TPTP Problem File: SWW667_2.p
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%------------------------------------------------------------------------------
% File : SWW667_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Vstte10 max sum-T-WP parameter max sum
% 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 : vstte10_max_sum-T-WP_parameter_max_sum [Fil14]
% Status : Theorem
% Rating : 0.75 v8.2.0, 0.62 v8.1.0, 0.75 v7.5.0, 0.80 v7.4.0, 0.50 v7.0.0, 0.57 v6.4.0, 0.33 v6.3.0, 0.71 v6.2.0, 0.75 v6.1.0
% Syntax : Number of formulae : 86 ( 29 unt; 40 typ; 0 def)
% Number of atoms : 107 ( 40 equ)
% Maximal formula atoms : 21 ( 1 avg)
% Number of connectives : 63 ( 2 ~; 3 |; 24 &)
% ( 0 <=>; 34 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 5 avg)
% Maximal term depth : 6 ( 1 avg)
% Number arithmetic : 116 ( 39 atm; 13 fun; 21 num; 43 var)
% Number of types : 8 ( 6 usr; 1 ari)
% Number of type conns : 60 ( 26 >; 34 *; 0 +; 0 <<)
% Number of predicates : 5 ( 2 usr; 0 prp; 2-4 aty)
% Number of functors : 37 ( 32 usr; 10 con; 0-5 aty)
% Number of variables : 132 ( 130 !; 2 ?; 132 :)
% 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(ref,type,
ref: ty > ty ).
tff(mk_ref,type,
mk_ref: ( ty * uni ) > uni ).
tff(mk_ref_sort1,axiom,
! [A: ty,X: uni] : sort1(ref(A),mk_ref(A,X)) ).
tff(contents,type,
contents: ( ty * uni ) > uni ).
tff(contents_sort1,axiom,
! [A: ty,X: uni] : sort1(A,contents(A,X)) ).
tff(contents_def1,axiom,
! [A: ty,U: uni] :
( sort1(A,U)
=> ( contents(A,mk_ref(A,U)) = U ) ) ).
tff(ref_inversion1,axiom,
! [A: ty,U: uni] :
( sort1(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] : sort1(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] : sort1(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] :
( sort1(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] :
( sort1(A,A1)
=> ( sort1(A,A2)
=> ! [B1: uni] :
( ( A1 != A2 )
=> ( get(B,A,set(B,A,M,A1,B1),A2) = get(B,A,M,A2) ) ) ) ) ).
tff(const1,type,
const: ( ty * ty * uni ) > uni ).
tff(const_sort1,axiom,
! [A: ty,B: ty,X: uni] : sort1(map(A,B),const(B,A,X)) ).
tff(const,axiom,
! [A: ty,B: ty,B1: uni,A1: uni] :
( sort1(B,B1)
=> ( get(B,A,const(B,A,B1),A1) = B1 ) ) ).
tff(array,type,
array: ty > ty ).
tff(mk_array,type,
mk_array1: ( ty * $int * uni ) > uni ).
tff(mk_array_sort1,axiom,
! [A: ty,X: $int,X1: uni] : sort1(array(A),mk_array1(A,X,X1)) ).
tff(length,type,
length1: ( ty * uni ) > $int ).
tff(length_def1,axiom,
! [A: ty,U: $int,U1: uni] : length1(A,mk_array1(A,U,U1)) = U ).
tff(elts,type,
elts: ( ty * uni ) > uni ).
tff(elts_sort1,axiom,
! [A: ty,X: uni] : sort1(map(int,A),elts(A,X)) ).
tff(elts_def1,axiom,
! [A: ty,U: $int,U1: uni] :
( sort1(map(int,A),U1)
=> ( elts(A,mk_array1(A,U,U1)) = U1 ) ) ).
tff(array_inversion1,axiom,
! [A: ty,U: uni] : U = mk_array1(A,length1(A,U),elts(A,U)) ).
tff(get1,type,
get2: ( ty * uni * $int ) > uni ).
tff(get_sort3,axiom,
! [A: ty,X: uni,X1: $int] : sort1(A,get2(A,X,X1)) ).
tff(t2tb,type,
t2tb: $int > uni ).
tff(t2tb_sort,axiom,
! [X: $int] : sort1(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] : get2(A,A1,I) = get(A,int,elts(A,A1),t2tb(I)) ).
tff(set1,type,
set2: ( ty * uni * $int * uni ) > uni ).
tff(set_sort3,axiom,
! [A: ty,X: uni,X1: $int,X2: uni] : sort1(array(A),set2(A,X,X1,X2)) ).
tff(set_def,axiom,
! [A: ty,A1: uni,I: $int,V: uni] : set2(A,A1,I,V) = mk_array1(A,length1(A,A1),set(A,int,elts(A,A1),t2tb(I),V)) ).
tff(make,type,
make1: ( ty * $int * uni ) > uni ).
tff(make_sort1,axiom,
! [A: ty,X: $int,X1: uni] : sort1(array(A),make1(A,X,X1)) ).
tff(make_def,axiom,
! [A: ty,N: $int,V: uni] : make1(A,N,V) = mk_array1(A,N,const(A,int,V)) ).
tff(map_int_int,type,
map_int_int: $tType ).
tff(sum,type,
sum2: ( map_int_int * $int * $int ) > $int ).
tff(sum_def_empty,axiom,
! [C: map_int_int,I: $int,J: $int] :
( $lesseq(J,I)
=> ( sum2(C,I,J) = 0 ) ) ).
tff(t2tb1,type,
t2tb1: map_int_int > uni ).
tff(t2tb_sort1,axiom,
! [X: map_int_int] : sort1(map(int,int),t2tb1(X)) ).
tff(tb2t1,type,
tb2t1: uni > map_int_int ).
tff(bridgeL1,axiom,
! [I: map_int_int] : tb2t1(t2tb1(I)) = I ).
tff(bridgeR1,axiom,
! [J: uni] : t2tb1(tb2t1(J)) = J ).
tff(sum_def_non_empty,axiom,
! [C: map_int_int,I: $int,J: $int] :
( $less(I,J)
=> ( sum2(C,I,J) = $sum(tb2t(get(int,int,t2tb1(C),t2tb(I))),sum2(C,$sum(I,1),J)) ) ) ).
tff(sum_right_extension,axiom,
! [C: map_int_int,I: $int,J: $int] :
( $less(I,J)
=> ( sum2(C,I,J) = $sum(sum2(C,I,$difference(J,1)),tb2t(get(int,int,t2tb1(C),t2tb($difference(J,1))))) ) ) ).
tff(sum_transitivity,axiom,
! [C: map_int_int,I: $int,K: $int,J: $int] :
( ( $lesseq(I,K)
& $lesseq(K,J) )
=> ( sum2(C,I,J) = $sum(sum2(C,I,K),sum2(C,K,J)) ) ) ).
tff(sum_eq,axiom,
! [C1: map_int_int,C2: map_int_int,I: $int,J: $int] :
( ! [K: $int] :
( ( $lesseq(I,K)
& $less(K,J) )
=> ( tb2t(get(int,int,t2tb1(C1),t2tb(K))) = tb2t(get(int,int,t2tb1(C2),t2tb(K))) ) )
=> ( sum2(C1,I,J) = sum2(C2,I,J) ) ) ).
tff(array_int,type,
array_int: $tType ).
tff(sum1,type,
sum3: ( array_int * $int * $int ) > $int ).
tff(t2tb2,type,
t2tb2: array_int > uni ).
tff(t2tb_sort2,axiom,
! [X: array_int] : sort1(array(int),t2tb2(X)) ).
tff(tb2t2,type,
tb2t2: uni > array_int ).
tff(bridgeL2,axiom,
! [I: array_int] : tb2t2(t2tb2(I)) = I ).
tff(bridgeR2,axiom,
! [J: uni] : t2tb2(tb2t2(J)) = J ).
tff(sum_def,axiom,
! [A: array_int,L: $int,H: $int] : sum3(A,L,H) = sum2(tb2t1(elts(int,t2tb2(A))),L,H) ).
tff(is_max,type,
is_max1: ( array_int * $int * $int * $int ) > $o ).
tff(is_max_def,axiom,
! [A: array_int,L: $int,H: $int,M: $int] :
( ( is_max1(A,L,H,M)
=> ( ! [K: $int] :
( ( $lesseq(L,K)
& $less(K,H) )
=> $lesseq(tb2t(get2(int,t2tb2(A),K)),M) )
& ( ( $lesseq(H,L)
& ( M = 0 ) )
| ( $less(L,H)
& ? [K: $int] :
( $lesseq(L,K)
& $less(K,H)
& ( M = tb2t(get2(int,t2tb2(A),K)) ) ) ) ) ) )
& ( ( ! [K: $int] :
( ( $lesseq(L,K)
& $less(K,H) )
=> $lesseq(tb2t(get2(int,t2tb2(A),K)),M) )
& ( ( $lesseq(H,L)
& ( M = 0 ) )
| ( $less(L,H)
& ? [K: $int] :
( $lesseq(L,K)
& $less(K,H)
& ( M = tb2t(get2(int,t2tb2(A),K)) ) ) ) ) )
=> is_max1(A,L,H,M) ) ) ).
tff(wP_parameter_max_sum1,conjecture,
! [A: $int,A1: map_int_int,N: $int] :
( ( $lesseq(0,A)
& ( N = A )
& ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,N) )
=> $lesseq(0,tb2t(get(int,int,t2tb1(A1),t2tb(I)))) ) )
=> ( $lesseq(0,$difference(N,1))
=> ! [M: $int,S: $int,I: $int] :
( ( $lesseq(0,I)
& $lesseq(I,$difference(N,1)) )
=> ( ( ( S = sum2(A1,0,I) )
& is_max1(tb2t2(mk_array1(int,A,t2tb1(A1))),0,I,M)
& $lesseq(S,$product(I,M)) )
=> ( ( $lesseq(0,I)
& $less(I,A) )
=> ( $less(M,tb2t(get(int,int,t2tb1(A1),t2tb(I))))
=> ( ( $lesseq(0,I)
& $less(I,A) )
=> ! [M1: $int] :
( ( M1 = tb2t(get(int,int,t2tb1(A1),t2tb(I))) )
=> ( ( $lesseq(0,I)
& $less(I,A) )
=> ! [S1: $int] :
( ( S1 = $sum(S,tb2t(get(int,int,t2tb1(A1),t2tb(I)))) )
=> is_max1(tb2t2(mk_array1(int,A,t2tb1(A1))),0,$sum(I,1),M1) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------