@@ -15,6 +15,87 @@ read first.
 
 ---
 
+## Unreleased — since 2.3.0
+
+### At a glance
+
+| Silent? | What | Was | Is |
+|---|---|---|---|
+| **Silent** | `"a implies (b implies c)".ToEntity().Stringize()` | `a implies b implies c`, which reads back as `(a implies b) implies c` | `a implies (b implies c)` |
+| | `"(a implies b) implies c".ToEntity().Stringize()` | `(a implies b) implies c` | `a implies b implies c` |
+| **Silent** | `@"A \ (B \ C)".ToEntity().Stringize()`, and `Latexize` | `A \ B \ C` / `A \setminus B \setminus C` | `A \ (B \ C)` / `A \setminus \left(B \setminus C\right)` |
+| **Silent** | `@"A \ (B \/ C)".ToEntity().Stringize()`, and `Latexize` | `A \ B \/ C` / `A \setminus B \cup C` | `A \ (B \/ C)` / `A \setminus \left(B \cup C\right)` |
+| **Silent** | `@"A \/ (B \ C)".ToEntity().Stringize()`, and `Latexize` | `A \/ B \ C` / `A \cup B \setminus C` | `A \/ (B \ C)` / `A \cup \left(B \setminus C\right)` |
+| **Silent** | `"(x provided p) provided q".ToEntity().Stringize()`, and `Latexize` | `x provided p provided q`, which reads back as `x provided (p provided q)` | `(x provided p) provided q` |
+| **Silent** | `"a in (b in c)".ToEntity().Stringize()`, and `Latexize` | `a in b in c` / `a \in b \in c` | `a in (b in c)` / `a \in \left(b \in c\right)` |
+| **Silent** | `"x * (y mod z)".ToEntity().Stringize()`, and `Latexize` | `x * y mod z` / `x y \bmod z` | `x * (y mod z)` / `x \left(y \bmod z\right)` |
+| **Silent** | `"-1 * (y mod z)".ToEntity().Stringize()` | `-y mod z` | `-(y mod z)` |
+
+### A printed operator that is not associative keeps its brackets
+
+`Stringize` is the library's own input format: parsing what it prints has to give back the
+expression it printed, and that is what [`StringizeRoundTripTest`](Sources/Tests/UnitTests/Convenience/StringizeRoundTripTest.cs)
+enforces. Six operators broke it, five of them in the same way — the printer left the **right**
+operand unbracketed at its own precedence level, while the grammar folds that level to the left, so
+the printed text came back re-associated. The sixth, `provided`, is the mirror: it is the one infix
+operator the grammar folds to the **right**, so there it is the *left* operand that mis-associates.
+
+Four of the six are not associative, so the re-association changed the answer and not merely
+the shape. Measured on a build of 2.3.0 and a build of this branch:
+
+| input | 2.3.0 printed | 2.3.0 read that back as | value moved |
+|---|---|---|---|
+| `false implies (true implies false)` | `False implies True implies False` | `(False implies True) implies False` | `True` → `False` |
+| `{ 1, 2, 3 } \ ({ 2, 3 } \ { 3 })` | `{ 1, 2, 3 } \ { 2, 3 } \ { 3 }` | `({ 1, 2, 3 } \ { 2, 3 }) \ { 3 }` | `{ 1, 3 }` → `{ 1 }` |
+| `{ 1, 2 } \/ ({ 3 } \ { 1, 2 })` | `{ 1, 2 } \/ { 3 } \ { 1, 2 }` | `({ 1, 2 } \/ { 3 }) \ { 1, 2 }` | `{ 1, 2, 3 }` → `{ 3 }` |
+| `{ 1, 2 } \ ({ 2 } \/ { 3 })` | `{ 1, 2 } \ { 2 } \/ { 3 }` | `({ 1, 2 } \ { 2 }) \/ { 3 }` | `{ 1 }` → `{ 1, 3 }` |
+| `2 * (3 mod 2)` | `2 * 3 mod 2` | `(2 * 3) mod 2` | `2` → `0` |
+
+On this branch each of those prints its brackets and reads back as itself, so the value column
+no longer moves.
+
+`implies` also changes in the other direction, because its printer had the rule the wrong way
+round rather than missing: it bracketed the **assumption** — the operand that never needs it under
+a left fold — and not the conclusion. So `(a implies b) implies c`, which used to print as
+`(a implies b) implies c`, now prints flat as `a implies b implies c`. Both read back as the same
+expression; the brackets were redundant, and printing them is what made the genuinely ambiguous
+case look no different.
+
+`\/` and `mod` are bracketed only against the operator that makes them ambiguous, because both
+share a precedence level with an operator that is not associative while being associative
+themselves:
+
+- `A \/ (B \/ C)` still prints `A \/ B \/ C`; `A \/ (B \ C)` now prints `A \/ (B \ C)`.
+- `x * (y * z)` still prints `x * y * z` and `x * (y / z)` still prints `x * y / z`;
+  `x * (y mod z)` now prints `x * (y mod z)`.
+- `-1 * (y mod z)` printed `-y mod z` and now prints `-(y mod z)`.
+
+`provided` is bracketed on the other side, and it is the case where the value is *not* the test.
+`(x provided p) provided q` and `x provided (p provided q)` are both `x` exactly when `p` and `q`
+hold, so no answer moves — and they are different expressions, which is what the round trip is
+about. It printed flat and read back as the right-nested one.
+
+**What has not changed, deliberately.** An operator that *is* associative still prints flat, so
+`x + (y + z)` still prints `x + y + z`, and likewise for `*`, `and`, `or`, `xor`, `unite` and
+`intersect`. Those come back as a different `Entity` — a left-nested tree instead of
+a right-nested one — and as the same number, because the bracketing carries no mathematics. The
+alternative would print every expanded polynomial as a right-nested pile of parentheses:
+`"(a + b + c + d) ^ 2".ToEntity().Expand()` prints, on both versions,
+`d ^ 2 + 2 * c * d + c ^ 2 + 2 * b * d + 2 * b * c + b ^ 2 + 2 * a * d + 2 * a * c + 2 * a * b + a ^ 2`,
+and its sum is right-nested throughout. `StringizeRoundTripTest` pins that decision so that it
+stays one.
+
+**LaTeX.** `Latexize` moves for the four set and `mod` cases above and **not** for `implies`.
+[CSharpMath.Evaluation](https://github.com/verybadcat/CSharpMath/blob/master/CSharpMath.Evaluation/Evaluation.cs),
+which reads our LaTeX back, folds `\cup`, `\setminus`, `\in`, `\cdot` and `\bmod` to the left at
+the same precedences this grammar uses — so those needed the same brackets — but folds `\to` to the
+**right**, which is the usual convention for implication and is what `Latexize` was already
+bracketing for. The change only ever adds `\left(`/`\right)` groups, which CSharpMath already
+parses, so nothing downstream needs a matching change
+([#822](https://github.com/asc-community/AngouriMath/issues/822)).
+
+---
+
 ## 2.3.0 — since 2.2.0
 
 ### At a glance
 
@@ -1,4 +1,4 @@
-# Expression syntax
+# Expression syntax
 
 What `MathS.FromString` accepts, and what `Entity.Stringize` produces. The authority is
 [`Core/Antlr/AngouriMath.g`](../../Core/Antlr/AngouriMath.g); this page is a reading of it. To
@@ -22,23 +22,28 @@ will say so ([#822](https://github.com/asc-community/AngouriMath/issues/822)).
 
 ## Operators, loosest first
 
-Everything on one line has the same precedence and groups to the left, except `^`, which groups
-to the right.
+Everything on one line has the same precedence and groups to the left, except `^` and `provided`,
+which group to the right.
+
+Grouping is not decoration: an operator that is not associative prints the brackets it has, so
+`a implies (b implies c)` prints with them and `(a implies b) implies c` prints without. Where the
+operator *is* associative the brackets carry no mathematics and are not printed, so `x + (y + z)`
+prints as `x + y + z` and reads back as `(x + y) + z` — the same number, a differently shaped tree.
 
 | | operators | notes |
 |---|---|---|
-| 1 | `provided` | |
+| 1 | `provided` | **groups to the right**: `a provided b provided c` is `a provided (b provided c)` |
 | 2 | `implies` | |
 | 3 | `or` `\|` | |
 | 4 | `xor` | |
 | 5 | `and` `&` | |
 | 6 | `not` | prefix |
 | 7 | `=` `<>` `>` `>=` `<` `<=` | chained: `a < b < c` means `a < b and b < c` |
 | 8 | `in` | |
-| 9 | `unite` `\/`, `setsubtract` `\` | |
+| 9 | `unite` `\/`, `setsubtract` `\` | one level, so `A \/ B \ C` is `(A \/ B) \ C` |
 | 10 | `intersect` `/\` | |
 | 11 | `+` `-` | |
-| 12 | `*` `/` `mod` | |
+| 12 | `*` `/` `mod` | one level, so `x * y mod z` is `(x * y) mod z` |
 | 13 | `+` `-` | prefix |
 | 14 | `^` | **groups to the right**: `a ^ b ^ c` is `a ^ (b ^ c)` |
 | 15 | `!` | postfix factorial |
 
@@ -49,10 +49,13 @@ public override string Latexize()
                                 };
                             case 1:
                                 if (longArray[index - 1] is Integer(-1))
-                                    return $"-{currIn.Latexize(currIn.LatexPriority < LatexPriority)}"; // display "-1 * x * y" as "-x \cdot y", only for the first -1
+                                    return $"-{currIn.Latexize(currIn.LatexPriority < LatexPriority || currIn is Modf)}"; // display "-1 * x * y" as "-x \cdot y", only for the first -1
                                 break;
                         }
-                        var currOut = currIn.Latexize(currIn.LatexPriority < LatexPriority);
+                        // A factor other than the first that is a `mod` needs brackets: \bmod sits
+                        // at this precedence level and is not associative, so "x \cdot y \bmod z"
+                        // reads as "(x y) mod z". 2 * (3 mod 2) is 2 and (2 * 3) mod 2 is 0.
+                        var currOut = currIn.Latexize(currIn.LatexPriority < LatexPriority || currIn is Modf);
 
                         return (longArray[index - 1], currIn) switch // whether we use juxtaposition and omit \cdot
                         {
 
@@ -50,8 +50,12 @@ public override string Latexize()
             partial record Unionf
             {
                 /// <inheritdoc/>
+                // CSharpMath.Evaluation reads \cup and \setminus at one precedence level and folds
+                // them to the left, exactly as the grammar here does, so a \setminus on the right
+                // needs bracketing or it comes back as the outer operator. Union with union is
+                // associative and stays flat.
                 public override string Latexize()
-                    => $@"{Left.Latexize(Left.LatexPriority < LatexPriority)} \cup {Right.Latexize(Right.LatexPriority < LatexPriority)}";
+                    => $@"{Left.Latexize(Left.LatexPriority < LatexPriority)} \cup {Right.Latexize(Right.LatexPriority < LatexPriority || Right is SetMinusf)}";
             }
 
             partial record Intersectionf
@@ -64,22 +68,27 @@ public override string Latexize()
             partial record SetMinusf
             {
                 /// <inheritdoc/>
+                // Not associative, and sharing a precedence level with union: anything at that
+                // level on the right is bracketed, the same rule \frac's text form follows.
                 public override string Latexize()
-                    => $@"{Left.Latexize(Left.LatexPriority < LatexPriority)} \setminus {Right.Latexize(Right.LatexPriority < LatexPriority)}";
+                    => $@"{Left.Latexize(Left.LatexPriority < LatexPriority)} \setminus {Right.Latexize(Right.LatexPriority <= LatexPriority)}";
             }
 
             partial record Inf
             {
                 /// <inheritdoc/>
+                // \in is left-folded by CSharpMath too, and membership is not associative.
                 public override string Latexize()
-                    => $@"{Element.Latexize(Element.LatexPriority < LatexPriority)} \in {SupSet.Latexize(SupSet.LatexPriority < LatexPriority)}";
+                    => $@"{Element.Latexize(Element.LatexPriority < LatexPriority)} \in {SupSet.Latexize(SupSet.LatexPriority <= LatexPriority)}";
             }
         }
 
         partial record Providedf
         {
             /// <inheritdoc/>
-            public override string Latexize() => $@"{Expression.Latexize(Expression.LatexPriority < LatexPriority)} \quad \text{{for}} \quad {Predicate.Latexize(Predicate.LatexPriority < LatexPriority)}";
+            // Right-folding, so the left operand is the one that needs bracketing -- see the
+            // remark on Stringize.
+            public override string Latexize() => $@"{Expression.Latexize(Expression.LatexPriority <= LatexPriority)} \quad \text{{for}} \quad {Predicate.Latexize(Predicate.LatexPriority < LatexPriority)}";
         }
 
         partial record Piecewise
 
@@ -30,10 +30,14 @@ public override string Stringize() =>
         public partial record Mulf
         {
             /// <inheritdoc/>
+            // Multiplication is associative, so a product or a quotient on the right may stay
+            // unbracketed -- but `mod` shares this precedence level and is not associative, so a
+            // `mod` on the right must be bracketed or it is re-read as the outer operator:
+            // `2 * (3 mod 2)` is 2 and `(2 * 3) mod 2` is 0.
             public override string Stringize() =>
                 (Multiplier is Integer(-1) && !MathS.Diagnostic.OutputExplicit ? "-"
                     : Multiplier.Stringize(Multiplier.Priority < Priority) + " * ")
-                + Multiplicand.Stringize(Multiplicand.Priority < Priority);
+                + Multiplicand.Stringize(Multiplicand.Priority < Priority || Multiplicand is Modf);
             /// <inheritdoc/>
             public override string ToString() => Stringize();
         }
 
@@ -55,8 +55,13 @@ public override string Stringize()
         partial record Impliesf
         {
             /// <inheritdoc/>
+            // The grammar folds `implies` to the left, like everything but `^` and `provided`, so
+            // it is the *conclusion* that needs bracketing when it is an implication of its own.
+            // The two are not interchangeable: `false implies (true implies false)` is true, and
+            // `(false implies true) implies false` is false, so printing the first without its
+            // brackets said the opposite of what it meant.
             public override string Stringize()
-                => $"{Assumption.Stringize(Assumption.Priority <= Priority)} implies {Conclusion.Stringize(Conclusion.Priority < Priority)}";
+                => $"{Assumption.Stringize(Assumption.Priority < Priority)} implies {Conclusion.Stringize(Conclusion.Priority <= Priority)}";
             /// <inheritdoc/>
             public override string ToString() => Stringize();
         }
 
@@ -88,8 +88,13 @@ partial record Complexes
             partial record Unionf
             {
                 /// <inheritdoc/>
+                // Union is associative, so a union on the right may stay unbracketed -- but `\/`
+                // and `\` share one precedence level and are folded by one loop in the grammar, so
+                // a *set difference* on the right must be bracketed or it is re-read as the outer
+                // operator: `{ 1, 2 } \/ ({ 3 } \ { 1, 2 })` is { 1, 2, 3 } and
+                // `({ 1, 2 } \/ { 3 }) \ { 1, 2 }` is { 3 }.
                 public override string Stringize()
-                    => $@"{Left.Stringize(Left.Priority < Priority)} \/ {Right.Stringize(Right.Priority < Priority)}";
+                    => $@"{Left.Stringize(Left.Priority < Priority)} \/ {Right.Stringize(Right.Priority < Priority || Right is SetMinusf)}";
                 /// <inheritdoc/>
                 public override string ToString() => Stringize();
             }
@@ -106,17 +111,23 @@ public override string Stringize()
             partial record SetMinusf
             {
                 /// <inheritdoc/>
+                // Set difference is not associative, and it shares its precedence level with
+                // union, so anything at that level on the right needs bracketing -- the same rule
+                // `-` and `/` follow. `{1,2,3} \ ({2,3} \ {3})` is { 1, 3 } where
+                // `({1,2,3} \ {2,3}) \ {3}` is { 1 }.
                 public override string Stringize()
-                    => $@"{Left.Stringize(Left.Priority < Priority)} \ {Right.Stringize(Right.Priority < Priority)}";
+                    => $@"{Left.Stringize(Left.Priority < Priority)} \ {Right.Stringize(Right.Priority <= Priority)}";
                 /// <inheritdoc/>
                 public override string ToString() => Stringize();
             }
 
             partial record Inf
             {
                 /// <inheritdoc/>
+                // `in` is folded to the left and is not associative: `(a in b) in c` asks whether
+                // a truth value is an element of c, `a in (b in c)` whether a is an element of one.
                 public override string Stringize()
-                    => $@"{Element.Stringize(Element.Priority < Priority)} in {SupSet.Stringize(SupSet.Priority < Priority)}";
+                    => $@"{Element.Stringize(Element.Priority < Priority)} in {SupSet.Stringize(SupSet.Priority <= Priority)}";
                 /// <inheritdoc/>
                 public override string ToString() => Stringize();
             }
@@ -127,7 +138,14 @@ public override string Stringize()
         partial record Providedf
         {
             /// <inheritdoc/>
-            public override string Stringize() => $@"{Expression.Stringize(Expression.Priority < Priority.Provided)} provided {Predicate.Stringize(Predicate.Priority < Priority.Provided)}";
+            // `provided` is the one infix operator the grammar folds to the *right*, so this is
+            // the mirror of the `implies` rule: the flat form is read as
+            // `a provided (b provided c)`, and it is the left operand that has to say when it is
+            // an attached condition of its own. `(x provided p) provided q` printed flat came
+            // back as `x provided (p provided q)` -- the same value, since both are `x` exactly
+            // when `p` and `q` hold, and a different expression, which is what the round trip is
+            // about.
+            public override string Stringize() => $@"{Expression.Stringize(Expression.Priority <= Priority.Provided)} provided {Predicate.Stringize(Predicate.Priority < Priority.Provided)}";
             /// <inheritdoc/>
             public override string ToString() => Stringize();
         }
 
@@ -216,7 +216,20 @@ public void TestFormula8()
         [Fact] public void TestProvided1() => Assert.Equal(MathS.Provided("a", "b"), FromString("a provided b"));
         [Fact] public void TestProvided2() => Assert.Equal(MathS.Provided("a", MathS.Provided("b", "c")), FromString("a provided b provided c"));
         [Fact] public void TestProvided3() => Assert.Equal(MathS.Provided(MathS.Provided("a", "b"), "c"), FromString("(a provided b) provided c"));
-        [Fact] public void TestProvided4() => Assert.Equal(FromString("a provided (b provided c)").Stringize(), FromString("(a provided b) provided c").Stringize());
+        // These two are different expressions -- `provided` folds to the right, so only the first
+        // is what the flat spelling means -- and this used to assert that they print alike. That
+        // is the round trip failing, written down as an expectation: the printed form of the
+        // second came back as the first. They must print differently, and each must read back as
+        // itself.
+        [Fact] public void TestProvided4()
+        {
+            var rightNested = FromString("a provided (b provided c)");
+            var leftNested = FromString("(a provided b) provided c");
+            Assert.NotEqual(rightNested, leftNested);
+            Assert.NotEqual(rightNested.Stringize(), leftNested.Stringize());
+            Assert.Equal(rightNested, FromString(rightNested.Stringize()));
+            Assert.Equal(leftNested, FromString(leftNested.Stringize()));
+        }
 
         [Theory]
         [InlineData("sh", "Sinh")]
Original file line number	Diff line number	Diff line change
`@@ -50,8 +50,12 @@ public override string Latexize()`
`50`	`50`	`partial record Unionf`
`51`	`51`	`{`
`52`	`52`	`/// <inheritdoc/>`
	`53`	`+ // CSharpMath.Evaluation reads \cup and \setminus at one precedence level and folds`
	`54`	`+ // them to the left, exactly as the grammar here does, so a \setminus on the right`
	`55`	`+ // needs bracketing or it comes back as the outer operator. Union with union is`
	`56`	`+ // associative and stays flat.`
`53`	`57`	`public override string Latexize()`
`54`		`- => $@"{Left.Latexize(Left.LatexPriority < LatexPriority)} \cup {Right.Latexize(Right.LatexPriority < LatexPriority)}";`
	`58`	`+ => $@"{Left.Latexize(Left.LatexPriority < LatexPriority)} \cup {Right.Latexize(Right.LatexPriority < LatexPriority \|\| Right is SetMinusf)}";`
`55`	`59`	`}`
`56`	`60`
`57`	`61`	`partial record Intersectionf`
`@@ -64,22 +68,27 @@ public override string Latexize()`
`64`	`68`	`partial record SetMinusf`
`65`	`69`	`{`
`66`	`70`	`/// <inheritdoc/>`
	`71`	`+ // Not associative, and sharing a precedence level with union: anything at that`
	`72`	`+ // level on the right is bracketed, the same rule \frac's text form follows.`
`67`	`73`	`public override string Latexize()`
`68`		`- => $@"{Left.Latexize(Left.LatexPriority < LatexPriority)} \setminus {Right.Latexize(Right.LatexPriority < LatexPriority)}";`
	`74`	`+ => $@"{Left.Latexize(Left.LatexPriority < LatexPriority)} \setminus {Right.Latexize(Right.LatexPriority <= LatexPriority)}";`
`69`	`75`	`}`
`70`	`76`
`71`	`77`	`partial record Inf`
`72`	`78`	`{`
`73`	`79`	`/// <inheritdoc/>`
	`80`	`+ // \in is left-folded by CSharpMath too, and membership is not associative.`
`74`	`81`	`public override string Latexize()`
`75`		`- => $@"{Element.Latexize(Element.LatexPriority < LatexPriority)} \in {SupSet.Latexize(SupSet.LatexPriority < LatexPriority)}";`
	`82`	`+ => $@"{Element.Latexize(Element.LatexPriority < LatexPriority)} \in {SupSet.Latexize(SupSet.LatexPriority <= LatexPriority)}";`
`76`	`83`	`}`
`77`	`84`	`}`
`78`	`85`
`79`	`86`	`partial record Providedf`
`80`	`87`	`{`
`81`	`88`	`/// <inheritdoc/>`
`82`		`- public override string Latexize() => $@"{Expression.Latexize(Expression.LatexPriority < LatexPriority)} \quad \text{{for}} \quad {Predicate.Latexize(Predicate.LatexPriority < LatexPriority)}";`
	`89`	`+ // Right-folding, so the left operand is the one that needs bracketing -- see the`
	`90`	`+ // remark on Stringize.`
	`91`	`+ public override string Latexize() => $@"{Expression.Latexize(Expression.LatexPriority <= LatexPriority)} \quad \text{{for}} \quad {Predicate.Latexize(Predicate.LatexPriority < LatexPriority)}";`
`83`	`92`	`}`
`84`	`93`
`85`	`94`	`partial record Piecewise`