TPTP Problem File: SWW531_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW531_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Huffman's Algorithm line 587
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla09] Blanchette (2009), Proof Pearl: Mechanizing the Textbo
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : huff_587 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 157 ( 51 unt; 41 typ; 0 def)
% Number of atoms : 238 ( 79 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 155 ( 33 ~; 2 |; 15 &)
% ( 41 <=>; 64 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 10 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 35 ( 20 >; 15 *; 0 +; 0 <<)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-3 aty)
% Number of functors : 25 ( 25 usr; 9 con; 0-5 aty)
% Number of variables : 250 ( 215 !; 8 ?; 250 :)
% ( 27 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:15:51
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
tff(ty_t_a,type,
a: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Huffman__Mirabelle__lalbadcutu_Otree,type,
huffma1450048681e_tree: $tType > $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (35)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[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_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Oalphabet,type,
huffma675207370phabet:
!>[A: $tType] : ( huffma1450048681e_tree(A) > fun(A,bool) ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Oconsistent,type,
huffma1518433673istent:
!>[A: $tType] : ( huffma1450048681e_tree(A) > $o ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Odepth,type,
huffma410068972_depth:
!>[A: $tType] : ( ( huffma1450048681e_tree(A) * A ) > nat ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Oheight,type,
huffma945805758height:
!>[A: $tType] : ( huffma1450048681e_tree(A) > nat ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Otree_OInnerNode,type,
huffma1146269203erNode:
!>[A: $tType] : ( ( nat * huffma1450048681e_tree(A) * huffma1450048681e_tree(A) ) > huffma1450048681e_tree(A) ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Otree_OLeaf,type,
huffma2021818691e_Leaf:
!>[A: $tType] : ( ( nat * A ) > huffma1450048681e_tree(A) ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Otree_Otree__case,type,
huffma107959123e_case:
!>[A: $tType,T1: $tType] : ( ( fun(nat,fun(A,T1)) * fun(nat,fun(huffma1450048681e_tree(A),fun(huffma1450048681e_tree(A),T1))) * huffma1450048681e_tree(A) ) > T1 ) ).
tff(sy_c_Huffman__Mirabelle__lalbadcutu_Otree_Otree__rec,type,
huffma1280178957ee_rec:
!>[A: $tType,T1: $tType] : ( ( fun(nat,fun(A,T1)) * fun(nat,fun(huffma1450048681e_tree(A),fun(huffma1450048681e_tree(A),fun(T1,fun(T1,T1))))) * huffma1450048681e_tree(A) ) > T1 ) ).
tff(sy_c_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_Int_Onat,type,
nat1: int > nat ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[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_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_b____,type,
b: a ).
tff(sy_v_c____,type,
c: a ).
tff(sy_v_t,type,
t: huffma1450048681e_tree(a) ).
tff(sy_v_t_092_060_094isub_0621____,type,
t_1: huffma1450048681e_tree(a) ).
tff(sy_v_t_092_060_094isub_0622____,type,
t_2: huffma1450048681e_tree(a) ).
tff(sy_v_u,type,
u: huffma1450048681e_tree(a) ).
tff(sy_v_w____,type,
w: nat ).
%----Relevant facts (99)
tff(fact_0_assms_I2_J,axiom,
huffma1518433673istent(a,t) ).
tff(fact_1_assms_I1_J,axiom,
ord_less(nat,zero_zero(nat),huffma945805758height(a,t)) ).
tff(fact_2_assms_I3_J,axiom,
huffma675207370phabet(a,t) = huffma675207370phabet(a,u) ).
tff(fact_3_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_4_less__nat__zero__code,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_5_less__zeroE,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_6_not__less0,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_7_gr__implies__not0,axiom,
! [N1: nat,M: nat] :
( ord_less(nat,M,N1)
=> ( N1 != zero_zero(nat) ) ) ).
tff(fact_8_gr0I,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N1) ) ).
tff(fact_9_height_Osimps_I1_J,axiom,
! [A: $tType,A4: A,W: nat] : huffma945805758height(A,huffma2021818691e_Leaf(A,W,A4)) = zero_zero(nat) ).
tff(fact_10_less__not__refl,axiom,
! [N1: nat] : ~ ord_less(nat,N1,N1) ).
tff(fact_11_nat__neq__iff,axiom,
! [N: nat,M1: nat] :
( ( M1 != N )
<=> ( ord_less(nat,M1,N)
| ord_less(nat,N,M1) ) ) ).
tff(fact_12_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_13_c,axiom,
member(a,c,huffma675207370phabet(a,t_2)) ).
tff(fact_14_b,axiom,
member(a,b,huffma675207370phabet(a,t_1)) ).
tff(fact_15_tree_Osimps_I1_J,axiom,
! [B: $tType,A5: B,Nat4: nat,A2: B,Nat: nat] :
( ( huffma2021818691e_Leaf(B,Nat,A2) = huffma2021818691e_Leaf(B,Nat4,A5) )
<=> ( ( Nat = Nat4 )
& ( A2 = A5 ) ) ) ).
tff(fact_16_consistent_Osimps_I1_J,axiom,
! [A: $tType,A4: A,W: nat] : huffma1518433673istent(A,huffma2021818691e_Leaf(A,W,A4)) ).
tff(fact_17_zero__reorient,axiom,
! [B: $tType] :
( zero(B)
=> ! [X1: B] :
( ( zero_zero(B) = X1 )
<=> ( X1 = zero_zero(B) ) ) ) ).
tff(fact_18_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),N: nat,M1: nat] :
( ( ord_less(nat,M1,N)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N),M1)) )
=> ( ( ( M1 = N )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N),M1)) )
=> ( ( ord_less(nat,N,M1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N),M1)) ) ) ) ).
tff(fact_19_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_20_less__not__refl2,axiom,
! [M: nat,N1: nat] :
( ord_less(nat,N1,M)
=> ( M != N1 ) ) ).
tff(fact_21_less__irrefl__nat,axiom,
! [N1: nat] : ~ ord_less(nat,N1,N1) ).
tff(fact_22__096_B_Bthesis_O_A_I_B_Bb_O_Ab_A_058_Aalphabet_At_092_060_094isub_0621_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [B2: a] : ~ member(a,B2,huffma675207370phabet(a,t_1)) ).
tff(fact_23__096_B_Bthesis_O_A_I_B_Bc_O_Ac_A_058_Aalphabet_At_092_060_094isub_0622_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [C1: a] : ~ member(a,C1,huffma675207370phabet(a,t_2)) ).
tff(fact_24_tree_Osize_I3_J,axiom,
! [A: $tType,A4: A,Nat2: nat] : size_size(huffma1450048681e_tree(A),huffma2021818691e_Leaf(A,Nat2,A4)) = zero_zero(nat) ).
tff(fact_25_depth_Osimps_I1_J,axiom,
! [A: $tType,A4: A,B1: A,W: nat] : huffma410068972_depth(A,huffma2021818691e_Leaf(A,W,B1),A4) = zero_zero(nat) ).
tff(fact_26_tree_Osimps_I5_J,axiom,
! [B: $tType,C: $tType,A2: C,Nat: nat,F2: fun(nat,fun(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),B))),F1: fun(nat,fun(C,B))] : huffma107959123e_case(C,B,F1,F2,huffma2021818691e_Leaf(C,Nat,A2)) = aa(C,B,aa(nat,fun(C,B),F1,Nat),A2) ).
tff(fact_27_exists__in__alphabet,axiom,
! [B: $tType,Ta: huffma1450048681e_tree(B)] :
? [A3: B] : member(B,A3,huffma675207370phabet(B,Ta)) ).
tff(fact_28_of__nat__0__less__iff,axiom,
! [B: $tType] :
( linordered_semidom(B)
=> ! [N: nat] :
( ord_less(B,zero_zero(B),semiring_1_of_nat(B,N))
<=> ord_less(nat,zero_zero(nat),N) ) ) ).
tff(fact_29_exists__at__height,axiom,
! [B: $tType,Ta: huffma1450048681e_tree(B)] :
( huffma1518433673istent(B,Ta)
=> ? [X3: B] :
( member(B,X3,huffma675207370phabet(B,Ta))
& ( huffma410068972_depth(B,Ta,X3) = huffma945805758height(B,Ta) ) ) ) ).
tff(fact_30_InnerNode,axiom,
t = huffma1146269203erNode(a,w,t_1,t_2) ).
tff(fact_31_of__nat__eq__iff,axiom,
! [B: $tType] :
( semiring_char_0(B)
=> ! [N: nat,M1: nat] :
( ( semiring_1_of_nat(B,M1) = semiring_1_of_nat(B,N) )
<=> ( M1 = N ) ) ) ).
tff(fact_32_tree_Osimps_I2_J,axiom,
! [B: $tType,Tree23: huffma1450048681e_tree(B),Tree13: huffma1450048681e_tree(B),Nat4: nat,Tree2: huffma1450048681e_tree(B),Tree1: huffma1450048681e_tree(B),Nat: nat] :
( ( huffma1146269203erNode(B,Nat,Tree1,Tree2) = huffma1146269203erNode(B,Nat4,Tree13,Tree23) )
<=> ( ( Nat = Nat4 )
& ( Tree1 = Tree13 )
& ( Tree2 = Tree23 ) ) ) ).
tff(fact_33_bc,axiom,
b != c ).
tff(fact_34_of__nat__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,zero_zero(nat)) = zero_zero(A) ) ) ).
tff(fact_35_of__nat__less__iff,axiom,
! [B: $tType] :
( linordered_semidom(B)
=> ! [N: nat,M1: nat] :
( ord_less(B,semiring_1_of_nat(B,M1),semiring_1_of_nat(B,N))
<=> ord_less(nat,M1,N) ) ) ).
tff(fact_36_tree_Osimps_I6_J,axiom,
! [B: $tType,C: $tType,Tree2: huffma1450048681e_tree(C),Tree1: huffma1450048681e_tree(C),Nat: nat,F2: fun(nat,fun(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),B))),F1: fun(nat,fun(C,B))] : huffma107959123e_case(C,B,F1,F2,huffma1146269203erNode(C,Nat,Tree1,Tree2)) = aa(huffma1450048681e_tree(C),B,aa(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),B),aa(nat,fun(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),B)),F2,Nat),Tree1),Tree2) ).
tff(fact_37_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_38_less__imp__of__nat__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N1: nat,M: nat] :
( ord_less(nat,M,N1)
=> ord_less(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N1)) ) ) ).
tff(fact_39_of__nat__less__imp__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N1: nat,M: nat] :
( ord_less(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N1))
=> ord_less(nat,M,N1) ) ) ).
tff(fact_40_tree_Osimps_I4_J,axiom,
! [A: $tType,A4: A,Nat2: nat,Tree22: huffma1450048681e_tree(A),Tree12: huffma1450048681e_tree(A),Nat3: nat] : huffma1146269203erNode(A,Nat3,Tree12,Tree22) != huffma2021818691e_Leaf(A,Nat2,A4) ).
tff(fact_41_tree_Osimps_I3_J,axiom,
! [A: $tType,Tree22: huffma1450048681e_tree(A),Tree12: huffma1450048681e_tree(A),Nat3: nat,A4: A,Nat2: nat] : huffma2021818691e_Leaf(A,Nat2,A4) != huffma1146269203erNode(A,Nat3,Tree12,Tree22) ).
tff(fact_42_zero__less__int__conv,axiom,
! [N: nat] :
( ord_less(int,zero_zero(int),semiring_1_of_nat(int,N))
<=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_43_tree_Oexhaust,axiom,
! [A: $tType,Y: huffma1450048681e_tree(A)] :
( ! [Nat1: nat,A3: A] : Y != huffma2021818691e_Leaf(A,Nat1,A3)
=> ~ ! [Nat1: nat,Tree11: huffma1450048681e_tree(A),Tree21: huffma1450048681e_tree(A)] : Y != huffma1146269203erNode(A,Nat1,Tree11,Tree21) ) ).
tff(fact_44_infinite__descent0,axiom,
! [N: nat,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,zero_zero(nat)))
=> ( ! [N3: nat] :
( ord_less(nat,zero_zero(nat),N3)
=> ( ~ pp(aa(nat,bool,P1,N3))
=> ? [M2: nat] :
( ord_less(nat,M2,N3)
& ~ pp(aa(nat,bool,P1,M2)) ) ) )
=> pp(aa(nat,bool,P1,N)) ) ) ).
tff(fact_45_tree_Orecs_I1_J,axiom,
! [B: $tType,C: $tType,A2: C,Nat: nat,F2: fun(nat,fun(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),fun(B,fun(B,B))))),F1: fun(nat,fun(C,B))] : huffma1280178957ee_rec(C,B,F1,F2,huffma2021818691e_Leaf(C,Nat,A2)) = aa(C,B,aa(nat,fun(C,B),F1,Nat),A2) ).
tff(fact_46_tree_Orecs_I2_J,axiom,
! [B: $tType,C: $tType,Tree2: huffma1450048681e_tree(C),Tree1: huffma1450048681e_tree(C),Nat: nat,F2: fun(nat,fun(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),fun(B,fun(B,B))))),F1: fun(nat,fun(C,B))] : huffma1280178957ee_rec(C,B,F1,F2,huffma1146269203erNode(C,Nat,Tree1,Tree2)) = aa(B,B,aa(B,fun(B,B),aa(huffma1450048681e_tree(C),fun(B,fun(B,B)),aa(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),fun(B,fun(B,B))),aa(nat,fun(huffma1450048681e_tree(C),fun(huffma1450048681e_tree(C),fun(B,fun(B,B)))),F2,Nat),Tree1),Tree2),huffma1280178957ee_rec(C,B,F1,F2,Tree1)),huffma1280178957ee_rec(C,B,F1,F2,Tree2)) ).
tff(fact_47_transfer__int__nat__relations_I2_J,axiom,
! [Y1: nat,X1: nat] :
( ord_less(int,semiring_1_of_nat(int,X1),semiring_1_of_nat(int,Y1))
<=> ord_less(nat,X1,Y1) ) ).
tff(fact_48_int__less__0__conv,axiom,
! [K: nat] : ~ ord_less(int,semiring_1_of_nat(int,K),zero_zero(int)) ).
tff(fact_49_int__if__cong,axiom,
! [Y1: nat,X1: nat,P1: bool] :
( ( pp(P1)
=> ( semiring_1_of_nat(int,X1) = semiring_1_of_nat(int,if(nat,P1,X1,Y1)) ) )
& ( ~ pp(P1)
=> ( semiring_1_of_nat(int,Y1) = semiring_1_of_nat(int,if(nat,P1,X1,Y1)) ) ) ) ).
tff(fact_50_transfer__int__nat__relations_I1_J,axiom,
! [Y1: nat,X1: nat] :
( ( semiring_1_of_nat(int,X1) = semiring_1_of_nat(int,Y1) )
<=> ( X1 = Y1 ) ) ).
tff(fact_51_int__int__eq,axiom,
! [N: nat,M1: nat] :
( ( semiring_1_of_nat(int,M1) = semiring_1_of_nat(int,N) )
<=> ( M1 = N ) ) ).
tff(fact_52_int__eq__0__conv,axiom,
! [N: nat] :
( ( semiring_1_of_nat(int,N) = zero_zero(int) )
<=> ( N = zero_zero(nat) ) ) ).
tff(fact_53_int__0,axiom,
semiring_1_of_nat(int,zero_zero(nat)) = zero_zero(int) ).
tff(fact_54_transfer__int__nat__numerals_I1_J,axiom,
zero_zero(int) = semiring_1_of_nat(int,zero_zero(nat)) ).
tff(fact_55_zless__int,axiom,
! [N: nat,M1: nat] :
( ord_less(int,semiring_1_of_nat(int,M1),semiring_1_of_nat(int,N))
<=> ord_less(nat,M1,N) ) ).
tff(fact_56_zero__less__imp__eq__int,axiom,
! [K: int] :
( ord_less(int,zero_zero(int),K)
=> ? [N3: nat] :
( ord_less(nat,zero_zero(nat),N3)
& ( K = semiring_1_of_nat(int,N3) ) ) ) ).
tff(fact_57_of__int__less__0__iff,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [Z1: int] :
( ord_less(B,ring_1_of_int(B,Z1),zero_zero(B))
<=> ord_less(int,Z1,zero_zero(int)) ) ) ).
tff(fact_58_of__int__0__less__iff,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [Z1: int] :
( ord_less(B,zero_zero(B),ring_1_of_int(B,Z1))
<=> ord_less(int,zero_zero(int),Z1) ) ) ).
tff(fact_59_zero__less__nat__eq,axiom,
! [Z1: int] :
( ord_less(nat,zero_zero(nat),nat1(Z1))
<=> ord_less(int,zero_zero(int),Z1) ) ).
tff(fact_60_of__int__eq__iff,axiom,
! [B: $tType] :
( ring_char_0(B)
=> ! [Z1: int,Wa: int] :
( ( ring_1_of_int(B,Wa) = ring_1_of_int(B,Z1) )
<=> ( Wa = Z1 ) ) ) ).
tff(fact_61_nat__0,axiom,
nat1(zero_zero(int)) = zero_zero(nat) ).
tff(fact_62_of__int__eq__0__iff,axiom,
! [B: $tType] :
( ring_char_0(B)
=> ! [Z1: int] :
( ( ring_1_of_int(B,Z1) = zero_zero(B) )
<=> ( Z1 = zero_zero(int) ) ) ) ).
tff(fact_63_of__int__0__eq__iff,axiom,
! [B: $tType] :
( ring_char_0(B)
=> ! [Z1: int] :
( ( zero_zero(B) = ring_1_of_int(B,Z1) )
<=> ( Z1 = zero_zero(int) ) ) ) ).
tff(fact_64_of__int__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,zero_zero(int)) = zero_zero(A) ) ) ).
tff(fact_65_of__int__less__iff,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [Z1: int,Wa: int] :
( ord_less(B,ring_1_of_int(B,Wa),ring_1_of_int(B,Z1))
<=> ord_less(int,Wa,Z1) ) ) ).
tff(fact_66_of__int__of__nat__eq,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N1: nat] : ring_1_of_int(A,semiring_1_of_nat(int,N1)) = semiring_1_of_nat(A,N1) ) ).
tff(fact_67_zless__nat__conj,axiom,
! [Z1: int,Wa: int] :
( ord_less(nat,nat1(Wa),nat1(Z1))
<=> ( ord_less(int,zero_zero(int),Z1)
& ord_less(int,Wa,Z1) ) ) ).
tff(fact_68_of__int__int__eq,axiom,
! [N1: nat] : ring_1_of_int(int,semiring_1_of_nat(int,N1)) = semiring_1_of_nat(int,N1) ).
tff(fact_69_nat__int,axiom,
! [N1: nat] : nat1(semiring_1_of_nat(int,N1)) = N1 ).
tff(fact_70_nat__if__cong,axiom,
! [Y1: int,X1: int,P1: bool] :
( ( pp(P1)
=> ( nat1(X1) = nat1(if(int,P1,X1,Y1)) ) )
& ( ~ pp(P1)
=> ( nat1(Y1) = nat1(if(int,P1,X1,Y1)) ) ) ) ).
tff(fact_71_transfer__nat__int__numerals_I1_J,axiom,
zero_zero(nat) = nat1(zero_zero(int)) ).
tff(fact_72_nat__mono__iff,axiom,
! [Wa: int,Z1: int] :
( ord_less(int,zero_zero(int),Z1)
=> ( ord_less(nat,nat1(Wa),nat1(Z1))
<=> ord_less(int,Wa,Z1) ) ) ).
tff(fact_73_zless__nat__eq__int__zless,axiom,
! [Z1: int,M1: nat] :
( ord_less(nat,M1,nat1(Z1))
<=> ord_less(int,semiring_1_of_nat(int,M1),Z1) ) ).
tff(fact_74_mem__def,axiom,
! [B: $tType,A1: fun(B,bool),X1: B] :
( member(B,X1,A1)
<=> pp(aa(B,bool,A1,X1)) ) ).
tff(fact_75_split__nat,axiom,
! [I: int,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,nat1(I)))
<=> ( ! [N2: nat] :
( ( I = semiring_1_of_nat(int,N2) )
=> pp(aa(nat,bool,P1,N2)) )
& ( ord_less(int,I,zero_zero(int))
=> pp(aa(nat,bool,P1,zero_zero(nat))) ) ) ) ).
tff(fact_76_nat__less__iff,axiom,
! [M1: nat,Wa: int] :
( ord_less_eq(int,zero_zero(int),Wa)
=> ( ord_less(nat,nat1(Wa),M1)
<=> ord_less(int,Wa,semiring_1_of_nat(int,M1)) ) ) ).
tff(fact_77_of__nat__nat,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z: int] :
( ord_less_eq(int,zero_zero(int),Z)
=> ( semiring_1_of_nat(A,nat1(Z)) = ring_1_of_int(A,Z) ) ) ) ).
tff(fact_78_nat__eq__iff,axiom,
! [M1: nat,Wa: int] :
( ( nat1(Wa) = M1 )
<=> ( ( ord_less_eq(int,zero_zero(int),Wa)
=> ( Wa = semiring_1_of_nat(int,M1) ) )
& ( ~ ord_less_eq(int,zero_zero(int),Wa)
=> ( M1 = zero_zero(nat) ) ) ) ) ).
tff(fact_79_of__nat__le__iff,axiom,
! [B: $tType] :
( linordered_semidom(B)
=> ! [N: nat,M1: nat] :
( ord_less_eq(B,semiring_1_of_nat(B,M1),semiring_1_of_nat(B,N))
<=> ord_less_eq(nat,M1,N) ) ) ).
tff(fact_80_of__int__le__iff,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [Z1: int,Wa: int] :
( ord_less_eq(B,ring_1_of_int(B,Wa),ring_1_of_int(B,Z1))
<=> ord_less_eq(int,Wa,Z1) ) ) ).
tff(fact_81_nat__0__iff,axiom,
! [I: int] :
( ( nat1(I) = zero_zero(nat) )
<=> ord_less_eq(int,I,zero_zero(int)) ) ).
tff(fact_82_int__nat__eq,axiom,
! [Z: int] :
( ( ord_less_eq(int,zero_zero(int),Z)
=> ( semiring_1_of_nat(int,nat1(Z)) = Z ) )
& ( ~ ord_less_eq(int,zero_zero(int),Z)
=> ( semiring_1_of_nat(int,nat1(Z)) = zero_zero(int) ) ) ) ).
tff(fact_83_of__int__le__0__iff,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [Z1: int] :
( ord_less_eq(B,ring_1_of_int(B,Z1),zero_zero(B))
<=> ord_less_eq(int,Z1,zero_zero(int)) ) ) ).
tff(fact_84_of__int__0__le__iff,axiom,
! [B: $tType] :
( linordered_idom(B)
=> ! [Z1: int] :
( ord_less_eq(B,zero_zero(B),ring_1_of_int(B,Z1))
<=> ord_less_eq(int,zero_zero(int),Z1) ) ) ).
tff(fact_85_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq(int,zero_zero(int),zero_zero(int)) ).
tff(fact_86_less__int__def,axiom,
! [Wa: int,Z1: int] :
( ord_less(int,Z1,Wa)
<=> ( ord_less_eq(int,Z1,Wa)
& ( Z1 != Wa ) ) ) ).
tff(fact_87_of__nat__0__le__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N1: nat] : ord_less_eq(A,zero_zero(A),semiring_1_of_nat(A,N1)) ) ).
tff(fact_88_zero__le__imp__of__nat,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M: nat] : ord_less_eq(A,zero_zero(A),semiring_1_of_nat(A,M)) ) ).
tff(fact_89_zero__zle__int,axiom,
! [N1: nat] : ord_less_eq(int,zero_zero(int),semiring_1_of_nat(int,N1)) ).
tff(fact_90_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z: nat] : ord_less_eq(int,zero_zero(int),semiring_1_of_nat(int,Z)) ).
tff(fact_91_transfer__int__nat__quantifiers_I2_J,axiom,
! [P1: fun(int,bool)] :
( ? [X2: int] :
( ord_less_eq(int,zero_zero(int),X2)
& pp(aa(int,bool,P1,X2)) )
<=> ? [X2: nat] : pp(aa(int,bool,P1,semiring_1_of_nat(int,X2))) ) ).
tff(fact_92_transfer__int__nat__quantifiers_I1_J,axiom,
! [P1: fun(int,bool)] :
( ! [X2: int] :
( ord_less_eq(int,zero_zero(int),X2)
=> pp(aa(int,bool,P1,X2)) )
<=> ! [X2: nat] : pp(aa(int,bool,P1,semiring_1_of_nat(int,X2))) ) ).
tff(fact_93_ex__nat,axiom,
! [P1: fun(nat,bool)] :
( ? [X11: nat] : pp(aa(nat,bool,P1,X11))
<=> ? [X2: int] :
( ord_less_eq(int,zero_zero(int),X2)
& pp(aa(nat,bool,P1,nat1(X2))) ) ) ).
tff(fact_94_all__nat,axiom,
! [P1: fun(nat,bool)] :
( ! [X11: nat] : pp(aa(nat,bool,P1,X11))
<=> ! [X2: int] :
( ord_less_eq(int,zero_zero(int),X2)
=> pp(aa(nat,bool,P1,nat1(X2))) ) ) ).
tff(fact_95_transfer__nat__int__relations_I1_J,axiom,
! [Y1: int,X1: int] :
( ord_less_eq(int,zero_zero(int),X1)
=> ( ord_less_eq(int,zero_zero(int),Y1)
=> ( ( nat1(X1) = nat1(Y1) )
<=> ( X1 = Y1 ) ) ) ) ).
tff(fact_96_eq__nat__nat__iff,axiom,
! [Z2: int,Z1: int] :
( ord_less_eq(int,zero_zero(int),Z1)
=> ( ord_less_eq(int,zero_zero(int),Z2)
=> ( ( nat1(Z1) = nat1(Z2) )
<=> ( Z1 = Z2 ) ) ) ) ).
tff(fact_97_int__le__0__conv,axiom,
! [N: nat] :
( ord_less_eq(int,semiring_1_of_nat(int,N),zero_zero(int))
<=> ( N = zero_zero(nat) ) ) ).
tff(fact_98_nat__le__0,axiom,
! [Z: int] :
( ord_less_eq(int,Z,zero_zero(int))
=> ( nat1(Z) = zero_zero(nat) ) ) ).
%----Arities (11)
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_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(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_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
%----Helper facts (5)
tff(help_If_1_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fTrue,X,Y) = X ).
tff(help_If_2_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fFalse,X,Y) = Y ).
tff(help_If_3_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
ord_less(nat,zero_zero(nat),huffma945805758height(a,u)) ).
%------------------------------------------------------------------------------