TPTP Problem File: SWW629_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW629_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Mergesort queue-T-WP parameter mergesort
% Version : Especial : Let and conditional terms encoded away.
% English :
% Refs : [Fil14] Filliatre (2014), Email to Geoff Sutcliffe
% : [BF+] Bobot et al. (URL), Toccata: Certified Programs and Cert
% Source : [Fil14]
% Names : mergesort_queue-T-WP_parameter_mergesort [Fil14]
% Status : Theorem
% Rating : 0.00 v8.2.0, 0.12 v7.5.0, 0.30 v7.4.0, 0.12 v7.3.0, 0.00 v7.0.0, 0.14 v6.4.0, 0.00 v6.3.0, 0.29 v6.2.0, 0.38 v6.1.0
% Syntax : Number of formulae : 114 ( 41 unt; 40 typ; 0 def)
% Number of atoms : 159 ( 60 equ)
% Maximal formula atoms : 23 ( 1 avg)
% Number of connectives : 91 ( 6 ~; 8 |; 22 &)
% ( 8 <=>; 47 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number arithmetic : 29 ( 6 atm; 8 fun; 12 num; 3 var)
% Number of types : 8 ( 6 usr; 1 ari)
% Number of type conns : 51 ( 25 >; 26 *; 0 +; 0 <<)
% Number of predicates : 8 ( 5 usr; 0 prp; 1-3 aty)
% Number of functors : 33 ( 29 usr; 11 con; 0-5 aty)
% Number of variables : 212 ( 206 !; 6 ?; 212 :)
% SPC : TF0_THM_EQU_ARI
% Comments :
%------------------------------------------------------------------------------
tff(uni,type,
uni: $tType ).
tff(ty,type,
ty: $tType ).
tff(sort,type,
sort1: ( ty * uni ) > $o ).
tff(witness,type,
witness1: ty > uni ).
tff(witness_sort1,axiom,
! [A: ty] : sort1(A,witness1(A)) ).
tff(int,type,
int: ty ).
tff(real,type,
real: ty ).
tff(bool,type,
bool1: $tType ).
tff(bool1,type,
bool: ty ).
tff(true,type,
true1: bool1 ).
tff(false,type,
false1: bool1 ).
tff(match_bool,type,
match_bool1: ( ty * bool1 * uni * uni ) > uni ).
tff(match_bool_sort1,axiom,
! [A: ty,X: bool1,X1: uni,X2: uni] : sort1(A,match_bool1(A,X,X1,X2)) ).
tff(match_bool_True,axiom,
! [A: ty,Z: uni,Z1: uni] :
( sort1(A,Z)
=> ( match_bool1(A,true1,Z,Z1) = Z ) ) ).
tff(match_bool_False,axiom,
! [A: ty,Z: uni,Z1: uni] :
( sort1(A,Z1)
=> ( match_bool1(A,false1,Z,Z1) = Z1 ) ) ).
tff(true_False,axiom,
true1 != false1 ).
tff(bool_inversion,axiom,
! [U: bool1] :
( ( U = true1 )
| ( U = false1 ) ) ).
tff(tuple0,type,
tuple02: $tType ).
tff(tuple01,type,
tuple0: ty ).
tff(tuple02,type,
tuple03: tuple02 ).
tff(tuple0_inversion,axiom,
! [U: tuple02] : U = tuple03 ).
tff(qtmark,type,
qtmark: ty ).
tff(compatOrderMult,axiom,
! [X: $int,Y: $int,Z: $int] :
( $lesseq(X,Y)
=> ( $lesseq(0,Z)
=> $lesseq($product(X,Z),$product(Y,Z)) ) ) ).
tff(list,type,
list: ty > ty ).
tff(nil,type,
nil: ty > uni ).
tff(nil_sort1,axiom,
! [A: ty] : sort1(list(A),nil(A)) ).
tff(cons,type,
cons: ( ty * uni * uni ) > uni ).
tff(cons_sort1,axiom,
! [A: ty,X: uni,X1: uni] : sort1(list(A),cons(A,X,X1)) ).
tff(match_list,type,
match_list1: ( ty * ty * uni * uni * uni ) > uni ).
tff(match_list_sort1,axiom,
! [A: ty,A1: ty,X: uni,X1: uni,X2: uni] : sort1(A1,match_list1(A1,A,X,X1,X2)) ).
tff(match_list_Nil1,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni] :
( sort1(A1,Z)
=> ( match_list1(A1,A,nil(A),Z,Z1) = Z ) ) ).
tff(match_list_Cons1,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni,U: uni,U1: uni] :
( sort1(A1,Z1)
=> ( match_list1(A1,A,cons(A,U,U1),Z,Z1) = Z1 ) ) ).
tff(nil_Cons1,axiom,
! [A: ty,V: uni,V1: uni] : nil(A) != cons(A,V,V1) ).
tff(cons_proj_1,type,
cons_proj_11: ( ty * uni ) > uni ).
tff(cons_proj_1_sort1,axiom,
! [A: ty,X: uni] : sort1(A,cons_proj_11(A,X)) ).
tff(cons_proj_1_def1,axiom,
! [A: ty,U: uni,U1: uni] :
( sort1(A,U)
=> ( cons_proj_11(A,cons(A,U,U1)) = U ) ) ).
tff(cons_proj_2,type,
cons_proj_21: ( ty * uni ) > uni ).
tff(cons_proj_2_sort1,axiom,
! [A: ty,X: uni] : sort1(list(A),cons_proj_21(A,X)) ).
tff(cons_proj_2_def1,axiom,
! [A: ty,U: uni,U1: uni] : cons_proj_21(A,cons(A,U,U1)) = U1 ).
tff(list_inversion1,axiom,
! [A: ty,U: uni] :
( ( U = nil(A) )
| ( U = cons(A,cons_proj_11(A,U),cons_proj_21(A,U)) ) ) ).
tff(mem,type,
mem: ( ty * uni * uni ) > $o ).
tff(mem_def,axiom,
! [A: ty,X: uni] :
( sort1(A,X)
=> ( ~ mem(A,X,nil(A))
& ! [X1: uni,X2: uni] :
( sort1(A,X1)
=> ( mem(A,X,cons(A,X1,X2))
<=> ( ( X = X1 )
| mem(A,X,X2) ) ) ) ) ) ).
tff(length,type,
length2: ( ty * uni ) > $int ).
tff(length_def,axiom,
! [A: ty] :
( ( length2(A,nil(A)) = 0 )
& ! [X: uni,X1: uni] : length2(A,cons(A,X,X1)) = $sum(1,length2(A,X1)) ) ).
tff(length_nonnegative,axiom,
! [A: ty,L: uni] : $lesseq(0,length2(A,L)) ).
tff(length_nil,axiom,
! [A: ty,L: uni] :
( ( length2(A,L) = 0 )
<=> ( L = nil(A) ) ) ).
tff(infix_plpl,type,
infix_plpl: ( ty * uni * uni ) > uni ).
tff(infix_plpl_sort1,axiom,
! [A: ty,X: uni,X1: uni] : sort1(list(A),infix_plpl(A,X,X1)) ).
tff(infix_plpl_def,axiom,
! [A: ty,L2: uni] :
( ( infix_plpl(A,nil(A),L2) = L2 )
& ! [X: uni,X1: uni] : infix_plpl(A,cons(A,X,X1),L2) = cons(A,X,infix_plpl(A,X1,L2)) ) ).
tff(append_assoc,axiom,
! [A: ty,L1: uni,L2: uni,L3: uni] : infix_plpl(A,L1,infix_plpl(A,L2,L3)) = infix_plpl(A,infix_plpl(A,L1,L2),L3) ).
tff(append_l_nil,axiom,
! [A: ty,L: uni] : infix_plpl(A,L,nil(A)) = L ).
tff(append_length,axiom,
! [A: ty,L1: uni,L2: uni] : length2(A,infix_plpl(A,L1,L2)) = $sum(length2(A,L1),length2(A,L2)) ).
tff(mem_append,axiom,
! [A: ty,X: uni,L1: uni,L2: uni] :
( mem(A,X,infix_plpl(A,L1,L2))
<=> ( mem(A,X,L1)
| mem(A,X,L2) ) ) ).
tff(mem_decomp,axiom,
! [A: ty,X: uni,L: uni] :
( mem(A,X,L)
=> ? [L1: uni,L2: uni] :
( sort1(list(A),L1)
& sort1(list(A),L2)
& ( L = infix_plpl(A,L1,cons(A,X,L2)) ) ) ) ).
tff(num_occ,type,
num_occ1: ( ty * uni * uni ) > $int ).
tff(num_occ_def,axiom,
! [A: ty,X: uni] :
( sort1(A,X)
=> ( ( num_occ1(A,X,nil(A)) = 0 )
& ! [X1: uni,X2: uni] :
( sort1(A,X1)
=> ( ( ( X = X1 )
=> ( num_occ1(A,X,cons(A,X1,X2)) = $sum(1,num_occ1(A,X,X2)) ) )
& ( ( X != X1 )
=> ( num_occ1(A,X,cons(A,X1,X2)) = $sum(0,num_occ1(A,X,X2)) ) ) ) ) ) ) ).
tff(mem_Num_Occ,axiom,
! [A: ty,X: uni,L: uni] :
( mem(A,X,L)
<=> $less(0,num_occ1(A,X,L)) ) ).
tff(append_Num_Occ,axiom,
! [A: ty,X: uni,L1: uni,L2: uni] : num_occ1(A,X,infix_plpl(A,L1,L2)) = $sum(num_occ1(A,X,L1),num_occ1(A,X,L2)) ).
tff(reverse,type,
reverse: ( ty * uni ) > uni ).
tff(reverse_sort1,axiom,
! [A: ty,X: uni] : sort1(list(A),reverse(A,X)) ).
tff(reverse_def,axiom,
! [A: ty] :
( ( reverse(A,nil(A)) = nil(A) )
& ! [X: uni,X1: uni] : reverse(A,cons(A,X,X1)) = infix_plpl(A,reverse(A,X1),cons(A,X,nil(A))) ) ).
tff(reverse_append,axiom,
! [A: ty,L1: uni,L2: uni,X: uni] : infix_plpl(A,reverse(A,cons(A,X,L1)),L2) = infix_plpl(A,reverse(A,L1),cons(A,X,L2)) ).
tff(reverse_cons,axiom,
! [A: ty,L: uni,X: uni] : reverse(A,cons(A,X,L)) = infix_plpl(A,reverse(A,L),cons(A,X,nil(A))) ).
tff(reverse_reverse,axiom,
! [A: ty,L: uni] : reverse(A,reverse(A,L)) = L ).
tff(reverse_mem,axiom,
! [A: ty,L: uni,X: uni] :
( mem(A,X,L)
<=> mem(A,X,reverse(A,L)) ) ).
tff(reverse_length,axiom,
! [A: ty,L: uni] : length2(A,reverse(A,L)) = length2(A,L) ).
tff(reverse_num_occ,axiom,
! [A: ty,X: uni,L: uni] : num_occ1(A,X,L) = num_occ1(A,X,reverse(A,L)) ).
tff(permut,type,
permut: ( ty * uni * uni ) > $o ).
tff(permut_def,axiom,
! [A: ty,L1: uni,L2: uni] :
( ( permut(A,L1,L2)
=> ! [X: uni] : num_occ1(A,X,L1) = num_occ1(A,X,L2) )
& ( ! [X: uni] :
( sort1(A,X)
=> ( num_occ1(A,X,L1) = num_occ1(A,X,L2) ) )
=> permut(A,L1,L2) ) ) ).
tff(permut_refl,axiom,
! [A: ty,L: uni] : permut(A,L,L) ).
tff(permut_sym,axiom,
! [A: ty,L1: uni,L2: uni] :
( permut(A,L1,L2)
=> permut(A,L2,L1) ) ).
tff(permut_trans,axiom,
! [A: ty,L1: uni,L2: uni,L3: uni] :
( permut(A,L1,L2)
=> ( permut(A,L2,L3)
=> permut(A,L1,L3) ) ) ).
tff(permut_cons,axiom,
! [A: ty,X: uni,L1: uni,L2: uni] :
( permut(A,L1,L2)
=> permut(A,cons(A,X,L1),cons(A,X,L2)) ) ).
tff(permut_swap,axiom,
! [A: ty,X: uni,Y: uni,L: uni] : permut(A,cons(A,X,cons(A,Y,L)),cons(A,Y,cons(A,X,L))) ).
tff(permut_cons_append,axiom,
! [A: ty,X: uni,L1: uni,L2: uni] : permut(A,infix_plpl(A,cons(A,X,L1),L2),infix_plpl(A,L1,cons(A,X,L2))) ).
tff(permut_assoc,axiom,
! [A: ty,L1: uni,L2: uni,L3: uni] : permut(A,infix_plpl(A,infix_plpl(A,L1,L2),L3),infix_plpl(A,L1,infix_plpl(A,L2,L3))) ).
tff(permut_append,axiom,
! [A: ty,L1: uni,L2: uni,K1: uni,K2: uni] :
( permut(A,L1,K1)
=> ( permut(A,L2,K2)
=> permut(A,infix_plpl(A,L1,L2),infix_plpl(A,K1,K2)) ) ) ).
tff(permut_append_swap,axiom,
! [A: ty,L1: uni,L2: uni] : permut(A,infix_plpl(A,L1,L2),infix_plpl(A,L2,L1)) ).
tff(permut_mem,axiom,
! [A: ty,X: uni,L1: uni,L2: uni] :
( permut(A,L1,L2)
=> ( mem(A,X,L1)
=> mem(A,X,L2) ) ) ).
tff(permut_length,axiom,
! [A: ty,L1: uni,L2: uni] :
( permut(A,L1,L2)
=> ( length2(A,L1) = length2(A,L2) ) ) ).
tff(t,type,
t: ty > ty ).
tff(mk_t,type,
mk_t: ( ty * uni ) > uni ).
tff(mk_t_sort1,axiom,
! [A: ty,X: uni] : sort1(t(A),mk_t(A,X)) ).
tff(elts,type,
elts: ( ty * uni ) > uni ).
tff(elts_sort1,axiom,
! [A: ty,X: uni] : sort1(list(A),elts(A,X)) ).
tff(elts_def1,axiom,
! [A: ty,U: uni] : elts(A,mk_t(A,U)) = U ).
tff(t_inversion1,axiom,
! [A: ty,U: uni] : U = mk_t(A,elts(A,U)) ).
tff(length1,type,
length3: ( ty * uni ) > $int ).
tff(length_def1,axiom,
! [A: ty,Q: uni] : length3(A,Q) = length2(A,elts(A,Q)) ).
tff(elt,type,
elt1: $tType ).
tff(elt1,type,
elt: ty ).
tff(le,type,
le1: ( elt1 * elt1 ) > $o ).
tff(refl1,axiom,
! [X: elt1] : le1(X,X) ).
tff(trans1,axiom,
! [X: elt1,Y: elt1,Z: elt1] :
( le1(X,Y)
=> ( le1(Y,Z)
=> le1(X,Z) ) ) ).
tff(total1,axiom,
! [X: elt1,Y: elt1] :
( le1(X,Y)
| le1(Y,X) ) ).
tff(list_elt,type,
list_elt: $tType ).
tff(sorted,type,
sorted1: list_elt > $o ).
tff(t2tb,type,
t2tb: list_elt > uni ).
tff(t2tb_sort,axiom,
! [X: list_elt] : sort1(list(elt),t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > list_elt ).
tff(bridgeL,axiom,
! [I: list_elt] : tb2t(t2tb(I)) = I ).
tff(bridgeR,axiom,
! [J: uni] : t2tb(tb2t(J)) = J ).
tff(sorted_Nil,axiom,
sorted1(tb2t(nil(elt))) ).
tff(t2tb1,type,
t2tb1: elt1 > uni ).
tff(t2tb_sort1,axiom,
! [X: elt1] : sort1(elt,t2tb1(X)) ).
tff(tb2t1,type,
tb2t1: uni > elt1 ).
tff(bridgeL1,axiom,
! [I: elt1] : tb2t1(t2tb1(I)) = I ).
tff(bridgeR1,axiom,
! [J: uni] :
( sort1(elt,J)
=> ( t2tb1(tb2t1(J)) = J ) ) ).
tff(sorted_One,axiom,
! [X: elt1] : sorted1(tb2t(cons(elt,t2tb1(X),nil(elt)))) ).
tff(sorted_Two,axiom,
! [X: elt1,Y: elt1,L: list_elt] :
( le1(X,Y)
=> ( sorted1(tb2t(cons(elt,t2tb1(Y),t2tb(L))))
=> sorted1(tb2t(cons(elt,t2tb1(X),cons(elt,t2tb1(Y),t2tb(L))))) ) ) ).
tff(sorted_inversion,axiom,
! [Z: list_elt] :
( sorted1(Z)
=> ( ( Z = tb2t(nil(elt)) )
| ? [X: elt1] : Z = tb2t(cons(elt,t2tb1(X),nil(elt)))
| ? [X: elt1,Y: elt1,L: list_elt] :
( le1(X,Y)
& sorted1(tb2t(cons(elt,t2tb1(Y),t2tb(L))))
& ( Z = tb2t(cons(elt,t2tb1(X),cons(elt,t2tb1(Y),t2tb(L)))) ) ) ) ) ).
tff(sorted_mem,axiom,
! [X: elt1,L: list_elt] :
( ( ! [Y: elt1] :
( mem(elt,t2tb1(Y),t2tb(L))
=> le1(X,Y) )
& sorted1(L) )
<=> sorted1(tb2t(cons(elt,t2tb1(X),t2tb(L)))) ) ).
tff(sorted_append,axiom,
! [L1: list_elt,L2: list_elt] :
( ( sorted1(L1)
& sorted1(L2)
& ! [X: elt1,Y: elt1] :
( mem(elt,t2tb1(X),t2tb(L1))
=> ( mem(elt,t2tb1(Y),t2tb(L2))
=> le1(X,Y) ) ) )
<=> sorted1(tb2t(infix_plpl(elt,t2tb(L1),t2tb(L2)))) ) ).
tff(wP_parameter_mergesort,conjecture,
! [Q: list_elt] :
( $less(1,length2(elt,t2tb(Q)))
=> ! [Q1: list_elt] :
( ( Q1 = tb2t(nil(elt)) )
=> ! [Q2: list_elt] :
( ( Q2 = tb2t(nil(elt)) )
=> ! [Q21: list_elt,Q11: list_elt,Q3: list_elt] :
( ( permut(elt,infix_plpl(elt,infix_plpl(elt,t2tb(Q11),t2tb(Q21)),t2tb(Q3)),t2tb(Q))
& ( ( length2(elt,t2tb(Q11)) = length2(elt,t2tb(Q21)) )
| ( ( length2(elt,t2tb(Q3)) = 0 )
& ( length2(elt,t2tb(Q11)) = $sum(length2(elt,t2tb(Q21)),1) ) ) ) )
=> ! [O: bool1] :
( ( ( O = true1 )
<=> ( Q3 = tb2t(nil(elt)) ) )
=> ( ~ ( ( O != true1 ) )
=> ( ( Q3 = tb2t(nil(elt)) )
=> ( permut(elt,infix_plpl(elt,t2tb(Q11),t2tb(Q21)),t2tb(Q))
=> ! [Q12: list_elt] :
( ( sorted1(Q12)
& permut(elt,t2tb(Q12),t2tb(Q11)) )
=> ! [Q22: list_elt] :
( ( sorted1(Q22)
& permut(elt,t2tb(Q22),t2tb(Q21)) )
=> ( ( ( Q3 = tb2t(nil(elt)) )
& sorted1(Q12)
& sorted1(Q22) )
=> ! [Q4: list_elt] :
( ( sorted1(Q4)
& permut(elt,t2tb(Q4),infix_plpl(elt,t2tb(Q12),t2tb(Q22))) )
=> ( sorted1(Q4)
& permut(elt,t2tb(Q4),t2tb(Q)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------