TPTP Problem File: SWW583_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW583_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Counting sort-T-WP parameter counting 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 : counting_sort-T-WP_parameter_counting_sort [Fil14]
% Status : Theorem
% Rating : 0.62 v7.5.0, 0.60 v7.4.0, 0.38 v7.3.0, 0.33 v7.0.0, 0.43 v6.4.0, 0.33 v6.3.0, 0.71 v6.2.0, 0.88 v6.1.0
% Syntax : Number of formulae : 141 ( 37 unt; 56 typ; 0 def)
% Number of atoms : 247 ( 61 equ)
% Maximal formula atoms : 48 ( 1 avg)
% Number of connectives : 172 ( 10 ~; 1 |; 55 &)
% ( 7 <=>; 99 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number arithmetic : 343 ( 116 atm; 35 fun; 74 num; 118 var)
% Number of types : 9 ( 7 usr; 1 ari)
% Number of type conns : 92 ( 40 >; 52 *; 0 +; 0 <<)
% Number of predicates : 11 ( 8 usr; 0 prp; 1-3 aty)
% Number of functors : 46 ( 41 usr; 11 con; 0-5 aty)
% Number of variables : 269 ( 269 !; 0 ?; 269 :)
% 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_sort,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(map,type,
map: ( ty * ty ) > ty ).
tff(get,type,
get: ( ty * ty * uni * uni ) > uni ).
tff(get_sort,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_sort,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_sort,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_sort,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_sort,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_sort1,axiom,
! [A: ty,X: uni,X1: $int] : sort(A,get1(A,X,X1)) ).
tff(t2tb,type,
t2tb: $int > uni ).
tff(t2tb_sort,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_sort1,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_sort,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(map_int_int,type,
map_int_int: $tType ).
tff(sorted_sub,type,
sorted_sub: ( map_int_int * $int * $int ) > $o ).
tff(t2tb1,type,
t2tb1: map_int_int > uni ).
tff(t2tb_sort1,axiom,
! [X: map_int_int] : sort(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(sorted_sub_def,axiom,
! [A: map_int_int,L: $int,U: $int] :
( sorted_sub(A,L,U)
<=> ! [I1: $int,I2: $int] :
( ( $lesseq(L,I1)
& $lesseq(I1,I2)
& $less(I2,U) )
=> $lesseq(tb2t(get(int,int,t2tb1(A),t2tb(I1))),tb2t(get(int,int,t2tb1(A),t2tb(I2)))) ) ) ).
tff(array_int,type,
array_int: $tType ).
tff(sorted_sub1,type,
sorted_sub1: ( array_int * $int * $int ) > $o ).
tff(t2tb2,type,
t2tb2: array_int > uni ).
tff(t2tb_sort2,axiom,
! [X: array_int] : sort(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(sorted_sub_def1,axiom,
! [A: array_int,L: $int,U: $int] :
( sorted_sub1(A,L,U)
<=> sorted_sub(tb2t1(elts(int,t2tb2(A))),L,U) ) ).
tff(sorted,type,
sorted: array_int > $o ).
tff(sorted_def,axiom,
! [A: array_int] :
( sorted(A)
<=> sorted_sub(tb2t1(elts(int,t2tb2(A))),0,length(int,t2tb2(A))) ) ).
tff(k,type,
k: $int ).
tff(k_positive,axiom,
$less(0,k) ).
tff(k_values,type,
k_values: array_int > $o ).
tff(k_values_def,axiom,
! [A: array_int] :
( k_values(A)
<=> ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,length(int,t2tb2(A))) )
=> ( $lesseq(0,tb2t(get1(int,t2tb2(A),I)))
& $less(tb2t(get1(int,t2tb2(A),I)),k) ) ) ) ).
tff(tuple2,type,
tuple2: ( ty * ty ) > ty ).
tff(tuple21,type,
tuple21: ( ty * ty * uni * uni ) > uni ).
tff(tuple2_sort,axiom,
! [A: ty,A1: ty,X: uni,X1: uni] : sort(tuple2(A1,A),tuple21(A1,A,X,X1)) ).
tff(tuple2_proj_1,type,
tuple2_proj_1: ( ty * ty * uni ) > uni ).
tff(tuple2_proj_1_sort,axiom,
! [A: ty,A1: ty,X: uni] : sort(A1,tuple2_proj_1(A1,A,X)) ).
tff(tuple2_proj_1_def,axiom,
! [A: ty,A1: ty,U: uni,U1: uni] :
( sort(A1,U)
=> ( tuple2_proj_1(A1,A,tuple21(A1,A,U,U1)) = U ) ) ).
tff(tuple2_proj_2,type,
tuple2_proj_2: ( ty * ty * uni ) > uni ).
tff(tuple2_proj_2_sort,axiom,
! [A: ty,A1: ty,X: uni] : sort(A,tuple2_proj_2(A1,A,X)) ).
tff(tuple2_proj_2_def,axiom,
! [A: ty,A1: ty,U: uni,U1: uni] :
( sort(A,U1)
=> ( tuple2_proj_2(A1,A,tuple21(A1,A,U,U1)) = U1 ) ) ).
tff(tuple2_inversion,axiom,
! [A: ty,A1: ty,U: uni] :
( sort(tuple2(A1,A),U)
=> ( U = tuple21(A1,A,tuple2_proj_1(A1,A,U),tuple2_proj_2(A1,A,U)) ) ) ).
tff(lparray_intcm_intrp,type,
lparray_intcm_intrp: $tType ).
tff(eq,type,
eq: ( lparray_intcm_intrp * $int ) > $o ).
tff(t2tb3,type,
t2tb3: lparray_intcm_intrp > uni ).
tff(t2tb_sort3,axiom,
! [X: lparray_intcm_intrp] : sort(tuple2(array(int),int),t2tb3(X)) ).
tff(tb2t3,type,
tb2t3: uni > lparray_intcm_intrp ).
tff(bridgeL3,axiom,
! [I: lparray_intcm_intrp] : tb2t3(t2tb3(I)) = I ).
tff(bridgeR3,axiom,
! [J: uni] : t2tb3(tb2t3(J)) = J ).
tff(eq_def,axiom,
! [I: $int,A: array_int,V: $int] :
( eq(tb2t3(tuple21(array(int),int,t2tb2(A),t2tb(V))),I)
<=> ( tb2t(get1(int,t2tb2(A),I)) = V ) ) ).
tff(num_of,type,
num_of: ( lparray_intcm_intrp * $int * $int ) > $int ).
tff(num_of_empty,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $lesseq(B,A)
=> ( num_of(P,A,B) = 0 ) ) ).
tff(num_of_right_no_add,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( ~ eq(P,$difference(B,1))
=> ( num_of(P,A,B) = num_of(P,A,$difference(B,1)) ) ) ) ).
tff(num_of_right_add,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( eq(P,$difference(B,1))
=> ( num_of(P,A,B) = $sum(1,num_of(P,A,$difference(B,1))) ) ) ) ).
tff(num_of_bounds,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( $lesseq(0,num_of(P,A,B))
& $lesseq(num_of(P,A,B),$difference(B,A)) ) ) ).
tff(num_of_append,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int,C: $int] :
( ( $lesseq(A,B)
& $lesseq(B,C) )
=> ( num_of(P,A,C) = $sum(num_of(P,A,B),num_of(P,B,C)) ) ) ).
tff(num_of_left_no_add,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( ~ eq(P,A)
=> ( num_of(P,A,B) = num_of(P,$sum(A,1),B) ) ) ) ).
tff(num_of_left_add,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( eq(P,A)
=> ( num_of(P,A,B) = $sum(1,num_of(P,$sum(A,1),B)) ) ) ) ).
tff(empty,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( ! [N: $int] :
( ( $lesseq(A,N)
& $less(N,B) )
=> ~ eq(P,N) )
=> ( num_of(P,A,B) = 0 ) ) ).
tff(full,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $lesseq(A,B)
=> ( ! [N: $int] :
( ( $lesseq(A,N)
& $less(N,B) )
=> eq(P,N) )
=> ( num_of(P,A,B) = $difference(B,A) ) ) ) ).
tff(num_of_increasing,axiom,
! [P: lparray_intcm_intrp,I: $int,J: $int,K: $int] :
( ( $lesseq(I,J)
& $lesseq(J,K) )
=> $lesseq(num_of(P,I,J),num_of(P,I,K)) ) ).
tff(num_of_strictly_increasing,axiom,
! [P: lparray_intcm_intrp,I: $int,J: $int,K: $int,L: $int] :
( ( $lesseq(I,J)
& $lesseq(J,K)
& $less(K,L) )
=> ( eq(P,K)
=> $less(num_of(P,I,J),num_of(P,I,L)) ) ) ).
tff(num_of_change_any,axiom,
! [P1: lparray_intcm_intrp,P2: lparray_intcm_intrp,A: $int,B: $int] :
( ! [J: $int] :
( ( $lesseq(A,J)
& $less(J,B) )
=> ( eq(P1,J)
=> eq(P2,J) ) )
=> $lesseq(num_of(P1,A,B),num_of(P2,A,B)) ) ).
tff(num_of_change_some,axiom,
! [P1: lparray_intcm_intrp,P2: lparray_intcm_intrp,A: $int,B: $int,I: $int] :
( ( $lesseq(A,I)
& $less(I,B) )
=> ( ! [J: $int] :
( ( $lesseq(A,J)
& $less(J,B) )
=> ( eq(P1,J)
=> eq(P2,J) ) )
=> ( ~ eq(P1,I)
=> ( eq(P2,I)
=> $less(num_of(P1,A,B),num_of(P2,A,B)) ) ) ) ) ).
tff(numeq,type,
numeq: ( array_int * $int * $int * $int ) > $int ).
tff(numeq_def,axiom,
! [A: array_int,V: $int,I: $int,J: $int] : numeq(A,V,I,J) = num_of(tb2t3(tuple21(array(int),int,t2tb2(A),t2tb(V))),I,J) ).
tff(lt,type,
lt: ( lparray_intcm_intrp * $int ) > $o ).
tff(lt_def,axiom,
! [I: $int,A: array_int,V: $int] :
( lt(tb2t3(tuple21(array(int),int,t2tb2(A),t2tb(V))),I)
<=> $less(tb2t(get1(int,t2tb2(A),I)),V) ) ).
tff(num_of1,type,
num_of1: ( lparray_intcm_intrp * $int * $int ) > $int ).
tff(num_of_empty1,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $lesseq(B,A)
=> ( num_of1(P,A,B) = 0 ) ) ).
tff(num_of_right_no_add1,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( ~ lt(P,$difference(B,1))
=> ( num_of1(P,A,B) = num_of1(P,A,$difference(B,1)) ) ) ) ).
tff(num_of_right_add1,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( lt(P,$difference(B,1))
=> ( num_of1(P,A,B) = $sum(1,num_of1(P,A,$difference(B,1))) ) ) ) ).
tff(num_of_bounds1,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( $lesseq(0,num_of1(P,A,B))
& $lesseq(num_of1(P,A,B),$difference(B,A)) ) ) ).
tff(num_of_append1,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int,C: $int] :
( ( $lesseq(A,B)
& $lesseq(B,C) )
=> ( num_of1(P,A,C) = $sum(num_of1(P,A,B),num_of1(P,B,C)) ) ) ).
tff(num_of_left_no_add1,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( ~ lt(P,A)
=> ( num_of1(P,A,B) = num_of1(P,$sum(A,1),B) ) ) ) ).
tff(num_of_left_add1,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $less(A,B)
=> ( lt(P,A)
=> ( num_of1(P,A,B) = $sum(1,num_of1(P,$sum(A,1),B)) ) ) ) ).
tff(empty1,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( ! [N: $int] :
( ( $lesseq(A,N)
& $less(N,B) )
=> ~ lt(P,N) )
=> ( num_of1(P,A,B) = 0 ) ) ).
tff(full1,axiom,
! [P: lparray_intcm_intrp,A: $int,B: $int] :
( $lesseq(A,B)
=> ( ! [N: $int] :
( ( $lesseq(A,N)
& $less(N,B) )
=> lt(P,N) )
=> ( num_of1(P,A,B) = $difference(B,A) ) ) ) ).
tff(num_of_increasing1,axiom,
! [P: lparray_intcm_intrp,I: $int,J: $int,K: $int] :
( ( $lesseq(I,J)
& $lesseq(J,K) )
=> $lesseq(num_of1(P,I,J),num_of1(P,I,K)) ) ).
tff(num_of_strictly_increasing1,axiom,
! [P: lparray_intcm_intrp,I: $int,J: $int,K: $int,L: $int] :
( ( $lesseq(I,J)
& $lesseq(J,K)
& $less(K,L) )
=> ( lt(P,K)
=> $less(num_of1(P,I,J),num_of1(P,I,L)) ) ) ).
tff(num_of_change_any1,axiom,
! [P1: lparray_intcm_intrp,P2: lparray_intcm_intrp,A: $int,B: $int] :
( ! [J: $int] :
( ( $lesseq(A,J)
& $less(J,B) )
=> ( lt(P1,J)
=> lt(P2,J) ) )
=> $lesseq(num_of1(P1,A,B),num_of1(P2,A,B)) ) ).
tff(num_of_change_some1,axiom,
! [P1: lparray_intcm_intrp,P2: lparray_intcm_intrp,A: $int,B: $int,I: $int] :
( ( $lesseq(A,I)
& $less(I,B) )
=> ( ! [J: $int] :
( ( $lesseq(A,J)
& $less(J,B) )
=> ( lt(P1,J)
=> lt(P2,J) ) )
=> ( ~ lt(P1,I)
=> ( lt(P2,I)
=> $less(num_of1(P1,A,B),num_of1(P2,A,B)) ) ) ) ) ).
tff(numlt,type,
numlt: ( array_int * $int * $int * $int ) > $int ).
tff(numlt_def,axiom,
! [A: array_int,V: $int,I: $int,J: $int] : numlt(A,V,I,J) = num_of1(tb2t3(tuple21(array(int),int,t2tb2(A),t2tb(V))),I,J) ).
tff(eqlt,axiom,
! [A: array_int] :
( k_values(A)
=> ! [V: $int] :
( ( $lesseq(0,V)
& $less(V,k) )
=> ! [L: $int,U: $int] :
( ( $lesseq(0,L)
& $less(L,U)
& $lesseq(U,length(int,t2tb2(A))) )
=> ( $sum(numlt(A,V,L,U),numeq(A,V,L,U)) = numlt(A,$sum(V,1),L,U) ) ) ) ) ).
tff(permut,type,
permut: ( array_int * array_int ) > $o ).
tff(permut_def,axiom,
! [A: array_int,B: array_int] :
( permut(A,B)
<=> ( ( length(int,t2tb2(A)) = length(int,t2tb2(B)) )
& ! [V: $int] :
( ( $lesseq(0,V)
& $less(V,k) )
=> ( numeq(A,V,0,length(int,t2tb2(A))) = numeq(B,V,0,length(int,t2tb2(B))) ) ) ) ) ).
tff(ref,type,
ref: ty > ty ).
tff(mk_ref,type,
mk_ref: ( ty * uni ) > uni ).
tff(mk_ref_sort,axiom,
! [A: ty,X: uni] : sort(ref(A),mk_ref(A,X)) ).
tff(contents,type,
contents: ( ty * uni ) > uni ).
tff(contents_sort,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(wP_parameter_counting_sort,conjecture,
! [A: $int,A1: map_int_int,B: $int,B1: map_int_int] :
( ( $lesseq(0,A)
& $lesseq(0,B)
& k_values(tb2t2(mk_array(int,A,t2tb1(A1))))
& ( A = B ) )
=> ( $lesseq(0,k)
=> ( $lesseq(0,k)
=> ( $lesseq(0,$difference(A,1))
=> ! [C: map_int_int] :
( ! [V: $int] :
( ( $lesseq(0,V)
& $less(V,k) )
=> ( tb2t(get(int,int,t2tb1(C),t2tb(V))) = num_of(tb2t3(tuple21(array(int),int,mk_array(int,A,t2tb1(A1)),t2tb(V))),0,$sum($difference(A,1),1)) ) )
=> ( $lesseq(0,$difference(k,1))
=> ! [J: $int,B2: map_int_int,V: $int] :
( ( $lesseq(0,V)
& $lesseq(V,$difference(k,1)) )
=> ( ( ( J = num_of1(tb2t3(tuple21(array(int),int,mk_array(int,A,t2tb1(A1)),t2tb(V))),0,A) )
& sorted_sub(B2,0,J)
& ! [E: $int] :
( ( $lesseq(0,E)
& $less(E,J) )
=> ( $lesseq(0,tb2t(get(int,int,t2tb1(B2),t2tb(E))))
& $less(tb2t(get(int,int,t2tb1(B2),t2tb(E))),V) ) )
& ! [F: $int] :
( ( $lesseq(0,F)
& $less(F,V) )
=> ( num_of(tb2t3(tuple21(array(int),int,mk_array(int,B,t2tb1(B2)),t2tb(F))),0,J) = num_of(tb2t3(tuple21(array(int),int,mk_array(int,A,t2tb1(A1)),t2tb(F))),0,A) ) ) )
=> ( ( $lesseq(0,k)
& $lesseq(0,V)
& $less(V,k) )
=> ( $lesseq(1,tb2t(get(int,int,t2tb1(C),t2tb(V))))
=> ! [J1: $int,B3: map_int_int,I: $int] :
( ( $lesseq(1,I)
& $lesseq(I,tb2t(get(int,int,t2tb1(C),t2tb(V)))) )
=> ( ( ( $sum($difference(J1,I),1) = num_of1(tb2t3(tuple21(array(int),int,mk_array(int,A,t2tb1(A1)),t2tb(V))),0,A) )
& sorted_sub(B3,0,J1)
& ! [E: $int] :
( ( $lesseq(0,E)
& $less(E,J1) )
=> ( $lesseq(0,tb2t(get(int,int,t2tb1(B3),t2tb(E))))
& $lesseq(tb2t(get(int,int,t2tb1(B3),t2tb(E))),V) ) )
& ! [F: $int] :
( ( $lesseq(0,F)
& $less(F,V) )
=> ( num_of(tb2t3(tuple21(array(int),int,mk_array(int,B,t2tb1(B3)),t2tb(F))),0,J1) = num_of(tb2t3(tuple21(array(int),int,mk_array(int,A,t2tb1(A1)),t2tb(F))),0,A) ) )
& ( num_of(tb2t3(tuple21(array(int),int,mk_array(int,B,t2tb1(B3)),t2tb(V))),0,J1) = $difference(I,1) ) )
=> ( ( $lesseq(0,B)
& $lesseq(0,J1)
& $less(J1,B) )
=> ! [B4: map_int_int] :
( ( $lesseq(0,B)
& ( B4 = tb2t1(set(int,int,t2tb1(B3),t2tb(J1),t2tb(V))) ) )
=> ! [J2: $int] :
( ( J2 = $sum(J1,1) )
=> ! [E: $int] :
( ( $lesseq(0,E)
& $less(E,J2) )
=> ( $lesseq(0,tb2t(get(int,int,t2tb1(B4),t2tb(E))))
& $lesseq(tb2t(get(int,int,t2tb1(B4),t2tb(E))),V) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------