TPTP Problem File: SWW612_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW612_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Knuth prime numbers-T-WP parameter prime numbers
% 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 : knuth_prime_numbers-T-WP_parameter_prime_numbers [Fil14]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.62 v7.5.0, 0.60 v7.4.0, 0.38 v7.3.0, 0.33 v7.1.0, 0.50 v7.0.0, 0.57 v6.4.0, 0.33 v6.3.0, 0.71 v6.2.0, 0.62 v6.1.0
% Syntax : Number of formulae : 175 ( 47 unt; 55 typ; 0 def)
% Number of atoms : 299 ( 77 equ)
% Maximal formula atoms : 39 ( 1 avg)
% Number of connectives : 201 ( 22 ~; 8 |; 56 &)
% ( 10 <=>; 105 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 5 avg)
% Maximal term depth : 6 ( 1 avg)
% Number arithmetic : 452 ( 89 atm; 74 fun; 128 num; 161 var)
% Number of types : 9 ( 7 usr; 1 ari)
% Number of type conns : 82 ( 40 >; 42 *; 0 +; 0 <<)
% Number of predicates : 12 ( 9 usr; 0 prp; 1-2 aty)
% Number of functors : 49 ( 39 usr; 12 con; 0-5 aty)
% Number of variables : 273 ( 262 !; 11 ?; 273 :)
% 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(abs,type,
abs1: $int > $int ).
tff(abs_def,axiom,
! [X: $int] :
( ( $lesseq(0,X)
=> ( abs1(X) = X ) )
& ( ~ $lesseq(0,X)
=> ( abs1(X) = $uminus(X) ) ) ) ).
tff(abs_le,axiom,
! [X: $int,Y: $int] :
( $lesseq(abs1(X),Y)
<=> ( $lesseq($uminus(Y),X)
& $lesseq(X,Y) ) ) ).
tff(abs_pos,axiom,
! [X: $int] : $lesseq(0,abs1(X)) ).
tff(div,type,
div2: ( $int * $int ) > $int ).
tff(mod,type,
mod2: ( $int * $int ) > $int ).
tff(div_mod,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> ( X = $sum($product(Y,div2(X,Y)),mod2(X,Y)) ) ) ).
tff(div_bound,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> ( $lesseq(0,div2(X,Y))
& $lesseq(div2(X,Y),X) ) ) ).
tff(mod_bound,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> ( $less($uminus(abs1(Y)),mod2(X,Y))
& $less(mod2(X,Y),abs1(Y)) ) ) ).
tff(div_sign_pos,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> $lesseq(0,div2(X,Y)) ) ).
tff(div_sign_neg,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
& $less(0,Y) )
=> $lesseq(div2(X,Y),0) ) ).
tff(mod_sign_pos,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& ( Y != 0 ) )
=> $lesseq(0,mod2(X,Y)) ) ).
tff(mod_sign_neg,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
& ( Y != 0 ) )
=> $lesseq(mod2(X,Y),0) ) ).
tff(rounds_toward_zero,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> $lesseq(abs1($product(div2(X,Y),Y)),abs1(X)) ) ).
tff(div_1,axiom,
! [X: $int] : div2(X,1) = X ).
tff(mod_1,axiom,
! [X: $int] : mod2(X,1) = 0 ).
tff(div_inf,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(X,Y) )
=> ( div2(X,Y) = 0 ) ) ).
tff(mod_inf,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(X,Y) )
=> ( mod2(X,Y) = X ) ) ).
tff(div_mult,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $less(0,X)
& $lesseq(0,Y)
& $lesseq(0,Z) )
=> ( div2($sum($product(X,Y),Z),X) = $sum(Y,div2(Z,X)) ) ) ).
tff(mod_mult,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $less(0,X)
& $lesseq(0,Y)
& $lesseq(0,Z) )
=> ( mod2($sum($product(X,Y),Z),X) = mod2(Z,X) ) ) ).
tff(lt_nat,type,
lt_nat1: ( $int * $int ) > $o ).
tff(lt_nat_def,axiom,
! [X: $int,Y: $int] :
( lt_nat1(X,Y)
<=> ( $lesseq(0,Y)
& $less(X,Y) ) ) ).
tff(tuple2,type,
tuple2: ( ty * ty ) > ty ).
tff(tuple21,type,
tuple21: ( ty * ty * uni * uni ) > uni ).
tff(tuple2_sort1,axiom,
! [A: ty,A1: ty,X: uni,X1: uni] : sort1(tuple2(A1,A),tuple21(A1,A,X,X1)) ).
tff(tuple2_proj_1,type,
tuple2_proj_11: ( ty * ty * uni ) > uni ).
tff(tuple2_proj_1_sort1,axiom,
! [A: ty,A1: ty,X: uni] : sort1(A1,tuple2_proj_11(A1,A,X)) ).
tff(tuple2_proj_1_def1,axiom,
! [A: ty,A1: ty,U: uni,U1: uni] :
( sort1(A1,U)
=> ( tuple2_proj_11(A1,A,tuple21(A1,A,U,U1)) = U ) ) ).
tff(tuple2_proj_2,type,
tuple2_proj_21: ( ty * ty * uni ) > uni ).
tff(tuple2_proj_2_sort1,axiom,
! [A: ty,A1: ty,X: uni] : sort1(A,tuple2_proj_21(A1,A,X)) ).
tff(tuple2_proj_2_def1,axiom,
! [A: ty,A1: ty,U: uni,U1: uni] :
( sort1(A,U1)
=> ( tuple2_proj_21(A1,A,tuple21(A1,A,U,U1)) = U1 ) ) ).
tff(tuple2_inversion1,axiom,
! [A: ty,A1: ty,U: uni] :
( sort1(tuple2(A1,A),U)
=> ( U = tuple21(A1,A,tuple2_proj_11(A1,A,U),tuple2_proj_21(A1,A,U)) ) ) ).
tff(lpintcm_intrp,type,
lpintcm_intrp: $tType ).
tff(lex,type,
lex1: ( lpintcm_intrp * lpintcm_intrp ) > $o ).
tff(t2tb,type,
t2tb: lpintcm_intrp > uni ).
tff(t2tb_sort,axiom,
! [X: lpintcm_intrp] : sort1(tuple2(int,int),t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > lpintcm_intrp ).
tff(bridgeL,axiom,
! [I: lpintcm_intrp] : 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(lex_1,axiom,
! [X1: $int,X2: $int,Y1: $int,Y2: $int] :
( lt_nat1(X1,X2)
=> lex1(tb2t(tuple21(int,int,t2tb1(X1),t2tb1(Y1))),tb2t(tuple21(int,int,t2tb1(X2),t2tb1(Y2)))) ) ).
tff(lex_2,axiom,
! [X: $int,Y1: $int,Y2: $int] :
( lt_nat1(Y1,Y2)
=> lex1(tb2t(tuple21(int,int,t2tb1(X),t2tb1(Y1))),tb2t(tuple21(int,int,t2tb1(X),t2tb1(Y2)))) ) ).
tff(lex_inversion,axiom,
! [Z: lpintcm_intrp,Z1: lpintcm_intrp] :
( lex1(Z,Z1)
=> ( ? [X1: $int,X2: $int,Y1: $int,Y2: $int] :
( lt_nat1(X1,X2)
& ( Z = tb2t(tuple21(int,int,t2tb1(X1),t2tb1(Y1))) )
& ( Z1 = tb2t(tuple21(int,int,t2tb1(X2),t2tb1(Y2))) ) )
| ? [X: $int,Y1: $int,Y2: $int] :
( lt_nat1(Y1,Y2)
& ( Z = tb2t(tuple21(int,int,t2tb1(X),t2tb1(Y1))) )
& ( Z1 = tb2t(tuple21(int,int,t2tb1(X),t2tb1(Y2))) ) ) ) ) ).
tff(even,type,
even1: $int > $o ).
tff(even_def,axiom,
! [N: $int] :
( even1(N)
<=> ? [K: $int] : N = $product(2,K) ) ).
tff(odd,type,
odd1: $int > $o ).
tff(odd_def,axiom,
! [N: $int] :
( odd1(N)
<=> ? [K: $int] : N = $sum($product(2,K),1) ) ).
tff(even_or_odd,axiom,
! [N: $int] :
( even1(N)
| odd1(N) ) ).
tff(even_not_odd,axiom,
! [N: $int] :
( even1(N)
=> ~ odd1(N) ) ).
tff(odd_not_even,axiom,
! [N: $int] :
( odd1(N)
=> ~ even1(N) ) ).
tff(even_odd,axiom,
! [N: $int] :
( even1(N)
=> odd1($sum(N,1)) ) ).
tff(odd_even,axiom,
! [N: $int] :
( odd1(N)
=> even1($sum(N,1)) ) ).
tff(even_even,axiom,
! [N: $int] :
( even1(N)
=> even1($sum(N,2)) ) ).
tff(odd_odd,axiom,
! [N: $int] :
( odd1(N)
=> odd1($sum(N,2)) ) ).
tff(even_2k,axiom,
! [K: $int] : even1($product(2,K)) ).
tff(odd_2k1,axiom,
! [K: $int] : odd1($sum($product(2,K),1)) ).
tff(divides,type,
divides1: ( $int * $int ) > $o ).
tff(divides_def,axiom,
! [D: $int,N: $int] :
( divides1(D,N)
<=> ? [Q: $int] : N = $product(Q,D) ) ).
tff(divides_refl,axiom,
! [N: $int] : divides1(N,N) ).
tff(divides_1_n,axiom,
! [N: $int] : divides1(1,N) ).
tff(divides_0,axiom,
! [N: $int] : divides1(N,0) ).
tff(divides_left,axiom,
! [A: $int,B: $int,C: $int] :
( divides1(A,B)
=> divides1($product(C,A),$product(C,B)) ) ).
tff(divides_right,axiom,
! [A: $int,B: $int,C: $int] :
( divides1(A,B)
=> divides1($product(A,C),$product(B,C)) ) ).
tff(divides_oppr,axiom,
! [A: $int,B: $int] :
( divides1(A,B)
=> divides1(A,$uminus(B)) ) ).
tff(divides_oppl,axiom,
! [A: $int,B: $int] :
( divides1(A,B)
=> divides1($uminus(A),B) ) ).
tff(divides_oppr_rev,axiom,
! [A: $int,B: $int] :
( divides1($uminus(A),B)
=> divides1(A,B) ) ).
tff(divides_oppl_rev,axiom,
! [A: $int,B: $int] :
( divides1(A,$uminus(B))
=> divides1(A,B) ) ).
tff(divides_plusr,axiom,
! [A: $int,B: $int,C: $int] :
( divides1(A,B)
=> ( divides1(A,C)
=> divides1(A,$sum(B,C)) ) ) ).
tff(divides_minusr,axiom,
! [A: $int,B: $int,C: $int] :
( divides1(A,B)
=> ( divides1(A,C)
=> divides1(A,$difference(B,C)) ) ) ).
tff(divides_multl,axiom,
! [A: $int,B: $int,C: $int] :
( divides1(A,B)
=> divides1(A,$product(C,B)) ) ).
tff(divides_multr,axiom,
! [A: $int,B: $int,C: $int] :
( divides1(A,B)
=> divides1(A,$product(B,C)) ) ).
tff(divides_factorl,axiom,
! [A: $int,B: $int] : divides1(A,$product(B,A)) ).
tff(divides_factorr,axiom,
! [A: $int,B: $int] : divides1(A,$product(A,B)) ).
tff(divides_n_1,axiom,
! [N: $int] :
( divides1(N,1)
=> ( ( N = 1 )
| ( N = $uminus(1) ) ) ) ).
tff(divides_antisym,axiom,
! [A: $int,B: $int] :
( divides1(A,B)
=> ( divides1(B,A)
=> ( ( A = B )
| ( A = $uminus(B) ) ) ) ) ).
tff(divides_trans,axiom,
! [A: $int,B: $int,C: $int] :
( divides1(A,B)
=> ( divides1(B,C)
=> divides1(A,C) ) ) ).
tff(divides_bounds,axiom,
! [A: $int,B: $int] :
( divides1(A,B)
=> ( ( B != 0 )
=> $lesseq(abs1(A),abs1(B)) ) ) ).
tff(div_mult1,axiom,
! [X: $int,Y: $int,Z: $int] :
( $less(0,X)
=> ( $quotient_e($sum($product(X,Y),Z),X) = $sum(Y,$quotient_e(Z,X)) ) ) ).
tff(mod_mult1,axiom,
! [X: $int,Y: $int,Z: $int] :
( $less(0,X)
=> ( $remainder_e($sum($product(X,Y),Z),X) = $remainder_e(Z,X) ) ) ).
tff(mod_divides_euclidean,axiom,
! [A: $int,B: $int] :
( ( B != 0 )
=> ( ( $remainder_e(A,B) = 0 )
=> divides1(B,A) ) ) ).
tff(divides_mod_euclidean,axiom,
! [A: $int,B: $int] :
( ( B != 0 )
=> ( divides1(B,A)
=> ( $remainder_e(A,B) = 0 ) ) ) ).
tff(mod_divides_computer,axiom,
! [A: $int,B: $int] :
( ( B != 0 )
=> ( ( mod2(A,B) = 0 )
=> divides1(B,A) ) ) ).
tff(divides_mod_computer,axiom,
! [A: $int,B: $int] :
( ( B != 0 )
=> ( divides1(B,A)
=> ( mod2(A,B) = 0 ) ) ) ).
tff(even_divides,axiom,
! [A: $int] :
( even1(A)
<=> divides1(2,A) ) ).
tff(odd_divides,axiom,
! [A: $int] :
( odd1(A)
<=> ~ divides1(2,A) ) ).
tff(prime,type,
prime1: $int > $o ).
tff(prime_def,axiom,
! [P: $int] :
( prime1(P)
<=> ( $lesseq(2,P)
& ! [N: $int] :
( ( $less(1,N)
& $less(N,P) )
=> ~ divides1(N,P) ) ) ) ).
tff(not_prime_1,axiom,
~ prime1(1) ).
tff(prime_2,axiom,
prime1(2) ).
tff(prime_3,axiom,
prime1(3) ).
tff(prime_divisors,axiom,
! [P: $int] :
( prime1(P)
=> ! [D: $int] :
( divides1(D,P)
=> ( ( D = 1 )
| ( D = $uminus(1) )
| ( D = P )
| ( D = $uminus(P) ) ) ) ) ).
tff(small_divisors,axiom,
! [P: $int] :
( $lesseq(2,P)
=> ( ! [D: $int] :
( $lesseq(2,D)
=> ( prime1(D)
=> ( ( $less(1,$product(D,D))
& $lesseq($product(D,D),P) )
=> ~ divides1(D,P) ) ) )
=> prime1(P) ) ) ).
tff(even_prime,axiom,
! [P: $int] :
( prime1(P)
=> ( even1(P)
=> ( P = 2 ) ) ) ).
tff(odd_prime,axiom,
! [P: $int] :
( prime1(P)
=> ( $lesseq(3,P)
=> odd1(P) ) ) ).
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(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(no_prime_in,type,
no_prime_in1: ( $int * $int ) > $o ).
tff(no_prime_in_def,axiom,
! [L: $int,U: $int] :
( no_prime_in1(L,U)
<=> ! [X: $int] :
( ( $less(L,X)
& $less(X,U) )
=> ~ prime1(X) ) ) ).
tff(array_int,type,
array_int: $tType ).
tff(first_primes,type,
first_primes1: ( array_int * $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(first_primes_def,axiom,
! [P: array_int,U: $int] :
( first_primes1(P,U)
<=> ( ( tb2t1(get2(int,t2tb2(P),0)) = 2 )
& ! [I: $int,J: $int] :
( ( $lesseq(0,I)
& $less(I,J)
& $less(J,U) )
=> $less(tb2t1(get2(int,t2tb2(P),I)),tb2t1(get2(int,t2tb2(P),J))) )
& ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,U) )
=> prime1(tb2t1(get2(int,t2tb2(P),I))) )
& ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,$difference(U,1)) )
=> no_prime_in1(tb2t1(get2(int,t2tb2(P),I)),tb2t1(get2(int,t2tb2(P),$sum(I,1)))) ) ) ) ).
tff(exists_prime,axiom,
! [P: array_int,U: $int] :
( $lesseq(1,U)
=> ( first_primes1(P,U)
=> ! [D: $int] :
( ( $lesseq(2,D)
& $lesseq(D,tb2t1(get2(int,t2tb2(P),$difference(U,1)))) )
=> ( prime1(D)
=> ? [I: $int] :
( $lesseq(0,I)
& $less(I,U)
& ( D = tb2t1(get2(int,t2tb2(P),I)) ) ) ) ) ) ) ).
tff(bertrand_postulate,axiom,
! [P: $int] :
( prime1(P)
=> ~ no_prime_in1(P,$product(2,P)) ) ).
tff(map_int_int,type,
map_int_int: $tType ).
tff(t2tb3,type,
t2tb3: map_int_int > uni ).
tff(t2tb_sort3,axiom,
! [X: map_int_int] : sort1(map(int,int),t2tb3(X)) ).
tff(tb2t3,type,
tb2t3: uni > map_int_int ).
tff(bridgeL3,axiom,
! [I: map_int_int] : tb2t3(t2tb3(I)) = I ).
tff(bridgeR3,axiom,
! [J: uni] : t2tb3(tb2t3(J)) = J ).
tff(wP_parameter_prime_numbers,conjecture,
! [M: $int] :
( $lesseq(2,M)
=> ( $lesseq(0,M)
=> ( $lesseq(0,M)
=> ( ( $lesseq(0,0)
& $less(0,M) )
=> ! [P: map_int_int] :
( ( $lesseq(0,M)
& ( P = tb2t3(set(int,int,const(int,int,t2tb1(0)),t2tb1(0),t2tb1(2))) ) )
=> ( ( $lesseq(0,1)
& $less(1,M) )
=> ! [P1: map_int_int] :
( ( $lesseq(0,M)
& ( P1 = tb2t3(set(int,int,t2tb3(P),t2tb1(1),t2tb1(3))) ) )
=> ( $lesseq(2,$difference(M,1))
=> ! [N: $int,P2: map_int_int,J: $int] :
( ( $lesseq(2,J)
& $lesseq(J,$difference(M,1)) )
=> ( ( first_primes1(tb2t2(mk_array1(int,M,t2tb3(P2))),J)
& $less(tb2t1(get(int,int,t2tb3(P2),t2tb1($difference(J,1)))),N)
& $less(N,$product(2,tb2t1(get(int,int,t2tb3(P2),t2tb1($difference(J,1))))))
& odd1(N)
& no_prime_in1(tb2t1(get(int,int,t2tb3(P2),t2tb1($difference(J,1)))),N) )
=> ( ( $lesseq(0,M)
& $lesseq(1,1)
& $less(1,J)
& first_primes1(tb2t2(mk_array1(int,M,t2tb3(P2))),J)
& $less(tb2t1(get(int,int,t2tb3(P2),t2tb1($difference(J,1)))),N)
& $less(N,$product(2,tb2t1(get(int,int,t2tb3(P2),t2tb1($difference(J,1))))))
& odd1(N)
& no_prime_in1(tb2t1(get(int,int,t2tb3(P2),t2tb1($difference(J,1)))),N)
& ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,1) )
=> ~ divides1(tb2t1(get(int,int,t2tb3(P2),t2tb1(I))),N) ) )
=> ! [N1: $int] :
( ( $less(tb2t1(get(int,int,t2tb3(P2),t2tb1($difference(J,1)))),N1)
& prime1(N1)
& no_prime_in1(tb2t1(get(int,int,t2tb3(P2),t2tb1($difference(J,1)))),N1) )
=> ( ( $lesseq(0,J)
& $less(J,M) )
=> ! [P3: map_int_int] :
( ( $lesseq(0,M)
& ( P3 = tb2t3(set(int,int,t2tb3(P2),t2tb1(J),t2tb1(N1))) ) )
=> ! [N2: $int] :
( ( N2 = $sum(N1,2) )
=> no_prime_in1(tb2t1(get(int,int,t2tb3(P3),t2tb1($difference($sum(J,1),1)))),N2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------