TPTP Problem File: SWW638_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW638_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Residual-T-WP parameter accepts epsilon
% 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 : residual-T-WP_parameter_accepts_epsilon [Fil14]
% Status : Theorem
% Rating : 0.38 v8.2.0, 0.50 v7.5.0, 0.60 v7.4.0, 0.50 v7.3.0, 0.33 v7.0.0, 0.43 v6.4.0, 0.33 v6.3.0, 0.57 v6.2.0, 0.50 v6.1.0
% Syntax : Number of formulae : 121 ( 46 unt; 47 typ; 0 def)
% Number of atoms : 140 ( 79 equ)
% Maximal formula atoms : 21 ( 1 avg)
% Number of connectives : 84 ( 18 ~; 15 |; 18 &)
% ( 5 <=>; 28 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number arithmetic : 16 ( 4 atm; 4 fun; 5 num; 3 var)
% Number of types : 9 ( 7 usr; 1 ari)
% Number of type conns : 55 ( 28 >; 27 *; 0 +; 0 <<)
% Number of predicates : 5 ( 3 usr; 0 prp; 2-3 aty)
% Number of functors : 41 ( 37 usr; 14 con; 0-8 aty)
% Number of variables : 225 ( 208 !; 17 ?; 225 :)
% 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(char,type,
char2: $tType ).
tff(char1,type,
char1: ty ).
tff(regexp,type,
regexp1: $tType ).
tff(regexp1,type,
regexp: ty ).
tff(empty,type,
empty1: regexp1 ).
tff(epsilon,type,
epsilon1: regexp1 ).
tff(char2,type,
char3: char2 > regexp1 ).
tff(alt,type,
alt1: ( regexp1 * regexp1 ) > regexp1 ).
tff(concat,type,
concat1: ( regexp1 * regexp1 ) > regexp1 ).
tff(star,type,
star1: regexp1 > regexp1 ).
tff(match_regexp,type,
match_regexp1: ( ty * regexp1 * uni * uni * uni * uni * uni * uni ) > uni ).
tff(match_regexp_sort1,axiom,
! [A: ty,X: regexp1,X1: uni,X2: uni,X3: uni,X4: uni,X5: uni,X6: uni] : sort1(A,match_regexp1(A,X,X1,X2,X3,X4,X5,X6)) ).
tff(match_regexp_Empty,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni,Z4: uni,Z5: uni] :
( sort1(A,Z)
=> ( match_regexp1(A,empty1,Z,Z1,Z2,Z3,Z4,Z5) = Z ) ) ).
tff(match_regexp_Epsilon,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni,Z4: uni,Z5: uni] :
( sort1(A,Z1)
=> ( match_regexp1(A,epsilon1,Z,Z1,Z2,Z3,Z4,Z5) = Z1 ) ) ).
tff(match_regexp_Char,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni,Z4: uni,Z5: uni,U: char2] :
( sort1(A,Z2)
=> ( match_regexp1(A,char3(U),Z,Z1,Z2,Z3,Z4,Z5) = Z2 ) ) ).
tff(match_regexp_Alt,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni,Z4: uni,Z5: uni,U: regexp1,U1: regexp1] :
( sort1(A,Z3)
=> ( match_regexp1(A,alt1(U,U1),Z,Z1,Z2,Z3,Z4,Z5) = Z3 ) ) ).
tff(match_regexp_Concat,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni,Z4: uni,Z5: uni,U: regexp1,U1: regexp1] :
( sort1(A,Z4)
=> ( match_regexp1(A,concat1(U,U1),Z,Z1,Z2,Z3,Z4,Z5) = Z4 ) ) ).
tff(match_regexp_Star,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni,Z4: uni,Z5: uni,U: regexp1] :
( sort1(A,Z5)
=> ( match_regexp1(A,star1(U),Z,Z1,Z2,Z3,Z4,Z5) = Z5 ) ) ).
tff(empty_Epsilon,axiom,
empty1 != epsilon1 ).
tff(empty_Char,axiom,
! [V: char2] : empty1 != char3(V) ).
tff(empty_Alt,axiom,
! [V: regexp1,V1: regexp1] : empty1 != alt1(V,V1) ).
tff(empty_Concat,axiom,
! [V: regexp1,V1: regexp1] : empty1 != concat1(V,V1) ).
tff(empty_Star,axiom,
! [V: regexp1] : empty1 != star1(V) ).
tff(epsilon_Char,axiom,
! [V: char2] : epsilon1 != char3(V) ).
tff(epsilon_Alt,axiom,
! [V: regexp1,V1: regexp1] : epsilon1 != alt1(V,V1) ).
tff(epsilon_Concat,axiom,
! [V: regexp1,V1: regexp1] : epsilon1 != concat1(V,V1) ).
tff(epsilon_Star,axiom,
! [V: regexp1] : epsilon1 != star1(V) ).
tff(char_Alt,axiom,
! [U: char2,V: regexp1,V1: regexp1] : char3(U) != alt1(V,V1) ).
tff(char_Concat,axiom,
! [U: char2,V: regexp1,V1: regexp1] : char3(U) != concat1(V,V1) ).
tff(char_Star,axiom,
! [U: char2,V: regexp1] : char3(U) != star1(V) ).
tff(alt_Concat,axiom,
! [U: regexp1,U1: regexp1,V: regexp1,V1: regexp1] : alt1(U,U1) != concat1(V,V1) ).
tff(alt_Star,axiom,
! [U: regexp1,U1: regexp1,V: regexp1] : alt1(U,U1) != star1(V) ).
tff(concat_Star,axiom,
! [U: regexp1,U1: regexp1,V: regexp1] : concat1(U,U1) != star1(V) ).
tff(char_proj_1,type,
char_proj_11: regexp1 > char2 ).
tff(char_proj_1_def,axiom,
! [U: char2] : char_proj_11(char3(U)) = U ).
tff(alt_proj_1,type,
alt_proj_11: regexp1 > regexp1 ).
tff(alt_proj_1_def,axiom,
! [U: regexp1,U1: regexp1] : alt_proj_11(alt1(U,U1)) = U ).
tff(alt_proj_2,type,
alt_proj_21: regexp1 > regexp1 ).
tff(alt_proj_2_def,axiom,
! [U: regexp1,U1: regexp1] : alt_proj_21(alt1(U,U1)) = U1 ).
tff(concat_proj_1,type,
concat_proj_11: regexp1 > regexp1 ).
tff(concat_proj_1_def,axiom,
! [U: regexp1,U1: regexp1] : concat_proj_11(concat1(U,U1)) = U ).
tff(concat_proj_2,type,
concat_proj_21: regexp1 > regexp1 ).
tff(concat_proj_2_def,axiom,
! [U: regexp1,U1: regexp1] : concat_proj_21(concat1(U,U1)) = U1 ).
tff(star_proj_1,type,
star_proj_11: regexp1 > regexp1 ).
tff(star_proj_1_def,axiom,
! [U: regexp1] : star_proj_11(star1(U)) = U ).
tff(regexp_inversion,axiom,
! [U: regexp1] :
( ( U = empty1 )
| ( U = epsilon1 )
| ( U = char3(char_proj_11(U)) )
| ( U = alt1(alt_proj_11(U),alt_proj_21(U)) )
| ( U = concat1(concat_proj_11(U),concat_proj_21(U)) )
| ( U = star1(star_proj_11(U)) ) ) ).
tff(list,type,
list: ty > ty ).
tff(nil,type,
nil: ty > uni ).
tff(nil_sort1,axiom,
! [A: ty] : sort1(list(A),nil(A)) ).
tff(cons,type,
cons: ( ty * uni * uni ) > uni ).
tff(cons_sort1,axiom,
! [A: ty,X: uni,X1: uni] : sort1(list(A),cons(A,X,X1)) ).
tff(match_list,type,
match_list: ( ty * ty * uni * uni * uni ) > uni ).
tff(match_list_sort1,axiom,
! [A: ty,A1: ty,X: uni,X1: uni,X2: uni] : sort1(A1,match_list(A1,A,X,X1,X2)) ).
tff(match_list_Nil,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni] :
( sort1(A1,Z)
=> ( match_list(A1,A,nil(A),Z,Z1) = Z ) ) ).
tff(match_list_Cons,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni,U: uni,U1: uni] :
( sort1(A1,Z1)
=> ( match_list(A1,A,cons(A,U,U1),Z,Z1) = Z1 ) ) ).
tff(nil_Cons,axiom,
! [A: ty,V: uni,V1: uni] : nil(A) != cons(A,V,V1) ).
tff(cons_proj_1,type,
cons_proj_1: ( ty * uni ) > uni ).
tff(cons_proj_1_sort1,axiom,
! [A: ty,X: uni] : sort1(A,cons_proj_1(A,X)) ).
tff(cons_proj_1_def,axiom,
! [A: ty,U: uni,U1: uni] :
( sort1(A,U)
=> ( cons_proj_1(A,cons(A,U,U1)) = U ) ) ).
tff(cons_proj_2,type,
cons_proj_2: ( ty * uni ) > uni ).
tff(cons_proj_2_sort1,axiom,
! [A: ty,X: uni] : sort1(list(A),cons_proj_2(A,X)) ).
tff(cons_proj_2_def,axiom,
! [A: ty,U: uni,U1: uni] : cons_proj_2(A,cons(A,U,U1)) = U1 ).
tff(list_inversion,axiom,
! [A: ty,U: uni] :
( ( U = nil(A) )
| ( U = cons(A,cons_proj_1(A,U),cons_proj_2(A,U)) ) ) ).
tff(infix_plpl,type,
infix_plpl: ( ty * uni * uni ) > uni ).
tff(infix_plpl_sort1,axiom,
! [A: ty,X: uni,X1: uni] : sort1(list(A),infix_plpl(A,X,X1)) ).
tff(infix_plpl_def,axiom,
! [A: ty,L2: uni] :
( ( infix_plpl(A,nil(A),L2) = L2 )
& ! [X: uni,X1: uni] : infix_plpl(A,cons(A,X,X1),L2) = cons(A,X,infix_plpl(A,X1,L2)) ) ).
tff(append_assoc,axiom,
! [A: ty,L1: uni,L2: uni,L3: uni] : infix_plpl(A,L1,infix_plpl(A,L2,L3)) = infix_plpl(A,infix_plpl(A,L1,L2),L3) ).
tff(append_l_nil,axiom,
! [A: ty,L: uni] : infix_plpl(A,L,nil(A)) = L ).
tff(compatOrderMult,axiom,
! [X: $int,Y: $int,Z: $int] :
( $lesseq(X,Y)
=> ( $lesseq(0,Z)
=> $lesseq($product(X,Z),$product(Y,Z)) ) ) ).
tff(length,type,
length1: ( ty * uni ) > $int ).
tff(length_def,axiom,
! [A: ty] :
( ( length1(A,nil(A)) = 0 )
& ! [X: uni,X1: uni] : length1(A,cons(A,X,X1)) = $sum(1,length1(A,X1)) ) ).
tff(length_nonnegative,axiom,
! [A: ty,L: uni] : $lesseq(0,length1(A,L)) ).
tff(length_nil,axiom,
! [A: ty,L: uni] :
( ( length1(A,L) = 0 )
<=> ( L = nil(A) ) ) ).
tff(append_length,axiom,
! [A: ty,L1: uni,L2: uni] : length1(A,infix_plpl(A,L1,L2)) = $sum(length1(A,L1),length1(A,L2)) ).
tff(mem,type,
mem: ( ty * uni * uni ) > $o ).
tff(mem_def,axiom,
! [A: ty,X: uni] :
( sort1(A,X)
=> ( ~ mem(A,X,nil(A))
& ! [X1: uni,X2: uni] :
( sort1(A,X1)
=> ( mem(A,X,cons(A,X1,X2))
<=> ( ( X = X1 )
| mem(A,X,X2) ) ) ) ) ) ).
tff(mem_append,axiom,
! [A: ty,X: uni,L1: uni,L2: uni] :
( mem(A,X,infix_plpl(A,L1,L2))
<=> ( mem(A,X,L1)
| mem(A,X,L2) ) ) ).
tff(mem_decomp,axiom,
! [A: ty,X: uni,L: uni] :
( mem(A,X,L)
=> ? [L1: uni,L2: uni] :
( sort1(list(A),L1)
& sort1(list(A),L2)
& ( L = infix_plpl(A,L1,cons(A,X,L2)) ) ) ) ).
tff(list_char,type,
list_char: $tType ).
tff(mem1,type,
mem2: ( list_char * regexp1 ) > $o ).
tff(t2tb,type,
t2tb: list_char > uni ).
tff(t2tb_sort,axiom,
! [X: list_char] : sort1(list(char1),t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > list_char ).
tff(bridgeL,axiom,
! [I: list_char] : tb2t(t2tb(I)) = I ).
tff(bridgeR,axiom,
! [J: uni] : t2tb(tb2t(J)) = J ).
tff(mem_eps,axiom,
mem2(tb2t(nil(char1)),epsilon1) ).
tff(t2tb1,type,
t2tb1: char2 > uni ).
tff(t2tb_sort1,axiom,
! [X: char2] : sort1(char1,t2tb1(X)) ).
tff(tb2t1,type,
tb2t1: uni > char2 ).
tff(bridgeL1,axiom,
! [I: char2] : tb2t1(t2tb1(I)) = I ).
tff(bridgeR1,axiom,
! [J: uni] :
( sort1(char1,J)
=> ( t2tb1(tb2t1(J)) = J ) ) ).
tff(mem_char,axiom,
! [C: char2] : mem2(tb2t(cons(char1,t2tb1(C),nil(char1))),char3(C)) ).
tff(mem_altl,axiom,
! [W: list_char,R1: regexp1,R2: regexp1] :
( mem2(W,R1)
=> mem2(W,alt1(R1,R2)) ) ).
tff(mem_altr,axiom,
! [W: list_char,R1: regexp1,R2: regexp1] :
( mem2(W,R2)
=> mem2(W,alt1(R1,R2)) ) ).
tff(mem_concat,axiom,
! [W1: list_char,W2: list_char,R1: regexp1,R2: regexp1] :
( mem2(W1,R1)
=> ( mem2(W2,R2)
=> mem2(tb2t(infix_plpl(char1,t2tb(W1),t2tb(W2))),concat1(R1,R2)) ) ) ).
tff(mems1,axiom,
! [R: regexp1] : mem2(tb2t(nil(char1)),star1(R)) ).
tff(mems2,axiom,
! [W1: list_char,W2: list_char,R: regexp1] :
( mem2(W1,R)
=> ( mem2(W2,star1(R))
=> mem2(tb2t(infix_plpl(char1,t2tb(W1),t2tb(W2))),star1(R)) ) ) ).
tff(mem_inversion,axiom,
! [Z: list_char,Z1: regexp1] :
( mem2(Z,Z1)
=> ( ( ( Z = tb2t(nil(char1)) )
& ( Z1 = epsilon1 ) )
| ? [C: char2] :
( ( Z = tb2t(cons(char1,t2tb1(C),nil(char1))) )
& ( Z1 = char3(C) ) )
| ? [W: list_char,R1: regexp1,R2: regexp1] :
( mem2(W,R1)
& ( Z = W )
& ( Z1 = alt1(R1,R2) ) )
| ? [W: list_char,R1: regexp1,R2: regexp1] :
( mem2(W,R2)
& ( Z = W )
& ( Z1 = alt1(R1,R2) ) )
| ? [W1: list_char,W2: list_char,R1: regexp1,R2: regexp1] :
( mem2(W1,R1)
& mem2(W2,R2)
& ( Z = tb2t(infix_plpl(char1,t2tb(W1),t2tb(W2))) )
& ( Z1 = concat1(R1,R2) ) )
| ? [R: regexp1] :
( ( Z = tb2t(nil(char1)) )
& ( Z1 = star1(R) ) )
| ? [W1: list_char,W2: list_char,R: regexp1] :
( mem2(W1,R)
& mem2(W2,star1(R))
& ( Z = tb2t(infix_plpl(char1,t2tb(W1),t2tb(W2))) )
& ( Z1 = star1(R) ) ) ) ) ).
tff(wP_parameter_accepts_epsilon,conjecture,
! [X: regexp1,X1: regexp1,Result: bool1] :
( ( ( Result = true1 )
<=> mem2(tb2t(nil(char1)),X) )
=> ( ( Result = true1 )
=> ! [Result1: bool1] :
( ( ( Result1 = true1 )
<=> mem2(tb2t(nil(char1)),X1) )
=> ( mem2(tb2t(nil(char1)),concat1(X,X1))
=> ( Result1 = true1 ) ) ) ) ) ).
%------------------------------------------------------------------------------