != suc(M) ).
tff(fact_17_nat_Osimps_I3_J,axiom,
! [Nat2: nat] : suc(Nat2) != zero_zero(nat) ).
tff(fact_18_Suc__not__Zero,axiom,
! [M: nat] : suc(M) != zero_zero(nat) ).
tff(fact_19_nat_Osimps_I2_J,axiom,
! [Nat1: nat] : zero_zero(nat) != suc(Nat1) ).
tff(fact_20_Zero__not__Suc,axiom,
! [M: nat] : zero_zero(nat) != suc(M) ).
tff(fact_21_evaln__Suc,axiom,
! [S5: state,N: nat,S: state,C: com] :
( evaln(C,S,N,S5)
=> evaln(C,S,suc(N),S5) ) ).
tff(fact_22_triple__valid__def2,axiom,
! [B: $tType,Qa: fun(B,fun(state,bool)),C3: com,Pa: fun(B,fun(state,bool)),Na: nat] :
( hoare_1633586161_valid(B,Na,hoare_1841697145triple(B,Pa,C3,Qa))
<=> ! [Z: B,S3: state] :
( pp(aa(state,bool,aa(B,fun(state,bool),Pa,Z),S3))
=> ! [S4: state] :
( evaln(C3,S3,Na,S4)
=> pp(aa(state,bool,aa(B,fun(state,bool),Qa,Z),S4)) ) ) ) ).
tff(fact_23_Suc__inject,axiom,
! [Y: nat,X: nat] :
( ( suc(X) = suc(Y) )
=> ( X = Y ) ) ).
tff(fact_24_Suc__n__not__n,axiom,
! [N: nat] : suc(N) != N ).
tff(fact_25_triple_Osize_I1_J,axiom,
! [B: $tType,Fun2: fun(B,fun(state,bool)),Com: com,Fun1: fun(B,fun(state,bool)),Fa: fun(B,nat)] : hoare_47506394e_size(B,Fa,hoare_1841697145triple(B,Fun1,Com,Fun2)) = zero_zero(nat) ).
tff(fact_26_triple_Osize_I2_J,axiom,
! [B: $tType,Fun2: fun(B,fun(state,bool)),Com: com,Fun1: fun(B,fun(state,bool))] : size_size(hoare_28830079triple(B),hoare_1841697145triple(B,Fun1,Com,Fun2)) = zero_zero(nat) ).
tff(fact_27_evalc__evaln,axiom,
! [T: state,S: state,C: com] :
( evalc(C,S,T)
=> ? [N1: nat] : evaln(C,S,N1,T) ) ).
tff(fact_28_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero(nat) )
=> ~ ! [Nat: nat] : Y != suc(Nat) ) ).
tff(fact_29_zero__induct,axiom,
! [K1: nat,Pa: fun(nat,bool)] :
( pp(aa(nat,bool,Pa,K1))
=> ( ! [N1: nat] :
( pp(aa(nat,bool,Pa,suc(N1)))
=> pp(aa(nat,bool,Pa,N1)) )
=> pp(aa(nat,bool,Pa,zero_zero(nat))) ) ) ).
tff(fact_30_nat__induct,axiom,
! [Na: nat,Pa: fun(nat,bool)] :
( pp(aa(nat,bool,Pa,zero_zero(nat)))
=> ( ! [N1: nat] :
( pp(aa(nat,bool,Pa,N1))
=> pp(aa(nat,bool,Pa,suc(N1))) )
=> pp(aa(nat,bool,Pa,Na)) ) ) ).
tff(fact_31_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ? [M2: nat] : N = suc(M2) ) ).
tff(fact_32_of__nat__aux_Osimps_I2_J,axiom,
! [B: $tType] :
( semiring_1(B)
=> ! [I1: B,Na: nat,Inc: fun(B,B)] : semiri532925092at_aux(B,Inc,suc(Na),I1) = semiri532925092at_aux(B,Inc,Na,aa(B,B,Inc,I1)) ) ).
tff(fact_33_of__nat__aux_Osimps_I1_J,axiom,
! [B: $tType] :
( semiring_1(B)
=> ! [I1: B,Inc: fun(B,B)] : semiri532925092at_aux(B,Inc,zero_zero(nat),I1) = I1 ) ).
tff(fact_34_zero__reorient,axiom,
! [B: $tType] :
( zero(B)
=> ! [X1: B] :
( ( zero_zero(B) = X1 )
<=> ( X1 = zero_zero(B) ) ) ) ).
tff(fact_35_evaln__max2,axiom,
! [T2: state,N21: nat,S2: state,C2: com,T1: state,N11: nat,S1: state,C1: com] :
( evaln(C1,S1,N11,T1)
=> ( evaln(C2,S2,N21,T2)
=> ? [N1: nat] :
( evaln(C1,S1,N1,T1)
& evaln(C2,S2,N1,T2) ) ) ) ).
tff(fact_36_com_Osize_I7_J,axiom,
! [Pname: pname] : com_size(body(Pname)) = zero_zero(nat) ).
tff(fact_37_com_Osize_I15_J,axiom,
! [Pname: pname] : size_size(com,body(Pname)) = zero_zero(nat) ).
tff(fact_38_One__nat__def,axiom,
one_one(nat) = suc(zero_zero(nat)) ).
tff(fact_39_less__Suc0,axiom,
! [Na: nat] :
( ord_less(nat,Na,suc(zero_zero(nat)))
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_40_neq0__conv,axiom,
! [Na: nat] :
( ( Na != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_41_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_42_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_43_Suc__mono,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ord_less(nat,suc(M),suc(N)) ) ).
tff(fact_44_Suc__less__eq,axiom,
! [Na: nat,M1: nat] :
( ord_less(nat,suc(M1),suc(Na))
<=> ord_less(nat,M1,Na) ) ).
tff(fact_45_lessI,axiom,
! [N: nat] : ord_less(nat,N,suc(N)) ).
tff(fact_46_zero__less__Suc,axiom,
! [N: nat] : ord_less(nat,zero_zero(nat),suc(N)) ).
tff(fact_47_one__reorient,axiom,
! [B: $tType] :
( one(B)
=> ! [X1: B] :
( ( one_one(B) = X1 )
<=> ( X1 = one_one(B) ) ) ) ).
tff(fact_48_nat__less__cases,axiom,
! [Pa: fun(nat,fun(nat,bool)),Na: nat,M1: nat] :
( ( ord_less(nat,M1,Na)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),Pa,Na),M1)) )
=> ( ( ( M1 = Na )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),Pa,Na),M1)) )
=> ( ( ord_less(nat,Na,M1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),Pa,Na),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),Pa,Na),M1)) ) ) ) ).
tff(fact_49_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_50_less__not__refl2,axiom,
! [M: nat,N: nat] :
( ord_less(nat,N,M)
=> ( M != N ) ) ).
tff(fact_51_less__irrefl__nat,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_52_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_53_nat__neq__iff,axiom,
! [Na: nat,M1: nat] :
( ( M1 != Na )
<=> ( ord_less(nat,M1,Na)
| ord_less(nat,Na,M1) ) ) ).
tff(fact_54_less__not__refl,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_55_not__less0,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_56_gr__implies__not0,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ( N != zero_zero(nat) ) ) ).
tff(fact_57_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_58_Suc__less__SucD,axiom,
! [N: nat,M: nat] :
( ord_less(nat,suc(M),suc(N))
=> ord_less(nat,M,N) ) ).
tff(fact_59_Suc__lessD,axiom,
! [N: nat,M: nat] :
( ord_less(nat,suc(M),N)
=> ord_less(nat,M,N) ) ).
tff(fact_60_less__SucE,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,suc(N))
=> ( ~ ord_less(nat,M,N)
=> ( M = N ) ) ) ).
tff(fact_61_less__trans__Suc,axiom,
! [K: nat,J2: nat,I: nat] :
( ord_less(nat,I,J2)
=> ( ord_less(nat,J2,K)
=> ord_less(nat,suc(I),K) ) ) ).
tff(fact_62_Suc__lessI,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ( ( suc(M) != N )
=> ord_less(nat,suc(M),N) ) ) ).
tff(fact_63_less__SucI,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ord_less(nat,M,suc(N)) ) ).
tff(fact_64_less__antisym,axiom,
! [M: nat,N: nat] :
( ~ ord_less(nat,N,M)
=> ( ord_less(nat,N,suc(M))
=> ( M = N ) ) ) ).
tff(fact_65_not__less__less__Suc__eq,axiom,
! [M1: nat,Na: nat] :
( ~ ord_less(nat,Na,M1)
=> ( ord_less(nat,Na,suc(M1))
<=> ( Na = M1 ) ) ) ).
tff(fact_66_less__Suc__eq,axiom,
! [Na: nat,M1: nat] :
( ord_less(nat,M1,suc(Na))
<=> ( ord_less(nat,M1,Na)
| ( M1 = Na ) ) ) ).
tff(fact_67_not__less__eq,axiom,
! [Na: nat,M1: nat] :
( ~ ord_less(nat,M1,Na)
<=> ord_less(nat,Na,suc(M1)) ) ).
tff(fact_68_gr0__conv__Suc,axiom,
! [Na: nat] :
( ord_less(nat,zero_zero(nat),Na)
<=> ? [M3: nat] : Na = suc(M3) ) ).
tff(fact_69_less__Suc__eq__0__disj,axiom,
! [Na: nat,M1: nat] :
( ord_less(nat,M1,suc(Na))
<=> ( ( M1 = zero_zero(nat) )
| ? [J1: nat] :
( ( M1 = suc(J1) )
& ord_less(nat,J1,Na) ) ) ) ).
tff(fact_70_gr0__implies__Suc,axiom,
! [N: nat] :
( ord_less(nat,zero_zero(nat),N)
=> ? [M2: nat] : N = suc(M2) ) ).
tff(fact_71_lift__Suc__mono__less,axiom,
! [B: $tType] :
( order(B)
=> ! [N2: nat,Na: nat,F: fun(nat,B)] :
( ! [N1: nat] : ord_less(B,aa(nat,B,F,N1),aa(nat,B,F,suc(N1)))
=> ( ord_less(nat,Na,N2)
=> ord_less(B,aa(nat,B,F,Na),aa(nat,B,F,N2)) ) ) ) ).
tff(fact_72_lift__Suc__mono__less__iff,axiom,
! [B: $tType] :
( order(B)
=> ! [M1: nat,Na: nat,F: fun(nat,B)] :
( ! [N1: nat] : ord_less(B,aa(nat,B,F,N1),aa(nat,B,F,suc(N1)))
=> ( ord_less(B,aa(nat,B,F,Na),aa(nat,B,F,M1))
<=> ord_less(nat,Na,M1) ) ) ) ).
tff(fact_73_not__one__less__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less(A,one_one(A),zero_zero(A)) ) ).
tff(fact_74_zero__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less(A,zero_zero(A),one_one(A)) ) ).
tff(fact_75_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ ord_less(A,X,Y)
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_76_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_77_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_78_Suc__lessE,axiom,
! [K: nat,I: nat] :
( ord_less(nat,suc(I),K)
=> ~ ! [J: nat] :
( ord_less(nat,I,J)
=> ( K != suc(J) ) ) ) ).
tff(fact_79_lessE,axiom,
! [K: nat,I: nat] :
( ord_less(nat,I,K)
=> ( ( K != suc(I) )
=> ~ ! [J: nat] :
( ord_less(nat,I,J)
=> ( K != suc(J) ) ) ) ) ).
tff(fact_80_of__nat__0__less__iff,axiom,
! [B: $tType] :
( linordered_semidom(B)
=> ! [Na: nat] :
( ord_less(B,zero_zero(B),semiring_1_of_nat(B,Na))
<=> ord_less(nat,zero_zero(nat),Na) ) ) ).
tff(fact_81_sgn__1__pos,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [A2: B] :
( ( sgn_sgn(B,A2) = one_one(B) )
<=> ord_less(B,zero_zero(B),A2) ) ) ).
tff(fact_82_of__nat__eq__iff,axiom,
! [B: $tType] :
( semiring_char_0(B)
=> ! [Na: nat,M1: nat] :
( ( semiring_1_of_nat(B,M1) = semiring_1_of_nat(B,Na) )
<=> ( M1 = Na ) ) ) ).
tff(fact_83_sgn__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : sgn_sgn(A,sgn_sgn(A,A1)) = sgn_sgn(A,A1) ) ).
tff(fact_84_sgn0,axiom,
! [A: $tType] :
( sgn_if(A)
=> ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_85_of__nat__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,zero_zero(nat)) = zero_zero(A) ) ) ).
tff(fact_86_of__nat__less__iff,axiom,
! [B: $tType] :
( linordered_semidom(B)
=> ! [Na: nat,M1: nat] :
( ord_less(B,semiring_1_of_nat(B,M1),semiring_1_of_nat(B,Na))
<=> ord_less(nat,M1,Na) ) ) ).
tff(fact_87_of__nat__1,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,one_one(nat)) = one_one(A) ) ) ).
tff(fact_88_sgn__less,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [A2: B] :
( ord_less(B,sgn_sgn(B,A2),zero_zero(B))
<=> ord_less(B,A2,zero_zero(B)) ) ) ).
tff(fact_89_sgn__greater,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [A2: B] :
( ord_less(B,zero_zero(B),sgn_sgn(B,A2))
<=> ord_less(B,zero_zero(B),A2) ) ) ).
tff(fact_90_of__nat__less__imp__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M: nat] :
( ord_less(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N))
=> ord_less(nat,M,N) ) ) ).
tff(fact_91_less__imp__of__nat__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ord_less(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N)) ) ) ).
tff(fact_92_of__nat__less__0__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M: nat] : ~ ord_less(A,semiring_1_of_nat(A,M),zero_zero(A)) ) ).
tff(fact_93_sgn__0__0,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [A2: B] :
( ( sgn_sgn(B,A2) = zero_zero(B) )
<=> ( A2 = zero_zero(B) ) ) ) ).
tff(fact_94_sgn__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( sgn_sgn(A,A1) = one_one(A) ) ) ) ).
tff(fact_95_zero__less__int__conv,axiom,
! [Na: nat] :
( ord_less(int,zero_zero(int),semiring_1_of_nat(int,Na))
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_96_int__Suc0__eq__1,axiom,
semiring_1_of_nat(int,suc(zero_zero(nat))) = one_one(int) ).
%----Arities (18)
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
tff(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Orderings_Oorder,axiom,
order(int) ).
tff(arity_Int_Oint___Groups_Osgn__if,axiom,
sgn_if(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0(nat) ).
tff(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
( hoare_1633586161_valid(a,n,hoare_1841697145triple(a,p,the(com,body1(pn)),q))
<=> hoare_1633586161_valid(a,suc(n),hoare_1841697145triple(a,p,body(pn),q)) ) ).
%------------------------------------------------------------------------------