YES(O(1),O(1))

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Strict Trs:
  { not(x) -> xor(x, true())
  , implies(x, y) -> xor(and(x, y), xor(x, true()))
  , or(x, y) -> xor(and(x, y), xor(x, y))
  , =(x, y) -> xor(x, xor(y, true())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add following weak dependency pairs:

Strict DPs:
  { not^#(x) -> c_1()
  , implies^#(x, y) -> c_2()
  , or^#(x, y) -> c_3()
  , =^#(x, y) -> c_4() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Strict DPs:
  { not^#(x) -> c_1()
  , implies^#(x, y) -> c_2()
  , or^#(x, y) -> c_3()
  , =^#(x, y) -> c_4() }
Strict Trs:
  { not(x) -> xor(x, true())
  , implies(x, y) -> xor(and(x, y), xor(x, true()))
  , or(x, y) -> xor(and(x, y), xor(x, y))
  , =(x, y) -> xor(x, xor(y, true())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

No rule is usable, rules are removed from the input problem.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Strict DPs:
  { not^#(x) -> c_1()
  , implies^#(x, y) -> c_2()
  , or^#(x, y) -> c_3()
  , =^#(x, y) -> c_4() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  none

TcT has computed following constructor-restricted matrix
interpretation.

          [not^#](x1) = [1]
                           
                [c_1] = [0]
                           
  [implies^#](x1, x2) = [2]
                           
                [c_2] = [1]
                           
       [or^#](x1, x2) = [1]
                           
                [c_3] = [0]
                           
        [=^#](x1, x2) = [1]
                           
                [c_4] = [0]

This order satisfies following ordering constraints:


Further, it can be verified that all rules not oriented are covered by the weightgap condition.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Weak DPs:
  { not^#(x) -> c_1()
  , implies^#(x, y) -> c_2()
  , or^#(x, y) -> c_3()
  , =^#(x, y) -> c_4() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ not^#(x) -> c_1()
, implies^#(x, y) -> c_2()
, or^#(x, y) -> c_3()
, =^#(x, y) -> c_4() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

Hurray, we answered YES(O(1),O(1))