TPTP Problem File: SWW606_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW606_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Insertion sort naive-T-WP parameter sort
% 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 : insertion_sort_naive-T-WP_parameter_sort [Fil14]
% Status : Theorem
% Rating : 0.62 v8.2.0, 0.75 v7.5.0, 0.80 v7.4.0, 0.62 v7.3.0, 0.50 v7.0.0, 0.71 v6.4.0, 0.33 v6.3.0, 0.71 v6.2.0, 0.88 v6.1.0
% Syntax : Number of formulae : 127 ( 30 unt; 56 typ; 0 def)
% Number of atoms : 218 ( 55 equ)
% Maximal formula atoms : 38 ( 1 avg)
% Number of connectives : 155 ( 8 ~; 1 |; 62 &)
% ( 10 <=>; 74 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number arithmetic : 249 ( 84 atm; 25 fun; 55 num; 85 var)
% Number of types : 10 ( 8 usr; 1 ari)
% Number of type conns : 109 ( 38 >; 71 *; 0 +; 0 <<)
% Number of predicates : 16 ( 13 usr; 0 prp; 2-7 aty)
% Number of functors : 40 ( 35 usr; 12 con; 0-5 aty)
% Number of variables : 261 ( 259 !; 2 ?; 261 :)
% SPC : TF0_THM_EQU_ARI
% Comments :
%------------------------------------------------------------------------------
tff(uni,type,
uni: $tType ).
tff(ty,type,
ty: $tType ).
tff(sort,type,
sort: ( ty * uni ) > $o ).
tff(witness,type,
witness: ty > uni ).
tff(witness_sort,axiom,
! [A: ty] : sort(A,witness(A)) ).
tff(int,type,
int: ty ).
tff(real,type,
real: ty ).
tff(bool,type,
bool: $tType ).
tff(bool1,type,
bool1: ty ).
tff(true,type,
true: bool ).
tff(false,type,
false: bool ).
tff(match_bool,type,
match_bool: ( ty * bool * uni * uni ) > uni ).
tff(match_bool_sort1,axiom,
! [A: ty,X: bool,X1: uni,X2: uni] : sort(A,match_bool(A,X,X1,X2)) ).
tff(match_bool_True,axiom,
! [A: ty,Z: uni,Z1: uni] :
( sort(A,Z)
=> ( match_bool(A,true,Z,Z1) = Z ) ) ).
tff(match_bool_False,axiom,
! [A: ty,Z: uni,Z1: uni] :
( sort(A,Z1)
=> ( match_bool(A,false,Z,Z1) = Z1 ) ) ).
tff(true_False,axiom,
true != false ).
tff(bool_inversion,axiom,
! [U: bool] :
( ( U = true )
| ( U = false ) ) ).
tff(tuple0,type,
tuple0: $tType ).
tff(tuple01,type,
tuple01: ty ).
tff(tuple02,type,
tuple02: tuple0 ).
tff(tuple0_inversion,axiom,
! [U: tuple0] : U = tuple02 ).
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(ref,type,
ref: ty > ty ).
tff(mk_ref,type,
mk_ref: ( ty * uni ) > uni ).
tff(mk_ref_sort1,axiom,
! [A: ty,X: uni] : sort(ref(A),mk_ref(A,X)) ).
tff(contents,type,
contents: ( ty * uni ) > uni ).
tff(contents_sort1,axiom,
! [A: ty,X: uni] : sort(A,contents(A,X)) ).
tff(contents_def,axiom,
! [A: ty,U: uni] :
( sort(A,U)
=> ( contents(A,mk_ref(A,U)) = U ) ) ).
tff(ref_inversion,axiom,
! [A: ty,U: uni] :
( sort(ref(A),U)
=> ( U = mk_ref(A,contents(A,U)) ) ) ).
tff(map,type,
map: ( ty * ty ) > ty ).
tff(get,type,
get: ( ty * ty * uni * uni ) > uni ).
tff(get_sort2,axiom,
! [A: ty,B: ty,X: uni,X1: uni] : sort(B,get(B,A,X,X1)) ).
tff(set,type,
set: ( ty * ty * uni * uni * uni ) > uni ).
tff(set_sort2,axiom,
! [A: ty,B: ty,X: uni,X1: uni,X2: uni] : sort(map(A,B),set(B,A,X,X1,X2)) ).
tff(select_eq,axiom,
! [A: ty,B: ty,M: uni,A1: uni,A2: uni,B1: uni] :
( sort(B,B1)
=> ( ( A1 = A2 )
=> ( get(B,A,set(B,A,M,A1,B1),A2) = B1 ) ) ) ).
tff(select_neq,axiom,
! [A: ty,B: ty,M: uni,A1: uni,A2: uni] :
( sort(A,A1)
=> ( sort(A,A2)
=> ! [B1: uni] :
( ( A1 != A2 )
=> ( get(B,A,set(B,A,M,A1,B1),A2) = get(B,A,M,A2) ) ) ) ) ).
tff(const,type,
const: ( ty * ty * uni ) > uni ).
tff(const_sort1,axiom,
! [A: ty,B: ty,X: uni] : sort(map(A,B),const(B,A,X)) ).
tff(const1,axiom,
! [A: ty,B: ty,B1: uni,A1: uni] :
( sort(B,B1)
=> ( get(B,A,const(B,A,B1),A1) = B1 ) ) ).
tff(array,type,
array: ty > ty ).
tff(mk_array,type,
mk_array: ( ty * $int * uni ) > uni ).
tff(mk_array_sort1,axiom,
! [A: ty,X: $int,X1: uni] : sort(array(A),mk_array(A,X,X1)) ).
tff(length,type,
length: ( ty * uni ) > $int ).
tff(length_def,axiom,
! [A: ty,U: $int,U1: uni] : length(A,mk_array(A,U,U1)) = U ).
tff(elts,type,
elts: ( ty * uni ) > uni ).
tff(elts_sort1,axiom,
! [A: ty,X: uni] : sort(map(int,A),elts(A,X)) ).
tff(elts_def,axiom,
! [A: ty,U: $int,U1: uni] :
( sort(map(int,A),U1)
=> ( elts(A,mk_array(A,U,U1)) = U1 ) ) ).
tff(array_inversion,axiom,
! [A: ty,U: uni] : U = mk_array(A,length(A,U),elts(A,U)) ).
tff(get1,type,
get1: ( ty * uni * $int ) > uni ).
tff(get_sort3,axiom,
! [A: ty,X: uni,X1: $int] : sort(A,get1(A,X,X1)) ).
tff(t2tb,type,
t2tb: $int > uni ).
tff(t2tb_sort3,axiom,
! [X: $int] : sort(int,t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > $int ).
tff(bridgeL,axiom,
! [I: $int] : tb2t(t2tb(I)) = I ).
tff(bridgeR,axiom,
! [J: uni] : t2tb(tb2t(J)) = J ).
tff(get_def,axiom,
! [A: ty,A1: uni,I: $int] : get1(A,A1,I) = get(A,int,elts(A,A1),t2tb(I)) ).
tff(set1,type,
set1: ( ty * uni * $int * uni ) > uni ).
tff(set_sort3,axiom,
! [A: ty,X: uni,X1: $int,X2: uni] : sort(array(A),set1(A,X,X1,X2)) ).
tff(set_def,axiom,
! [A: ty,A1: uni,I: $int,V: uni] : set1(A,A1,I,V) = mk_array(A,length(A,A1),set(A,int,elts(A,A1),t2tb(I),V)) ).
tff(make,type,
make: ( ty * $int * uni ) > uni ).
tff(make_sort1,axiom,
! [A: ty,X: $int,X1: uni] : sort(array(A),make(A,X,X1)) ).
tff(make_def,axiom,
! [A: ty,N: $int,V: uni] : make(A,N,V) = mk_array(A,N,const(A,int,V)) ).
tff(occ,type,
occ: ( ty * uni * uni * $int * $int ) > $int ).
tff(occ_empty,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( $lesseq(U,L)
=> ( occ(A,V,M,L,U) = 0 ) ) ).
tff(occ_right_no_add,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( sort(A,V)
=> ( $less(L,U)
=> ( ( get(A,int,M,t2tb($difference(U,1))) != V )
=> ( occ(A,V,M,L,U) = occ(A,V,M,L,$difference(U,1)) ) ) ) ) ).
tff(occ_right_add,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( $less(L,U)
=> ( ( get(A,int,M,t2tb($difference(U,1))) = V )
=> ( occ(A,V,M,L,U) = $sum(1,occ(A,V,M,L,$difference(U,1))) ) ) ) ).
tff(occ_bounds,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( $lesseq(L,U)
=> ( $lesseq(0,occ(A,V,M,L,U))
& $lesseq(occ(A,V,M,L,U),$difference(U,L)) ) ) ).
tff(occ_append,axiom,
! [A: ty,V: uni,M: uni,L: $int,Mid: $int,U: $int] :
( ( $lesseq(L,Mid)
& $lesseq(Mid,U) )
=> ( occ(A,V,M,L,U) = $sum(occ(A,V,M,L,Mid),occ(A,V,M,Mid,U)) ) ) ).
tff(occ_neq,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( sort(A,V)
=> ( ! [I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( get(A,int,M,t2tb(I)) != V ) )
=> ( occ(A,V,M,L,U) = 0 ) ) ) ).
tff(occ_exists,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( sort(A,V)
=> ( $less(0,occ(A,V,M,L,U))
=> ? [I: $int] :
( $lesseq(L,I)
& $less(I,U)
& ( get(A,int,M,t2tb(I)) = V ) ) ) ) ).
tff(occ_pos,axiom,
! [A: ty,M: uni,L: $int,U: $int,I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> $less(0,occ(A,get(A,int,M,t2tb(I)),M,L,U)) ) ).
tff(occ_eq,axiom,
! [A: ty,V: uni,M1: uni,M2: uni,L: $int,U: $int] :
( ! [I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( get(A,int,M1,t2tb(I)) = get(A,int,M2,t2tb(I)) ) )
=> ( occ(A,V,M1,L,U) = occ(A,V,M2,L,U) ) ) ).
tff(permut,type,
permut: ( ty * uni * uni * $int * $int ) > $o ).
tff(permut_def,axiom,
! [A: ty,M1: uni,M2: uni,L: $int,U: $int] :
( ( permut(A,M1,M2,L,U)
=> ! [V: uni] : occ(A,V,M1,L,U) = occ(A,V,M2,L,U) )
& ( ! [V: uni] :
( sort(A,V)
=> ( occ(A,V,M1,L,U) = occ(A,V,M2,L,U) ) )
=> permut(A,M1,M2,L,U) ) ) ).
tff(permut_trans,axiom,
! [A: ty,A1: uni,A2: uni,A3: uni,L: $int,U: $int] :
( permut(A,A1,A2,L,U)
=> ( permut(A,A2,A3,L,U)
=> permut(A,A1,A3,L,U) ) ) ).
tff(permut_exists,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int,I: $int] :
( permut(A,A1,A2,L,U)
=> ( ( $lesseq(L,I)
& $less(I,U) )
=> ? [J: $int] :
( $lesseq(L,J)
& $less(J,U)
& ( get(A,int,A1,t2tb(J)) = get(A,int,A2,t2tb(I)) ) ) ) ) ).
tff(map_eq_sub,type,
map_eq_sub: ( ty * uni * uni * $int * $int ) > $o ).
tff(map_eq_sub_def,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
( map_eq_sub(A,A1,A2,L,U)
<=> ! [I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( get(A,int,A1,t2tb(I)) = get(A,int,A2,t2tb(I)) ) ) ) ).
tff(array_eq_sub,type,
array_eq_sub: ( ty * uni * uni * $int * $int ) > $o ).
tff(array_eq_sub_def,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
( array_eq_sub(A,A1,A2,L,U)
<=> ( ( length(A,A1) = length(A,A2) )
& $lesseq(0,L)
& $lesseq(L,length(A,A1))
& $lesseq(0,U)
& $lesseq(U,length(A,A1))
& map_eq_sub(A,elts(A,A1),elts(A,A2),L,U) ) ) ).
tff(array_eq,type,
array_eq: ( ty * uni * uni ) > $o ).
tff(array_eq_def,axiom,
! [A: ty,A1: uni,A2: uni] :
( array_eq(A,A1,A2)
<=> ( ( length(A,A1) = length(A,A2) )
& map_eq_sub(A,elts(A,A1),elts(A,A2),0,length(A,A1)) ) ) ).
tff(exchange,type,
exchange: ( ty * uni * uni * $int * $int * $int * $int ) > $o ).
tff(exchange_def,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int,I: $int,J: $int] :
( exchange(A,A1,A2,L,U,I,J)
<=> ( $lesseq(L,I)
& $less(I,U)
& $lesseq(L,J)
& $less(J,U)
& ( get(A,int,A1,t2tb(I)) = get(A,int,A2,t2tb(J)) )
& ( get(A,int,A1,t2tb(J)) = get(A,int,A2,t2tb(I)) )
& ! [K: $int] :
( ( $lesseq(L,K)
& $less(K,U) )
=> ( ( K != I )
=> ( ( K != J )
=> ( get(A,int,A1,t2tb(K)) = get(A,int,A2,t2tb(K)) ) ) ) ) ) ) ).
tff(exchange_set,axiom,
! [A: ty,A1: uni,L: $int,U: $int,I: $int,J: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( ( $lesseq(L,J)
& $less(J,U) )
=> exchange(A,A1,set(A,int,set(A,int,A1,t2tb(I),get(A,int,A1,t2tb(J))),t2tb(J),get(A,int,A1,t2tb(I))),L,U,I,J) ) ) ).
tff(exchange1,type,
exchange1: ( ty * uni * uni * $int * $int ) > $o ).
tff(exchange_def1,axiom,
! [A: ty,A1: uni,A2: uni,I: $int,J: $int] :
( exchange1(A,A1,A2,I,J)
<=> ( ( length(A,A1) = length(A,A2) )
& exchange(A,elts(A,A1),elts(A,A2),0,length(A,A1),I,J) ) ) ).
tff(permut1,type,
permut1: ( ty * uni * uni * $int * $int ) > $o ).
tff(permut_def1,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
( permut1(A,A1,A2,L,U)
<=> ( ( length(A,A1) = length(A,A2) )
& $lesseq(0,L)
& $lesseq(L,length(A,A1))
& $lesseq(0,U)
& $lesseq(U,length(A,A1))
& permut(A,elts(A,A1),elts(A,A2),L,U) ) ) ).
tff(permut_sub,type,
permut_sub: ( ty * uni * uni * $int * $int ) > $o ).
tff(permut_sub_def,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
( permut_sub(A,A1,A2,L,U)
<=> ( map_eq_sub(A,elts(A,A1),elts(A,A2),0,L)
& permut1(A,A1,A2,L,U)
& map_eq_sub(A,elts(A,A1),elts(A,A2),U,length(A,A1)) ) ) ).
tff(permut_all,type,
permut_all: ( ty * uni * uni ) > $o ).
tff(permut_all_def,axiom,
! [A: ty,A1: uni,A2: uni] :
( permut_all(A,A1,A2)
<=> ( ( length(A,A1) = length(A,A2) )
& permut(A,elts(A,A1),elts(A,A2),0,length(A,A1)) ) ) ).
tff(exchange_permut_sub,axiom,
! [A: ty,A1: uni,A2: uni,I: $int,J: $int,L: $int,U: $int] :
( exchange1(A,A1,A2,I,J)
=> ( ( $lesseq(L,I)
& $less(I,U) )
=> ( ( $lesseq(L,J)
& $less(J,U) )
=> ( $lesseq(0,L)
=> ( $lesseq(U,length(A,A1))
=> permut_sub(A,A1,A2,L,U) ) ) ) ) ) ).
tff(permut_sub_weakening,axiom,
! [A: ty,A1: uni,A2: uni,L1: $int,U1: $int,L2: $int,U2: $int] :
( permut_sub(A,A1,A2,L1,U1)
=> ( ( $lesseq(0,L2)
& $lesseq(L2,L1) )
=> ( ( $lesseq(U1,U2)
& $lesseq(U2,length(A,A1)) )
=> permut_sub(A,A1,A2,L2,U2) ) ) ) ).
tff(exchange_permut_all,axiom,
! [A: ty,A1: uni,A2: uni,I: $int,J: $int] :
( exchange1(A,A1,A2,I,J)
=> permut_all(A,A1,A2) ) ).
tff(param,type,
param: $tType ).
tff(param1,type,
param1: ty ).
tff(elt,type,
elt: $tType ).
tff(elt1,type,
elt1: ty ).
tff(le,type,
le: ( param * elt * elt ) > $o ).
tff(le_refl,axiom,
! [P: param,X: elt] : le(P,X,X) ).
tff(le_asym,axiom,
! [P: param,X: elt,Y: elt] :
( ~ le(P,X,Y)
=> le(P,Y,X) ) ).
tff(le_trans,axiom,
! [P: param,X: elt,Y: elt,Z: elt] :
( ( le(P,X,Y)
& le(P,Y,Z) )
=> le(P,X,Z) ) ).
tff(array_elt,type,
array_elt: $tType ).
tff(sorted_sub2,type,
sorted_sub2: ( param * array_elt * $int * $int ) > $o ).
tff(t2tb3,type,
t2tb3: array_elt > uni ).
tff(t2tb_sort4,axiom,
! [X: array_elt] : sort(array(elt1),t2tb3(X)) ).
tff(tb2t3,type,
tb2t3: uni > array_elt ).
tff(bridgeL3,axiom,
! [I: array_elt] : tb2t3(t2tb3(I)) = I ).
tff(bridgeR3,axiom,
! [J: uni] : t2tb3(tb2t3(J)) = J ).
tff(t2tb4,type,
t2tb4: elt > uni ).
tff(t2tb_sort5,axiom,
! [X: elt] : sort(elt1,t2tb4(X)) ).
tff(tb2t4,type,
tb2t4: uni > elt ).
tff(bridgeL4,axiom,
! [I: elt] : tb2t4(t2tb4(I)) = I ).
tff(bridgeR4,axiom,
! [J: uni] :
( sort(elt1,J)
=> ( t2tb4(tb2t4(J)) = J ) ) ).
tff(sorted_sub_def2,axiom,
! [P: param,A: array_elt,L: $int,U: $int] :
( sorted_sub2(P,A,L,U)
<=> ! [I1: $int,I2: $int] :
( ( $lesseq(L,I1)
& $lesseq(I1,I2)
& $less(I2,U) )
=> le(P,tb2t4(get1(elt1,t2tb3(A),I1)),tb2t4(get1(elt1,t2tb3(A),I2))) ) ) ).
tff(sorted1,type,
sorted1: ( param * array_elt ) > $o ).
tff(sorted_def1,axiom,
! [P: param,A: array_elt] :
( sorted1(P,A)
<=> sorted_sub2(P,A,0,length(elt1,t2tb3(A))) ) ).
tff(map_int_elt,type,
map_int_elt: $tType ).
tff(t2tb5,type,
t2tb5: map_int_elt > uni ).
tff(t2tb_sort6,axiom,
! [X: map_int_elt] : sort(map(int,elt1),t2tb5(X)) ).
tff(tb2t5,type,
tb2t5: uni > map_int_elt ).
tff(bridgeL5,axiom,
! [I: map_int_elt] : tb2t5(t2tb5(I)) = I ).
tff(bridgeR5,axiom,
! [J: uni] :
( sort(map(int,elt1),J)
=> ( t2tb5(tb2t5(J)) = J ) ) ).
tff(wP_parameter_sort1,conjecture,
! [P: param,A: $int,A1: map_int_elt] :
( $lesseq(0,A)
=> ( $lesseq(0,$difference(A,1))
=> ! [A2: map_int_elt,I: $int] :
( ( $lesseq(0,I)
& $lesseq(I,$difference(A,1)) )
=> ( ( permut_all(elt1,mk_array(elt1,A,t2tb5(A1)),mk_array(elt1,A,t2tb5(A2)))
& sorted_sub2(P,tb2t3(mk_array(elt1,A,t2tb5(A2))),0,I) )
=> ! [J: $int,A3: map_int_elt] :
( ( $lesseq(0,J)
& $lesseq(J,I)
& permut_all(elt1,mk_array(elt1,A,t2tb5(A1)),mk_array(elt1,A,t2tb5(A3)))
& sorted_sub2(P,tb2t3(mk_array(elt1,A,t2tb5(A3))),0,J)
& sorted_sub2(P,tb2t3(mk_array(elt1,A,t2tb5(A3))),J,$sum(I,1))
& ! [K1: $int,K2: $int] :
( ( $lesseq(0,K1)
& $less(K1,J)
& $lesseq($sum(J,1),K2)
& $lesseq(K2,I) )
=> le(P,tb2t4(get(elt1,int,t2tb5(A3),t2tb(K1))),tb2t4(get(elt1,int,t2tb5(A3),t2tb(K2)))) ) )
=> ( $less(0,J)
=> ( ( $lesseq(0,A)
& $lesseq(0,J)
& $less(J,A) )
=> ( ( $lesseq(0,$difference(J,1))
& $less($difference(J,1),A) )
=> ( ~ le(P,tb2t4(get(elt1,int,t2tb5(A3),t2tb($difference(J,1)))),tb2t4(get(elt1,int,t2tb5(A3),t2tb(J))))
=> ( ( $lesseq(0,J)
& $less(J,A) )
=> ( ( $lesseq(0,$difference(J,1))
& $less($difference(J,1),A) )
=> ( ( $lesseq(0,J)
& $less(J,A) )
=> ! [A4: map_int_elt] :
( ( $lesseq(0,A)
& ( A4 = tb2t5(set(elt1,int,t2tb5(A3),t2tb(J),get(elt1,int,t2tb5(A3),t2tb($difference(J,1))))) ) )
=> ( ( $lesseq(0,$difference(J,1))
& $less($difference(J,1),A) )
=> ! [A5: map_int_elt] :
( ( $lesseq(0,A)
& ( A5 = tb2t5(set(elt1,int,t2tb5(A4),t2tb($difference(J,1)),get(elt1,int,t2tb5(A3),t2tb(J)))) ) )
=> ( exchange1(elt1,mk_array(elt1,A,t2tb5(A3)),mk_array(elt1,A,t2tb5(A5)),$difference(J,1),J)
=> ! [J1: $int] :
( ( J1 = $difference(J,1) )
=> sorted_sub2(P,tb2t3(mk_array(elt1,A,t2tb5(A5))),J1,$sum(I,1)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------