Feat: implement buses

air-script/src/cli/transpile.rs

@@ -77,6 +77,7 @@ impl Transpile {
                                 .chain(mir::passes::AstToMir::new(&diagnostics))
                                 .chain(mir::passes::Inlining::new(&diagnostics))
                                 .chain(mir::passes::Unrolling::new(&diagnostics))
+                                .chain(mir::passes::BusOpExpand::new(&diagnostics))
                                 .chain(air_ir::passes::MirToAir::new(&diagnostics));
                         pipeline.run(ast)
                     })

air-script/tests/buses/buses_complex.air

@@ -0,0 +1,39 @@
+def BusesAir
+
+trace_columns {
+    main: [a, b, s1, s2, d],
+}
+
+buses {
+    unit p,
+    mult q,
+}
+
+public_inputs {
+    inputs: [2],
+}
+
+boundary_constraints {
+    enf p.first = null;
+    enf q.first = null;
+    enf p.last = null;
+    enf q.last = null;
+    # TODO: to be used when we have support for variable-length public inputs
+    #enf p.last = inputs;
+    #enf q.last = inputs;
+}
+
+integrity_constraints {
+    enf s1^2 = s1;
+    enf s2^2 = s2;
+
+    p.add(1, a) when s1;
+    p.rem(1, b) when s2;
+
+    p.add(2, b) when 1 - s1;
+    p.rem(2, a) when 1 - s2;
+
+    q.add(3, a) when s1;
+    q.add(3, a) when s1;
+    q.rem(4, b) for d;
+}

air-script/tests/buses/buses_complex.masm

@@ -0,0 +1,179 @@
+# Procedure to efficiently compute the required exponentiations of the out-of-domain point `z` and cache them for later use.
+#
+# This computes the power of `z` needed to evaluate the periodic polynomials and the constraint divisors
+#
+# Input: [...]
+# Output: [...]
+proc.cache_z_exp
+    padw mem_loadw.4294903304 drop drop # load z
+    # => [z_1, z_0, ...]
+    # Exponentiate z trace_len times
+    mem_load.4294903307 neg
+    # => [count, z_1, z_0, ...] where count = -log2(trace_len)
+    dup.0 neq.0
+    while.true
+        movdn.2 dup.1 dup.1 ext2mul
+        # => [(e_1, e_0)^n, i, ...]
+        movup.2 add.1 dup.0 neq.0
+        # => [b, i+1, (e_1, e_0)^n, ...]
+    end # END while
+    push.0 mem_storew.500000100 # z^trace_len
+    # => [0, 0, (z_1, z_0)^trace_len, ...]
+    dropw # Clean stack
+end # END PROC cache_z_exp
+
+# Procedure to compute the exemption points.
+#
+# Input: [...]
+# Output: [g^{-2}, g^{-1}, ...]
+proc.get_exemptions_points
+    mem_load.4294799999
+    # => [g, ...]
+    push.1 swap div
+    # => [g^{-1}, ...]
+    dup.0 dup.0 mul
+    # => [g^{-2}, g^{-1}, ...]
+end # END PROC get_exemptions_points
+
+# Procedure to compute the integrity constraint divisor.
+#
+# The divisor is defined as `(z^trace_len - 1) / ((z - g^{trace_len-2}) * (z - g^{trace_len-1}))`
+# Procedure `cache_z_exp` must have been called prior to this.
+#
+# Input: [...]
+# Output: [divisor_1, divisor_0, ...]
+proc.compute_integrity_constraint_divisor
+    padw mem_loadw.500000100 drop drop # load z^trace_len
+    # Comments below use zt = `z^trace_len`
+    # => [zt_1, zt_0, ...]
+    push.1 push.0 ext2sub
+    # => [zt_1-1, zt_0-1, ...]
+    padw mem_loadw.4294903304 drop drop # load z
+    # => [z_1, z_0, zt_1-1, zt_0-1, ...]
+    exec.get_exemptions_points
+    # => [g^{trace_len-2}, g^{trace_len-1}, z_1, z_0, zt_1-1, zt_0-1, ...]
+    dup.0 mem_store.500000101 # Save a copy of `g^{trace_len-2} to be used by the boundary divisor
+    dup.3 dup.3 movup.3 push.0 ext2sub
+    # => [e_1, e_0, g^{trace_len-1}, z_1, z_0, zt_1-1, zt_0-1, ...]
+    movup.4 movup.4 movup.4 push.0 ext2sub
+    # => [e_3, e_2, e_1, e_0, zt_1-1, zt_0-1, ...]
+    ext2mul
+    # => [denominator_1, denominator_0, zt_1-1, zt_0-1, ...]
+    ext2div
+    # => [divisor_1, divisor_0, ...]
+end # END PROC compute_integrity_constraint_divisor
+
+# Procedure to evaluate numerators of all integrity constraints.
+#
+# All the 2 main and 2 auxiliary constraints are evaluated.
+# The result of each evaluation is kept on the stack, with the top of the stack
+# containing the evaluations for the auxiliary trace (if any) followed by the main trace.
+#
+# Input: [...]
+# Output: [(r_1, r_0)*, ...]
+# where: (r_1, r_0) is the quadratic extension element resulting from the integrity constraint evaluation.
+#        This procedure pushes 4 quadratic extension field elements to the stack
+proc.compute_integrity_constraints
+    # integrity constraint 0 for main
+    padw mem_loadw.4294900002 movdn.3 movdn.3 drop drop padw mem_loadw.4294900002 movdn.3 movdn.3 drop drop ext2mul padw mem_loadw.4294900002 movdn.3 movdn.3 drop drop ext2sub
+    # Multiply by the composition coefficient
+    padw mem_loadw.4294900200 movdn.3 movdn.3 drop drop ext2mul
+    # integrity constraint 1 for main
+    padw mem_loadw.4294900003 movdn.3 movdn.3 drop drop padw mem_loadw.4294900003 movdn.3 movdn.3 drop drop ext2mul padw mem_loadw.4294900003 movdn.3 movdn.3 drop drop ext2sub
+    # Multiply by the composition coefficient
+    padw mem_loadw.4294900200 drop drop ext2mul
+    # integrity constraint 0 for aux
+    padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.1 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900000 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add padw mem_loadw.4294900002 movdn.3 movdn.3 drop drop ext2mul push.1 push.0 padw mem_loadw.4294900002 movdn.3 movdn.3 drop drop ext2sub ext2add padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.2 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900001 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add push.1 push.0 padw mem_loadw.4294900002 movdn.3 movdn.3 drop drop ext2sub ext2mul push.1 push.0 push.1 push.0 padw mem_loadw.4294900002 movdn.3 movdn.3 drop drop ext2sub ext2sub ext2add ext2mul padw mem_loadw.4294900072 movdn.3 movdn.3 drop drop ext2mul padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.1 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900001 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add padw mem_loadw.4294900003 movdn.3 movdn.3 drop drop ext2mul push.1 push.0 padw mem_loadw.4294900003 movdn.3 movdn.3 drop drop ext2sub ext2add padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.2 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900000 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add push.1 push.0 padw mem_loadw.4294900003 movdn.3 movdn.3 drop drop ext2sub ext2mul push.1 push.0 push.1 push.0 padw mem_loadw.4294900003 movdn.3 movdn.3 drop drop ext2sub ext2sub ext2add ext2mul padw mem_loadw.4294900072 drop drop ext2mul ext2sub
+    # Multiply by the composition coefficient
+    padw mem_loadw.4294900201 movdn.3 movdn.3 drop drop ext2mul
+    # integrity constraint 1 for aux
+    padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.3 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900000 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.3 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900000 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add ext2mul padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.4 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900001 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add ext2mul padw mem_loadw.4294900073 movdn.3 movdn.3 drop drop ext2mul padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.3 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900000 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.4 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900001 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add ext2mul padw mem_loadw.4294900002 movdn.3 movdn.3 drop drop ext2mul padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.3 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900000 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.4 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900001 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add ext2mul padw mem_loadw.4294900002 movdn.3 movdn.3 drop drop ext2mul ext2add ext2add padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.3 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900000 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.3 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900000 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add ext2mul padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.4 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900001 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add ext2mul padw mem_loadw.4294900073 drop drop ext2mul padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.3 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900000 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add padw mem_loadw.4294900150 movdn.3 movdn.3 drop drop push.3 push.0 padw mem_loadw.4294900150 drop drop ext2mul ext2add padw mem_loadw.4294900000 movdn.3 movdn.3 drop drop padw mem_loadw.4294900151 movdn.3 movdn.3 drop drop ext2mul ext2add ext2mul padw mem_loadw.4294900004 movdn.3 movdn.3 drop drop ext2mul ext2add ext2sub
+    # Multiply by the composition coefficient
+    padw mem_loadw.4294900201 drop drop ext2mul
+end # END PROC compute_integrity_constraints
+
+# Procedure to evaluate the boundary constraint numerator for the first row of the auxiliary trace
+#
+# Input: [...]
+# Output: [(r_1, r_0)*, ...]
+# Where: (r_1, r_0) is one quadratic extension field element for each constraint
+proc.compute_boundary_constraints_aux_first
+    # boundary constraint 0 for aux
+    padw mem_loadw.4294900072 movdn.3 movdn.3 drop drop push.1 push.0 ext2sub
+    # Multiply by the composition coefficient
+    padw mem_loadw.4294900202 movdn.3 movdn.3 drop drop ext2mul
+    # boundary constraint 1 for aux
+    padw mem_loadw.4294900073 movdn.3 movdn.3 drop drop push.0 push.0 ext2sub
+    # Multiply by the composition coefficient
+    padw mem_loadw.4294900202 drop drop ext2mul
+end # END PROC compute_boundary_constraints_aux_first
+
+# Procedure to evaluate the boundary constraint numerator for the last row of the auxiliary trace
+#
+# Input: [...]
+# Output: [(r_1, r_0)*, ...]
+# Where: (r_1, r_0) is one quadratic extension field element for each constraint
+proc.compute_boundary_constraints_aux_last
+    # boundary constraint 2 for aux
+    padw mem_loadw.4294900072 movdn.3 movdn.3 drop drop push.1 push.0 ext2sub
+    # Multiply by the composition coefficient
+    padw mem_loadw.4294900203 movdn.3 movdn.3 drop drop ext2mul
+    # boundary constraint 3 for aux
+    padw mem_loadw.4294900073 movdn.3 movdn.3 drop drop push.0 push.0 ext2sub
+    # Multiply by the composition coefficient
+    padw mem_loadw.4294900203 drop drop ext2mul
+end # END PROC compute_boundary_constraints_aux_last
+
+# Procedure to evaluate all integrity constraints.
+#
+# Input: [...]
+# Output: [(r_1, r_0), ...]
+# Where: (r_1, r_0) is the final result with the divisor applied
+proc.evaluate_integrity_constraints
+    exec.compute_integrity_constraints
+    # Numerator of the transition constraint polynomial
+    ext2add ext2add ext2add ext2add
+    # Divisor of the transition constraint polynomial
+    exec.compute_integrity_constraint_divisor
+    ext2div # divide the numerator by the divisor
+end # END PROC evaluate_integrity_constraints
+
+# Procedure to evaluate all boundary constraints.
+#
+# Input: [...]
+# Output: [(r_1, r_0), ...]
+# Where: (r_1, r_0) is the final result with the divisor applied
+proc.evaluate_boundary_constraints
+    exec.compute_boundary_constraints_aux_last
+    # Accumulate the numerator for segment 1 LastRow
+    ext2add ext2add
+    # => [(aux_last1, aux_last0), ...]
+    # Compute the denominator for domain LastRow
+    padw mem_loadw.4294903304 drop drop # load z
+    mem_load.500000101 push.0 ext2sub
+    # Compute numerator/denominator for last row
+    ext2div
+    exec.compute_boundary_constraints_aux_first
+    # Accumulate the numerator for segment 1 FirstRow
+    ext2add ext2add
+    # => [(aux_first1, aux_first0), ...]
+    # Compute the denominator for domain FirstRow
+    padw mem_loadw.4294903304 drop drop # load z
+    push.1 push.0 ext2sub
+    # Compute numerator/denominator for first row
+    ext2div
+    # Add first and last row groups
+    ext2add
+end # END PROC evaluate_boundary_constraints
+
+# Procedure to evaluate the integrity and boundary constraints.
+#
+# Input: [...]
+# Output: [(r_1, r_0), ...]
+export.evaluate_constraints
+    exec.cache_z_exp
+    exec.evaluate_integrity_constraints
+    exec.evaluate_boundary_constraints
+    ext2add
+end # END PROC evaluate_constraints
+

air-script/tests/buses/buses_complex.rs

@@ -0,0 +1,96 @@
+use winter_air::{Air, AirContext, Assertion, AuxTraceRandElements, EvaluationFrame, ProofOptions as WinterProofOptions, TransitionConstraintDegree, TraceInfo};
+use winter_math::fields::f64::BaseElement as Felt;
+use winter_math::{ExtensionOf, FieldElement};
+use winter_utils::collections::Vec;
+use winter_utils::{ByteWriter, Serializable};
+
+pub struct PublicInputs {
+    inputs: [Felt; 2],
+}
+
+impl PublicInputs {
+    pub fn new(inputs: [Felt; 2]) -> Self {
+        Self { inputs }
+    }
+}
+
+impl Serializable for PublicInputs {
+    fn write_into<W: ByteWriter>(&self, target: &mut W) {
+        target.write(self.inputs.as_slice());
+    }
+}
+
+pub struct BusesAir {
+    context: AirContext<Felt>,
+    inputs: [Felt; 2],
+}
+
+impl BusesAir {
+    pub fn last_step(&self) -> usize {
+        self.trace_length() - self.context().num_transition_exemptions()
+    }
+}
+
+impl Air for BusesAir {
+    type BaseField = Felt;
+    type PublicInputs = PublicInputs;
+
+    fn context(&self) -> &AirContext<Felt> {
+        &self.context
+    }
+
+    fn new(trace_info: TraceInfo, public_inputs: PublicInputs, options: WinterProofOptions) -> Self {
+        let main_degrees = vec![TransitionConstraintDegree::new(2), TransitionConstraintDegree::new(2)];
+        let aux_degrees = vec![TransitionConstraintDegree::new(5), TransitionConstraintDegree::new(4)];
+        let num_main_assertions = 0;
+        let num_aux_assertions = 4;
+
+        let context = AirContext::new_multi_segment(
+            trace_info,
+            main_degrees,
+            aux_degrees,
+            num_main_assertions,
+            num_aux_assertions,
+            options,
+        )
+        .set_num_transition_exemptions(2);
+        Self { context, inputs: public_inputs.inputs }
+    }
+
+    fn get_periodic_column_values(&self) -> Vec<Vec<Felt>> {
+        vec![]
+    }
+
+    fn get_assertions(&self) -> Vec<Assertion<Felt>> {
+        let mut result = Vec::new();
+        result
+    }
+
+    fn get_aux_assertions<E: FieldElement<BaseField = Felt>>(&self, aux_rand_elements: &AuxTraceRandElements<E>) -> Vec<Assertion<E>> {
+        let mut result = Vec::new();
+        result.push(Assertion::single(0, 0, E::ONE));
+        result.push(Assertion::single(1, 0, E::ZERO));
+        result.push(Assertion::single(0, self.last_step(), E::ONE));
+        result.push(Assertion::single(1, self.last_step(), E::ZERO));
+        result
+    }
+
+    fn evaluate_transition<E: FieldElement<BaseField = Felt>>(&self, frame: &EvaluationFrame<E>, periodic_values: &[E], result: &mut [E]) {
+        let main_current = frame.current();
+        let main_next = frame.next();
+        result[0] = main_current[2] * main_current[2] - main_current[2];
+        result[1] = main_current[3] * main_current[3] - main_current[3];
+    }
+
+    fn evaluate_aux_transition<F, E>(&self, main_frame: &EvaluationFrame<F>, aux_frame: &EvaluationFrame<E>, _periodic_values: &[F], aux_rand_elements: &AuxTraceRandElements<E>, result: &mut [E])
+    where F: FieldElement<BaseField = Felt>,
+          E: FieldElement<BaseField = Felt> + ExtensionOf<F>,
+    {
+        let main_current = main_frame.current();
+        let main_next = main_frame.next();
+        let aux_current = aux_frame.current();
+        let aux_next = aux_frame.next();
+        result[0] = ((aux_rand_elements.get_segment_elements(0)[0] + E::ONE * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[0]) * aux_rand_elements.get_segment_elements(0)[2]) * E::from(main_current[2]) + E::ONE - E::from(main_current[2])) * ((aux_rand_elements.get_segment_elements(0)[0] + E::from(2_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[1]) * aux_rand_elements.get_segment_elements(0)[2]) * (E::ONE - E::from(main_current[2])) + E::ONE - (E::ONE - E::from(main_current[2]))) * aux_current[0] - ((aux_rand_elements.get_segment_elements(0)[0] + E::ONE * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[1]) * aux_rand_elements.get_segment_elements(0)[2]) * E::from(main_current[3]) + E::ONE - E::from(main_current[3])) * ((aux_rand_elements.get_segment_elements(0)[0] + E::from(2_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[0]) * aux_rand_elements.get_segment_elements(0)[2]) * (E::ONE - E::from(main_current[3])) + E::ONE - (E::ONE - E::from(main_current[3]))) * aux_next[0];
+        result[1] = (aux_rand_elements.get_segment_elements(0)[0] + E::from(3_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[0]) * aux_rand_elements.get_segment_elements(0)[2]) * (aux_rand_elements.get_segment_elements(0)[0] + E::from(3_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[0]) * aux_rand_elements.get_segment_elements(0)[2]) * (aux_rand_elements.get_segment_elements(0)[0] + E::from(4_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[1]) * aux_rand_elements.get_segment_elements(0)[2]) * aux_current[1] + (aux_rand_elements.get_segment_elements(0)[0] + E::from(3_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[0]) * aux_rand_elements.get_segment_elements(0)[2]) * (aux_rand_elements.get_segment_elements(0)[0] + E::from(4_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[1]) * aux_rand_elements.get_segment_elements(0)[2]) * E::from(main_current[2]) + (aux_rand_elements.get_segment_elements(0)[0] + E::from(3_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[0]) * aux_rand_elements.get_segment_elements(0)[2]) * (aux_rand_elements.get_segment_elements(0)[0] + E::from(4_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[1]) * aux_rand_elements.get_segment_elements(0)[2]) * E::from(main_current[2]) - ((aux_rand_elements.get_segment_elements(0)[0] + E::from(3_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[0]) * aux_rand_elements.get_segment_elements(0)[2]) * (aux_rand_elements.get_segment_elements(0)[0] + E::from(3_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[0]) * aux_rand_elements.get_segment_elements(0)[2]) * (aux_rand_elements.get_segment_elements(0)[0] + E::from(4_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[1]) * aux_rand_elements.get_segment_elements(0)[2]) * aux_next[1] + (aux_rand_elements.get_segment_elements(0)[0] + E::from(3_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[0]) * aux_rand_elements.get_segment_elements(0)[2]) * (aux_rand_elements.get_segment_elements(0)[0] + E::from(3_u64) * aux_rand_elements.get_segment_elements(0)[1] + E::from(main_current[0]) * aux_rand_elements.get_segment_elements(0)[2]) * E::from(main_current[4]));
+    }
+}