diff --git a/mechanization/props/StrictlyNullable.v b/mechanization/props/StrictlyNullable.v
index e545899..2db8d92 100644
--- a/mechanization/props/StrictlyNullable.v
+++ b/mechanization/props/StrictlyNullable.v
@@ -375,6 +375,8 @@ Proof.
                CompiledQuantifier_max (compileQuantifier q))%NoI); [ discriminate |].
     inversion COMPILE as [M].
     clear COMPILE M m.
+    change (countLeftCapturingParensBefore (Quantified r q) ctx)
+      with (countLeftCapturingParensBefore r ctx).
     apply repeat_matcher_sn with (root:=root); auto.
     unfold repeatMatcherFuel. lia.
   (* concatenation *)
@@ -417,7 +419,7 @@ Proof.
     rewrite NOFAIL. rewrite Nat.add_sub.
     (* the update is independent of the direction *)
     match goal with
-    | [ |- context[(if ?c then ?i else ?e)]] => replace (if c then i else e) with e
+    | [ |- context[(if ?c then ?i else ?e)]] => replace (if c then i else e) with i
     end.
     2: { destruct dir; auto. } simpl.
     destruct (List.Update.Nat.One.update (CaptureRange_or_undefined (CaptureRange.make (endIndex y) (endIndex y))) (captures y) (countLeftCapturingParensBefore (Group name r) ctx)) eqn:UPD.
@@ -567,6 +569,9 @@ Proof.
   unfold Coercions.wrap_Matcher in COMPILESTAR. inversion COMPILESTAR as [STAR]. clear COMPILESTAR.
   apply strictly_nullable_repeatmatcher' with (x:=x) (c:=c) (root:=root) in SUBSTAR; auto.
   simpl in SUBSTAR. rewrite PeanoNat.Nat.add_0_r in SUBSTAR.
-  unfold repeatMatcher. rewrite SUBSTAR. auto.
+  unfold repeatMatcher. 
+  change (countLeftCapturingParensBefore (Quantified r (Greedy Star)) ctx)
+    with (countLeftCapturingParensBefore r ctx).
+  rewrite SUBSTAR. auto.
 Qed.
 End StriclyNullable.
diff --git a/mechanization/spec/Semantics.v b/mechanization/spec/Semantics.v
index 00fbb9f..273038d 100644
--- a/mechanization/spec/Semantics.v
+++ b/mechanization/spec/Semantics.v
@@ -387,7 +387,7 @@ Module Semantics. Section main.
       (*>> 3. Return the union of CharSets A and B. <<*)
       CharSet.union A B
 
-  (** >> NonemptyClassRangesNoDash :: ClassAtomNoDash - ClassAtom ClassRanges <<*)
+  (** >> [OMITTED] NonemptyClassRangesNoDash :: ClassAtomNoDash - ClassAtom ClassRanges <<*)
       (*>> 1. Let A be CompileToCharSet of ClassAtomNoDash with argument rer. <<*)
       (*>> 2. Let B be CompileToCharSet of ClassAtom with argument rer. <<*)
       (*>> 3. Let C be CompileToCharSet of ClassRanges with argument rer. <<*)
@@ -512,7 +512,7 @@ Module Semantics. Section main.
     (*>> 9. If greedy is false, then <<*)
     if greedy is false then
       (*>> a. Let z be c(x). <<*)
-      let! z =<< c x in
+      let z := c x in
       (*>> b. If z is not failure, return z. <<*)
       if z != failure
         then z else
@@ -520,7 +520,7 @@ Module Semantics. Section main.
       m xr d
     else
     (*>> 10. Let z be m(xr, d). <<*)
-    let! z =<< m xr d in
+    let z := m xr d in
     (*>> 11. If z is not failure, return z. <<*)
     if z != failure
       then z else
@@ -614,7 +614,7 @@ Module Semantics. Section main.
       (*>> 3. Assert: q.[[Min]] ≤ q.[[Max]]. <<*)
       assert! (CompiledQuantifier_min q <=? CompiledQuantifier_max q)%NoI;
       (*>> 4. Let parenIndex be CountLeftCapturingParensBefore(Term). <<*)
-      let parenIndex := countLeftCapturingParensBefore r ctx in
+      let parenIndex := countLeftCapturingParensBefore self ctx in
       (*>> 5. Let parenCount be CountLeftCapturingParensWithin(Atom). <<*)
       let parenCount := countLeftCapturingParensWithin r (Quantified_inner qu :: ctx) in
       (*>> 6. Return a new Matcher with parameters (x, c) that captures m, q, parenIndex, and parenCount and performs the following steps when called: <<*)
@@ -647,7 +647,7 @@ Module Semantics. Section main.
           (*>> d. Let e be x's endIndex. <<*)
           let e := MatchState.endIndex x in
           (*>> e. If e = 0, or if rer.[[Multiline]] is true and the character Input[e - 1] is matched by LineTerminator, then <<*)
-          if! (e =? 0)%Z ||! ((RegExpRecord.multiline rer is true) &&! (let! c =<< input[(e-1)%Z] in CharSet.contains Characters.line_terminators c)) then
+          if! (e =? 0)%Z ||! ((RegExpRecord.multiline rer is true) &&! (let! d =<< input[(e-1)%Z] in CharSet.contains Characters.line_terminators d)) then
             (*>> i. Return c(x). <<*)
             c x
           else
@@ -667,7 +667,7 @@ Module Semantics. Section main.
           (*>> e. Let InputLength be the number of elements in Input. <<*)
           let inputLength := List.length input in
           (*>> f. If e = InputLength, or if rer.[[Multiline]] is true and the character Input[e] is matched by LineTerminator, then <<*)
-          if! (e =? inputLength)%Z ||! ((RegExpRecord.multiline rer is true) &&! (let! c =<< input[e] in CharSet.contains Characters.line_terminators c)) then
+          if! (e =? inputLength)%Z ||! ((RegExpRecord.multiline rer is true) &&! (let! d =<< input[e] in CharSet.contains Characters.line_terminators d)) then
             (*>> i. Return c(x). <<*)
             c x
           else
@@ -846,7 +846,7 @@ Module Semantics. Section main.
         (*>> 1. Let ch be the character matched by PatternCharacter. <<*)
         let ch := c in
         (*>> 2. Let A be a one-element CharSet containing the character ch. <<*)
-        let A := CharSet.singleton c in
+        let A := CharSet.singleton ch in
         (*>> 3. Return CharacterSetMatcher(rer, A, false, direction). <<*)
         characterSetMatcher rer A false direction
 
@@ -901,6 +901,7 @@ Module Semantics. Section main.
               (*>> vii. Else, <<*)
               else
                 (*>> 1. Assert: direction is backward. <<*)
+                  assert! (direction == backward) ;
                 (*>> 2. Assert: ye ≤ xe. <<*)
                 assert! (ye <=? xe)%Z ;
                 (*>> 3. Let r be the CaptureRange (ye, xe). <<*)
@@ -932,16 +933,16 @@ Module Semantics. Section main.
           (*>> 2. Let ch be the character whose character value is cv. <<*)
           let ch := Character.from_numeric_value cv in
           (*>> 3. Let A be a one-element CharSet containing the character ch. <<*)
-          let a := CharSet.singleton ch in
+          let A := CharSet.singleton ch in
           (*>> 4. Return CharacterSetMatcher(rer, A, false, direction). <<*)
-          characterSetMatcher rer a false direction
+          characterSetMatcher rer A false direction
 
       (** >> AtomEscape :: CharacterClassEscape <<*)
       | AtomEsc (ACharacterClassEsc cce) =>
           (*>> 1. Let A be CompileToCharSet of CharacterClassEscape with argument rer. <<*)
-          let! a =<< compileToCharSet cce rer in
+          let! A =<< compileToCharSet cce rer in
           (*>> 2. Return CharacterSetMatcher(rer, A, false, direction). <<*)
-          characterSetMatcher rer a false direction
+          characterSetMatcher rer A false direction
 
       (** >> AtomEscape :: k GroupName <<*)
       | AtomEsc (GroupEsc gn) =>