TPTP Problem File: SWW644_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW644_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Sudoku-T-WP parameter check valid
% 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 : sudoku-T-WP_parameter_check_valid [Fil14]
% Status : Theorem
% Rating : 0.88 v8.2.0, 0.75 v7.5.0, 0.80 v7.4.0, 0.75 v7.3.0, 0.67 v7.0.0, 0.71 v6.4.0, 0.33 v6.3.0, 0.86 v6.2.0, 1.00 v6.1.0
% Syntax : Number of formulae : 120 ( 33 unt; 59 typ; 0 def)
% Number of atoms : 195 ( 46 equ)
% Maximal formula atoms : 41 ( 1 avg)
% Number of connectives : 148 ( 14 ~; 1 |; 77 &)
% ( 16 <=>; 40 =>; 0 <=; 0 <~>)
% Maximal formula depth : 42 ( 6 avg)
% Maximal term depth : 8 ( 2 avg)
% Number arithmetic : 177 ( 50 atm; 18 fun; 59 num; 50 var)
% Number of types : 9 ( 7 usr; 1 ari)
% Number of type conns : 97 ( 43 >; 54 *; 0 +; 0 <<)
% Number of predicates : 20 ( 17 usr; 0 prp; 1-5 aty)
% Number of functors : 42 ( 35 usr; 14 con; 0-6 aty)
% Number of variables : 205 ( 205 !; 0 ?; 205 :)
% 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(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(is_index,type,
is_index1: $int > $o ).
tff(is_index_def,axiom,
! [I: $int] :
( is_index1(I)
<=> ( $lesseq(0,I)
& $less(I,81) ) ) ).
tff(map_int_int,type,
map_int_int: $tType ).
tff(valid_values,type,
valid_values1: map_int_int > $o ).
tff(t2tb,type,
t2tb: map_int_int > uni ).
tff(t2tb_sort,axiom,
! [X: map_int_int] : sort1(map(int,int),t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > map_int_int ).
tff(bridgeL,axiom,
! [I: map_int_int] : tb2t(t2tb(I)) = I ).
tff(bridgeR,axiom,
! [J: uni] : t2tb(tb2t(J)) = J ).
tff(t2tb1,type,
t2tb1: $int > uni ).
tff(t2tb_sort1,axiom,
! [X: $int] : sort1(int,t2tb1(X)) ).
tff(tb2t1,type,
tb2t1: uni > $int ).
tff(bridgeL1,axiom,
! [I: $int] : tb2t1(t2tb1(I)) = I ).
tff(bridgeR1,axiom,
! [J: uni] : t2tb1(tb2t1(J)) = J ).
tff(valid_values_def,axiom,
! [G: map_int_int] :
( valid_values1(G)
<=> ! [I: $int] :
( is_index1(I)
=> ( $lesseq(0,tb2t1(get(int,int,t2tb(G),t2tb1(I))))
& $lesseq(tb2t1(get(int,int,t2tb(G),t2tb1(I))),9) ) ) ) ).
tff(grid_eq_sub1,type,
grid_eq_sub1: ( map_int_int * map_int_int * $int * $int ) > $o ).
tff(grid_eq_sub_def,axiom,
! [G1: map_int_int,G2: map_int_int,A: $int,B: $int] :
( grid_eq_sub1(G1,G2,A,B)
<=> ! [J: $int] :
( ( $lesseq(A,J)
& $less(J,B) )
=> ( tb2t1(get(int,int,t2tb(G1),t2tb1(J))) = tb2t1(get(int,int,t2tb(G2),t2tb1(J))) ) ) ) ).
tff(grid_eq_sub,axiom,
! [G1: map_int_int,G2: map_int_int,A: $int,B: $int] :
( ( $lesseq(0,A)
& $lesseq(A,81)
& $lesseq(0,B)
& $lesseq(B,81)
& grid_eq_sub1(G1,G2,0,81) )
=> grid_eq_sub1(G1,G2,A,B) ) ).
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(get_def,axiom,
! [A: ty,A1: uni,I: $int] : get2(A,A1,I) = get(A,int,elts(A,A1),t2tb1(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),t2tb1(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(sudoku_chunks,type,
sudoku_chunks1: $tType ).
tff(sudoku_chunks1,type,
sudoku_chunks: ty ).
tff(array_int,type,
array_int: $tType ).
tff(mk_sudoku_chunks,type,
mk_sudoku_chunks1: ( array_int * array_int * array_int * array_int * array_int * array_int ) > sudoku_chunks1 ).
tff(column_start,type,
column_start1: sudoku_chunks1 > array_int ).
tff(column_start_def1,axiom,
! [U: array_int,U1: array_int,U2: array_int,U3: array_int,U4: array_int,U5: array_int] : column_start1(mk_sudoku_chunks1(U,U1,U2,U3,U4,U5)) = U ).
tff(column_offsets,type,
column_offsets1: sudoku_chunks1 > array_int ).
tff(column_offsets_def1,axiom,
! [U: array_int,U1: array_int,U2: array_int,U3: array_int,U4: array_int,U5: array_int] : column_offsets1(mk_sudoku_chunks1(U,U1,U2,U3,U4,U5)) = U1 ).
tff(row_start,type,
row_start1: sudoku_chunks1 > array_int ).
tff(row_start_def1,axiom,
! [U: array_int,U1: array_int,U2: array_int,U3: array_int,U4: array_int,U5: array_int] : row_start1(mk_sudoku_chunks1(U,U1,U2,U3,U4,U5)) = U2 ).
tff(row_offsets,type,
row_offsets1: sudoku_chunks1 > array_int ).
tff(row_offsets_def1,axiom,
! [U: array_int,U1: array_int,U2: array_int,U3: array_int,U4: array_int,U5: array_int] : row_offsets1(mk_sudoku_chunks1(U,U1,U2,U3,U4,U5)) = U3 ).
tff(square_start,type,
square_start1: sudoku_chunks1 > array_int ).
tff(square_start_def1,axiom,
! [U: array_int,U1: array_int,U2: array_int,U3: array_int,U4: array_int,U5: array_int] : square_start1(mk_sudoku_chunks1(U,U1,U2,U3,U4,U5)) = U4 ).
tff(square_offsets,type,
square_offsets1: sudoku_chunks1 > array_int ).
tff(square_offsets_def1,axiom,
! [U: array_int,U1: array_int,U2: array_int,U3: array_int,U4: array_int,U5: array_int] : square_offsets1(mk_sudoku_chunks1(U,U1,U2,U3,U4,U5)) = U5 ).
tff(sudoku_chunks_inversion1,axiom,
! [U: sudoku_chunks1] : U = mk_sudoku_chunks1(column_start1(U),column_offsets1(U),row_start1(U),row_offsets1(U),square_start1(U),square_offsets1(U)) ).
tff(chunk_valid_indexes,type,
chunk_valid_indexes1: ( array_int * array_int ) > $o ).
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(chunk_valid_indexes_def,axiom,
! [Start: array_int,Offsets: array_int] :
( chunk_valid_indexes1(Start,Offsets)
<=> ( ( length1(int,t2tb2(Start)) = 81 )
& ( length1(int,t2tb2(Offsets)) = 9 )
& ! [I: $int,O: $int] :
( ( is_index1(I)
& $lesseq(0,O)
& $less(O,9) )
=> is_index1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O)))) ) ) ) ).
tff(disjoint_chunks,type,
disjoint_chunks1: ( array_int * array_int ) > $o ).
tff(disjoint_chunks_def,axiom,
! [Start: array_int,Offsets: array_int] :
( disjoint_chunks1(Start,Offsets)
<=> ( ( length1(int,t2tb2(Start)) = 81 )
& ( length1(int,t2tb2(Offsets)) = 9 )
& ! [I1: $int,I2: $int,O: $int] :
( ( is_index1(I1)
& is_index1(I2)
& $lesseq(0,O)
& $less(O,9) )
=> ( ( tb2t1(get2(int,t2tb2(Start),I1)) != tb2t1(get2(int,t2tb2(Start),I2)) )
=> ( I1 != $sum(tb2t1(get2(int,t2tb2(Start),I2)),tb2t1(get2(int,t2tb2(Offsets),O))) ) ) ) ) ) ).
tff(well_formed_sudoku,type,
well_formed_sudoku1: sudoku_chunks1 > $o ).
tff(well_formed_sudoku_def,axiom,
! [S: sudoku_chunks1] :
( well_formed_sudoku1(S)
<=> ( chunk_valid_indexes1(column_start1(S),column_offsets1(S))
& chunk_valid_indexes1(row_start1(S),row_offsets1(S))
& chunk_valid_indexes1(square_start1(S),square_offsets1(S))
& disjoint_chunks1(column_start1(S),column_offsets1(S))
& disjoint_chunks1(row_start1(S),row_offsets1(S))
& disjoint_chunks1(square_start1(S),square_offsets1(S)) ) ) ).
tff(valid_chunk,type,
valid_chunk1: ( map_int_int * $int * array_int * array_int ) > $o ).
tff(valid_chunk_def,axiom,
! [G: map_int_int,I: $int,Start: array_int,Offsets: array_int] :
( valid_chunk1(G,I,Start,Offsets)
<=> ! [O1: $int,O2: $int] :
( ( $lesseq(0,O1)
& $less(O1,9)
& $lesseq(0,O2)
& $less(O2,9)
& ( O1 != O2 ) )
=> ( ( $lesseq(1,tb2t1(get(int,int,t2tb(G),t2tb1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O1)))))))
& $lesseq(tb2t1(get(int,int,t2tb(G),t2tb1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O1)))))),9)
& $lesseq(1,tb2t1(get(int,int,t2tb(G),t2tb1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O2)))))))
& $lesseq(tb2t1(get(int,int,t2tb(G),t2tb1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O2)))))),9) )
=> ( tb2t1(get(int,int,t2tb(G),t2tb1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O1)))))) != tb2t1(get(int,int,t2tb(G),t2tb1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O2)))))) ) ) ) ) ).
tff(valid_column,type,
valid_column1: ( sudoku_chunks1 * map_int_int * $int ) > $o ).
tff(valid_column_def,axiom,
! [S: sudoku_chunks1,G: map_int_int,I: $int] :
( valid_column1(S,G,I)
<=> valid_chunk1(G,I,column_start1(S),column_offsets1(S)) ) ).
tff(valid_row,type,
valid_row1: ( sudoku_chunks1 * map_int_int * $int ) > $o ).
tff(valid_row_def,axiom,
! [S: sudoku_chunks1,G: map_int_int,I: $int] :
( valid_row1(S,G,I)
<=> valid_chunk1(G,I,row_start1(S),row_offsets1(S)) ) ).
tff(valid_square,type,
valid_square1: ( sudoku_chunks1 * map_int_int * $int ) > $o ).
tff(valid_square_def,axiom,
! [S: sudoku_chunks1,G: map_int_int,I: $int] :
( valid_square1(S,G,I)
<=> valid_chunk1(G,I,square_start1(S),square_offsets1(S)) ) ).
tff(valid,type,
valid1: ( sudoku_chunks1 * map_int_int ) > $o ).
tff(valid_def,axiom,
! [S: sudoku_chunks1,G: map_int_int] :
( valid1(S,G)
<=> ! [I: $int] :
( is_index1(I)
=> ( valid_column1(S,G,I)
& valid_row1(S,G,I)
& valid_square1(S,G,I) ) ) ) ).
tff(full,type,
full1: map_int_int > $o ).
tff(full_def,axiom,
! [G: map_int_int] :
( full1(G)
<=> ! [I: $int] :
( is_index1(I)
=> ( $lesseq(1,tb2t1(get(int,int,t2tb(G),t2tb1(I))))
& $lesseq(tb2t1(get(int,int,t2tb(G),t2tb1(I))),9) ) ) ) ).
tff(included,type,
included1: ( map_int_int * map_int_int ) > $o ).
tff(included_def,axiom,
! [G1: map_int_int,G2: map_int_int] :
( included1(G1,G2)
<=> ! [I: $int] :
( ( is_index1(I)
& $lesseq(1,tb2t1(get(int,int,t2tb(G1),t2tb1(I))))
& $lesseq(tb2t1(get(int,int,t2tb(G1),t2tb1(I))),9) )
=> ( tb2t1(get(int,int,t2tb(G2),t2tb1(I))) = tb2t1(get(int,int,t2tb(G1),t2tb1(I))) ) ) ) ).
tff(subset_valid_chunk,axiom,
! [G: map_int_int,H: map_int_int] :
( included1(G,H)
=> ! [I: $int,S: array_int,O: array_int] :
( ( is_index1(I)
& chunk_valid_indexes1(S,O)
& valid_chunk1(H,I,S,O) )
=> valid_chunk1(G,I,S,O) ) ) ).
tff(subset_valid,axiom,
! [S: sudoku_chunks1,G: map_int_int,H: map_int_int] :
( ( well_formed_sudoku1(S)
& included1(G,H)
& valid1(S,H) )
=> valid1(S,G) ) ).
tff(is_solution_for,type,
is_solution_for1: ( sudoku_chunks1 * map_int_int * map_int_int ) > $o ).
tff(is_solution_for_def,axiom,
! [S: sudoku_chunks1,Sol: map_int_int,Data: map_int_int] :
( is_solution_for1(S,Sol,Data)
<=> ( included1(Data,Sol)
& full1(Sol)
& valid1(S,Sol) ) ) ).
tff(valid_chunk_up_to,type,
valid_chunk_up_to1: ( map_int_int * $int * array_int * array_int * $int ) > $o ).
tff(valid_chunk_up_to_def,axiom,
! [G: map_int_int,I: $int,Start: array_int,Offsets: array_int,Off: $int] :
( valid_chunk_up_to1(G,I,Start,Offsets,Off)
<=> ! [O1: $int,O2: $int] :
( ( $lesseq(0,O1)
& $less(O1,Off)
& $lesseq(0,O2)
& $less(O2,Off)
& ( O1 != O2 ) )
=> ( ( $lesseq(1,tb2t1(get(int,int,t2tb(G),t2tb1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O1)))))))
& $lesseq(tb2t1(get(int,int,t2tb(G),t2tb1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O1)))))),9)
& $lesseq(1,tb2t1(get(int,int,t2tb(G),t2tb1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O2)))))))
& $lesseq(tb2t1(get(int,int,t2tb(G),t2tb1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O2)))))),9) )
=> ( tb2t1(get(int,int,t2tb(G),t2tb1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O1)))))) != tb2t1(get(int,int,t2tb(G),t2tb1($sum(tb2t1(get2(int,t2tb2(Start),I)),tb2t1(get2(int,t2tb2(Offsets),O2)))))) ) ) ) ) ).
tff(valid_up_to,type,
valid_up_to1: ( sudoku_chunks1 * map_int_int * $int ) > $o ).
tff(valid_up_to_def,axiom,
! [S: sudoku_chunks1,G: map_int_int,I: $int] :
( valid_up_to1(S,G,I)
<=> ! [J: $int] :
( ( $lesseq(0,J)
& $less(J,I) )
=> ( valid_column1(S,G,J)
& valid_row1(S,G,J)
& valid_square1(S,G,J) ) ) ) ).
tff(wP_parameter_check_valid,conjecture,
! [S: $int,S1: map_int_int,S2: $int,S3: map_int_int,S4: $int,S5: map_int_int,S6: $int,S7: map_int_int,S8: $int,S9: map_int_int,S10: $int,S11: map_int_int,G: $int,G1: map_int_int] :
( ( $lesseq(0,S)
& $lesseq(0,S2)
& $lesseq(0,S4)
& $lesseq(0,S6)
& $lesseq(0,S8)
& $lesseq(0,S10)
& $lesseq(0,G)
& well_formed_sudoku1(mk_sudoku_chunks1(tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))),tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))),tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11)))))
& ( G = 81 )
& valid_values1(G1) )
=> ( ( $less(80,0)
=> valid1(mk_sudoku_chunks1(tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))),tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))),tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11)))),G1) )
& ( $lesseq(0,80)
=> ( valid_up_to1(mk_sudoku_chunks1(tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))),tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))),tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11)))),G1,0)
& ! [I: $int] :
( ( $lesseq(0,I)
& $lesseq(I,80) )
=> ( valid_up_to1(mk_sudoku_chunks1(tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))),tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))),tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11)))),G1,I)
=> ( ( G = 81 )
& valid_values1(G1)
& is_index1(I)
& chunk_valid_indexes1(tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))))
& ( valid_chunk1(G1,I,tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))))
=> ( ( G = 81 )
& valid_values1(G1)
& is_index1(I)
& chunk_valid_indexes1(tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))))
& ( valid_chunk1(G1,I,tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))))
=> ( ( G = 81 )
& valid_values1(G1)
& is_index1(I)
& chunk_valid_indexes1(tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11))))
& ( valid_chunk1(G1,I,tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11))))
=> valid_up_to1(mk_sudoku_chunks1(tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))),tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))),tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11)))),G1,$sum(I,1)) )
& ( ~ valid_chunk1(G1,I,tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11))))
=> ~ valid1(mk_sudoku_chunks1(tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))),tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))),tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11)))),G1) ) ) )
& ( ~ valid_chunk1(G1,I,tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))))
=> ~ valid1(mk_sudoku_chunks1(tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))),tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))),tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11)))),G1) ) ) )
& ( ~ valid_chunk1(G1,I,tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))))
=> ~ valid1(mk_sudoku_chunks1(tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))),tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))),tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11)))),G1) ) ) ) )
& ( valid_up_to1(mk_sudoku_chunks1(tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))),tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))),tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11)))),G1,$sum(80,1))
=> valid1(mk_sudoku_chunks1(tb2t2(mk_array1(int,S,t2tb(S1))),tb2t2(mk_array1(int,S2,t2tb(S3))),tb2t2(mk_array1(int,S4,t2tb(S5))),tb2t2(mk_array1(int,S6,t2tb(S7))),tb2t2(mk_array1(int,S8,t2tb(S9))),tb2t2(mk_array1(int,S10,t2tb(S11)))),G1) ) ) ) ) ) ).
%------------------------------------------------------------------------------