TPTP Problem File: SWW651_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW651_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Toy compiler-T-WP parameter soundness gen
% 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 : toy_compiler-T-WP_parameter_soundness_gen [Fil14]
% Status : Theorem
% Rating : 0.88 v8.2.0, 1.00 v7.1.0, 0.83 v7.0.0, 0.86 v6.4.0, 1.00 v6.3.0, 0.86 v6.2.0, 1.00 v6.1.0
% Syntax : Number of formulae : 137 ( 50 unt; 59 typ; 0 def)
% Number of atoms : 147 ( 105 equ)
% Maximal formula atoms : 27 ( 1 avg)
% Number of connectives : 84 ( 15 ~; 10 |; 23 &)
% ( 3 <=>; 33 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 5 avg)
% Maximal term depth : 7 ( 2 avg)
% Number arithmetic : 42 ( 4 atm; 10 fun; 5 num; 23 var)
% Number of types : 10 ( 8 usr; 1 ari)
% Number of type conns : 69 ( 38 >; 31 *; 0 +; 0 <<)
% Number of predicates : 4 ( 2 usr; 0 prp; 2-3 aty)
% Number of functors : 54 ( 49 usr; 15 con; 0-6 aty)
% Number of variables : 225 ( 223 !; 2 ?; 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(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(expr,type,
expr1: $tType ).
tff(expr1,type,
expr: ty ).
tff(cte,type,
cte1: $int > expr1 ).
tff(plus,type,
plus1: ( expr1 * expr1 ) > expr1 ).
tff(minus,type,
minus1: ( expr1 * expr1 ) > expr1 ).
tff(mult,type,
mult1: ( expr1 * expr1 ) > expr1 ).
tff(match_expr,type,
match_expr1: ( ty * expr1 * uni * uni * uni * uni ) > uni ).
tff(match_expr_sort1,axiom,
! [A: ty,X: expr1,X1: uni,X2: uni,X3: uni,X4: uni] : sort1(A,match_expr1(A,X,X1,X2,X3,X4)) ).
tff(match_expr_Cte1,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni,U: $int] :
( sort1(A,Z)
=> ( match_expr1(A,cte1(U),Z,Z1,Z2,Z3) = Z ) ) ).
tff(match_expr_Plus1,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni,U: expr1,U1: expr1] :
( sort1(A,Z1)
=> ( match_expr1(A,plus1(U,U1),Z,Z1,Z2,Z3) = Z1 ) ) ).
tff(match_expr_Minus1,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni,U: expr1,U1: expr1] :
( sort1(A,Z2)
=> ( match_expr1(A,minus1(U,U1),Z,Z1,Z2,Z3) = Z2 ) ) ).
tff(match_expr_Mult1,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni,U: expr1,U1: expr1] :
( sort1(A,Z3)
=> ( match_expr1(A,mult1(U,U1),Z,Z1,Z2,Z3) = Z3 ) ) ).
tff(cte_Plus1,axiom,
! [U: $int,V: expr1,V1: expr1] : cte1(U) != plus1(V,V1) ).
tff(cte_Minus1,axiom,
! [U: $int,V: expr1,V1: expr1] : cte1(U) != minus1(V,V1) ).
tff(cte_Mult1,axiom,
! [U: $int,V: expr1,V1: expr1] : cte1(U) != mult1(V,V1) ).
tff(plus_Minus1,axiom,
! [U: expr1,U1: expr1,V: expr1,V1: expr1] : plus1(U,U1) != minus1(V,V1) ).
tff(plus_Mult1,axiom,
! [U: expr1,U1: expr1,V: expr1,V1: expr1] : plus1(U,U1) != mult1(V,V1) ).
tff(minus_Mult1,axiom,
! [U: expr1,U1: expr1,V: expr1,V1: expr1] : minus1(U,U1) != mult1(V,V1) ).
tff(cte_proj_1,type,
cte_proj_11: expr1 > $int ).
tff(cte_proj_1_def1,axiom,
! [U: $int] : cte_proj_11(cte1(U)) = U ).
tff(plus_proj_1,type,
plus_proj_11: expr1 > expr1 ).
tff(plus_proj_1_def1,axiom,
! [U: expr1,U1: expr1] : plus_proj_11(plus1(U,U1)) = U ).
tff(plus_proj_2,type,
plus_proj_21: expr1 > expr1 ).
tff(plus_proj_2_def1,axiom,
! [U: expr1,U1: expr1] : plus_proj_21(plus1(U,U1)) = U1 ).
tff(minus_proj_1,type,
minus_proj_11: expr1 > expr1 ).
tff(minus_proj_1_def1,axiom,
! [U: expr1,U1: expr1] : minus_proj_11(minus1(U,U1)) = U ).
tff(minus_proj_2,type,
minus_proj_21: expr1 > expr1 ).
tff(minus_proj_2_def1,axiom,
! [U: expr1,U1: expr1] : minus_proj_21(minus1(U,U1)) = U1 ).
tff(mult_proj_1,type,
mult_proj_11: expr1 > expr1 ).
tff(mult_proj_1_def1,axiom,
! [U: expr1,U1: expr1] : mult_proj_11(mult1(U,U1)) = U ).
tff(mult_proj_2,type,
mult_proj_21: expr1 > expr1 ).
tff(mult_proj_2_def1,axiom,
! [U: expr1,U1: expr1] : mult_proj_21(mult1(U,U1)) = U1 ).
tff(expr_inversion1,axiom,
! [U: expr1] :
( ( U = cte1(cte_proj_11(U)) )
| ( U = plus1(plus_proj_11(U),plus_proj_21(U)) )
| ( U = minus1(minus_proj_11(U),minus_proj_21(U)) )
| ( U = mult1(mult_proj_11(U),mult_proj_21(U)) ) ) ).
tff(eval_expr,type,
eval_expr1: expr1 > $int ).
tff(eval_expr_def,axiom,
( ! [X: $int] : eval_expr1(cte1(X)) = X
& ! [X: expr1,X1: expr1] : eval_expr1(plus1(X,X1)) = $sum(eval_expr1(X),eval_expr1(X1))
& ! [X: expr1,X1: expr1] : eval_expr1(minus1(X,X1)) = $difference(eval_expr1(X),eval_expr1(X1))
& ! [X: expr1,X1: expr1] : eval_expr1(mult1(X,X1)) = $product(eval_expr1(X),eval_expr1(X1)) ) ).
tff(asm,type,
asm1: $tType ).
tff(asm1,type,
asm: ty ).
tff(push,type,
push1: $int > asm1 ).
tff(add,type,
add1: asm1 ).
tff(sub,type,
sub1: asm1 ).
tff(mul,type,
mul1: asm1 ).
tff(match_asm,type,
match_asm1: ( ty * asm1 * uni * uni * uni * uni ) > uni ).
tff(match_asm_sort1,axiom,
! [A: ty,X: asm1,X1: uni,X2: uni,X3: uni,X4: uni] : sort1(A,match_asm1(A,X,X1,X2,X3,X4)) ).
tff(match_asm_Push1,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni,U: $int] :
( sort1(A,Z)
=> ( match_asm1(A,push1(U),Z,Z1,Z2,Z3) = Z ) ) ).
tff(match_asm_Add1,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni] :
( sort1(A,Z1)
=> ( match_asm1(A,add1,Z,Z1,Z2,Z3) = Z1 ) ) ).
tff(match_asm_Sub1,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni] :
( sort1(A,Z2)
=> ( match_asm1(A,sub1,Z,Z1,Z2,Z3) = Z2 ) ) ).
tff(match_asm_Mul1,axiom,
! [A: ty,Z: uni,Z1: uni,Z2: uni,Z3: uni] :
( sort1(A,Z3)
=> ( match_asm1(A,mul1,Z,Z1,Z2,Z3) = Z3 ) ) ).
tff(push_Add1,axiom,
! [U: $int] : push1(U) != add1 ).
tff(push_Sub1,axiom,
! [U: $int] : push1(U) != sub1 ).
tff(push_Mul1,axiom,
! [U: $int] : push1(U) != mul1 ).
tff(add_Sub1,axiom,
add1 != sub1 ).
tff(add_Mul1,axiom,
add1 != mul1 ).
tff(sub_Mul1,axiom,
sub1 != mul1 ).
tff(push_proj_1,type,
push_proj_11: asm1 > $int ).
tff(push_proj_1_def1,axiom,
! [U: $int] : push_proj_11(push1(U)) = U ).
tff(asm_inversion1,axiom,
! [U: asm1] :
( ( U = push1(push_proj_11(U)) )
| ( U = add1 )
| ( U = sub1 )
| ( U = mul1 ) ) ).
tff(list_asm,type,
list_asm: $tType ).
tff(list_int,type,
list_int: $tType ).
tff(compute,type,
compute1: ( list_int * list_asm ) > list_int ).
tff(t2tb1,type,
t2tb1: list_int > uni ).
tff(t2tb_sort1,axiom,
! [X: list_int] : sort1(list(int),t2tb1(X)) ).
tff(tb2t1,type,
tb2t1: uni > list_int ).
tff(bridgeL1,axiom,
! [I: list_int] : tb2t1(t2tb1(I)) = I ).
tff(bridgeR1,axiom,
! [J: uni] : t2tb1(tb2t1(J)) = J ).
tff(t2tb,type,
t2tb: list_asm > uni ).
tff(t2tb_sort,axiom,
! [X: list_asm] : sort1(list(asm),t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > list_asm ).
tff(bridgeL,axiom,
! [I: list_asm] : tb2t(t2tb(I)) = I ).
tff(bridgeR,axiom,
! [J: uni] : t2tb(tb2t(J)) = J ).
tff(t2tb2,type,
t2tb2: asm1 > uni ).
tff(t2tb_sort2,axiom,
! [X: asm1] : sort1(asm,t2tb2(X)) ).
tff(tb2t2,type,
tb2t2: uni > asm1 ).
tff(bridgeL2,axiom,
! [I: asm1] : tb2t2(t2tb2(I)) = I ).
tff(bridgeR2,axiom,
! [J: uni] : t2tb2(tb2t2(J)) = J ).
tff(t2tb3,type,
t2tb3: $int > uni ).
tff(t2tb_sort3,axiom,
! [X: $int] : sort1(int,t2tb3(X)) ).
tff(tb2t3,type,
tb2t3: uni > $int ).
tff(bridgeL3,axiom,
! [I: $int] : tb2t3(t2tb3(I)) = I ).
tff(bridgeR3,axiom,
! [J: uni] : t2tb3(tb2t3(J)) = J ).
tff(compute_def,axiom,
! [S: list_int] :
( ( compute1(S,tb2t(nil(asm))) = S )
& ! [X: asm1,X1: list_asm] :
( ! [X2: $int] :
( ( X = push1(X2) )
=> ( compute1(tb2t1(nil(int)),tb2t(cons(asm,t2tb2(X),t2tb(X1)))) = compute1(tb2t1(cons(int,t2tb3(X2),nil(int))),X1) ) )
& ( ( X = add1 )
=> ( compute1(tb2t1(nil(int)),tb2t(cons(asm,t2tb2(X),t2tb(X1)))) = tb2t1(nil(int)) ) )
& ( ( X = sub1 )
=> ( compute1(tb2t1(nil(int)),tb2t(cons(asm,t2tb2(X),t2tb(X1)))) = tb2t1(nil(int)) ) )
& ( ( X = mul1 )
=> ( compute1(tb2t1(nil(int)),tb2t(cons(asm,t2tb2(X),t2tb(X1)))) = tb2t1(nil(int)) ) )
& ! [X2: $int,X3: list_int] :
( ( ( X3 = tb2t1(nil(int)) )
=> ( ! [X4: $int] :
( ( X = push1(X4) )
=> ( compute1(tb2t1(cons(int,t2tb3(X2),t2tb1(X3))),tb2t(cons(asm,t2tb2(X),t2tb(X1)))) = compute1(tb2t1(cons(int,t2tb3(X4),cons(int,t2tb3(X2),t2tb1(X3)))),X1) ) )
& ( ( X = add1 )
=> ( compute1(tb2t1(cons(int,t2tb3(X2),t2tb1(X3))),tb2t(cons(asm,t2tb2(X),t2tb(X1)))) = tb2t1(cons(int,t2tb3(X2),t2tb1(X3))) ) )
& ( ( X = sub1 )
=> ( compute1(tb2t1(cons(int,t2tb3(X2),t2tb1(X3))),tb2t(cons(asm,t2tb2(X),t2tb(X1)))) = tb2t1(cons(int,t2tb3(X2),t2tb1(X3))) ) )
& ( ( X = mul1 )
=> ( compute1(tb2t1(cons(int,t2tb3(X2),t2tb1(X3))),tb2t(cons(asm,t2tb2(X),t2tb(X1)))) = tb2t1(cons(int,t2tb3(X2),t2tb1(X3))) ) ) ) )
& ! [X4: $int,X5: list_int] :
( ( X3 = tb2t1(cons(int,t2tb3(X4),t2tb1(X5))) )
=> ( ! [X6: $int] :
( ( X = push1(X6) )
=> ( compute1(tb2t1(cons(int,t2tb3(X2),t2tb1(X3))),tb2t(cons(asm,t2tb2(X),t2tb(X1)))) = compute1(tb2t1(cons(int,t2tb3(X6),cons(int,t2tb3(X2),t2tb1(X3)))),X1) ) )
& ( ( X = add1 )
=> ( compute1(tb2t1(cons(int,t2tb3(X2),t2tb1(X3))),tb2t(cons(asm,t2tb2(X),t2tb(X1)))) = compute1(tb2t1(cons(int,t2tb3($sum(X4,X2)),t2tb1(X5))),X1) ) )
& ( ( X = sub1 )
=> ( compute1(tb2t1(cons(int,t2tb3(X2),t2tb1(X3))),tb2t(cons(asm,t2tb2(X),t2tb(X1)))) = compute1(tb2t1(cons(int,t2tb3($difference(X4,X2)),t2tb1(X5))),X1) ) )
& ( ( X = mul1 )
=> ( compute1(tb2t1(cons(int,t2tb3(X2),t2tb1(X3))),tb2t(cons(asm,t2tb2(X),t2tb(X1)))) = compute1(tb2t1(cons(int,t2tb3($product(X4,X2)),t2tb1(X5))),X1) ) ) ) ) ) ) ) ).
tff(compile,type,
compile1: expr1 > list_asm ).
tff(compile_def,axiom,
( ! [X: $int] : compile1(cte1(X)) = tb2t(cons(asm,t2tb2(push1(X)),nil(asm)))
& ! [X: expr1,X1: expr1] : compile1(plus1(X,X1)) = tb2t(infix_plpl(asm,t2tb(compile1(X)),infix_plpl(asm,t2tb(compile1(X1)),cons(asm,t2tb2(add1),nil(asm)))))
& ! [X: expr1,X1: expr1] : compile1(minus1(X,X1)) = tb2t(infix_plpl(asm,t2tb(compile1(X)),infix_plpl(asm,t2tb(compile1(X1)),cons(asm,t2tb2(sub1),nil(asm)))))
& ! [X: expr1,X1: expr1] : compile1(mult1(X,X1)) = tb2t(infix_plpl(asm,t2tb(compile1(X)),infix_plpl(asm,t2tb(compile1(X1)),cons(asm,t2tb2(mul1),nil(asm))))) ) ).
tff(wP_parameter_soundness_gen,conjecture,
! [S: list_int,R: list_asm,X: $int] :
( ( tb2t(infix_plpl(asm,t2tb(compile1(cte1(X))),t2tb(R))) = tb2t(cons(asm,t2tb2(push1(X)),t2tb(R))) )
=> ( compute1(S,tb2t(infix_plpl(asm,t2tb(compile1(cte1(X))),t2tb(R)))) = compute1(tb2t1(cons(int,t2tb3(eval_expr1(cte1(X))),t2tb1(S))),R) ) ) ).
%------------------------------------------------------------------------------