TPTP Problem File: SWW660_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW660_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Verifythis fm2012 LRS-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 : verifythis_fm2012_LRS-T-WP_parameter_sort [Fil14]
% Status : Theorem
% Rating : 0.75 v8.2.0, 0.88 v8.1.0, 1.00 v7.5.0, 0.90 v7.4.0, 0.88 v7.3.0, 0.67 v7.1.0, 0.50 v7.0.0, 0.71 v6.4.0, 0.67 v6.3.0, 0.71 v6.2.0, 1.00 v6.1.0
% Syntax : Number of formulae : 136 ( 28 unt; 57 typ; 0 def)
% Number of atoms : 288 ( 63 equ)
% Maximal formula atoms : 57 ( 2 avg)
% Number of connectives : 221 ( 12 ~; 3 |; 98 &)
% ( 19 <=>; 89 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number arithmetic : 376 ( 126 atm; 44 fun; 83 num; 123 var)
% Number of types : 8 ( 6 usr; 1 ari)
% Number of type conns : 125 ( 43 >; 82 *; 0 +; 0 <<)
% Number of predicates : 23 ( 20 usr; 0 prp; 1-7 aty)
% Number of functors : 36 ( 31 usr; 10 con; 0-5 aty)
% Number of variables : 301 ( 297 !; 4 ?; 301 :)
% 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_sort2,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_def3,axiom,
! [A: ty,U: uni] :
( sort1(A,U)
=> ( contents(A,mk_ref(A,U)) = U ) ) ).
tff(ref_inversion3,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_sort4,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_sort4,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_sort2,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_sort2,axiom,
! [A: ty,X: $int,X1: uni] : sort1(array(A),mk_array1(A,X,X1)) ).
tff(length,type,
length1: ( ty * uni ) > $int ).
tff(length_def3,axiom,
! [A: ty,U: $int,U1: uni] : length1(A,mk_array1(A,U,U1)) = U ).
tff(elts,type,
elts: ( ty * uni ) > uni ).
tff(elts_sort2,axiom,
! [A: ty,X: uni] : sort1(map(int,A),elts(A,X)) ).
tff(elts_def3,axiom,
! [A: ty,U: $int,U1: uni] :
( sort1(map(int,A),U1)
=> ( elts(A,mk_array1(A,U,U1)) = U1 ) ) ).
tff(array_inversion3,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_sort5,axiom,
! [A: ty,X: uni,X1: $int] : sort1(A,get2(A,X,X1)) ).
tff(t2tb,type,
t2tb: $int > uni ).
tff(t2tb_sort2,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_sort5,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_sort2,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(array_int,type,
array_int: $tType ).
tff(is_common_prefix,type,
is_common_prefix1: ( array_int * $int * $int * $int ) > $o ).
tff(t2tb1,type,
t2tb1: array_int > uni ).
tff(t2tb_sort3,axiom,
! [X: array_int] : sort1(array(int),t2tb1(X)) ).
tff(tb2t1,type,
tb2t1: uni > array_int ).
tff(bridgeL1,axiom,
! [I: array_int] : tb2t1(t2tb1(I)) = I ).
tff(bridgeR1,axiom,
! [J: uni] : t2tb1(tb2t1(J)) = J ).
tff(is_common_prefix_def,axiom,
! [A: array_int,X: $int,Y: $int,L: $int] :
( is_common_prefix1(A,X,Y,L)
<=> ( $lesseq(0,L)
& $lesseq($sum(X,L),length1(int,t2tb1(A)))
& $lesseq($sum(Y,L),length1(int,t2tb1(A)))
& ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,L) )
=> ( tb2t(get2(int,t2tb1(A),$sum(X,I))) = tb2t(get2(int,t2tb1(A),$sum(Y,I))) ) ) ) ) ).
tff(not_common_prefix_if_last_char_are_different,axiom,
! [A: array_int,X: $int,Y: $int,L: $int] :
( ( $lesseq(0,L)
& $less($sum(X,L),length1(int,t2tb1(A)))
& $less($sum(Y,L),length1(int,t2tb1(A)))
& ( tb2t(get2(int,t2tb1(A),$sum(X,L))) != tb2t(get2(int,t2tb1(A),$sum(Y,L))) ) )
=> ~ is_common_prefix1(A,X,Y,$sum(L,1)) ) ).
tff(is_longest_common_prefix,type,
is_longest_common_prefix1: ( array_int * $int * $int * $int ) > $o ).
tff(is_longest_common_prefix_def,axiom,
! [A: array_int,X: $int,Y: $int,L: $int] :
( is_longest_common_prefix1(A,X,Y,L)
<=> ( is_common_prefix1(A,X,Y,L)
& ! [M: $int] :
( $less(L,M)
=> ~ is_common_prefix1(A,X,Y,M) ) ) ) ).
tff(longest_common_prefix_succ,axiom,
! [A: array_int,X: $int,Y: $int,L: $int] :
( ( is_common_prefix1(A,X,Y,L)
& ~ is_common_prefix1(A,X,Y,$sum(L,1)) )
=> is_longest_common_prefix1(A,X,Y,L) ) ).
tff(lt,type,
lt1: ( array_int * $int * $int ) > $o ).
tff(lt_def,axiom,
! [A: array_int,X: $int,Y: $int] :
( lt1(A,X,Y)
<=> ( $lesseq(0,X)
& $lesseq(X,length1(int,t2tb1(A)))
& $lesseq(0,Y)
& $lesseq(Y,length1(int,t2tb1(A)))
& ? [L: $int] :
( is_common_prefix1(A,X,Y,L)
& $less($sum(Y,L),length1(int,t2tb1(A)))
& ( ( $sum(X,L) = length1(int,t2tb1(A)) )
| $less(tb2t(get2(int,t2tb1(A),$sum(X,L))),tb2t(get2(int,t2tb1(A),$sum(Y,L)))) ) ) ) ) ).
tff(map_int_int,type,
map_int_int: $tType ).
tff(injective,type,
injective1: ( map_int_int * $int ) > $o ).
tff(t2tb2,type,
t2tb2: map_int_int > uni ).
tff(t2tb_sort4,axiom,
! [X: map_int_int] : sort1(map(int,int),t2tb2(X)) ).
tff(tb2t2,type,
tb2t2: uni > map_int_int ).
tff(bridgeL2,axiom,
! [I: map_int_int] : tb2t2(t2tb2(I)) = I ).
tff(bridgeR2,axiom,
! [J: uni] : t2tb2(tb2t2(J)) = J ).
tff(injective_def,axiom,
! [A: map_int_int,N: $int] :
( injective1(A,N)
<=> ! [I: $int,J: $int] :
( ( $lesseq(0,I)
& $less(I,N) )
=> ( ( $lesseq(0,J)
& $less(J,N) )
=> ( ( I != J )
=> ( tb2t(get(int,int,t2tb2(A),t2tb(I))) != tb2t(get(int,int,t2tb2(A),t2tb(J))) ) ) ) ) ) ).
tff(surjective,type,
surjective1: ( map_int_int * $int ) > $o ).
tff(surjective_def,axiom,
! [A: map_int_int,N: $int] :
( surjective1(A,N)
<=> ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,N) )
=> ? [J: $int] :
( $lesseq(0,J)
& $less(J,N)
& ( tb2t(get(int,int,t2tb2(A),t2tb(J))) = I ) ) ) ) ).
tff(range,type,
range2: ( map_int_int * $int ) > $o ).
tff(range_def,axiom,
! [A: map_int_int,N: $int] :
( range2(A,N)
<=> ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,N) )
=> ( $lesseq(0,tb2t(get(int,int,t2tb2(A),t2tb(I))))
& $less(tb2t(get(int,int,t2tb2(A),t2tb(I))),N) ) ) ) ).
tff(injective_surjective,axiom,
! [A: map_int_int,N: $int] :
( injective1(A,N)
=> ( range2(A,N)
=> surjective1(A,N) ) ) ).
tff(occ,type,
occ1: ( ty * uni * uni * $int * $int ) > $int ).
tff(occ_empty,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( $lesseq(U,L)
=> ( occ1(A,V,M,L,U) = 0 ) ) ).
tff(occ_right_no_add,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( sort1(A,V)
=> ( $less(L,U)
=> ( ( get(A,int,M,t2tb($difference(U,1))) != V )
=> ( occ1(A,V,M,L,U) = occ1(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 )
=> ( occ1(A,V,M,L,U) = $sum(1,occ1(A,V,M,L,$difference(U,1))) ) ) ) ).
tff(occ_bounds,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( $lesseq(L,U)
=> ( $lesseq(0,occ1(A,V,M,L,U))
& $lesseq(occ1(A,V,M,L,U),$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) )
=> ( occ1(A,V,M,L,U) = $sum(occ1(A,V,M,L,Mid),occ1(A,V,M,Mid,U)) ) ) ).
tff(occ_neq,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( sort1(A,V)
=> ( ! [I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( get(A,int,M,t2tb(I)) != V ) )
=> ( occ1(A,V,M,L,U) = 0 ) ) ) ).
tff(occ_exists,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( sort1(A,V)
=> ( $less(0,occ1(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,occ1(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)) ) )
=> ( occ1(A,V,M1,L,U) = occ1(A,V,M2,L,U) ) ) ).
tff(injection_occ,axiom,
! [M: map_int_int,N: $int] :
( injective1(M,N)
<=> ! [V: $int] : $lesseq(occ1(int,t2tb(V),t2tb2(M),0,N),1) ) ).
tff(range1,type,
range3: array_int > $o ).
tff(range_def1,axiom,
! [A: array_int] :
( range3(A)
<=> range2(tb2t2(elts(int,t2tb1(A))),length1(int,t2tb1(A))) ) ).
tff(permut,type,
permut2: ( ty * uni * uni * $int * $int ) > $o ).
tff(permut_def,axiom,
! [A: ty,M1: uni,M2: uni,L: $int,U: $int] :
( ( permut2(A,M1,M2,L,U)
=> ! [V: uni] : occ1(A,V,M1,L,U) = occ1(A,V,M2,L,U) )
& ( ! [V: uni] :
( sort1(A,V)
=> ( occ1(A,V,M1,L,U) = occ1(A,V,M2,L,U) ) )
=> permut2(A,M1,M2,L,U) ) ) ).
tff(permut_trans,axiom,
! [A: ty,A1: uni,A2: uni,A3: uni,L: $int,U: $int] :
( permut2(A,A1,A2,L,U)
=> ( permut2(A,A2,A3,L,U)
=> permut2(A,A1,A3,L,U) ) ) ).
tff(permut_exists,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int,I: $int] :
( permut2(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_sub1: ( ty * uni * uni * $int * $int ) > $o ).
tff(map_eq_sub_def,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
( map_eq_sub1(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_sub1: ( ty * uni * uni * $int * $int ) > $o ).
tff(array_eq_sub_def,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
( array_eq_sub1(A,A1,A2,L,U)
<=> ( ( length1(A,A1) = length1(A,A2) )
& $lesseq(0,L)
& $lesseq(L,length1(A,A1))
& $lesseq(0,U)
& $lesseq(U,length1(A,A1))
& map_eq_sub1(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)
<=> ( ( length1(A,A1) = length1(A,A2) )
& map_eq_sub1(A,elts(A,A1),elts(A,A2),0,length1(A,A1)) ) ) ).
tff(exchange,type,
exchange2: ( 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] :
( exchange2(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) )
=> exchange2(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,
exchange3: ( ty * uni * uni * $int * $int ) > $o ).
tff(exchange_def1,axiom,
! [A: ty,A1: uni,A2: uni,I: $int,J: $int] :
( exchange3(A,A1,A2,I,J)
<=> ( ( length1(A,A1) = length1(A,A2) )
& exchange2(A,elts(A,A1),elts(A,A2),0,length1(A,A1),I,J) ) ) ).
tff(permut1,type,
permut3: ( ty * uni * uni * $int * $int ) > $o ).
tff(permut_def1,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
( permut3(A,A1,A2,L,U)
<=> ( ( length1(A,A1) = length1(A,A2) )
& $lesseq(0,L)
& $lesseq(L,length1(A,A1))
& $lesseq(0,U)
& $lesseq(U,length1(A,A1))
& permut2(A,elts(A,A1),elts(A,A2),L,U) ) ) ).
tff(permut_sub,type,
permut_sub1: ( ty * uni * uni * $int * $int ) > $o ).
tff(permut_sub_def,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
( permut_sub1(A,A1,A2,L,U)
<=> ( map_eq_sub1(A,elts(A,A1),elts(A,A2),0,L)
& permut3(A,A1,A2,L,U)
& map_eq_sub1(A,elts(A,A1),elts(A,A2),U,length1(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)
<=> ( ( length1(A,A1) = length1(A,A2) )
& permut2(A,elts(A,A1),elts(A,A2),0,length1(A,A1)) ) ) ).
tff(exchange_permut_sub,axiom,
! [A: ty,A1: uni,A2: uni,I: $int,J: $int,L: $int,U: $int] :
( exchange3(A,A1,A2,I,J)
=> ( ( $lesseq(L,I)
& $less(I,U) )
=> ( ( $lesseq(L,J)
& $less(J,U) )
=> ( $lesseq(0,L)
=> ( $lesseq(U,length1(A,A1))
=> permut_sub1(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_sub1(A,A1,A2,L1,U1)
=> ( ( $lesseq(0,L2)
& $lesseq(L2,L1) )
=> ( ( $lesseq(U1,U2)
& $lesseq(U2,length1(A,A1)) )
=> permut_sub1(A,A1,A2,L2,U2) ) ) ) ).
tff(exchange_permut_all,axiom,
! [A: ty,A1: uni,A2: uni,I: $int,J: $int] :
( exchange3(A,A1,A2,I,J)
=> permut_all(A,A1,A2) ) ).
tff(le,type,
le1: ( array_int * $int * $int ) > $o ).
tff(le_def,axiom,
! [A: array_int,X: $int,Y: $int] :
( le1(A,X,Y)
<=> ( ( X = Y )
| lt1(A,X,Y) ) ) ).
tff(lcp_same_index,axiom,
! [A: array_int,X: $int] :
( ( $lesseq(0,X)
& $lesseq(X,length1(int,t2tb1(A))) )
=> is_longest_common_prefix1(A,X,X,$difference(length1(int,t2tb1(A)),X)) ) ).
tff(le_trans,axiom,
! [A: array_int,X: $int,Y: $int,Z: $int] :
( ( le1(A,X,Y)
& le1(A,Y,Z) )
=> le1(A,X,Z) ) ).
tff(sorted_sub,type,
sorted_sub1: ( array_int * map_int_int * $int * $int ) > $o ).
tff(sorted_sub_def,axiom,
! [A: array_int,Data: map_int_int,L: $int,U: $int] :
( sorted_sub1(A,Data,L,U)
<=> ! [I1: $int,I2: $int] :
( ( $lesseq(L,I1)
& $lesseq(I1,I2)
& $less(I2,U) )
=> le1(A,tb2t(get(int,int,t2tb2(Data),t2tb(I1))),tb2t(get(int,int,t2tb2(Data),t2tb(I2)))) ) ) ).
tff(sorted,type,
sorted1: ( array_int * array_int ) > $o ).
tff(sorted_def,axiom,
! [A: array_int,Data: array_int] :
( sorted1(A,Data)
<=> sorted_sub1(A,tb2t2(elts(int,t2tb1(Data))),0,length1(int,t2tb1(Data))) ) ).
tff(wP_parameter_sort,conjecture,
! [A: $int,A1: map_int_int,Data: $int,Data1: map_int_int] :
( ( $lesseq(0,A)
& $lesseq(0,Data)
& ( Data = A )
& range2(Data1,Data) )
=> ( $lesseq(0,$difference(Data,1))
=> ! [Data2: map_int_int,I: $int] :
( ( $lesseq(0,I)
& $lesseq(I,$difference(Data,1)) )
=> ( ( permut_all(int,mk_array1(int,Data,t2tb2(Data1)),mk_array1(int,Data,t2tb2(Data2)))
& sorted_sub1(tb2t1(mk_array1(int,A,t2tb2(A1))),Data2,0,I)
& range2(Data2,Data) )
=> ! [J: $int,Data3: map_int_int] :
( ( $lesseq(0,J)
& $lesseq(J,I)
& range2(Data3,Data)
& permut_all(int,mk_array1(int,Data,t2tb2(Data1)),mk_array1(int,Data,t2tb2(Data3)))
& sorted_sub1(tb2t1(mk_array1(int,A,t2tb2(A1))),Data3,0,J)
& sorted_sub1(tb2t1(mk_array1(int,A,t2tb2(A1))),Data3,J,$sum(I,1))
& ! [K1: $int,K2: $int] :
( ( $lesseq(0,K1)
& $less(K1,J)
& $lesseq($sum(J,1),K2)
& $lesseq(K2,I) )
=> le1(tb2t1(mk_array1(int,A,t2tb2(A1))),tb2t(get(int,int,t2tb2(Data3),t2tb(K1))),tb2t(get(int,int,t2tb2(Data3),t2tb(K2)))) ) )
=> ( $less(0,J)
=> ( ( $lesseq(0,Data)
& $lesseq(0,J)
& $less(J,Data) )
=> ( ( $lesseq(0,$difference(J,1))
& $less($difference(J,1),Data) )
=> ( ( $lesseq(0,tb2t(get(int,int,t2tb2(Data3),t2tb($difference(J,1)))))
& $lesseq(tb2t(get(int,int,t2tb2(Data3),t2tb($difference(J,1)))),A)
& $lesseq(0,tb2t(get(int,int,t2tb2(Data3),t2tb(J))))
& $lesseq(tb2t(get(int,int,t2tb2(Data3),t2tb(J))),A) )
=> ! [O: $int] :
( ( ( ( O = 0 )
=> ( tb2t(get(int,int,t2tb2(Data3),t2tb($difference(J,1)))) = tb2t(get(int,int,t2tb2(Data3),t2tb(J))) ) )
& ( $less(O,0)
=> lt1(tb2t1(mk_array1(int,A,t2tb2(A1))),tb2t(get(int,int,t2tb2(Data3),t2tb($difference(J,1)))),tb2t(get(int,int,t2tb2(Data3),t2tb(J)))) )
& ( $less(0,O)
=> lt1(tb2t1(mk_array1(int,A,t2tb2(A1))),tb2t(get(int,int,t2tb2(Data3),t2tb(J))),tb2t(get(int,int,t2tb2(Data3),t2tb($difference(J,1))))) ) )
=> ( $less(0,O)
=> ( ( $lesseq(0,J)
& $less(J,Data) )
=> ( ( $lesseq(0,$difference(J,1))
& $less($difference(J,1),Data) )
=> ( ( $lesseq(0,J)
& $less(J,Data) )
=> ! [Data4: map_int_int] :
( ( $lesseq(0,Data)
& ( Data4 = tb2t2(set(int,int,t2tb2(Data3),t2tb(J),get(int,int,t2tb2(Data3),t2tb($difference(J,1))))) ) )
=> ( ( $lesseq(0,$difference(J,1))
& $less($difference(J,1),Data) )
=> ! [Data5: map_int_int] :
( ( $lesseq(0,Data)
& ( Data5 = tb2t2(set(int,int,t2tb2(Data4),t2tb($difference(J,1)),get(int,int,t2tb2(Data3),t2tb(J)))) ) )
=> ( exchange3(int,mk_array1(int,Data,t2tb2(Data3)),mk_array1(int,Data,t2tb2(Data5)),$difference(J,1),J)
=> ! [J1: $int] :
( ( J1 = $difference(J,1) )
=> ! [K1: $int,K2: $int] :
( ( $lesseq(0,K1)
& $less(K1,J1)
& $lesseq($sum(J1,1),K2)
& $lesseq(K2,I) )
=> le1(tb2t1(mk_array1(int,A,t2tb2(A1))),tb2t(get(int,int,t2tb2(Data5),t2tb(K1))),tb2t(get(int,int,t2tb2(Data5),t2tb(K2)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------