TPTP Problem File: SWW573_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW573_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Algo63-T-WP parameter partition
% 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 : algo63-T-WP_parameter_partition_ [Fil14]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.75 v7.5.0, 0.80 v7.4.0, 0.50 v7.3.0, 0.33 v7.0.0, 0.43 v6.4.0, 0.33 v6.3.0, 0.71 v6.2.0, 0.75 v6.1.0
% Syntax : Number of formulae : 138 ( 35 unt; 51 typ; 0 def)
% Number of atoms : 303 ( 90 equ)
% Maximal formula atoms : 59 ( 2 avg)
% Number of connectives : 228 ( 12 ~; 5 |; 100 &)
% ( 12 <=>; 99 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number arithmetic : 353 ( 136 atm; 18 fun; 43 num; 156 var)
% Number of types : 8 ( 6 usr; 1 ari)
% Number of type conns : 110 ( 37 >; 73 *; 0 +; 0 <<)
% Number of predicates : 15 ( 12 usr; 0 prp; 2-7 aty)
% Number of functors : 38 ( 33 usr; 10 con; 0-5 aty)
% Number of variables : 344 ( 340 !; 4 ?; 344 :)
% 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_sort3,axiom,
! [A: ty,B: ty,X: uni,X1: uni] : sort1(B,get(B,A,X,X1)) ).
tff(map_int_int,type,
map_int_int: $tType ).
tff(get1,type,
get3: ( map_int_int * $int ) > $int ).
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(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(select_eq2,axiom,
! [M: map_int_int,A1: $int,A2: $int,B: $int] :
( ( A1 = A2 )
=> ( get3(tb2t(set(int,int,t2tb(M),t2tb1(A1),t2tb1(B))),A2) = B ) ) ).
tff(select_eq3,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_neq2,axiom,
! [M: map_int_int,A1: $int,A2: $int,B: $int] :
( ( A1 != A2 )
=> ( get3(tb2t(set(int,int,t2tb(M),t2tb1(A1),t2tb1(B))),A2) = get3(M,A2) ) ) ).
tff(select_neq3,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(const2,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(const3,axiom,
! [B: $int,A: $int] : get3(tb2t(const(int,int,t2tb1(B))),A) = B ).
tff(const4,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_array2: ( ty * $int * uni ) > uni ).
tff(mk_array_sort2,axiom,
! [A: ty,X: $int,X1: uni] : sort1(array(A),mk_array2(A,X,X1)) ).
tff(array_int,type,
array_int: $tType ).
tff(mk_array1,type,
mk_array3: ( $int * map_int_int ) > array_int ).
tff(length,type,
length2: ( ty * uni ) > $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(length_def2,axiom,
! [U: $int,U1: map_int_int] : length2(int,t2tb2(mk_array3(U,U1))) = U ).
tff(length_def3,axiom,
! [A: ty,U: $int,U1: uni] : length2(A,mk_array2(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_def2,axiom,
! [U: $int,U1: map_int_int] : tb2t(elts(int,t2tb2(mk_array3(U,U1)))) = U1 ).
tff(elts_def3,axiom,
! [A: ty,U: $int,U1: uni] :
( sort1(map(int,A),U1)
=> ( elts(A,mk_array2(A,U,U1)) = U1 ) ) ).
tff(array_inversion2,axiom,
! [U: array_int] : U = mk_array3(length2(int,t2tb2(U)),tb2t(elts(int,t2tb2(U)))) ).
tff(array_inversion3,axiom,
! [A: ty,U: uni] : U = mk_array2(A,length2(A,U),elts(A,U)) ).
tff(get2,type,
get4: ( ty * uni * $int ) > uni ).
tff(get_sort4,axiom,
! [A: ty,X: uni,X1: $int] : sort1(A,get4(A,X,X1)) ).
tff(get_def2,axiom,
! [A: array_int,I: $int] : tb2t1(get4(int,t2tb2(A),I)) = get3(tb2t(elts(int,t2tb2(A))),I) ).
tff(get_def3,axiom,
! [A: ty,A1: uni,I: $int] : get4(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_def2,axiom,
! [A: array_int,I: $int,V: $int] : tb2t2(set2(int,t2tb2(A),I,t2tb1(V))) = mk_array3(length2(int,t2tb2(A)),tb2t(set(int,int,elts(int,t2tb2(A)),t2tb1(I),t2tb1(V)))) ).
tff(set_def3,axiom,
! [A: ty,A1: uni,I: $int,V: uni] : set2(A,A1,I,V) = mk_array2(A,length2(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_def2,axiom,
! [N: $int,V: $int] : tb2t2(make1(int,N,t2tb1(V))) = mk_array3(N,tb2t(const(int,int,t2tb1(V)))) ).
tff(make_def3,axiom,
! [A: ty,N: $int,V: uni] : make1(A,N,V) = mk_array2(A,N,const(A,int,V)) ).
tff(occ,type,
occ2: ( ty * uni * uni * $int * $int ) > $int ).
tff(occ_empty2,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( $lesseq(U,L)
=> ( occ2(A,V,M,L,U) = 0 ) ) ).
tff(occ_right_no_add2,axiom,
! [V: $int,M: map_int_int,L: $int,U: $int] :
( $less(L,U)
=> ( ( get3(M,$difference(U,1)) != V )
=> ( occ2(int,t2tb1(V),t2tb(M),L,U) = occ2(int,t2tb1(V),t2tb(M),L,$difference(U,1)) ) ) ) ).
tff(occ_right_no_add3,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( sort1(A,V)
=> ( $less(L,U)
=> ( ( get(A,int,M,t2tb1($difference(U,1))) != V )
=> ( occ2(A,V,M,L,U) = occ2(A,V,M,L,$difference(U,1)) ) ) ) ) ).
tff(occ_right_add2,axiom,
! [V: $int,M: map_int_int,L: $int,U: $int] :
( $less(L,U)
=> ( ( get3(M,$difference(U,1)) = V )
=> ( occ2(int,t2tb1(V),t2tb(M),L,U) = $sum(1,occ2(int,t2tb1(V),t2tb(M),L,$difference(U,1))) ) ) ) ).
tff(occ_right_add3,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( $less(L,U)
=> ( ( get(A,int,M,t2tb1($difference(U,1))) = V )
=> ( occ2(A,V,M,L,U) = $sum(1,occ2(A,V,M,L,$difference(U,1))) ) ) ) ).
tff(occ_bounds2,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( $lesseq(L,U)
=> ( $lesseq(0,occ2(A,V,M,L,U))
& $lesseq(occ2(A,V,M,L,U),$difference(U,L)) ) ) ).
tff(occ_append2,axiom,
! [A: ty,V: uni,M: uni,L: $int,Mid: $int,U: $int] :
( ( $lesseq(L,Mid)
& $lesseq(Mid,U) )
=> ( occ2(A,V,M,L,U) = $sum(occ2(A,V,M,L,Mid),occ2(A,V,M,Mid,U)) ) ) ).
tff(occ_neq2,axiom,
! [V: $int,M: map_int_int,L: $int,U: $int] :
( ! [I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( get3(M,I) != V ) )
=> ( occ2(int,t2tb1(V),t2tb(M),L,U) = 0 ) ) ).
tff(occ_neq3,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,t2tb1(I)) != V ) )
=> ( occ2(A,V,M,L,U) = 0 ) ) ) ).
tff(occ_exists2,axiom,
! [V: $int,M: map_int_int,L: $int,U: $int] :
( $less(0,occ2(int,t2tb1(V),t2tb(M),L,U))
=> ? [I: $int] :
( $lesseq(L,I)
& $less(I,U)
& ( get3(M,I) = V ) ) ) ).
tff(occ_exists3,axiom,
! [A: ty,V: uni,M: uni,L: $int,U: $int] :
( sort1(A,V)
=> ( $less(0,occ2(A,V,M,L,U))
=> ? [I: $int] :
( $lesseq(L,I)
& $less(I,U)
& ( get(A,int,M,t2tb1(I)) = V ) ) ) ) ).
tff(occ_pos2,axiom,
! [M: map_int_int,L: $int,U: $int,I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> $less(0,occ2(int,t2tb1(get3(M,I)),t2tb(M),L,U)) ) ).
tff(occ_pos3,axiom,
! [A: ty,M: uni,L: $int,U: $int,I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> $less(0,occ2(A,get(A,int,M,t2tb1(I)),M,L,U)) ) ).
tff(occ_eq2,axiom,
! [V: $int,M1: map_int_int,M2: map_int_int,L: $int,U: $int] :
( ! [I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( get3(M1,I) = get3(M2,I) ) )
=> ( occ2(int,t2tb1(V),t2tb(M1),L,U) = occ2(int,t2tb1(V),t2tb(M2),L,U) ) ) ).
tff(occ_eq3,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,t2tb1(I)) = get(A,int,M2,t2tb1(I)) ) )
=> ( occ2(A,V,M1,L,U) = occ2(A,V,M2,L,U) ) ) ).
tff(permut,type,
permut2: ( ty * uni * uni * $int * $int ) > $o ).
tff(permut_def4,axiom,
! [A: ty,M1: uni,M2: uni,L: $int,U: $int] :
( ( permut2(A,M1,M2,L,U)
=> ! [V: uni] : occ2(A,V,M1,L,U) = occ2(A,V,M2,L,U) )
& ( ! [V: uni] :
( sort1(A,V)
=> ( occ2(A,V,M1,L,U) = occ2(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_exists2,axiom,
! [A1: map_int_int,A2: map_int_int,L: $int,U: $int,I: $int] :
( permut2(int,t2tb(A1),t2tb(A2),L,U)
=> ( ( $lesseq(L,I)
& $less(I,U) )
=> ? [J: $int] :
( $lesseq(L,J)
& $less(J,U)
& ( get3(A1,J) = get3(A2,I) ) ) ) ) ).
tff(permut_exists3,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,t2tb1(J)) = get(A,int,A2,t2tb1(I)) ) ) ) ) ).
tff(map_eq_sub,type,
map_eq_sub1: ( ty * uni * uni * $int * $int ) > $o ).
tff(map_eq_sub_def2,axiom,
! [A1: map_int_int,A2: map_int_int,L: $int,U: $int] :
( map_eq_sub1(int,t2tb(A1),t2tb(A2),L,U)
<=> ! [I: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( get3(A1,I) = get3(A2,I) ) ) ) ).
tff(map_eq_sub_def3,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,t2tb1(I)) = get(A,int,A2,t2tb1(I)) ) ) ) ).
tff(array_eq_sub,type,
array_eq_sub1: ( ty * uni * uni * $int * $int ) > $o ).
tff(array_eq_sub_def2,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
( array_eq_sub1(A,A1,A2,L,U)
<=> ( ( length2(A,A1) = length2(A,A2) )
& $lesseq(0,L)
& $lesseq(L,length2(A,A1))
& $lesseq(0,U)
& $lesseq(U,length2(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_def2,axiom,
! [A: ty,A1: uni,A2: uni] :
( array_eq(A,A1,A2)
<=> ( ( length2(A,A1) = length2(A,A2) )
& map_eq_sub1(A,elts(A,A1),elts(A,A2),0,length2(A,A1)) ) ) ).
tff(exchange,type,
exchange2: ( ty * uni * uni * $int * $int * $int * $int ) > $o ).
tff(exchange_def4,axiom,
! [A1: map_int_int,A2: map_int_int,L: $int,U: $int,I: $int,J: $int] :
( exchange2(int,t2tb(A1),t2tb(A2),L,U,I,J)
<=> ( $lesseq(L,I)
& $less(I,U)
& $lesseq(L,J)
& $less(J,U)
& ( get3(A1,I) = get3(A2,J) )
& ( get3(A1,J) = get3(A2,I) )
& ! [K: $int] :
( ( $lesseq(L,K)
& $less(K,U) )
=> ( ( K != I )
=> ( ( K != J )
=> ( get3(A1,K) = get3(A2,K) ) ) ) ) ) ) ).
tff(exchange_def5,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,t2tb1(I)) = get(A,int,A2,t2tb1(J)) )
& ( get(A,int,A1,t2tb1(J)) = get(A,int,A2,t2tb1(I)) )
& ! [K: $int] :
( ( $lesseq(L,K)
& $less(K,U) )
=> ( ( K != I )
=> ( ( K != J )
=> ( get(A,int,A1,t2tb1(K)) = get(A,int,A2,t2tb1(K)) ) ) ) ) ) ) ).
tff(exchange_set2,axiom,
! [A: map_int_int,L: $int,U: $int,I: $int,J: $int] :
( ( $lesseq(L,I)
& $less(I,U) )
=> ( ( $lesseq(L,J)
& $less(J,U) )
=> exchange2(int,t2tb(A),set(int,int,set(int,int,t2tb(A),t2tb1(I),t2tb1(get3(A,J))),t2tb1(J),t2tb1(get3(A,I))),L,U,I,J) ) ) ).
tff(exchange_set3,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,t2tb1(I),get(A,int,A1,t2tb1(J))),t2tb1(J),get(A,int,A1,t2tb1(I))),L,U,I,J) ) ) ).
tff(exchange1,type,
exchange3: ( ty * uni * uni * $int * $int ) > $o ).
tff(exchange2,type,
exchange4: ( array_int * array_int * $int * $int ) > $o ).
tff(exchange_def6,axiom,
! [A1: array_int,A2: array_int,I: $int,J: $int] :
( exchange4(A1,A2,I,J)
<=> ( ( length2(int,t2tb2(A1)) = length2(int,t2tb2(A2)) )
& exchange2(int,elts(int,t2tb2(A1)),elts(int,t2tb2(A2)),0,length2(int,t2tb2(A1)),I,J) ) ) ).
tff(exchange_def7,axiom,
! [A: ty,A1: uni,A2: uni,I: $int,J: $int] :
( exchange3(A,A1,A2,I,J)
<=> ( ( length2(A,A1) = length2(A,A2) )
& exchange2(A,elts(A,A1),elts(A,A2),0,length2(A,A1),I,J) ) ) ).
tff(permut1,type,
permut3: ( ty * uni * uni * $int * $int ) > $o ).
tff(permut_def5,axiom,
! [A: ty,A1: uni,A2: uni,L: $int,U: $int] :
( permut3(A,A1,A2,L,U)
<=> ( ( length2(A,A1) = length2(A,A2) )
& $lesseq(0,L)
& $lesseq(L,length2(A,A1))
& $lesseq(0,U)
& $lesseq(U,length2(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_sub1,type,
permut_sub2: ( array_int * array_int * $int * $int ) > $o ).
tff(permut_sub_def2,axiom,
! [A1: array_int,A2: array_int,L: $int,U: $int] :
( permut_sub2(A1,A2,L,U)
<=> ( map_eq_sub1(int,elts(int,t2tb2(A1)),elts(int,t2tb2(A2)),0,L)
& permut3(int,t2tb2(A1),t2tb2(A2),L,U)
& map_eq_sub1(int,elts(int,t2tb2(A1)),elts(int,t2tb2(A2)),U,length2(int,t2tb2(A1))) ) ) ).
tff(permut_sub_def3,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,length2(A,A1)) ) ) ).
tff(permut_all,type,
permut_all: ( ty * uni * uni ) > $o ).
tff(permut_all_def2,axiom,
! [A: ty,A1: uni,A2: uni] :
( permut_all(A,A1,A2)
<=> ( ( length2(A,A1) = length2(A,A2) )
& permut2(A,elts(A,A1),elts(A,A2),0,length2(A,A1)) ) ) ).
tff(exchange_permut_sub2,axiom,
! [A1: array_int,A2: array_int,I: $int,J: $int,L: $int,U: $int] :
( exchange4(A1,A2,I,J)
=> ( ( $lesseq(L,I)
& $less(I,U) )
=> ( ( $lesseq(L,J)
& $less(J,U) )
=> ( $lesseq(0,L)
=> ( $lesseq(U,length2(int,t2tb2(A1)))
=> permut_sub2(A1,A2,L,U) ) ) ) ) ) ).
tff(exchange_permut_sub3,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,length2(A,A1))
=> permut_sub1(A,A1,A2,L,U) ) ) ) ) ) ).
tff(permut_sub_weakening2,axiom,
! [A1: array_int,A2: array_int,L1: $int,U1: $int,L2: $int,U2: $int] :
( permut_sub2(A1,A2,L1,U1)
=> ( ( $lesseq(0,L2)
& $lesseq(L2,L1) )
=> ( ( $lesseq(U1,U2)
& $lesseq(U2,length2(int,t2tb2(A1))) )
=> permut_sub2(A1,A2,L2,U2) ) ) ) ).
tff(permut_sub_weakening3,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,length2(A,A1)) )
=> permut_sub1(A,A1,A2,L2,U2) ) ) ) ).
tff(exchange_permut_all1,axiom,
! [A1: array_int,A2: array_int,I: $int,J: $int] :
( exchange4(A1,A2,I,J)
=> permut_all(int,t2tb2(A1),t2tb2(A2)) ) ).
tff(exchange_permut_all2,axiom,
! [A: ty,A1: uni,A2: uni,I: $int,J: $int] :
( exchange3(A,A1,A2,I,J)
=> permut_all(A,A1,A2) ) ).
tff(wP_parameter_partition_,conjecture,
! [A: $int,A1: map_int_int,M: $int,N: $int] :
( ( $lesseq(0,A)
& $lesseq(0,M)
& $less(M,N)
& $less(N,A) )
=> ! [F: $int] :
( ( $lesseq(M,F)
& $lesseq(F,N) )
=> ( ( $lesseq(0,F)
& $less(F,A) )
=> ! [Ghx: $int] :
( ( Ghx = get3(A1,F) )
=> ! [I: $int] :
( ( I = M )
=> ! [J: $int] :
( ( J = N )
=> ( ( $lesseq(M,J)
& $lesseq(J,N)
& $lesseq(M,I)
& $lesseq(I,N)
& permut_sub2(mk_array3(A,A1),mk_array3(A,A1),M,$sum(N,1))
& ! [R: $int] :
( ( $lesseq(M,R)
& $less(R,I) )
=> $lesseq(get3(A1,R),get3(A1,F)) )
& ! [R: $int] :
( ( $less(J,R)
& $lesseq(R,N) )
=> $lesseq(get3(A1,F),get3(A1,R)) )
& ( get3(A1,F) = get3(A1,F) ) )
=> ! [J1: $int,I1: $int,A2: map_int_int] :
( ( $lesseq(0,A)
& $lesseq(M,J1)
& $lesseq(J1,I1)
& $lesseq(I1,N)
& permut_sub2(mk_array3(A,A1),mk_array3(A,A2),M,$sum(N,1))
& ( ( I1 = N )
| $less(get3(A1,F),get3(A2,I1)) )
& ( ( J1 = M )
| $less(get3(A2,J1),get3(A1,F)) )
& ( get3(A2,F) = get3(A1,F) )
& ! [R: $int] :
( ( $lesseq(M,R)
& $less(R,I1) )
=> $lesseq(get3(A2,R),get3(A1,F)) )
& ! [R: $int] :
( ( $less(J1,R)
& $lesseq(R,N) )
=> $lesseq(get3(A1,F),get3(A2,R)) ) )
=> ( ( $less(J1,I1)
| ( ( J1 = I1 )
& ( I1 = M ) )
| ( ( J1 = I1 )
& ( I1 = N ) ) )
=> ( ~ $less(I1,F)
=> ( $less(F,J1)
=> ( ( $lesseq(0,M)
& $lesseq(M,F)
& $lesseq(F,N)
& $less(N,A)
& $lesseq(M,J1)
& $lesseq(J1,N) )
=> ! [A3: map_int_int] :
( ( $lesseq(0,A)
& exchange4(mk_array3(A,A2),mk_array3(A,A3),F,J1)
& permut_sub2(mk_array3(A,A2),mk_array3(A,A3),M,$sum(N,1)) )
=> ! [J2: $int] :
( ( J2 = $difference(J1,1) )
=> ! [R: $int] :
( ( $lesseq(I1,R)
& $lesseq(R,N) )
=> $lesseq(Ghx,get3(A3,R)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------