/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(?, O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 188 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
(7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(8) CdtProblem
(9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CdtProblem
(11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(12) CdtProblem
(13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
(15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 344 ms]
(16) CdtProblem
(17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 247 ms]
(18) CdtProblem
(19) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) BOUNDS(1, 1)


----------------------------------------

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   +(x, +(y, z)) -> +(+(x, y), z)
   +(*(x, y), +(x, z)) -> *(x, +(y, z))
   +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))


----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   +(x, +(y, z)) -> +(+(x, y), z)
   +(*(x, y), +(x, z)) -> *(x, +(y, z))
   +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

The (relative) TRS S consists of the following rules:

   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   +(x, +(y, z)) -> +(+(x, y), z)
   +(*(x, y), +(x, z)) -> *(x, +(y, z))
   +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

The (relative) TRS S consists of the following rules:

   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   +(z0, +(z1, z2)) -> +(+(z0, z1), z2)
   +(*(z0, z1), +(z0, z2)) -> *(z0, +(z1, z2))
   +(*(z0, z1), +(*(z0, z2), z3)) -> +(*(z0, +(z1, z2)), z3)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_+(z0, z1) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_*(z0, z1) -> c3(ENCARG(z0), ENCARG(z1))
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
S tuples:
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
K tuples:none
Defined Rule Symbols:   +_2, encArg_1, encode_+_2, encode_*_2

Defined Pair Symbols:   ENCARG_1, ENCODE_+_2, ENCODE_*_2, +'_2

Compound Symbols:   c_2, c1_3, c2_3, c3_2, c4_2, c5_1, c6_2


----------------------------------------

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   ENCODE_*(z0, z1) -> c3(ENCARG(z0), ENCARG(z1))

----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   +(z0, +(z1, z2)) -> +(+(z0, z1), z2)
   +(*(z0, z1), +(z0, z2)) -> *(z0, +(z1, z2))
   +(*(z0, z1), +(*(z0, z2), z3)) -> +(*(z0, +(z1, z2)), z3)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_+(z0, z1) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
S tuples:
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
K tuples:none
Defined Rule Symbols:   +_2, encArg_1, encode_+_2, encode_*_2

Defined Pair Symbols:   ENCARG_1, ENCODE_+_2, +'_2

Compound Symbols:   c_2, c1_3, c2_3, c4_2, c5_1, c6_2


----------------------------------------

(9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   +(z0, +(z1, z2)) -> +(+(z0, z1), z2)
   +(*(z0, z1), +(z0, z2)) -> *(z0, +(z1, z2))
   +(*(z0, z1), +(*(z0, z2), z3)) -> +(*(z0, +(z1, z2)), z3)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)))
   ENCODE_+(z0, z1) -> c3(ENCARG(z0))
   ENCODE_+(z0, z1) -> c3(ENCARG(z1))
S tuples:
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
K tuples:none
Defined Rule Symbols:   +_2, encArg_1, encode_+_2, encode_*_2

Defined Pair Symbols:   ENCARG_1, +'_2, ENCODE_+_2

Compound Symbols:   c_2, c1_3, c4_2, c5_1, c6_2, c3_1


----------------------------------------

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_+(z0, z1) -> c3(ENCARG(z0))
   ENCODE_+(z0, z1) -> c3(ENCARG(z1))

----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))
   +(z0, +(z1, z2)) -> +(+(z0, z1), z2)
   +(*(z0, z1), +(z0, z2)) -> *(z0, +(z1, z2))
   +(*(z0, z1), +(*(z0, z2), z3)) -> +(*(z0, +(z1, z2)), z3)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)))
S tuples:
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
K tuples:none
Defined Rule Symbols:   +_2, encArg_1, encode_+_2, encode_*_2

Defined Pair Symbols:   ENCARG_1, +'_2, ENCODE_+_2

Compound Symbols:   c_2, c1_3, c4_2, c5_1, c6_2, c3_1


----------------------------------------

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_+(z0, z1) -> +(encArg(z0), encArg(z1))
   encode_*(z0, z1) -> *(encArg(z0), encArg(z1))

----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   +(z0, +(z1, z2)) -> +(+(z0, z1), z2)
   +(*(z0, z1), +(z0, z2)) -> *(z0, +(z1, z2))
   +(*(z0, z1), +(*(z0, z2), z3)) -> +(*(z0, +(z1, z2)), z3)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)))
S tuples:
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
K tuples:none
Defined Rule Symbols:   encArg_1, +_2

Defined Pair Symbols:   ENCARG_1, +'_2, ENCODE_+_2

Compound Symbols:   c_2, c1_3, c4_2, c5_1, c6_2, c3_1


----------------------------------------

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
We considered the (Usable) Rules:
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   +(z0, +(z1, z2)) -> +(+(z0, z1), z2)
   +(*(z0, z1), +(z0, z2)) -> *(z0, +(z1, z2))
   +(*(z0, z1), +(*(z0, z2), z3)) -> +(*(z0, +(z1, z2)), z3)
And the Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = [1] + x_1 + x_2
   POL(+(x_1, x_2)) = x_1 + x_2
   POL(+'(x_1, x_2)) = x_2
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_+(x_1, x_2)) = [2] + x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(cons_+(x_1, x_2)) = [1] + x_1 + x_2
   POL(encArg(x_1)) = x_1

----------------------------------------

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   +(z0, +(z1, z2)) -> +(+(z0, z1), z2)
   +(*(z0, z1), +(z0, z2)) -> *(z0, +(z1, z2))
   +(*(z0, z1), +(*(z0, z2), z3)) -> +(*(z0, +(z1, z2)), z3)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)))
S tuples:
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
K tuples:
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
Defined Rule Symbols:   encArg_1, +_2

Defined Pair Symbols:   ENCARG_1, +'_2, ENCODE_+_2

Compound Symbols:   c_2, c1_3, c4_2, c5_1, c6_2, c3_1


----------------------------------------

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
We considered the (Usable) Rules:
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   +(z0, +(z1, z2)) -> +(+(z0, z1), z2)
   +(*(z0, z1), +(z0, z2)) -> *(z0, +(z1, z2))
   +(*(z0, z1), +(*(z0, z2), z3)) -> +(*(z0, +(z1, z2)), z3)
And the Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = x_1 + x_2
   POL(+(x_1, x_2)) = [2] + x_1 + x_2
   POL(+'(x_1, x_2)) = x_2
   POL(ENCARG(x_1)) = x_1^2
   POL(ENCODE_+(x_1, x_2)) = [1] + [2]x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(cons_+(x_1, x_2)) = [2] + x_1 + x_2
   POL(encArg(x_1)) = x_1

----------------------------------------

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1))
   encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1))
   +(z0, +(z1, z2)) -> +(+(z0, z1), z2)
   +(*(z0, z1), +(z0, z2)) -> *(z0, +(z1, z2))
   +(*(z0, z1), +(*(z0, z2), z3)) -> +(*(z0, +(z1, z2)), z3)
Tuples:
   ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1))
   ENCARG(cons_+(z0, z1)) -> c1(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
   ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)))
S tuples:none
K tuples:
   +'(*(z0, z1), +(*(z0, z2), z3)) -> c6(+'(*(z0, +(z1, z2)), z3), +'(z1, z2))
   +'(z0, +(z1, z2)) -> c4(+'(+(z0, z1), z2), +'(z0, z1))
   +'(*(z0, z1), +(z0, z2)) -> c5(+'(z1, z2))
Defined Rule Symbols:   encArg_1, +_2

Defined Pair Symbols:   ENCARG_1, +'_2, ENCODE_+_2

Compound Symbols:   c_2, c1_3, c4_2, c5_1, c6_2, c3_1


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(19) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(20)
BOUNDS(1, 1)