[proof-chunk] check rwtable fingerprint equality in last chunk (#1674)

### Description

Depends on #1641 with extra commit: adding fingerprint equality check on
chronological/by address rw_table
Fingerprint check gate will be enable in last chunk last row

### instance columns, top down order match instance array order

chunk ctx
- [current chunk index, total chunk, initial rwc] // equal with
chunk_{i-1}
- [next chunk index, total chunk, next rwc] equal with chunk_{i+1}

pi circuit
- [pi digest lo, pi digest hi] // same across all chunks

state circuit
- [prev permutation fingerprint] // equal with chunk_{i-1}
- [next permutation fingerprint] // equal with chunk_{i+1}
- [alpha, gamma] // same across all chunks

evm circuit
- [prev permutation fingerprint] // equal with chunk_{i-1}
- [next permutation fingerprint] // equal with chunk_{i+1}
- [alpha, gamma] // same across all chunks

### Type of change

- [ ] Bug fix (non-breaking change which fixes an issue)
- [x] New feature (non-breaking change which adds functionality)
- [ ] Breaking change (fix or feature that would cause existing
functionality to not work as expected)
- [ ] This change requires a documentation update

gadgets/src/permutation.rs

@@ -3,11 +3,11 @@
 //! power of gamma are defined in columns to trade more columns with less degrees
 use std::iter;
 #[rustfmt::skip]
-// | q_row_non_first | q_row_enable | alpha     | gamma     | gamma power 2   | ... | row fingerprint | accmulated fingerprint |
-// |-----------------|--------------|-----------|-----------|-----------------|     | --------------- | ---------------------- |
-// | 0               |1             |alpha      | gamma     | gamma **2       | ... |  F              |  F                     |
-// | 1               |1             |alpha      | gamma     | gamma **2       | ... |  F              |  F                     |
-// | 1               |1             |alpha      | gamma     | gamma **2       | ... |  F              |  F                     |
+// | q_row_non_first | q_row_enable | q_row_last | alpha     | gamma     | gamma power 2   | ... | row fingerprint | accmulated fingerprint |
+// |-----------------|--------------|------------|-----------|-----------|-----------------|     | --------------- | ---------------------- |
+// | 0               |1             |0           |alpha      | gamma     | gamma **2       | ... |  F              |  F                     |
+// | 1               |1             |0           |alpha      | gamma     | gamma **2       | ... |  F              |  F                     |
+// | 1               |1             |1           |alpha      | gamma     | gamma **2       | ... |  F              |  F                     |
 
 use std::marker::PhantomData;
 
@@ -23,16 +23,20 @@ use itertools::Itertools;
 /// Config for PermutationChipConfig
 #[derive(Clone, Debug)]
 pub struct PermutationChipConfig<F> {
-    // column
-    acc_fingerprints: Column<Advice>,
+    /// acc_fingerprints
+    pub acc_fingerprints: Column<Advice>,
     row_fingerprints: Column<Advice>,
     alpha: Column<Advice>,
     power_of_gamma: Vec<Column<Advice>>,
     // selector
     q_row_non_first: Selector, // 1 between (first, end], exclude first
     q_row_enable: Selector,    // 1 for all rows (including first)
+    /// q_row_last
+    pub q_row_last: Selector, // 1 in the last row
 
     _phantom: PhantomData<F>,
+
+    acc_fingerprints_cur_expr: Expression<F>,
 }
 
 /// (alpha, gamma, prev_acc_fingerprints, next_acc_fingerprints)
@@ -122,6 +126,7 @@ impl<F: Field> PermutationChipConfig<F> {
             // last offset
             if offset == fingerprints.len() - 1 {
                 last_fingerprint_cell = Some(row_acc_fingerprint_cell);
+                self.q_row_last.enable(region, offset)?;
             }
         }
 
@@ -159,6 +164,11 @@ impl<F: Field> PermutationChipConfig<F> {
         }))
         .for_each(|(col, ann)| region.name_column(|| format!("{}_{}", prefix, ann), col));
     }
+
+    /// acc_fingerprints_cur_expr
+    pub fn acc_fingerprints_cur_expr(&self) -> Expression<F> {
+        self.acc_fingerprints_cur_expr.clone()
+    }
 }
 
 /// permutation fingerprint gadget
@@ -185,11 +195,14 @@ impl<F: Field> PermutationChip<F> {
 
         let q_row_non_first = meta.selector();
         let q_row_enable = meta.selector();
+        let q_row_last = meta.selector();
 
         meta.enable_equality(acc_fingerprints);
         meta.enable_equality(alpha);
         meta.enable_equality(power_of_gamma[0]);
 
+        let mut acc_fingerprints_cur_expr: Expression<F> = 0.expr();
+
         meta.create_gate(
             "acc_fingerprints_cur = acc_fingerprints_prev * row_fingerprints_cur",
             |meta| {
@@ -198,6 +211,8 @@ impl<F: Field> PermutationChip<F> {
                 let acc_fingerprints_cur = meta.query_advice(acc_fingerprints, Rotation::cur());
                 let row_fingerprints_cur = meta.query_advice(row_fingerprints, Rotation::cur());
 
+                acc_fingerprints_cur_expr = acc_fingerprints_cur.clone();
+
                 [q_row_non_first
                     * (acc_fingerprints_cur - acc_fingerprints_prev * row_fingerprints_cur)]
             },
@@ -266,9 +281,11 @@ impl<F: Field> PermutationChip<F> {
 
         PermutationChipConfig {
             acc_fingerprints,
+            acc_fingerprints_cur_expr,
             row_fingerprints,
             q_row_non_first,
             q_row_enable,
+            q_row_last,
             alpha,
             power_of_gamma,
             _phantom: PhantomData::<F> {},