TPTP Problem File: SWW628=2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SWW628=2 : TPTP v7.4.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Mergesort queue-T-WP parameter merge
% 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_merge [Fil14]
% Status : Theorem
% Rating : 0.60 v7.4.0, 0.38 v7.3.0, 0.17 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.29 v6.2.0, 0.75 v6.1.0
% Syntax : Number of formulae : 114 ( 41 unit; 40 type)
% Number of atoms : 165 ( 64 equality)
% Maximal formula depth : 27 ( 4 average)
% Number of connectives : 101 ( 10 ~; 7 |; 25 &)
% ( 7 <=>; 52 =>; 0 <=; 0 <~>)
% ( 0 ~|; 0 ~&)
% Number of type conns : 51 ( 25 >; 26 *; 0 +; 0 <<)
% Number of predicates : 51 ( 43 propositional; 0-3 arity)
% Number of functors : 28 ( 6 constant; 0-5 arity)
% Number of variables : 223 ( 0 sgn; 211 !; 12 ?)
% ( 223 :; 0 !>; 0 ?*)
% Maximal term depth : 5 ( 2 average)
% Arithmetic symbols : 9 ( 2 prd; 2 fun; 2 num; 3 var)
% 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_merge,conjecture,(
! [Q1: list_elt,Q2: list_elt,Q: list_elt] :
( ( Q = tb2t(nil(elt))
& sorted1(Q1)
& sorted1(Q2) )
=> ! [Q3: list_elt,Q21: list_elt,Q11: list_elt] :
( ( sorted1(Q11)
& sorted1(Q21)
& sorted1(Q3)
& ! [X: elt1,Y: elt1] :
( mem(elt,t2tb1(X),t2tb(Q3))
=> ( mem(elt,t2tb1(Y),t2tb(Q11))
=> le1(X,Y) ) )
& ! [X: elt1,Y: elt1] :
( mem(elt,t2tb1(X),t2tb(Q3))
=> ( mem(elt,t2tb1(Y),t2tb(Q21))
=> le1(X,Y) ) )
& permut(elt,infix_plpl(elt,infix_plpl(elt,t2tb(Q3),t2tb(Q11)),t2tb(Q21)),infix_plpl(elt,t2tb(Q1),t2tb(Q2))) )
=> ( $less(0,length2(elt,t2tb(Q11)))
=> ( length2(elt,t2tb(Q11)) != 0
=> ( length2(elt,t2tb(Q21)) != 0
=> ( Q11 != tb2t(nil(elt))
=> ! [X1: elt1] :
( ? [X: elt1,X2: list_elt] :
( Q11 = tb2t(cons(elt,t2tb1(X),t2tb(X2)))
& X1 = X )
=> ( Q21 != tb2t(nil(elt))
=> ! [X2: elt1] :
( ? [X: elt1,X3: list_elt] :
( Q21 = tb2t(cons(elt,t2tb1(X),t2tb(X3)))
& X2 = X )
=> ( ~ le1(X1,X2)
=> ( Q21 != tb2t(nil(elt))
=> ! [Q22: list_elt,O: elt1] :
( ? [X: elt1,X3: list_elt] :
( Q21 = tb2t(cons(elt,t2tb1(X),t2tb(X3)))
& O = X
& Q22 = X3 )
=> ! [Q4: list_elt] :
( Q4 = tb2t(infix_plpl(elt,t2tb(Q3),cons(elt,t2tb1(O),nil(elt))))
=> permut(elt,infix_plpl(elt,infix_plpl(elt,t2tb(Q4),t2tb(Q11)),t2tb(Q22)),infix_plpl(elt,t2tb(Q1),t2tb(Q2))) ) ) ) ) ) ) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------