TPTP Problem File: SWW547_5.p
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%------------------------------------------------------------------------------
% File : SWW547_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Huffman's Algorithm line 1165
% 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_1165 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 143 ( 29 unt; 28 typ; 0 def)
% Number of atoms : 317 ( 82 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 258 ( 56 ~; 12 |; 14 &)
% ( 20 <=>; 156 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 25 ( 15 >; 10 *; 0 +; 0 <<)
% Number of predicates : 11 ( 10 usr; 0 prp; 1-3 aty)
% Number of functors : 15 ( 15 usr; 4 con; 0-4 aty)
% Number of variables : 340 ( 320 !; 1 ?; 340 :)
% ( 19 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:22:19
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_t_a,type,
a1: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Huffman__Mirabelle__lalbadcutu_Otree,type,
huffma1450048681e_tree: $tType > $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (23)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[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_Osibling,type,
huffma1401021291ibling:
!>[A: $tType] : ( ( huffma1450048681e_tree(A) * A ) > A ) ).
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_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_a,type,
a: a1 ).
tff(sy_v_t,type,
t: huffma1450048681e_tree(a1) ).
%----Relevant facts (99)
tff(fact_0_sibling__sibling__id,axiom,
! [A: $tType,A1: A,T: huffma1450048681e_tree(A)] :
( huffma1518433673istent(A,T)
=> ( huffma1401021291ibling(A,T,huffma1401021291ibling(A,T,A1)) = A1 ) ) ).
tff(fact_1_sibling__reciprocal,axiom,
! [A: $tType,B1: A,A1: A,T: huffma1450048681e_tree(A)] :
( huffma1518433673istent(A,T)
=> ( ( huffma1401021291ibling(A,T,A1) = B1 )
=> ( huffma1401021291ibling(A,T,B1) = A1 ) ) ) ).
tff(fact_2_consistent_Osimps_I1_J,axiom,
! [A: $tType,A1: A,W1: nat] : huffma1518433673istent(A,huffma2021818691e_Leaf(A,W1,A1)) ).
tff(fact_3_sibling_Osimps_I1_J,axiom,
! [A: $tType,A1: A,B1: A,W_b: nat] : huffma1401021291ibling(A,huffma2021818691e_Leaf(A,W_b,B1),A1) = A1 ).
tff(fact_4_notin__alphabet__imp__sibling__id,axiom,
! [B: $tType,Ta: huffma1450048681e_tree(B),Aa: B] :
( ~ member(B,Aa,huffma675207370phabet(B,Ta))
=> ( huffma1401021291ibling(B,Ta,Aa) = Aa ) ) ).
tff(fact_5_sibling__ne__imp__sibling__in__alphabet,axiom,
! [B: $tType,Aa: B,Ta: huffma1450048681e_tree(B)] :
( ( huffma1401021291ibling(B,Ta,Aa) != Aa )
=> member(B,huffma1401021291ibling(B,Ta,Aa),huffma675207370phabet(B,Ta)) ) ).
tff(fact_6_in__alphabet__imp__sibling__in__alphabet,axiom,
! [B: $tType,Ta: huffma1450048681e_tree(B),Aa: B] :
( member(B,Aa,huffma675207370phabet(B,Ta))
=> member(B,huffma1401021291ibling(B,Ta,Aa),huffma675207370phabet(B,Ta)) ) ).
tff(fact_7_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_8_depth__height__imp__sibling__ne,axiom,
! [B: $tType,Aa: B,Ta: huffma1450048681e_tree(B)] :
( huffma1518433673istent(B,Ta)
=> ( ( huffma410068972_depth(B,Ta,Aa) = huffma945805758height(B,Ta) )
=> ( ord_less(nat,zero_zero(nat),huffma945805758height(B,Ta))
=> ( member(B,Aa,huffma675207370phabet(B,Ta))
=> ( huffma1401021291ibling(B,Ta,Aa) != Aa ) ) ) ) ) ).
tff(fact_9_depth_Osimps_I1_J,axiom,
! [A: $tType,A1: A,B1: A,W1: nat] : huffma410068972_depth(A,huffma2021818691e_Leaf(A,W1,B1),A1) = zero_zero(nat) ).
tff(fact_10_sibling_Osimps_I2_J,axiom,
! [A: $tType,C: A,W_c: nat,W_b: nat,W1: nat,B1: A,A1: A] :
( ( ( A1 = B1 )
=> ( huffma1401021291ibling(A,huffma1146269203erNode(A,W1,huffma2021818691e_Leaf(A,W_b,B1),huffma2021818691e_Leaf(A,W_c,C)),A1) = C ) )
& ( ( A1 != B1 )
=> ( ( ( A1 = C )
=> ( huffma1401021291ibling(A,huffma1146269203erNode(A,W1,huffma2021818691e_Leaf(A,W_b,B1),huffma2021818691e_Leaf(A,W_c,C)),A1) = B1 ) )
& ( ( A1 != C )
=> ( huffma1401021291ibling(A,huffma1146269203erNode(A,W1,huffma2021818691e_Leaf(A,W_b,B1),huffma2021818691e_Leaf(A,W_c,C)),A1) = A1 ) ) ) ) ) ).
tff(fact_11_sibling_Osimps_I4_J,axiom,
! [B: $tType,Vb: huffma1450048681e_tree(B),Va: huffma1450048681e_tree(B),V: nat,W: nat,T_11: huffma1450048681e_tree(B),Aa: B] :
( ( member(B,Aa,huffma675207370phabet(B,T_11))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,W,T_11,huffma1146269203erNode(B,V,Va,Vb)),Aa) = huffma1401021291ibling(B,T_11,Aa) ) )
& ( ~ member(B,Aa,huffma675207370phabet(B,T_11))
=> ( ( member(B,Aa,huffma675207370phabet(B,huffma1146269203erNode(B,V,Va,Vb)))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,W,T_11,huffma1146269203erNode(B,V,Va,Vb)),Aa) = huffma1401021291ibling(B,huffma1146269203erNode(B,V,Va,Vb),Aa) ) )
& ( ~ member(B,Aa,huffma675207370phabet(B,huffma1146269203erNode(B,V,Va,Vb)))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,W,T_11,huffma1146269203erNode(B,V,Va,Vb)),Aa) = Aa ) ) ) ) ) ).
tff(fact_12_sibling_Osimps_I3_J,axiom,
! [B: $tType,T_21: huffma1450048681e_tree(B),W: nat,Vb: huffma1450048681e_tree(B),Va: huffma1450048681e_tree(B),V: nat,Aa: B] :
( ( member(B,Aa,huffma675207370phabet(B,huffma1146269203erNode(B,V,Va,Vb)))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,W,huffma1146269203erNode(B,V,Va,Vb),T_21),Aa) = huffma1401021291ibling(B,huffma1146269203erNode(B,V,Va,Vb),Aa) ) )
& ( ~ member(B,Aa,huffma675207370phabet(B,huffma1146269203erNode(B,V,Va,Vb)))
=> ( ( member(B,Aa,huffma675207370phabet(B,T_21))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,W,huffma1146269203erNode(B,V,Va,Vb),T_21),Aa) = huffma1401021291ibling(B,T_21,Aa) ) )
& ( ~ member(B,Aa,huffma675207370phabet(B,T_21))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,W,huffma1146269203erNode(B,V,Va,Vb),T_21),Aa) = Aa ) ) ) ) ) ).
tff(fact_13_depth__le__height,axiom,
! [A: $tType,A1: A,T: huffma1450048681e_tree(A)] : ord_less_eq(nat,huffma410068972_depth(A,T,A1),huffma945805758height(A,T)) ).
tff(fact_14_tree_Osimps_I2_J,axiom,
! [B: $tType,Tree22: huffma1450048681e_tree(B),Tree12: huffma1450048681e_tree(B),Nat3: nat,Tree21: huffma1450048681e_tree(B),Tree11: huffma1450048681e_tree(B),Nat2: nat] :
( ( huffma1146269203erNode(B,Nat2,Tree11,Tree21) = huffma1146269203erNode(B,Nat3,Tree12,Tree22) )
<=> ( ( Nat2 = Nat3 )
& ( Tree11 = Tree12 )
& ( Tree21 = Tree22 ) ) ) ).
tff(fact_15_tree_Osimps_I1_J,axiom,
! [B: $tType,A3: B,Nat3: nat,Aa: B,Nat2: nat] :
( ( huffma2021818691e_Leaf(B,Nat2,Aa) = huffma2021818691e_Leaf(B,Nat3,A3) )
<=> ( ( Nat2 = Nat3 )
& ( Aa = A3 ) ) ) ).
tff(fact_16_height_Osimps_I1_J,axiom,
! [A: $tType,A1: A,W1: nat] : huffma945805758height(A,huffma2021818691e_Leaf(A,W1,A1)) = zero_zero(nat) ).
tff(fact_17_tree_Osimps_I3_J,axiom,
! [A: $tType,Tree2: huffma1450048681e_tree(A),Tree1: huffma1450048681e_tree(A),Nat1: nat,A1: A,Nat: nat] : huffma2021818691e_Leaf(A,Nat,A1) != huffma1146269203erNode(A,Nat1,Tree1,Tree2) ).
tff(fact_18_tree_Osimps_I4_J,axiom,
! [A: $tType,A1: A,Nat: nat,Tree2: huffma1450048681e_tree(A),Tree1: huffma1450048681e_tree(A),Nat1: nat] : huffma1146269203erNode(A,Nat1,Tree1,Tree2) != huffma2021818691e_Leaf(A,Nat,A1) ).
tff(fact_19_height__gt__0__alphabet__eq__imp__height__gt__0,axiom,
! [B: $tType,U: huffma1450048681e_tree(B),Ta: huffma1450048681e_tree(B)] :
( ord_less(nat,zero_zero(nat),huffma945805758height(B,Ta))
=> ( huffma1518433673istent(B,Ta)
=> ( ( huffma675207370phabet(B,Ta) = huffma675207370phabet(B,U) )
=> ord_less(nat,zero_zero(nat),huffma945805758height(B,U)) ) ) ) ).
tff(fact_20_height__gt__0__notin__alphabet__imp__sibling__left,axiom,
! [B: $tType,T_21: huffma1450048681e_tree(B),W: nat,Aa: B,T_11: huffma1450048681e_tree(B)] :
( ord_less(nat,zero_zero(nat),huffma945805758height(B,T_11))
=> ( ~ member(B,Aa,huffma675207370phabet(B,T_11))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,W,T_11,T_21),Aa) = huffma1401021291ibling(B,T_21,Aa) ) ) ) ).
tff(fact_21_height__gt__0__notin__alphabet__imp__sibling__right,axiom,
! [B: $tType,W: nat,T_11: huffma1450048681e_tree(B),Aa: B,T_21: huffma1450048681e_tree(B)] :
( ord_less(nat,zero_zero(nat),huffma945805758height(B,T_21))
=> ( ~ member(B,Aa,huffma675207370phabet(B,T_11))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,W,T_11,T_21),Aa) = huffma1401021291ibling(B,T_21,Aa) ) ) ) ).
tff(fact_22_height__gt__0__in__alphabet__imp__sibling__left,axiom,
! [B: $tType,T_21: huffma1450048681e_tree(B),W: nat,Aa: B,T_11: huffma1450048681e_tree(B)] :
( ord_less(nat,zero_zero(nat),huffma945805758height(B,T_11))
=> ( member(B,Aa,huffma675207370phabet(B,T_11))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,W,T_11,T_21),Aa) = huffma1401021291ibling(B,T_11,Aa) ) ) ) ).
tff(fact_23_height__gt__0__in__alphabet__imp__sibling__right,axiom,
! [B: $tType,W: nat,T_11: huffma1450048681e_tree(B),Aa: B,T_21: huffma1450048681e_tree(B)] :
( ord_less(nat,zero_zero(nat),huffma945805758height(B,T_21))
=> ( member(B,Aa,huffma675207370phabet(B,T_11))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,W,T_11,T_21),Aa) = huffma1401021291ibling(B,T_11,Aa) ) ) ) ).
tff(fact_24_either__height__gt__0__imp__sibling,axiom,
! [B: $tType,W: nat,Aa: B,T_21: huffma1450048681e_tree(B),T_11: huffma1450048681e_tree(B)] :
( ( ord_less(nat,zero_zero(nat),huffma945805758height(B,T_11))
| ord_less(nat,zero_zero(nat),huffma945805758height(B,T_21)) )
=> ( ( member(B,Aa,huffma675207370phabet(B,T_11))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,W,T_11,T_21),Aa) = huffma1401021291ibling(B,T_11,Aa) ) )
& ( ~ member(B,Aa,huffma675207370phabet(B,T_11))
=> ( huffma1401021291ibling(B,huffma1146269203erNode(B,W,T_11,T_21),Aa) = huffma1401021291ibling(B,T_21,Aa) ) ) ) ) ).
tff(fact_25_height__0__imp__sibling__id,axiom,
! [A: $tType,A1: A,T: huffma1450048681e_tree(A)] :
( ( huffma945805758height(A,T) = zero_zero(nat) )
=> ( huffma1401021291ibling(A,T,A1) = A1 ) ) ).
tff(fact_26_le0,axiom,
! [N: nat] : ord_less_eq(nat,zero_zero(nat),N) ).
tff(fact_27_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ord_less_eq(nat,zero_zero(nat),N) ).
tff(fact_28_le__0__eq,axiom,
! [N1: nat] :
( ord_less_eq(nat,N1,zero_zero(nat))
<=> ( N1 = zero_zero(nat) ) ) ).
tff(fact_29_neq0__conv,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),N1) ) ).
tff(fact_30_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_31_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_32_tree_Osize_I3_J,axiom,
! [A: $tType,A1: A,Nat: nat] : size_size(huffma1450048681e_tree(A),huffma2021818691e_Leaf(A,Nat,A1)) = zero_zero(nat) ).
tff(fact_33_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X1: A] : ord_less_eq(A,X1,X1) ) ).
tff(fact_34_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less_eq(A,X1,Y1)
=> ord_less_eq(A,Y1,X1) ) ) ).
tff(fact_35_le__funE,axiom,
! [C1: $tType,B: $tType] :
( ord(C1)
=> ! [X: B,G: fun(B,C1),F: fun(B,C1)] :
( ord_less_eq(fun(B,C1),F,G)
=> ord_less_eq(C1,aa(B,C1,F,X),aa(B,C1,G,X)) ) ) ).
tff(fact_36_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
=> ( ord_less_eq(A,Y1,Z)
=> ord_less_eq(A,X1,Z) ) ) ) ).
tff(fact_37_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
=> ( ord_less_eq(A,Y1,X1)
=> ( X1 = Y1 ) ) ) ) ).
tff(fact_38_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ( ( B1 = C )
=> ord_less_eq(A,A1,C) ) ) ) ).
tff(fact_39_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B1: A,A1: A] :
( ( A1 = B1 )
=> ( ord_less_eq(A,B1,C)
=> ord_less_eq(A,A1,C) ) ) ) ).
tff(fact_40_order__antisym__conv,axiom,
! [B: $tType] :
( order(B)
=> ! [X: B,Y: B] :
( ord_less_eq(B,Y,X)
=> ( ord_less_eq(B,X,Y)
<=> ( X = Y ) ) ) ) ).
tff(fact_41_le__funD,axiom,
! [C1: $tType,B: $tType] :
( ord(C1)
=> ! [X: B,G: fun(B,C1),F: fun(B,C1)] :
( ord_less_eq(fun(B,C1),F,G)
=> ord_less_eq(C1,aa(B,C1,F,X),aa(B,C1,G,X)) ) ) ).
tff(fact_42_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ( X1 = Y1 )
=> ord_less_eq(A,X1,Y1) ) ) ).
tff(fact_43_order__eq__iff,axiom,
! [B: $tType] :
( order(B)
=> ! [Y: B,X: B] :
( ( X = Y )
<=> ( ord_less_eq(B,X,Y)
& ord_less_eq(B,Y,X) ) ) ) ).
tff(fact_44_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
| ord_less_eq(A,Y1,X1) ) ) ).
tff(fact_45_le__fun__def,axiom,
! [C1: $tType,B: $tType] :
( ord(C1)
=> ! [G: fun(B,C1),F: fun(B,C1)] :
( ord_less_eq(fun(B,C1),F,G)
<=> ! [X2: B] : ord_less_eq(C1,aa(B,C1,F,X2),aa(B,C1,G,X2)) ) ) ).
tff(fact_46_zero__reorient,axiom,
! [B: $tType] :
( zero(B)
=> ! [X: B] :
( ( zero_zero(B) = X )
<=> ( X = zero_zero(B) ) ) ) ).
tff(fact_47_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less(A,X1,Y1)
=> ( ( X1 != Y1 )
=> ord_less(A,Y1,X1) ) ) ) ).
tff(fact_48_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ~ ord_less(A,Y1,X1) ) ) ).
tff(fact_49_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( ord_less(A,Y1,Z)
=> ord_less(A,X1,Z) ) ) ) ).
tff(fact_50_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ( B1 = C )
=> ord_less(A,A1,C) ) ) ) ).
tff(fact_51_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B1: A,A1: A] :
( ( A1 = B1 )
=> ( ord_less(A,B1,C)
=> ord_less(A,A1,C) ) ) ) ).
tff(fact_52_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ~ ord_less(A,B1,A1) ) ) ).
tff(fact_53_order__less__imp__triv,axiom,
! [B: $tType] :
( preorder(B)
=> ! [P: bool,Y: B,X: B] :
( ord_less(B,X,Y)
=> ( ord_less(B,Y,X)
=> pp(P) ) ) ) ).
tff(fact_54_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( Y1 != X1 ) ) ) ).
tff(fact_55_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( X1 != Y1 ) ) ) ).
tff(fact_56_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ~ ord_less(A,Y1,X1) ) ) ).
tff(fact_57_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ~ ord_less(A,Y1,X1) ) ) ).
tff(fact_58_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( X1 != Y1 ) ) ) ).
tff(fact_59_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ( X1 != Y1 )
=> ( ~ ord_less(A,X1,Y1)
=> ord_less(A,Y1,X1) ) ) ) ).
tff(fact_60_linorder__antisym__conv3,axiom,
! [B: $tType] :
( linorder(B)
=> ! [X: B,Y: B] :
( ~ ord_less(B,Y,X)
=> ( ~ ord_less(B,X,Y)
<=> ( X = Y ) ) ) ) ).
tff(fact_61_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
| ( X1 = Y1 )
| ord_less(A,Y1,X1) ) ) ).
tff(fact_62_not__less__iff__gr__or__eq,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y: B,X: B] :
( ~ ord_less(B,X,Y)
<=> ( ord_less(B,Y,X)
| ( X = Y ) ) ) ) ).
tff(fact_63_linorder__neq__iff,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y: B,X: B] :
( ( X != Y )
<=> ( ord_less(B,X,Y)
| ord_less(B,Y,X) ) ) ) ).
tff(fact_64_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X1: A] : ~ ord_less(A,X1,X1) ) ).
tff(fact_65_nat__less__cases,axiom,
! [P: fun(nat,fun(nat,bool)),N1: nat,M1: nat] :
( ( ord_less(nat,M1,N1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N1),M1)) )
=> ( ( ( M1 = N1 )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N1),M1)) )
=> ( ( ord_less(nat,N1,M1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N1),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N1),M1)) ) ) ) ).
tff(fact_66_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_67_less__not__refl2,axiom,
! [M: nat,N: nat] :
( ord_less(nat,N,M)
=> ( M != N ) ) ).
tff(fact_68_less__irrefl__nat,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_69_linorder__neqE__nat,axiom,
! [Y1: nat,X1: nat] :
( ( X1 != Y1 )
=> ( ~ ord_less(nat,X1,Y1)
=> ord_less(nat,Y1,X1) ) ) ).
tff(fact_70_nat__neq__iff,axiom,
! [N1: nat,M1: nat] :
( ( M1 != N1 )
<=> ( ord_less(nat,M1,N1)
| ord_less(nat,N1,M1) ) ) ).
tff(fact_71_less__not__refl,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_72_le__antisym,axiom,
! [N: nat,M: nat] :
( ord_less_eq(nat,M,N)
=> ( ord_less_eq(nat,N,M)
=> ( M = N ) ) ) ).
tff(fact_73_le__trans,axiom,
! [K: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ( ord_less_eq(nat,J,K)
=> ord_less_eq(nat,I,K) ) ) ).
tff(fact_74_mem__def,axiom,
! [B: $tType,A2: fun(B,bool),X: B] :
( member(B,X,A2)
<=> pp(aa(B,bool,A2,X)) ) ).
tff(fact_75_eq__imp__le,axiom,
! [N: nat,M: nat] :
( ( M = N )
=> ord_less_eq(nat,M,N) ) ).
tff(fact_76_nat__le__linear,axiom,
! [N: nat,M: nat] :
( ord_less_eq(nat,M,N)
| ord_less_eq(nat,N,M) ) ).
tff(fact_77_le__refl,axiom,
! [N: nat] : ord_less_eq(nat,N,N) ).
tff(fact_78_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
=> ( ord_less(A,Y1,Z)
=> ord_less(A,X1,Z) ) ) ) ).
tff(fact_79_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( ord_less_eq(A,Y1,Z)
=> ord_less(A,X1,Z) ) ) ) ).
tff(fact_80_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ( ( A1 != B1 )
=> ord_less(A,A1,B1) ) ) ) ).
tff(fact_81_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
=> ( ord_less(A,X1,Y1)
| ( X1 = Y1 ) ) ) ) ).
tff(fact_82_linorder__antisym__conv2,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y: B,X: B] :
( ord_less_eq(B,X,Y)
=> ( ~ ord_less(B,X,Y)
<=> ( X = Y ) ) ) ) ).
tff(fact_83_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ord_less_eq(A,X1,Y1) ) ) ).
tff(fact_84_leD,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X1: A,Y1: A] :
( ord_less_eq(A,Y1,X1)
=> ~ ord_less(A,X1,Y1) ) ) ).
tff(fact_85_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B1: A,A1: A] :
( ( A1 != B1 )
=> ( ord_less_eq(A,A1,B1)
=> ord_less(A,A1,B1) ) ) ) ).
tff(fact_86_linorder__antisym__conv1,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y: B,X: B] :
( ~ ord_less(B,X,Y)
=> ( ord_less_eq(B,X,Y)
<=> ( X = Y ) ) ) ) ).
tff(fact_87_not__leE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X1: A,Y1: A] :
( ~ ord_less_eq(A,Y1,X1)
=> ord_less(A,X1,Y1) ) ) ).
tff(fact_88_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less(A,X1,Y1)
=> ord_less_eq(A,Y1,X1) ) ) ).
tff(fact_89_order__le__less,axiom,
! [B: $tType] :
( order(B)
=> ! [Y: B,X: B] :
( ord_less_eq(B,X,Y)
<=> ( ord_less(B,X,Y)
| ( X = Y ) ) ) ) ).
tff(fact_90_less__le__not__le,axiom,
! [B: $tType] :
( preorder(B)
=> ! [Y: B,X: B] :
( ord_less(B,X,Y)
<=> ( ord_less_eq(B,X,Y)
& ~ ord_less_eq(B,Y,X) ) ) ) ).
tff(fact_91_order__less__le,axiom,
! [B: $tType] :
( order(B)
=> ! [Y: B,X: B] :
( ord_less(B,X,Y)
<=> ( ord_less_eq(B,X,Y)
& ( X != Y ) ) ) ) ).
tff(fact_92_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
| ord_less(A,Y1,X1) ) ) ).
tff(fact_93_linorder__not__le,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y: B,X: B] :
( ~ ord_less_eq(B,X,Y)
<=> ord_less(B,Y,X) ) ) ).
tff(fact_94_linorder__not__less,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y: B,X: B] :
( ~ ord_less(B,X,Y)
<=> ord_less_eq(B,Y,X) ) ) ).
tff(fact_95_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_96_gr__implies__not0,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ( N != zero_zero(nat) ) ) ).
tff(fact_97_not__less0,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_98_less__or__eq__imp__le,axiom,
! [N: nat,M: nat] :
( ( ord_less(nat,M,N)
| ( M = N ) )
=> ord_less_eq(nat,M,N) ) ).
%----Arities (12)
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oord,axiom,
! [T_1: $tType,T_2: $tType] :
( ord(T_2)
=> ord(fun(T_1,T_2)) ) ).
tff(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
huffma1518433673istent(a1,t) ).
tff(conj_1,conjecture,
huffma410068972_depth(a1,t,huffma1401021291ibling(a1,t,a)) = huffma410068972_depth(a1,t,a) ).
%------------------------------------------------------------------------------